Practical Math Handbook by Sandra Moreano

Foreword This workbook came about from the contributions of many people over many years. Well worn handouts, powerful illustrations and simple explanations donated by teachers, tutors, and students alike were combined to reveal basic mathematical concepts. Our goal is to produce a step by step approach to basic mathematics from which you can make copies and give them to your friends, family and countrymen without fear of copyright infringement. We want to promote mathematical literacy for everyone. Take a moment to look over the Table of Contents found in front of each chapter. You will see that chapters are broken up into sections, followed by practice problems. Answers to practice problems are found at the end of each chapter. Within the lessons you will also encounter problems called YOUR TURN ‘s, for which the answers are not supplied. Complete them the best you can and check your answers with your instructor. If you are working through this book on your own, you can send an e-mail found on the book’s web site to obtain help. Find a pencil and write on the pages in this book. In most cases you will be able to show your work within the blank spaces provided, but there will be times you will learn more if work out the steps on an extra sheet of paper. Calculators are a useful tool for checking your work, however it is even more important that you get into the habit of mental calculation, so check with your instructor to determine when you can use them. Check the appendix for calculator practice exercises. We welcome any and all feedback so that this text can continue to be the best it can be. You will find the occasional mistake. Please feel free to point them out so that we can correct them for the following version. You may download and print a copy of this book at http://www.math550.com

Copyright ©2009 Central New Mexico Community College. Permission is granted to copy, distribute and/or modify this document, under the terms of the GNU, Free Documentation License. Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is available on http://www.gnu.org/copyleft/fdl.html

Math 550

Basic Mathematics Table of Contents

Chapter One

Whole Numbers

Chapter Two

Decimals

Chapter Three

Fractions

Chaper Four

Ratio and Proportions

147

Chapter Five

Percents

169

Appendix

195

iii

Whole Numbers

Chapter

Table of Contents Section 1.1 Face and Place Value.................................................. 8 Practice 1: Complete the Place Value Chart for the following numbers. . . . . 9 Practice 2: Expanded notation and writing out the numbers. . . . . . . . 12

Section 1.2 Rounding............................................................... 13 Practice 3: Round Whole Numbers. . .

Section 1.3 Whole Number Addition.............................................. 16 Practice 4: Complete the Addition Table. . . . . . . . . . . . . . 17 Practice 5: Add Single Digits . . . . . . . . . . . . . . . . . 18 Practice 6: Add and Carry as needed. . . . . . . . . . . . . . . 19 Practice 7: Add these Two and Three Digit Numbers . . . . . . . . . 20 Practice 8: Add 3 Digits and create Fourth Columns. . . . . . . . . . 21 Practice 9: Addition Word Problems . . . . . . . . . . . . . . 22

Section 1.4 Whole Number Subtraction......................................... 24 Practice 10: Subtract Single Digits. . . . . . . . . . . . . . . 24 Practice 11: Two Digit Subtraction with Borrowing. . . . . . . . . . 25 Practice 12: Subtraction with Borrowing. .. . . . . . . . . . . . . 27 Practice 13: Subtraction Word Problems. . . . . . . . . . . . . . 28

Section 1.5 Multiplication of Whole Numbers.................................... 30 Practice 14: Complete the Multiplication Table.. . . . . . . . . . . 30 Practice 15: Find the prime factors that multiply to the following: . . . . . 32 Practice 16: Multiply the one and two digit problems. . . . . . . . . . 34 Practice 17: Complete the Double and Triple Digit Multiplication.. . . . . . 35 Practice 18: Multiplication Problems. . . . . . . . . . . . . . . 36

Section 1.6 Division of Whole Numbers.......................................... 37 Practice 19: Single Digit Division.. . . . . . . . . . . . . . . . 38 Practice 20: Double Digit Division no Remainders. . . . . . . . . . . 38 Practice 21: Double Digit Division with Remainders. . . . . . . . . . . 39 Practice 22: Two and Three Digit Division with Remainders.. . . . . . . 40

Section 1.7 Area and Perimeter.................................................. 42 Practice 24: Calculate the Perimeter of the following objects.(P = 2L+ 2W). . . 43 Practice 25: Calculate the areas of the following spaces.(A = L x W). . . . 45

Suggested MALL activities. Planning a Trip, Carpeting a Room, Filling out the form 1040EZ

Question: What number do these groupings have in common?





♪♪♪

Whole Numbers

Do the Math

Answer: ___.

Appendix

The purpose of this book is to help provide you with a firm conceptual footing so that you can better understand what you may already have figured out

Percents

If you fast forward to the present day, not only is math essential in architecture, design, engineering, chemistry, aviation and the running of the government, but in navigating everyday life. At the supermarket you calculate whether you receive the correct change. And when you drive home you can figure out if it’s cheaper to take the bus next time. Without math, you can’t effectively budget your money, or have a good idea whether or not you can really afford major purchases. With math, you can figure out how much interest you can afford on a home loan so you don’t get surprised by variable rates. Calculators can figure out the results, but they cannot ask the questions you need to ask on a daily basis to enrich your life and live within your means.

Ratio and Proportion

In order to create monuments and immense public works, taxes had to be levied, but not so much as to create unrest. Economic planning was born. Thousands of laborers needed to be fed, so cooks had to be able to determine how much bread to cook and in what quantities. The pyramids of Egypt required highly advanced calculations in order to construct them according to the determination of the architects, who were also taking into account highly complex astrological computations. Wars were more often won on the logistical skills of the army behind the army than anything that happened on the field of battle. All such advancements would not have been possible without math.

Fractions

The development of the capacity for mathematical thinking caused one of the greatest shifts in human prehistory. For example, prehistoric man could calculate and communicate how long it would take seeds to sprout, and when the wheat or corn would be ready to harvest. This required developing number systems so that this information could be written down and passed along. The understanding of seasons, the cycles of rain and drought, and the construction of irrigation systems allowed for society-wide prosperity, a necessary basis for the emergence of great civilizations and empires.

Decimals

You hopefully answered “3.” There are both 3 items in each grouping, as well as 3 separate groupings. In each case 3 stands for a specific quantity, in this case, 3 scissors , 3 phones, and 3 musical notes. The numeral “3” is actually a symbol, and the understanding of what it represents is an example of abstract reasoning.

Whole Numbers

Chapter One

instinctively or from previous coursework.

Section 1.1 Face and Place Value

2 22

❤

Ratio and Proportion

She means that when he said “after work” he meant just that. Taking things at face value means understanding things as they are stated, with no elaboration or interpretation. In Math, face value means the number all alone, which is also called a digit, and not what it can represent in certain contexts.

Fractions

Decimals

Travis says to Sloane that he’ll be home right after work, and as it’s her turn to cook, she makes dinner for them both. But Travis doesn’t come home until 9 pm, having stopped for some beer with his coworkers. The couple has an argument over his late arrival, and Sloane says “I took what you told me at face value.” What does she mean?

It turns out Travis was not only drinking a few beers with the guys, he was playing poker with them. At one point, he was dealt a lone 2 of clubs. As anyone who plays poker can tell you, that’s not worth much. But later Travis also got the 2 of hearts, the 2 of spades, and the 2 of diamonds. (No wonder he didn’t want to go home yet, he was on a winning streak.) Though each card had a face value of 2, together with the other cards, it was also part of a winning hand. It represented something in addition to its exact value. Luckily for Travis, no one had a better four-of-a-kind, (like four 3s, or four 7s) so Travis won, appropriately enough, $2222. As you can imagine, this made Sloane a little less unhappy about his being late, especially when he gave her half his winnings. (This is called a bribe, and a smart one because Sloane was about to give Travis’s face an entirely different value.) Let’s break down the place value of the money Travis won.

Appendix

Percents

$ 2222 is actually:

2000 200 20 2

or or or or

The 2 in the 2000 is in the “thousands” place value. The 2 in the 200 is in the “hundreds” place value. The 2 in the 20 is in the “tens” place value. The 2 in the 2 is said to be in the “ones” place value. Up next is the are some exercises about place value.

Phone Pad 1

We call this set of numbers digits as they are used in infinite combinations to represent all our numbers.

Section 1.1 Face and Place Value

Whole Numbers

The Place-Value System Example One: Place the following number 4,321 into the Chart.

Tens

Ones

Practice 1: Complete the Place Value Chart for the following numbers.

Decimals

Hundreds

Thousands

Ten Thousands

Millions

Hundred Thousands

Pick up

The MLC Bookmark which includes the Times Table & Place Value Chart

a) 4,321 b) 159 c) 34,872 d) 9,330 e) 10,555 f) 45,350 g) 15,389 h) 310,302 i) 3,405,029 Fractions

Ones

Tens

Hundreds

Thousands

Ten Thousands

Hundred Thousands

Millions

Your Turn: What number is in the

following place values:

b) ones

c) ten thousands

d) thousands

e) thousands

f) hundreds

g) ones

h) hundred thousands

i) millions

Percents

a) tens

Ratio and Proportion

Appendix

Decimals

Whole Numbers

Chapter One

Reading Numbers Now that you have seen that the placement of digits represents a particular amount, this will help when you need to read, say, and spell out large numbers. Let start small. Read these out loud:

Fractions

843

You should say:

Ratio and Proportion

Three

Forty-Three

Eight hundred forty-three

Some of you probably said: “Eight-hundred AND forty-three.” This is incorrect as the word AND is reserved to indicate the decimal point. You will not be struck down by lightning for this error although you never know what will set off a Math teacher. They’re very unstable, you know.

As the numbers climb, the commas act as place markers. Every three digits, there’s another comma and each comma signals the name, Such as thousands, millions and billions. Example One: Read and write out 2,843 Two-thousand, eight-hundred forty-three Example Two: Read and write out 4,662,843 Four million, six hundred sixty-two thousand, eight hundred forty-three

Percents

Example Three: Read and Write out 23,409,620 Twenty three million, four hundred nine thousand, six hundred twenty Did you have any trouble? That’s where the commas can help. One comma means you’re into the thousands; two commas, into the millions. What do three commas mean? (Hint: How many people live on the planet?)

Appendix

6,000,000,000 That’s right. Six billion. Do you know approximately how many people live in the United States?

Section 1.1 Face and Place Value

B:. 300,000,000

C: 3,000,000,000

Your Turn: Write out your best guess __________________________ For practice reading numbers visit this web site, http://www.mathcats. com/explore/reallybignumbers.html

Expanded Notation

Decimals

If you want to return to the place value chart from Practice 1 and read those numbers, you will find the written words spelled out following the answers for Practice 1 found at the end of this chapter.

Whole Numbers

A. 30,000,000

Expanded notation takes a number apart so that the value of each digit is shown separately. Here is our friend 4,321 written out in expanded notation.

As you can see, the true value of each digit is taken out and placed into a simple equation.

Fractions

4,321 = 4000 + 300 + 20 + 1

Here are some more examples:

Forty five thousand, three hundred twenty seven. 40,000 + 5,000 + 300 + 20 + 7 Example Two: Write out and then record 9,043 in expanded notation.

Ratio and Proportion

Example One: Write out and then record 45,327 in expanded notation.

Nine thousand, forty three 9,000 + 0 + 40 + 3 Percents

Example Three: Write out and then record 1,324,500 in expanded notation. One million, three hundred twenty four thousand, five hundred. 1,000,000 + 300,000 + 20,000 + 4,000 + 500 + 0 + 0

Appendix

Decimals

Whole Numbers

Chapter One

Practice 2: Expanded notation and writing out the numbers. Translate the following population statistics for New Mexico towns into Expanded Notation and then into words. a) Madrid: pop 149

b) Placitas: pop 3,779

Fractions

c) Corrales: pop 7,893

Percents

Ratio and Proportion

d) Carlsbad: pop 25,410

e) Sante Fe: pop 72,056

f) Rio Rancho: pop 71,607

Appendix

g) Albuquerque: pop 518,271

h) New Mexico: pop 1,954,599

Section 1.2 Rounding

Whole Numbers

Section 1.2 Rounding When Rhonda goes shopping, she keeps a running tab in her head of how much she is spending. She says to herself, “These jeans I can’t live without cost $49.95, so that’s $50.00. The new DVD I’ve been waiting for is $24.05, so that’s $24.00 more dollars to add to my bill.” What she does is take the amount and bumps the value up or down to the nearest whole dollar. This is called rounding.

31 32

35 36

42 43

44 45

46 47 48

Five is our halfway number and that five

means we round up.

Fractions

Sometimes rounding will mean dropping down to the nearest 10. If we are asked to round 32 to the nearest 10 we can see that 32 is between 30 and 40 but closer to 30, so we round 32 down to 30.

Decimals

Take a look at this number line. If we are asked to round 49 to the nearest ten’s place, then we can see that although 49 is between forty and fifty, the nearest ten to would be 50, so we round 49 up to 50.

Steps to Rounding

First: Read carefully to determine what place value is indicated for rounding. Second: Look at the first number to the right of this place value. This number to the right is going to become a zero. But before it leaves it will have a say as to whether the marked number to the left will stay the same or bump up one number. If the digit to the right is less than 5, we don’t change the number . If the number to right is 5 or more, we bump the number up by one. The rounded digit stays the same or goes up, it never goes down.

Ratio and Proportion

Find out if it is the one’s, ten’s, hundred’s, thousand’s etc. Mark that digit. 

Third: We replace all the other numbers to the right of the round-off number with zeros.

To determine if 23 is closer to 20 or 30 we follow the steps.

Percents

Example One: Round the number 23 to the nearest ten. First: We note that 2 is in the tens place so we place a small pencil mark under the two. 

Third: now, we replace all the other numbers to the right of 2 with zeros; and we’re done! 23 ≈ 20

Appendix

Second: We look at the number to the right of the two. In our case it is a 3. Since 3 is less than 5, the 2 will stay the same.

Whole Numbers

Chapter One

Example Two: Round the number 459 to the nearest hundred. That’s 460. WRONG ANSWER ! I said to the nearest hundred. Four is in the hundred’s place. That’s our place value to round. The next digit on the right is a 5. The rule says for 5 or more we bump the four up by one, then use zeros to hold the other places so:

☟

This digit will stay the same or go up one.

The squiggly equal sign ≈ means approximately equal.

459 ≈ 500

Decimals

This digit tells the 4 to round up one and then becomes a zero along with the rest of the numbers to the right..

Fractions

Example Three: Round the number 6,349 to the nearest thousand. Six is in the thousand’s place so place a small pencil mark under the digit for reference. The next digit to the right is 3 and it is not big enough to change the 6 so we just drop the rest of the numbers and use the zeros to hold their places:

☟

This digit will stay the same or go up one.

6,349 ≈ 6,000

Ratio and Proportion

This digit tells the 6 to stay the same and then becomes a zero along with the rest of the numbers to the right..

Example Four: Round 67,928 to the nearest thousand. First: Place a pencil mark under the number in the thousands place. Second: Examine the digit to the right of the 7. It is a 9, so we round the 7 up to an 8. Finally: Drop the rest of the digits and replace them with zeros.

☟

This digit will stay the same or go up one.

Percents

67,928 ≈ 68,000 This digit tells the 7 to round up one and then becomes a zero along with the rest of the numbers to the right..

Example Five: Now, round 47,628 to the nearest hundred First: The number in the hundreds place is 6 so place a small pencil mark there for reference.

Second: Look at the digit to the right of the 6. It is a 2, which is less Appendix

than 5, so we do not change the 6.

Finally: We replace all the other numbers to the right of the 5 with zeros. (You underline and make your own notes for this one!) 47,628 ≈ 47,600

Section 1.2 Rounding

1. Round to the nearest ten: a) 83 b) 45 c) 65

d) 57

e) 138

f) 2,384

d) 7692

e) 1643

3. Round to the nearest thousand: a) 4,671 b) 27,863 c) 867

d) 375

e) 7,021

Decimals

2. Round to the nearest hundred: a) 247 b) 661 c) 2781

Whole Numbers

Practice 3: Round Whole Numbers.

Fractions

4. Round to the nearest ten thousand: a) 37,568 b) 51,923 c) 761,685

Ratio and Proportion

5. Round to the nearest hundred thousand: a) 137,568 b) 651,923

6. Round to the nearest million: b) 4,759,543

Percents

a) 1,037,568

The Rounding Poem

Appendix

Find the spot youâ&#x20AC;&#x2122;ve got to round. Check right next door, And see what youâ&#x20AC;&#x2122;ve found. 5 or more, you up the score. 4 or less, let it rest. All the numbers to the right Zero out and say goodnight.

Decimals

Whole Numbers

Chapter One

Section 1.3 Whole Number Addition Any preschooler with a cookie who sees that her brother also has a cookie discovers pretty early on the magic of addition. If she can take his cookie, she’ll have two, and if she can steal another one from the cookie jar, she’ll have three. (Years later, she will probably be able to calculate the calories in each cookie and how far she has to walk to work them off as well, but that math is for later chapters.) Your Turn: Translate this graphic into an equation: +

Fractions

_________________________

Your answer should be: 1 + 1 + 1 = 3 (3 is the total, or sum) Now add the fingers of your left

. This problem can be written 4 ways:

5+2=7

2+5=7

5 +2 7

2 +5 7

This line means equals

Let’s add some more, bearing in mind that if you don’t automatically know the answers you can literally find them by counting on your fingers. This is fine, but it will make adding larger numbers together more time-consuming. The following practice will give you a chance to memorize the addition table so that you no longer have to think twice when totaling small sums. This next table can help you to organize addition facts. Organize the facts in rows which run from left to right (horizontally) and columns which run from top to bottom (vertically). Record the results where any row crosses any column. Look at the example in the chart 5 + 2 = 7 . The seven is placed at the intersection of row 2 and column 5. As well as at the intersection of Row 5 and column 2.

Appendix

Percents

Ratio and Proportion

hand

to two sticking up from your right

Section 1.3 Whole Number Addition

Whole Numbers

Practice 4: Complete the Addition Table.

1 Decimals

3 4 7

Fractions

5 6 7

9 For larger totals you can group and add any pair then combine the total. Parentheses are notations which tell us to do that operation first. In the following number sentence you will see that we can re-group added numbers without changing the result. (3 + 9) + 5 =17

3 + 14 = 17

12 + 5 = 17

example of

associative property Percents

3 + (9 + 5) = 17

Ratio and Proportion

Your Turn: Can you find an easy way to add these numbers by grouping? 1 + 9 + 8 + 2 + 4 + 6 + 7 + 3 + 5 + 5 + 2+ 3 + 1 + 4 = ? When adding you can also change their order. 4 + 7 = 11 and 7 + 4 = 11

commutative property

Your Turn: Can you rearrange these numbers to add them faster? 1 + 2 + 3 +4 +5 + 5 + 6 + 7 + 8 + 9 = ?

Appendix

example of

Practice 5: Add Single Digits a) 3 + 5 + 1 =

b) 5 + 3 + 1 =

c) 4 2 +1

d) 2 4 +1

e) 1 + (3 + 2) =

f) (1 + 3) + 2 =

g) 9 +4

h) 4 +9

i) 7 + 3 + 4=

j) 4 + 7 + 3=

k) 8 + (2 + 5) =

l) (8 + 2) + 5 =

Fractions

Decimals

Whole Numbers

Chapter One

Percents

Ratio and Proportion

A lesson in Carrying When low digits are added together, the process is fairly straight forward.

But what happens when the digits sum to over 10? Well, if it’s the column on the left, it’s fairly simple. Just add 3 + 8 = 11, and write it down.

35 +84 119

But what the column is on the right? We know 8 + 4 = 12, but if we write that down, we’d have a total of 712, and that’s clearly not right. So we do something very simple, called CARRYING

Appendix

35 +34 69

We write down the 2, and “carry” the 10, which becomes a 1 when placed in the tens column. Then we add that 1 to the 3 and the 4 to equal 8.

+44

712

44 82

Section 1.3 Whole Number Addition

A) Step 1 1 ....

27 +8 5

Step 2 1....

27 +8 35

B) Step 1

Step 2

1....

46 +4 0

1....

46 +4 50

b) 13 + 2 + 9 =

16 +9

f) (13 + 7) + 5 =

Double-digit addition also uses CARRYING. The sum of any number greater than 10 must carry over its group of 10 to the next larger place value which is found in the column to the left. While its left over number that is less than 10 remains in the original column.

289 +332

This 1 is carried from 8+3+1 =12 and is added to the 2 and 3 in the third column.

Appendix

621

This 1 is carried from 9 + 2 = 11 and is added to the 8 and 3 in the second column

Percents

Example: Add these three digit numbers.

Ratio and Proportion

e) 13 + (7 + 5) =

9 +16

Fractions

a) 13 + 9 + 2 =

Decimals

Practice 6: Add and Carry as needed.

Whole Numbers

Examples: Addition with carrying

Practice 7: Add these Two and Three Digit Numbers a) 323 + 427 + 25 =

b) 25 + 323 + 427 =

c) 247 + 428

d) 428 + 247

e) 327 + (225 + 187) =

(327 + 225) + 187 =

Fractions

Decimals

Whole Numbers

Chapter One

Ratio and Proportion

Example One: Add 359 + 18 + 458

Arrange them in columns to get all the ones, tens and hundreds numbers right under each other.

1 2

359 18 +458

Appendix

Percents

835

This 2 is carried from first column and then added to the 5,5, and 1 of the second column

The one is carried from the second column and is added to the 3 + 4 in the third column

Example Two: Add 542 + 989 + 605 Sometimes, numbers carry into an â&#x20AC;&#x153;emptyâ&#x20AC;? column, like in this sum. Notice this sum has a number 2 in the thousands column. Here the 2 is carried from 3rd column to the empty 4th column

211

542 989 + 605 2,136

Section 1.3 Whole Number Addition

b) 250 + 326 + 427 =

247 428 + 350

350 247 + 428

e) 327 + (425 + 287) =

Decimals

a) 326 + 427 + 250 =

Whole Numbers

Practice 8: Add 3 Digits and create Fourth Columns.

f) (327 + 425) + 287 = Fractions

Friday, Saturday, and Sunday. Peggyâ&#x20AC;&#x2122;s job is to keep track of the number of tickets sold: Friday 3,543 tickets were sold, on Saturday they sold 5,350 more tickets, while on Sunday they sold only 2,475. In total, how many people should she say attended the event.

Ratio and Proportion

Example: A car show at the Convention Center ran for three days,

3,543 + 5,350 + 2,475 = the total sold which was 11,368 tickets which means 11,368 people. Percents

111

3543 5350 + 2475 11,368

Appendix

Whole Numbers

Chapter One

Practice 9: Addition

a) Alex has a checking account at the Bank of Albuquerque with an opening deposit of $100. He makes three more deposits for $50, $35, and $65. What is his balance after the three deposits? He needs to write a check for $143 dollars for his cable bill, will he have enough in his account?

Decimals Fractions Ratio and Proportion

Percents Appendix

Word Problems

Jennifer is an intern for the ABQ Transit Authority and has been asked to calculate the following for a study on commuter habits.

Travel Time Time spent traveling to work Less than 5 minutes

b) How many people have a travel time of 60 minutes or more?

87106 Number of people 605

5 to 9 minutes

2629

10 to 14 minutes

3415

15 to 19 minutes

2854

20 to 24 minutes

1770

25 to 29 minutes

399

30 to 34 minutes

872

35 to 39 minutes

40 to 44 minutes

176

45 to 59 minutes

292

60 to 89 minutes

192

90 or more minutes

c) How many people have a travel time of at least 5 minutes, but no more than 19 minutes?

d) In what time category do the largest number of people travel ?

e) What time category does your commute fit into? How many total minutes do you spend commuting in a week?

Section 1.3 Whole Number Addition

f) Candice takes a culinary arts master class in Las Vegas, Nevada. She travels round trip from Albuquerque, New Mexico on Untidy Airlines. One way is 671 miles. How many miles did Candice fly?

Whole Numbers

g) Candiceâ&#x20AC;&#x2122;s husband Nash is working for the forest service and needs to determine the risk of a forest fire in the Sandia Mountains. He hikes up the La Luz trail to locate an area of dry dead trees. The elevation at the bottom of the trail is 7078 feet above sea level. He climbs 2354 feet to find the dried up trees. What is the elevation at this location?

Decimals Fractions Percents

i) Marty lives at the Paseo del Volcan Apartments and pays $550 a month for a two bedroom apartment. The electric bill is $38. The cell phone bill is $89. Marty also has a cable that is $28. How much money does he need to make each month just to cover these bills?

Ratio and Proportion

h) Marty walks into Wallblues off San Mateo and Zuni. The Girl Scouts are selling cookies for $3.00 a box. He buys one box for his daughter and one box for his sister-in-law. Then goes inside to purchase $17.00 worth of dog food for Rimmie. How much did he spend on his trip to Wallblues?.

Appendix

Fractions

Decimals

Whole Numbers

Chapter One

Section 1.4 Whole Number Subtraction “Subtract” means to take away. Addition combines, subtraction removes. Hold up your hand. You should see 5 fingers. Now, bend down your thumb and forefinger. (like an “OK” sign.) That should leave 3 fingers sticking up. This is represented by this simple equation: 5

5 – 3 = 2 or -3 2

Factoid: Just as the total of two numbers added is called the sum, the result of one number subtracted from another is called the difference.

If you reverse the process, and put your fingers back up, you have checked your work by hand. (Yes, that was a joke. Mathematicians are extremely funny, as you’ll discover.) 5 -2 You simply cover the 5 with your thumb, and add back up 3 Addition comes more easily to many than subtraction. So, this is an easy way to use one process to check the results of the other.

Ratio and Proportion

b) 5 - 2

c) 9 - 7

d) 7 - 6

e) 9 - 3

f) 3 - 1

g) 6 - 3

h) 8 - 7

i) 9 -2

j) 4 - 3

k) 7 - 6

l) 8 -8

m) 7 - 4

n) 8 - 2

o) 9 - 6

p) 5 - 1

q) 9 - 2

r) 9 - 8

Appendix

a) 8 -5

Percents

Practice 10: Subtract Single Digits

Now let’s take a look at two-digit subtraction. This is fairly straight forward when the numerals

on top are both higher than the each numeral below, as in:

67 -24 43

Section 1.4 Whole Number Subtraction

Whole Numbers

Remember when we had to “carry” in addition? In subtraction, the equivalent is called “borrowing” Example: Let’s say that you have 34 dollars and you lend 5 to your friend. How much do you have left?

3¹4 2Y - 5 29

The 4 is changed to 14 by borrowing a 10 from the 30. Then adding the 10 to the four shown here by placing a small 1 to the upper-left of the 4.

Decimals

The 30 is then reduced to a 20 shown here by crossing out the 3 and placing a 2 directly by it.

Fractions

When the number in the first column is not high enough for subtraction you will need to borrow a group of 10 from the next largest column. In this case we changed the 34 to a 20 + 14. This allows us to subtract the 5 in the ones column.

3¹4 2Y - 5 9

b) 13 - 8

c) 23 - 7

d) 35 - 6

e) 11 - 3

f) 11 - 9

g) 21 - 3

h) 44 -7

i) 60 -2

j) 73 - 8

k) 45 - 6

l) 57 - 9

m) 12 - 4

n) 13 - 2

o) 76 - 7

p) 94 - 6

q) 83 - 4

r) 24 - 5

Appendix

Of course, this gets a little bit more complex when you have groups of higher numbers, because you sometimes have to borrow from the numbers twice. Here is a sample word problem that requires borrowing with two and three digits.

Percents

a) 22 - 9

Ratio and Proportion

Practice 11: Two Digit Subtraction with Borrowing

Decimals

Whole Numbers

Chapter One

Example One: Alice has $500 in the bank, but then she had to pay her PNM bill which was $180. What will her new bank balance be? The second 0 is changed to 10 by borrowing a 1 from the 5. Then add the 10 to the zero. Shown here by placing a small 1 to the upper-left of the 0.

โ ด ยน 5 00 Y - 1 80 320

The 500 is then reduced to a 400. Shown here by crossing the 5 and then placing a 4 to the upper left of the 5.

Example Two: Alice then buys $67 worth of groceries.

2 1 1

320 - 67

The 0 is changed to 10 by borrowing a 10 from the 20. Then add the 10 to the zero. Shown here by placing a small 1 to the upper-left of the 0.

The 300 is then reduced to 200 Shown here by placing a 2 next to the 3.

Now she has $253 left in the bank.

Appendix

Percents

Ratio and Proportion

Fractions

She has $320 left in the bank

Section 1.4 Whole Number Subtraction

cover a check she writes for $175 worth of books. After the check clears what will her balance be? 253 + 600 $853

7 14 1

853

-175 678

In this case we have had to place two marks over the 5 to represent two separate transactions. Look at the marks 1 and 4. The 4 represents the 10 we donated to the ones column and the 1 represents the 100 we borrowed from the 800.

Whole Numbers

Example Three: Then she deposits her $600 Pell Grant, just in time to

$678 is your answer

1 9 91

2000 -189 1811

$1811 is your answer

b) 1257 - 1168

c) 621 - 589

d) 5012 - 2113

e) 51 - 29

f) 9612 - 8166

g) 469 - 128

h) 2009 - 1257

i) 129 - 67

j) 5633 - 1241

k) 344 - 155

l) 913 - 28

m) 1008 - 41

n) 1241 - 452

o) 267 - 88

Percents

a) 85 - 29

Ratio and Proportion

Practice 12: Subtraction with Borrowing. .

Fractions

Taking a look from the left to right, 1000 is borrowed from the 2000. So the 2 becomes a 1. Then 100 is borrowed from the 1000, so the 0 becomes a 9. Then 10 is borrowed from the 100 so the 0 becomes a 9 and finally the 10 is represented with a 1 on top. Here is an equation representing the transaction. 2000 = 1000 + 900 + 90 + 10

Decimals

Example Four: After graduation, Sloane must begin paying back her student loan. She borrowed $2000, and her payment is $189 a month. How much will she still owe after making one payment?

Appendix

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter One

Practice 13: Subtraction Word Problems. Elena figures out her monthly budget. Elena is a court reporter who brings home $2082 a month. She immediately puts $82 in her rainy day fund and doesn’t touch it (except on a rainy day, of course). Elena is also a big party-girl, but she’s a very responsible one. Every month, she figures out all of her expenses before she decides what she can spend on going out. (She’s been doing this ever since she broke up with her stupid boyfriend, Hector, and hopes she can stop when she meets someone else who will pay his own way instead of that deadbeat.) So, Elena starts with 2000. Here are her bills: Rent : $540.12 Car Payment: $209.62 Student Loan Repayment:$103 Cable and Internet: $86.50 Gas & Electric: $36.92 a) Step 1: First she Rounds up those bills to the closest dollar!

$540 $210 $103 $87 $37

b) Step 2: Subtract this total from $2000 $2000

-____

Percents

c) Step 3:Then Elena allocates $120 a week for groceries. How much is that a month?

120 120 120 120

Appendix

d) Step 4:Subtract that Step 3 result from the result you got in Step 2 e) Step 5: Elena spends about $25 a week on toiletries and makeup Add that up

Section 1.4 Whole Number Subtraction

f) Step 6: Subtract that Step 5 result from the result you got in Step 4

Whole Numbers

25 25 25 + 25

She should have $443. If you didn’t get that, go back and check your math. If you do, Hot-Diggity for Elena! Not so fast, party-girl! g) Elena conveniently forgot that she spends at least $50 a week on gas, and that’s probably going to be about $60 a week this month, with prices going up. So how much more does she need to subtract?

Decimals

Subtract that from 443.

Fractions

60 60 60 + 60 240

443 - 240

Ratio and Proportion

h) Then Elena realizes it’s her sister’s birthday this month. She has to get her a present. Or maybe she’ll just take her out for drinks and work it into her party budget. She’ll have to get her some earrings or something though. Subtract $20.

i) And shoot, and there’s the $40 she borrowed from her cousin. Subtract that too. Percents

That leaves her $143, for movies, drinks, and cabs if she drinks too much and can’t drive home. How much is that a week? Elena won’t know until she divides by 4, and for that she has to read the next chapters and learn multiplication and division.

Appendix

Chapter One

Whole Numbers

Section 1.5 Multiplication of Whole Numbers In the previous section, we added the same number repeatedly, as in: 60 60 60 + 60 240

Ratio and Proportion

Fractions

Decimals

This repeated addition can be written out as the following multiplication problem: 4 x 60 = 240. We can deduce this answer from the multiplication of 6 x 4 = __ If you knew without thinking that the answer is 24, then you probably know your multiplication tables, and it wonâ&#x20AC;&#x2122;t be hard for you to fill out the following chart. If you had to spend some time figuring it out, then itâ&#x20AC;&#x2122;s even more important that you complete the chart and make sure you memorize the tables until they are second nature.

Practice 14: Complete the Multiplication Table. Helpful Tool Get your Times Table Bookmark at the

MLC

Percents

0 1 2 3 4

5 6

7 8 Appendix

Section 1.5 Multiplication of Whole Numbers

Letâ&#x20AC;&#x2122;s take a look at a simple rectangles that has been split into a square grid. What are the times table that represent the square?

A: 4 x 7 =28 squares

Decimals

A: 5 x 3 =15 squares

Whole Numbers

We are now going to take the multiplication table apart and look at the endless rectangles that can be drawn from the chart. Here we will see the connection between the geometry of rectangles and multiplication. This connection is called AREA. And everything built by man is constructed using this mathematical connection.

Your turn: complete the following:

Fractions

Percents

Your turn, In the grid below, complete the following: 1x3, 4x4, 2x10,10x2

Ratio and Proportion

Here we drew a rectangle around the appropriate amount of squares to represent the following times table 5x6, 1x10

Appendix

Decimals

Whole Numbers

Chapter One

Prime and Composite Numbers What are prime numbers? A prime number is a number that can only be divided by itself and 1. The first six prime numbers are 1, 2, 3, 5, 7 and 11. What are composite numbers? A composite number can be divided equally by a number other than one or itself. 10 is composite because it can be divided evenly by 5 and 2. Your Turn: Six is not a prime number because it can be broken up by which two factors? Your Turn: Fifteen is not a prime number because it can be broken up by which two factors?

Appendix

Percents

Ratio and Proportion

Fractions

Here are the three prime factors that multiply to 44. (2)(2)(11) = 44 Your Turn: Can find the prime factors for the number 24?

Practice 15: Find the prime factors that multiply to the following: a) ( ) (

c) (

) (

) = 15

b) (

) = 33

d) (

) (

) = 14

) (

) = 12

Find the correct composite numbers of the following prime factors. e) ( 2 ) ( 5 ) ( 3 ) =

f) ( 2 ) ( 11 ) =

g) ( 5 ) ( 7 ) =

h) ( 2 ) ( 3 ) ( 7 ) =

Section 1.5 Multiplication of Whole Numbers

In the following table, cirlce the prime numbers.

In the following table, cirlce the composite numbers.

Ratio and Proportion

Fractions

Your Turn:

Decimals

Whole Numbers

Your Turn:

Percents

Here are several more ways to show multiplication. See Glossary for definitions. Factors are considered the 13 and 3 while a product is another name for the answer, 39 â&#x20AC;&#x153;Thirteen times threeâ&#x20AC;?

13 x 3

(13)(3)

all equal 39

Appendix

Whole Numbers

Practice 16: Multiply the one and two digit problems.

Percents

Ratio and Proportion

Fractions

Example: Suppose you have 6 packages of paper. On the label it tells you each package has 25 sheets. How many sheets of paper do you have altogether

Decimals

Chapter One

25 x 6 150

a) 22 x 9

b) 13 x 8

c) 23 x 7

d) 35 x 6

e) 14 x 3

f) 12 x 9

g) 26 x 3

h) 44 x 7

i) 65 x2

j) 73 x 8

k) 45 x 6

l) 57 x9

m) 13 x 4

n) 37 x2

o) 76 x7

p) 94 x 6

q) 83 x 4

r) 53 x 8

The following example walks you through each step in the process of the more complex three digit multiplication problems.

Example: Multiply the following.

1 The 8 is up first. 8 x 6 is

48. That means 8 ones and 4 tens. Leave the 8 in the 1â&#x20AC;&#x2122;s place and carry the 4 to the tens place.

2 Next 8 x 4 is 32 + 4 is

36. Leave the 6 in the tens place and carry the 3 to the hundreds place. Next 8 x 3 = 24 add the 3. No place to carry so we write the result. 34

Appendix

The 3 from the product of 6 x 5 is carried to the 2. Notice the zero is placed in the answer column. Then the 3 is added to the product of 6 x 2

346 x 198 2768

3 Next up is the 9. First add a 5 Last up is the 1. Add two zero on the second because we are now in the tens place, then repeat the process.

zeros on the third row, then repeat the multiplication process.

4 Notice that your carrying

6 After multiplication, add the

marks change. Be careful! Erase the old ones if you need to.

346 x 198 2768 31140

three lines to obtain your total.

346 x 198 2768 31140 + 34600 68508

Section 1.5 Multiplication of Whole Numbers

a) 28 x 15

b) 245 x 69

c) 640 x 188

d) 2345 x 608

Whole Numbers

e) 1150 x 640

f) 3789 x 234

g) 50023 x 300

h) 20189 x 54

Decimals

Practice 17: Complete the Double and Triple Digit Multiplication.

i) 30

j) 5 ( 9002) = Fractions

. 269 =

9 x 24 x 15 = $3240 in a term (plus because it’s a work-study job, there’s no tax! ) $3240 is your answer

Ratio and Proportion

Example: Johnny earns 9 dollars an hour and he works 24 hours a week as a AV tech at school. How much does Johnny earn in a 15 week term? Because there are repeated numbers this can be determined by multiplying:

Your Turn This is an exercise that many of you may have already done in one form or another in real life. If not, you almost certainly will encounter something very similar in the future.

$12 job for 30h/wk

$10 job for 40h/wk Appendix

per week

Percents

You’ve been job hunting, and are finally offered two jobs at the same time: a part-time position that pays $12 per hour, for 30 hours a week, and a full-time position that offers $10 per hour but 40 hours a week. You like one job better, but you need above all to take home as much money as possible. Calculate how much each position pays:

per month (at 4 weeks/month) per year

Fractions

Decimals

Whole Numbers

Chapter One

Practice 18: Multiplication Problems a) Joey has a job driving a van for the Watermelon Prep Schoolâ&#x20AC;&#x2122;s field trip and is taking a group to Carlsbad Caverns. He works 20 hours a week and is paid 15 dollars an hour. What will he get paid for this 10 -week summer job? b) With the whole group in the van, he heads out to Carlsbad from Albuquerque. He checks the odometer when he stops for lunch and sees that he has traveled 139 miles from Albuquerque. If that is the halfway point, how far is Carlsbad from Albuquerque?

The table below shows the entrance fee to Carlsbad Caverns. Use the table to answer the following questions. Entrance Fee Adult

Children 6-15

Children 5 & Under

c) Watermelon Prep School will reimburse Joey for his group of 10 people to get into the caverns. He and two parents make up three adults while the rest are children aged 6-15. Two people would also like to take the Audio Tour. What will be the total cost for the tickets?

d) The group will be staying in Carlsbad for two nights at the Carlsbad Inn. The cost of one room is $39 per night. If he rents 3 rooms for two nights, how much should he expect to pay when you check out?

Appendix

Ratio and Proportion

Percents

Audio Tour Rental

Free

e) The motel gives a complementary breakfast so he will only be paying for two meals a day for two days for himself and each of the nine persons in your group. Lunches cost about $6 and dinners about $10.How much should he expect to pay for meals while they are in Carlsbad?

Section 1.6 Division of Whole Numbers

You’ve invited 3 friends over to watch the playoffs, and order a large pizza with 8 slices. If all four of you eat the same amount, how many slices can each of you have? If your answer is not “2,” you need to entertain more! If it was, then congratulations, you have done basic division.

8÷4=2

pairs with the multiplication problem 2 x 4 = 8,

24 ÷ 6

24 6

24/6

6)24

Order counts!

8÷2=4

but

2 ÷ 8 = .25

Ratio and Proportion

24 is called the dividend. 6 is called the divisor. The answer is called the quotient, which is 4.

Fractions

Here are four ways to show the same division problem. We read the first three “24 divided by 6”while the forth is read “ 6 into 24.

Decimals

Division Facts in a nut shell.

Whole Numbers

Section 1.6 Division of Whole Numbers

Your Turn: Label the following: Percents

7 9)63

Appendix

Decimals

Whole Numbers

Chapter One

Practice 19: Single Digit Division. a) 3)6

6 2

b) 8 2

c) 4)4

d) 9 ÷ 3

f) 4)8

g) 4 ÷ 2

h) 2)2

Example: Joanna a needs to bake 36 cupcakes for the school bake sale. She

wants to package them on paper plates, four in each plate. How many plates of cupcakes will Joanna have? 9

Appendix

Percents

Ratio and Proportion

Fractions

4)36 Looks like she has nine plates of 4 cupcakes. I’m sure they’ll be real good, too. But they’re for the bake sale, remember?

Practice 20: Double Digit Division no Remainders. a)

b) 18 ÷ 2

3)12

f) 2)16

28 4

4)16

d) 15 ÷ 3

2)14

h) 24 ÷ 3

Example: Just like a lot of things in life, division sometimes doesn’t come out evenly. Sometimes there’s some left over. If the cupcake batch had 38 cupcakes and you still wanted to package them 4 to a plate, you would have had 2 left over when you got done. I guess you could eat them yourself. Sometimes leftovers are not so bad.

9 4)38 -36 2

Those left over numbers are called ”remainders”. We write an “r” next to the answer that shows there was a remainder and write the left over amount with the “r”.

9r2 4)38 -36 2

Section 1.6 Division of Whole Numbers

3)11

h) 9)381

7)357

i) 5)207

4)15

2)3201

3)13

3)1625

6)235

2)19

8)4024

8)2960

l) 9)456

7)78

2)740

4)2084

3)513

Decimals

2)17

Whole Numbers

Practice 21: Double Digit Division with Remainders.

Example: Divide 135 by 10 2) Multiply: 1 times 10 and place the 10 right under the 13

3) Subtract: 13 minus 10 is 3 1 10)135 -10 3

4) Bring Down: the 5 and place it next to the 3

10)135

10)135 10

10)135 -10 35

7) Subtract: the 30 is subtracted from the 35 to yield 5

13 )135 10 -10 35 30 8) Remainder: is 5. So write the letter r with the remainder of 5 next to it.

13 r 5 10)135 -10 35 - 30 5

Appendix

13 )135 10 -10 35 - 30 5

6) Multiply: 3 times 10 is 30

Percents

13 10)135 -10 35

Ratio and Proportion

Begin the Process Again 5) Divide: 10 goes into 35 three times

Fractions

1) Divide: Ask yourself, how many 10’s are in a 13 ? One, so place the 1 directly over the 3.

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter One

Practice 22: Two and Three Digit Division with Remainders. a)

b) 12)135

e) 10)110

f) 15)155

g) 15)167

h) 19)234

k) 13)146

l) 10)87

p) 10)705

11)135

11)210

m) 10)237

19)334

12)123

11)237

13)135

15)152

d) 10)89

11)133

Division Word Problem Preparation

Percents

Example: A car can drive 340 miles on 17 gallons of gas. What is the cars Miles per Gallon also abbreviated MPG? â&#x20AC;&#x153;Perâ&#x20AC;? is one of those words that clue division, so we do a divide: 340 miles by 17 gallons.

Appendix

340 miles 17 gallons

20 17)340 34 0

The answer is 20 MPG

Section 1.6 Division of Whole Numbers

40 hours per week, the pay is $35,360 a year. You’re so used to getting paid by the hour, you’re not exactly sure how much more you’ll be making over your present salary. 40 hours = 1 week 1 year = 52 Weeks

Calculate your new salary:

Whole Numbers

Your Turn: You’ve been offered your first really good job working

$35,360 Decimals

per week per hour

Practice 23: Division Word Problems a) Sloane is working 40 hours a week and making $24,960 a year as a phlebotomist at Loveplace Hospital. Calculate what she makes an hour. Fractions

SO with the gas gauge on EMPTY, he pumps 20 gallons of gas and puts the adjustable trip odometer to 0. Over the 7 day week Nash travels 40 miles each day. At that point, the gas gauge points back to “E.” What can he estimate as his miles per gallon?

Ratio and Proportion

c) Candace manages a catering company and is in charge of the ordering food for an upcoming luncheon. She budgets of $1550 for 62 people. How much money can be spent on food for each attendee?

Percents

b) Nash figures he can afford new car payments if he gets at least 15 miles more per gallon than he’s getting now on his old truck. Since there are several car models to choose from with different mpg ratings, he wants to be absolutely sure about his present mileage.

Appendix

d) Mark has $1800 in credit card debt he has vowed to pay off over the next 9 months. How much should he pay off each month?

Section 1.7 Area and Perimeter

Perimeter

s rd ya

ya rd s

A baseball diamond is basically a square, as the distance between the bases are all the same length. The perimeter of this square is the length around it, or, the distance from home base to first, first base to second, second base to third, and third base to home. Basically, it is the length of the outline of the square. Does anyone know what the distance is between the bases? Seriously, does anyone know? Cause we don’t.

30 y

s rd

ya Fractions

The perimeter of a rectangle, such as a football field, would be measured differently, because all of the sides are not of the same length—the shorter sides being referred to as width You may be thinking, “But I thought a football field was 100 yards long! And I know a yard is 3 feet long, therefore 100 yards should make for a length of 300 feet! What do you think accounts for the extra 60 feet? Answer: The end zones.

Percents

160 feet

360 feet

End Zone

Ratio and Proportion

ar d

Okay, let’s say it’s 30 yards. The calculation of the perimeter of the square constituted by the 4 bases is fairly simple, as you need only multiply by 4 as in: 30 yds x 4 = 120 yards.

End Zone

Decimals

Whole Numbers

Chapter One

360 feet

Appendix

The football field is 360 feet long and 160 feet wide (this time we looked it up.). Calculating the perimeter, then, is a matter of simple addition. 360 + 360 + 160 + 160 = ______feet L + L + W + W = P Or if you want to get fancy try the formula 2L + 2W = P: (2

. 360 ) + (2 . 160) = _______ feet 42

Section 1.7 Area and Perimeter

of the following objects.(P = 2L+ 2W)

a) Count the grid marks to figure out the unit measure of each side. Then calculate the perimeter.

Whole Numbers

Practice 24: Calculate the Perimeter

Decimals

b) The square measures four units on each side. What is the perimeter?

Fractions Percents

d) What is the perimeter of this triangle?

Ratio and Proportion

c) stop sign - The total perimeter is 192 inches, what is the measure of each side?

pic

pic a

32 pica

Appendix

Area The area of our baseball diamond is basically the infield, or all that lies within the square itself. You may also think of area as the carpet that fills a room. For simple geometric shapes, calculating area is a matter of simple multiplication, For our square, in which each side was 30 yards, we multiply the number by itself:

30 ya rd s 30

s rd ya

Decimals

Whole Numbers

Chapter One

Can you see what we mean by 25 square feet?

When a rectangular space is involved, we multiply as well, Length times Width, or the formula uses the letters L x W = A For our football field, can you calculate the area?

Appendix

End Zone 360 feet

360 x 162 = _________ square feet.

162 feet

360 feet

End Zone

Percents

If you could draw a line down to mark each yard in both directions and then counted the resulting squares, you would get 900 squares! And each square is called a square yard. But donâ&#x20AC;&#x2122;t try to count that, it would take all day. Instead, let me give you a smaller dimension to count. How about a room measuring 5 ft. by 5 ft.. 5 feet 5 feet

Ratio and Proportion

Fractions

30 x 30 = _____ square yards.

Section 1.8 Decimal Preview

Whole Numbers

Practice 25: Calculate the areas of the following spaces.(A = L x W)

sq inches Decimals

sq inches

5 feet

sq feet

Ratio and Proportion

Fractions

sq inches

Percents

Your Turn: What is the perimeter of practice problem b) ? ________________________ Appendix

Decimals

Whole Numbers

Chapter One

Section 1.8 Decimal Preview Remember those division problems that had left over numbers called “remainders?” Another way to handle those leftovers is to use decimals. They look like any other number except they have a period (.) called a decimal point between the one’s place and the next number on the right which begins the decimal part. The last number on the right is often “rounded off.” In fact, you are probably more familiar with this type of leftover because that is the way many calculators handle them. The decimal form is also what we use for money quantities less than a dollar.

Example One: Gabe’s car went 200 miles on 11 gallons of gas. How many miles per gallon (mpg) does Gabe’s car average?

Fractions

“Per” is one of those words that clue division, so we do a divide: 200 miles by 11 gallons. 18.18

11)200.00

-1 1

-88

- 88 2

18.181818… ≈ 18.2 mpg means the car can go about 18.2 miles on a gallon of gas. We get those unstoppable numbers to stop, by selecting a position and rounding off the rest. (see page 14 for rounding rules). If you still have your MLC bookmark with the number place value chart, get it out. If you’ve lost it ask for another one. Check the number place value chart and you will see that it continues on to the right of the one’s position with even more positions. That transition point is marked with a decimal point (read AND). The numbers on the right have ever smaller and smaller values. You’re familiar with those parts in our money system. We call them coins and refer to those amounts as “CHANGE.” And you know change can add up to ever larger amounts. Look at any grocery receipt.

Appendix

Percents

Ratio and Proportion

-1 1

Notice that the division didn’t stop at the last zero of 200, but a decimal point was added and we continued dividing by adding zeros. This is the way your calculator handles division problems that don’t come out evenly. It will automatically create DECIMAL NUMBERS by adding a point and zeros and continuing to divide into ever smaller values.

Answers to the Whole Number Practice Problems

Answer to Practice Problems

Practice1:

million

hundred thousand

ten thousand

thousand

hundred

ten

one

b. c.

d. e.

h. i.

Just in case you might want to know, this is how they are written: a) four thousand three hundred twenty one b) one hundred fifty nine c) thirty four thousand eight hundred seventy two d) nine thousand three hundred thirty e) ten thousand five hundred fifty five f) forty five thousand three hundred fifty g) fifteen thousand three hundred eighty nine h) three hundred ten thousand three hundred two i) three million four hundred five thousand twenty nine

Practice 2: a) 100+40+9

One hundred forty-nine b) 3000+700+70+9 three thousand, seven hundred seventy-nine c) 7000+800+90+3 seven thousand, eight hundred ninety-three d) 20,000+5,000+400+10 twenty-five thousand, four hundred ten e) 70,000 + 2,000 + 50 + 6 seventy-two thousand, fifty-six

Whole Numbers

Chapter One

f) 70,000 + 1,000 + 600 + 7 seventy-one thousand, six hundred seven g) 500,000+10,000+8,000+200+70+1 five hundred eighteen thousand, two hundred seventy-one h) 1,000,000+900,000+50,000+4,000+500+90+9

Percents

Ratio and Proportion

Fractions

Decimals

one million, nine hundred fifty-four thousand, five hundred ninety-nine

Practice 3: 1. a) 80

b) 50

c) 70

2. a) 200

b) 700

c) 2,800

d) 7,700

e) 1,600

3. a) 5,000

b) 28,000

c) 1,000

d) 0

e) 7,000

4. a) 40,000

b) 50,000

c) 760,000

5. a) 100,000

b) 700,000

6. a) 1,000,000

b) 5,000,000

e)140

Practice 4:

Addition table answers +

Practice 5: Appendix

d) 60

a) 9 b) 9 c) 7 d) 7 e) 6 f) 6 g) 13 h) 13 i) 14 j) 14 k) 15 l) 15

f) 2380

Answers to the Whole Number Practice Problems

Practice 6:

a) 24 b) 24 c) 25 d) 25 e) 25 f) 25

Practice 7: a) 775

b) 775

c) 675

d) 675

Practice 8:

a) 1003 b) 1003 c) 1025 d)1025

e) 739

f) 739

e) 1039

f) 1039

Practice 9:

a) $250 , Yes! b) 274 c) 8,898 d) 10 to 14 minutes e) Answers will vary f) 1342 miles g) 9432 feet h) $23 i) $705

Practice 10:

a) 3 b) 3 c) 2 d) 1 e) 6 f) 2 g) 3

h) 1

i) 7

j) 1

k) 1

l) 0

m) 3

n) 6

o) 3

p) 4

q) 7

r) 1

Practice 11:

a) 13 b) 5 c) 16 d) 29 e) 8 f) 2 g) 18

h) 37

i) 58

j) 65

k) 39

l) 48

m) 8

n) 11

0) 69

p) 88

q) 79

r) 19

Practice 12: a) 56 f) 1446 k) 189

b) 89 g) 341 l) 885

Practice 13: a) $978 f) $443

b) $1022 g) $202

c) 32 h) 752 m) 967

d) 2899 i) 62 n) 789

e) 22 j) 4392 o) 179

c) $480 h) $182

d) $542 i) $142

e) $100

Fractions

Decimals

Whole Numbers

Chapter One

Practice 14: Multiplication Table Answers X

Percents

Ratio and Proportion

Practice 15: a) ( 3 ) ( 5 ) = 15

b) ( 2 ) ( 7 ) = 14

c) ( 3 ) ( 11 ) = 33

d) ( 2 ) ( 2 ) ( 3 ) = 12

e) ( 2 ) ( 5 ) ( 3 ) = 30 g) ( 5 ) ( 7 ) = 35

f) ( 2 ) ( 11 ) = 22 h) ( 2 ) ( 3 ) ( 7 ) = 42

Practice 16: a) 198

b) 104

c) 161

d) 210

e) 42

f) 108

g) 78

h) 308

i) 130

j) 584

k) 270

l) 513

m) 52

n) 74

o) 532

p) 564

q) 332

r) 424

Practice 17: a) 420 e) 736,000 i) 8,070

Appendix

b) 16,905 f) 886,626 j) 45,010

Practice 18: 1. a) $3,000

c) 120,320 d) 1,425,760 g) 15,006,900 h) 1,090,206

b) 278 miles

c) $43

d) $234

e) $320

Answers to the Whole Number Practice Problems

Practice 19: a) 2

b) 4

Practice 20: a) 4

b) 9

c) 1

d) 3

e) 3

f) 2

g) 2

h) 1

c) 4

d) 5

e) 8

f) 7

g) 7

h) 8

Practice 21: a) 3 r2

b) 8 r1

c) 3 r3

d) 4 r1

e) 9 r1

f) 11 r1

g) 42 r3

h) 41 r2

i) 503

j) 39r1

k) 50 r6

l) 521

m) 51

n) 1600 r1 o) 541 r2

p) 370

q) 370

r) 171

Practice 22: a) 12 r3

b) 11 r3

c) 10 r5

d) 8 r9

e) 11

f) 10 r5

g) 11 r2

h) 12 r6

i) 19 r1

j) 10 r3

k) 11 r3

l) 8 r7

m) 23 r7

n) 21 r6

o) 10 r2

p) 70 r5

q) 17 r11

r) 12 r 1 c) $25

d) $200

c) 24 units

d) 96 picas

Practice 23 a) $12.00

b) 14 mpg

Practice 24 a) 28 units

b) 16 units

Practice 25 a) 40 sq units d) 14 sq units

b) 24 sq units c) 16 sq units e) 25 sq feet

Chapter

Decimals Table of Contents

Section 2.1

Decimal Place Values............................................... 55

Practice 1: Words to Decimal Numbers. . . . . . . . . . . . . . 58 Practice 2: Decimal Place Value Practice. . . . . . . . . . . . . 59

Section 2.2 Rounding and Estimation.............................................. 59 Practice 3: Round to the nearest 1 or 0 . . . . . . . . . . . . . 60 Practice 4: Round to the nearest thousandths.. . . . . . . . . . . . 61 Practice 5: Round decimal numbers to the indicated place value. . . . . . . 63

Section 2.3 Comparing Decimals................................................... 64 Practice 6: Arrange decimal numbers from smallest to largest.. . . . . . 65 Practice 7: Arrange decimal numbers from largest to smallest... . . . . . 65

Section 2.4 Adding and Subtracting Decimals.................................... 66 Practice 8: Add or subtract decimal numbers as indicated. . .

Section 2.5 Multiplying Decimals.................................................. 70 Practice 9: Multiply the decimals numbers. . .

Section 2.6 Dividing Decimals..................................................... 71 Practice 10: Divide the decimal numbers. . . . . . . . . . . . . . 74 Practice 11: Divide and round to the hundredths place. . . . . . . . . . 75

Section 2.7 Decimal Word Problems.............................................. 75 Practice 12: Decimal Word Problems. . . . . . . . . . . . . . . 78

Section 2.8 Fraction Preview...................................................... 80 Preview A: Convert the following decimals to fractions.. . . . . . . . . 81 Preview B: Convert the fractions to decimal numbers. . . . . . . . . . 81

Answers to Practice Problems..................................................... 82 Copyright ÂŠ2009 Central New Mexico Community College. Permission is granted to copy, distribute and/or modify this document, under the terms of the GNU, Free Documentation License. Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is available on http://www.gnu.org/copyleft/fdl.html

The following MALL activities are available for class check out: Exploring Decimals, Using the Metric System , Writing Checks and Recording on a Check Register

Section 2.1 Decimal Place Values

Decimal Place Values

Remember the first time you watched the Olympics? You may have understood decimals before you even knew what to call them, when you figured out that a 9.9 was higher than a 9.8, but not by much. By a tenth.

Whole Numbers

Section 2.1

The word decimal comes from the Latin word ”decimus” which means “tenth.” In Math, these amounts are represented by what comes after the period or decimal point in a number. Decimals

Let’s say you’re getting paid $10.25 per hour. That .25 after the 10 represents a quarter, or 25 cents. (Guess what “Centesimus” means in Latin? Hundredth. Or two places to the right of the decimal point). Money is the most common example of the use of decimals in real life. It’s also an excellent way to grasp the meaning of that little period. First let’s try an experiment: Now add:

429 361 877 126

4.29 3.61 8.77 1.26

Fractions

Add:

1 2

$4.29 $3.61 $8.77 $1.26 $17.93

Percents

You would read 17.93 as “Seventeen and ninety-three hundredths” whereas you’d read $17.93 as “Seventeen dollars and ninety-three cents.” “Ninetythree cents” actually means ninety-three hundredths, and this reminds us of the role of place value in understanding decimals. Just as when you go left, you go up in multiples of ten, when you go right, you go down in tenths.

Ratio and Proportion

Yes, your instinct is correct.The answer is almost the same—at least as far as the digits if not the actual value of the total. The first answer is 1793, the second 17.93. If you got a little nervous about what the decimal point meant, then watch how clear it is when we add a certain sign.

Appendix

Whole Numbers

Chapter Two

Take the amount: $111.11 Broken down it’s:

100 dollars +10 dollars + 1 dollar decimal point + 10 cents + 1 cent

Fractions Ratio and Proportion

Hundredths

Tenths

Bookmark

Decimal

MLC

Ones

“Place Value”

Tens

Use this

Helpful Tool

Hundreds

Decimals

Below, we place these numbers in the place value chart.

Moving to the left, we multiply by groups of 10. For example, 10 ones are ten, 10 tens are 100, and 10 hundreds are one thousand. Moving to the right each value is divided by 10. For example: One divided by ten is .1 (1/10) and 1/10 divided by ten is .01 (1/100). Simple calculations were the reason behind the invention of the metric system, which is used in most of the world yet still lags in the United States. For instance, in the metric system, one meter is 10 decimeters, one decimeter is 10 centimeters, and one centimeter is 10 millimeters. Everything is divisible by 10. In the United States, although our monetary system is based on the decimal system ( 10 dimes = $1), it is difficult to make quick calculations with 12 inches equaling a foot, and 5280 feet equaling a mile. (For more details, take a look at the measurement chart in the glossary).

Appendix

Percents

How to speak Decimal We see numbers written as symbols, like “8” We also see numbers written in word form like “Eight” In each case they are spoken the same way. When we see this number “875, we could say “eight seven five” But if we said “eight-hundred seventy-five “ we would know a little more about the number. It would become a quantity and can no longer be something else, like a serial number or password. When we see a decimal number written as 8.75, we often say “ eight point seven five ,“ but we write this decimal number as “eight and seventy-five hundredths.” Spoken this way it tells us we are expressing a quantity. In addition the same words also represent the fractional equivalent:

75 100

Section 2.1 Decimal Place Values

Examples: Words translated into numbers

Look at .429 We see “four hundred twenty nine” then we pause to figure out the place value of our last digit 9. Counting from right after the decimal point are tens, hundreds, then thousandths. “four hundred twenty-nine thousandths.”

Your Turn: Draw an arrow from each of these numbers to their correct wording. 4.29 42.9

400.29

Four hundred twenty-nine thousandths Four and twenty-nine hundredths Forty-two and nine tenths

Appendix

.0429

Four hundred twenty-nine ten-thousandths Percents

0.429

Four hundred and twenty-nine hundredths

Ratio and Proportion

So our answer is:

Fractions

Fifty-eight.................................................................................................... 58 One-hundred twenty-five thousandths ...................................... 0.125 Fifty-eight and one hundred twenty five thousandths .............58.125 One hundred and twenty five thousandths ..............................100.025 Eleven and three hundredths ..........................................................11.03..

Decimals

In this text we use the traditional method of writing numbers. This means the “and” represents the decimal point and the place values such as ten, tenths and hundred, hundredths are named. 5.2 becomes five and two tenths, while 25.19 becomes twenty-five and nineteen hundredths.

Whole Numbers

The problem is that in our daily language the word “AND” is used for many different purposes and so we don’t always associate it with a decimal point. Also it is easy to miss the difference between the words “hundred” and “hundredths” or “ten” and “tenths.” Said out loud we almost have to spit out the “ths” to be understood. So we often just say “point” as in “5.2” or five point two.

Whole Numbers

Chapter Two

Practice 1: Words to Decimal Numbers a) Twenty-nine

_____

b) Eighty-one hundredths _____ c) Twenty- nine and eighty-one hundredths

_________

d) Nine thousand thirty-four point seven ______________

Decimals

e) One and four thousandths ___________________________ f) One hundred and sixty-two thousandths _________________ g) Forty-five thousandths

_________

h) Four thousand three hundred twenty-one ten thousandths _____________ i) One hundred twenty point five ________________

Fractions

j) Seventeen thousandths _________ k) One and seven tenths

_________

Here we take a closer look at place values.

3. 2 รท 10

Ratio and Proportion

10 x 5

9 รท 100

100 x 7

1 รท 1000

1000 x 3 3,753.291

Here each face value is multiplied by the place value:

ones

50 Percents

See your yellow bookmark.

.09

700

.001

3000 3,753. 291

Your Turn: Complete the chart with a label for each place value.

Appendix

Examples: Decimal Place Value In the number 2039.876, what digit is in the tenths place? In the number 2039.876, what digit is in the ones place? In the number 2039.876, what digit is in the tens place? In the number 2039.876, what digit is in the thousandths place?

8 9 3 6

Section 2.2 Rounding and Estimation

___

b) In the number 78.9 what digit (number) is in the ones place?

___

c) In the number 78.9 what digit (number) is in the tens place?

___

d) In the number 6174.903 what digit is in the thousands place?

___

e) In the number 6174.903 what digit is in the thousandths place?

___

f) In the number 6174.903 what digit is in the hundredths place?

___

g) In the number 6174.903 what digit is in the tenths place?

___

h) In the number 6174.903 what digit is in the ones place?

___

i) In the number 6174.903 what digit is in the tens place?

___

j) In the number 6174.903 what digit is in the hundreds place?

___

A decimal is an amount between two whole numbers. They are represented by the tic marks in between the numbers below. Look how the .4 is represented in this metric ruler.

3 4

6 7

9 10

Ratio and Proportion

Fractions

Section 2.2 Rounding and Estimation

Decimals

a) In the number 78.9 what digit (number) is in the tenths place?

Whole Numbers

Practice 2: Decimal Place Value Practice

To take a closer look we will need to magnify the portion.

Percents

Example One: Is 0.4 closer to 0 or 1?

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

Appendix

Now we can see that the .4 is closer to 0 than 1 and therefore is rounded down to 0.

Whole Numbers

Chapter Two

Example Two: Is 0.5 closer to 0 or 1?

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

Decimals

The answer is neither, but a mathematical rule was created for halfway points. And that is to always round the value up. So .5 is bumped up to 1. For the next example we will add tick marks in between the tenths to represent the hundredths place. Example Three: Is 0.07 closer to 0 or 1?

Note that 1 is the same thing as 1.0

Ratio and Proportion

Fractions

.07

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

0.07 rounds to .1 WRONG ANSWER. Be very cautious, we did not ask to choose between 0 and 0.1 but between 0 and 1.0 So 0.07 rounds to 0. Your Turn: Round these to 0 or 1 a) 0.6

b) 0.09

c)0 .30

Practice 3: Round to the nearest

d) 0.49

e) 0.67

1 or 0

Appendix

Percents

Place 1 or 0 in each box.

a) 0.11

b) 0.25

c) 0.22

d) 0.10

e) 0.99

f) 0.06

g) 0.34

h) 0.5O

i) 0.05

j) 0.71

k) 0.55

l) 0.7

m) 0.27

n) 0.90

o) 0.69

p) 0.39

q) 0.2

r) 0.80

s) 0.03

t) 0.82

Section 2.2 Rounding and Estimation

To figure this out we would multiply 54.67 x .0825 = 4.5102

Whole Numbers

Money and taxes are frequently rounded off to the nearest hundredths place. For example, if your restaurant bill comes to $54.67 and your sales tax is 8.25%, how much tax would you need to pay?

However, $4.5102 is not added to the bill, instead it is rounded to $4.51.

Practice 4: Round to the nearest thousandths.

Fractions

Step 1 Add up your hits. Say perhaps 112 Step2 Add up your at-bats. Say perhaps 436 Step3 Divide your hits by your at-bats. (H/B) 112 ÷ 436 = .25688 Step4 Round off the number to the thousandths place. .25688 ≈ .257

Decimals

Calculating baseball stats is one area in which we round to the nearest thousandths place. Here, for example, is how batting averages are calculated:

All time record batting averages. Average

Ty Cobb

0.36636

Rogers Hornsby

0.35850

Joe Jackson

0.35575

Pete Browning

0.34892

Ed Delahanty

0.34590

Tris Speaker

0.34468

Ted Williams

0.34441

Billy Hamilton

0.34429

Dan Brouthers

0.34213

Babe Ruth

0.34207

Rounded

When you hear the expression “he is batting a thousand” it is actually 1.000 that means the batter gets a hit every time he is up at bat. (Ofcourse this never happens in real life, which is why it’s an expression).

Percents

Name

Ratio and Proportion

Rank

Appendix

More Practice with Rounding

Example One: Round 199.99 to the nearest whole (ones) number

Answer: 200 Example Two: Round 5.6392 to the nearest tenth

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Two

Answer: 5.6

Example Three: Round .18999 to the nearest hundredth

Example Four: Round 28.05267 to the nearest thousandth

Answer: 28.053

Appendix

Percents

Answer: .19

Section 2.2 Rounding and Estimation

a) 0.1325 to thousandths b) 0.0091 to thousandths

Whole Numbers

Practice 5: Round decimal numbers to the indicated place value.

c) 0.0196 to thousandths Decimals

d) 5.1234 to thousandths e) 6.6666 to thousandths f) 40.61884 to thousandths g) 1.99999 to thousandths h) 0.1325 to hundredths

Fractions

i) 0.0091 to hundredths j) 0.3333 to hundredths k) 5.567 to hundredths

Ratio and Proportion

l) 48.001 to hundredths m) 1.32 to tenths n) 0.666 to tenths o) 7.987 to tenths p) 99.99 to tenths

Percents

q) 12.2 to ones r) 9.93 to ones s) 0.5 to ones t) 11.99 to ones u) 499 to the nearest hundred

Appendix

v) 999 to the nearest thousand

Whole Numbers

Chapter Two

Section 2.3 Comparing Decimals Next we are going to compare the size of decimal numbers. These symbols mean:

> <

greater or larger than less or smaller than

Hint: the tip always points to the smaller amount as in the samples below $ 6.48

$1.98

$202.59

$212.59

2.03

Decimals

Ratio and Proportion

Fractions

Example One: Which is larger 0.016 or 0.00899? For better comparison, stack and line up the numbers on top of each other. Then add unimportant zeros to help in your comparison.

Answer: 0.016

0.00899

Example Two: Arrange from the smallest to the largest: 3.018

3.18

3.1

3.018 3.180 3.100 3.080 0.318

Answer .318

3.08

3.018

0.318

3.08

Appendix

Percents

Use a straight edge to compare

0.01600 0.00899

3.1

3.18

Section 2.3 Comparing Decimal Numbers

You can use a separate piece of paper for this exercise. 8.7

8.2

7.96

8.014

0.15

0.01

0.1

0.0101

0.001

93.999

93.909

93.99901

94.0001

16.83

16.38

16.3

16.8

3.49

3.489

3.4899

3.48999

3.48989

You can use a separate piece of paper for this exercise. 2.061

2.612

0.66

6.21

14.01

140.1

1.401

14.1

14.11

0.0067

0.007

0.00618

0.00701

0.006

0.1

0.01

1.1

0.019

5.1

5.01

5.09

5.91

Appendix

2.62

Percents

Ratio and Proportion

Practice 7: Arrange decimal numbers from largest to smallest..

Fractions

7.8

Decimals

Whole Numbers

Practice 6: Arrange decimal numbers from smallest to largest.

Whole Numbers

Chapter Two

Section 2.4 Adding and Subtracting Decimals Rhonda Goes Shopping Rhonda’s grocery receipt provides a source for a multitude of exercises using decimals. First, let’s take a look at it. Why do you think almost all prices end in nine?

Ratio and Proportion

Fractions

Decimals

Item

Exact Price

Rounded Price to nearest whole number.

Orange Juice

1.99

Milk

1.99

Eggs

3.69

Bread

3.49

Deli Turkey

4.69

Peppers (2)

1.80

Bananas

1.19

Grapes

3.69

Cereal

2.69

Penne

2.99

Tomato Sauce

1.69

Noodle Bowl

2.49

Pasta

1.99

Cheddar Cheese

5.99

Apple

0.48

Frozen Pizza

7.99

Club Soda

1.19

Canned Fruit

0.89

Scalloped Potatoes

3.31

Can of Coffee

3.53

Percents

Total both columns. YOUR TURN: Let’s say Rhonda has exactly $60 in her wallet. As she shops, she notes the price of each item, and rounds each to the nearest dollar. That way, she’ll have a pretty good idea of when she is getting close to her limit.

Appendix

Will Rhonda have enough cash or will she need to put something back?

Section 2.4 Adding and Subtracting Decimals

4.29 36.1 87.7 + 1.26

Wrong answer! You probably know instinctively that this is not written down in a way that helps us add up the numbers.

1793

129.35

Instead, always line up the decimal points, adding zeros as necessary. Lined up the ones place will be on the left and the tenths will be on the right and everything else will line up too .

Tip: Graph paper can help you line up your numbers perfectly.

Fractions

Notice that the actual sum, 129.35 is quite different than 1793 or 17.93. The placement of the decimal is the key to getting the right answer.

Decimals

4.29 36.10 87.70 + 1.26

Whole Numbers

What’s wrong with this picture?

The Money Perspective: If you’ve ever counted a cash drawer or watched

This is the same reason we use columns to add or subtract decimal numbers; all the ones line up in the same column, all the tens in together etc. And so to the right of the decimal point, all the tenths are lined up and the hundredths are together, and so on.

Ratio and Proportion

someone count cash, most people begin by separating bills into piles of the same denominations, twenties, tens, fives, and ones. And they put their change into quarters, dimes, nickels and pennies. This just makes it easier to keep track of the money because they’re totaling likes with likes.

Percents Appendix

Appendix

Percents

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Two

Examples: When adding and subtracting decimal numbers, line up the decimal point of all the numbers. If a number does not show a decimal point, place a point to the right of the whole number. Add unimportant zeroes to keep the columns lined up. Add 13.6 and 42.18

13.60 +42.18 55.78

☜

1347.0000 + 0.0005 1347.0005

Add 1347 and 0.0005

Subtract 14.69 from 113.06

Add a zero

☜

Add

to right of 7 , then add four zeros.

2 1

113.06 -14.69 98.37

Notice how “subtract 14.69 from 113.06” means 113.06 - 14.69

Subtract 146 – 3.198

5991

146.000 -3.198 142.802

☜ Add three zeros

Practice 8: Add or subtract decimal numbers as indicated. a)

8.7 +5.4

38.64 - 8.87

11.00001 -1.11234

e) 98.01 102.49 + 5.98

346.8912 - 29.98764

f) 1234. - 0.1234

Section 2.4 Adding and Subtracting Decimals

h) 8416.28 - 1.489 =

i) 462 - 31.2 =

j) 0.1 + 1.9 + 13 =

Decimals

k) Add 0.001 to 87

Whole Numbers

g) 74.906 + 0.01 + 42 =

l) 143.012 +98.764

o) Subtract 6.8 from 14.2

p) Subtract 38.97 from 59

q) Add 5000 to 0.0186

r) Subtract 0.001 from 0.01

Ratio and Proportion

n) 6 + 132.89 + 80.14 =

Fractions

m) 20 + 14.8 + 0.018 =

Percents

Here are several ways to represent a multiplication problem. “Twenty-nine three hundredths times thirteen and four tenths.”numbers. Coming up and on the next page is the multiplication of decimal

29.03

13.4

29.03(13.4)

29.03 x 13.4

Appendix

29.03 X 13.4

Decimals

Whole Numbers

Chapter Two

Section 2.5 Multiplying Decimals When multiplying with decimal numbers you do NOT have to line the points up like you did for addition and subtraction. Instead the decimal point is placed after we complete the multiplication. We place the decimal point by counting how many digits are to the right of the decimal point in the two factors just multiplied. (Since the zeros to the left of the decimal point are not needed, we removed them.) Place the decimal in your answer that same number of places. Count starting on the right side. Fill in any empty spots with zeros. Example One: 0.02 x .0.008

Fractions

.02 x .008

Example Two: When multiplying three numbers together, multiply any two to get an answer, then multiply that answer by the third number. 1.02 x .3 x 46 =

Step 1 1.02 ☜ Two digits x .3 ☜ One digits .306 ☜ Three digits

Step 2 .306 ☜ Three digits x 46 ☜ No digits 1836 12240 14.076 ☜ Three digits

Practice 9: Multiply the decimals numbers. a) .8842 x .004

b) 5.76 x .25

c) 98.47 x .7

d) 8.04 x .02

e) .001 x .001

f) 8.88 x .88

g) 12.34 x 43.21

h) 4.095 x 2.2

Appendix

Percents

Ratio and Proportion

.008 ☜ Three digits X . 02 ☜ Two digits .00016 ☜ Five digits

Section 2.5 Multiplying Decimals

.612 =

k) 11.4 x 18 =

l) 2.7 x 8.3 x .04 =

m) 1.67 x 3.2 =

n) 3.6 ( 1.1) =

Decimals

j) 84.78

Whole Numbers

i) 0.1 x 0.1 x 0.1 =

Fractions

Rhonda’s Shopping Receipt

Your Turn: If there are four weeks in a month, how much will Rhonda spend on food? First use the estimated total of $60.00 . Then use the exact amount of the $57.77 reciept.

Percents

Your Turn: How much will Rhonda spend on food in a year ? (There are 52 weeks in a year. )

Ratio and Proportion

Let’s return to Ronda’s shopping receipt from Section 2.4. As Rhonda walks home from the grocery store, and realizes she carries just about this much in groceries about every 7 days. She decides to use this receipt to help calculate how much she spends on food.

Appendix

Decimals

Whole Numbers

Chapter Two

Section 2.6 Dividing Decimals In decimal multiplication we multiply first and then place the decimal point in the answer. With division, we get the decimal into position before we do the division. Here are the steps. Step One: In decimal division, first move the decimal point in the divisor all the way to the right, then move the decimal point in the dividend over the same number of times as you did in the divisor. See examples below. Step Two: Then place the decimal point in your quotient directly above where it is in the dividend. Step Three: Now just divide and keep adding zeroâ&#x20AC;&#x2122;s until you are done. Sometimes it helps to divide using straight vertical lines, you can do this by turning notebook paper on its side.

Examples: Fractions

6.1

.108

divisor

move decimal Ratio and Proportion

8.1

.02

Step One:

Step Two:

place decimal

Step Three:

Percents

840

work problem on notebook paper turned sideways.

.02)840.00

.108)8.100

. 02)84000.

. 108)8100.

4 2 00 0 2)8 4 0 0 0 -8 4 -4 0

..... 75. 1 0 8)81 0 0. - 756 540 - 540 0

4)6.1

.8)11.0

. 4)6.1

. 8)110.

(Nothing moved)

1.5 2 5 4)6.1 0 0

- 4

21 -20 10

-8

Appendix

20 -2 0 0

dividend

. 1 3.7 5 8)1 1 0.00

-8

- 24

60 - 56 40 - 40 0

Section 2.6 Dividing Decimals

Whole Numbers

Your Turn: Rhonda spends $8.05 on fruits and vegetables for the week. How much is that per day? Hint: divide by seven.

Your Turn: Alex is to bring some soda to a party and buys a case for $21.95. His friend Elena splits the cost. What does each one pay? Decimals Ratio and Proportion

Your Turn: Using Rhonda’s estimated weekly food bill of $60.00. What does she spend on food each day? Divide by 7.

Fractions

Your Turn: If Elena’s food bill is $67.32 a week, how much does she spend per day?

Percents

Decimal Division in a Nutshell

There are several ways to represent a division problem.

16.4 82

16.4 ÷ 82

16.4 / 82

.82 g16.4

Appendix

“Sixteen and four tenths divided by eighty-two hundredths.”

Fractions

Decimals

Whole Numbers

Chapter Two

Practice 10: Divide the decimal numbers. a)

e) 25 4

h) 25 ÷ 4

.82 g16.4

.3 g .69

1.4 g280

.6 ÷ 15

.002 g 4

20 g .1

.002

Ratio and Proportion

When dividing with decimals the numbers can often go on and on. We only need to continue dividing until we either find a repeating pattern or we reach just past the place value in which we have been asked to round. Examples: Round to the nearest hundredth. .6666 is called a repeating decimal . A line placed over the number indicates the number repeats.

Percents

2 2.1 8 0 9 2 1 )4 6 5.8 0 0 0 -4 2 45 -4 2 3 8 - 2 1 17 0 - 1 6 8 2 00 -1 8 9

≈22.18

.6 6 6 6 3)2.0 0 0 0 - 1 8 2 0 -1 8 20 -1 8 20 - 1 8 2

Appendix

≈.6 7

Section 2.7 Decimal Applications

37 g1.68

85 รท 0.3

.64).14208

.9).6354

9.1 6.3

.45)102.4

2.5)145.55

Fractions

g) 9.2 รท 2

Decimals

Whole Numbers

Practice 11: Divide and round to the hundredths place.

95 43 Ratio and Proportion

Section 2.7 Decimal Word Problems

12.3 13.2 + 11. 5 37.0

Percents

Example One: The annual rainfall for Espanola was 12.3 inches in 1960, 13.2 inches in 1961, and 11.5 in 1962. What was the total rainfall for the three years?

Appendix

Answer: 37 inches

Whole Numbers

Chapter Two

Example Two: What is the difference between Kevin’s salary of $523.86 per week and Marty’s salary of $318.90 per week ? 523.86 - 318.90 204.96

Fractions

Decimals

Answer: $204.96 Example Three: If you have a car that uses 19.2 gallons of gas to go 285 miles, how many miles per gallon (mpg) does the car get? (Round your answer to the nearest tenth). mpg = miles = per gallon

miles = gallon

14.84 192.)2850.00

; so divide 285 by 19.2

Be sure to include the units like feet, dollars, inches, miles per gallon, as they are part of the answer.

- 1 92 -

285 miles 19.2 gallon

930

768

1620

Appendix

Percents

Ratio and Proportion

- 1536

840 768 72

Answer: 14.8 mpg

Example Four: A construction company needs to order 3 hinges a piece for 15 doors. How much will it cost to buy all the needed hinges, if they are sold for $.75 a piece ? The total number of hinges is 3 x 15 = 45 hinges Then to find the total cost, multiply 45 x .75 = ?

$3375 WRONG ANSWER! Doesn’t that sound like too much? That would mean a cost of over three thousand dollars for some hinges! Always check to see if your answer makes sense. In this case we didn’t count the place values to the right of the decimal point. Answer: $33.75

That’s more like it.

Section 2.7 Decimal Applications

length

= 8.3 m

7.4 m =

The formula for Area is LxW=A

Decimals

Visually you can count the squares in the grid to determine the area. So you would need to estimate partial squares. It is more accurate to use the formula.

Whole Numbers

Example Five: What is the area of the classroom floor shown in the following diagram?

width

L = 8.3 W = 7.4 8.3 x 7.4 = 61.42 square meters Area = 61.42 square meters

Fractions

Example Six: What is the perimeter of the same classroom floor shown here again.

length

= 8.3 m You can count along the outside to determine the length of the perimeter. Or just use the formula.

7.4 m width

8.3 + 7.4 + 8.3 + 7.4 = 31.4 m or 2(8.3) + 2(7.4) = 31.4m

Percents

L+W+L+W=P or 2L + 2W = P

Ratio and Proportion

As you can see from the diagram the tiles do not fit evenly into the floor plan. The tile has been cut to cover the whole floor. This is what accounts for the decimal values.

We donâ&#x20AC;&#x2122;t square the meters because they are just lines.

Appendix

Practice 12: Decimal Word Problems a) During five days Marty drove 15.4 miles, 24.2 miles, 10.4 miles, 18.7 miles, and 7.5 miles. How many total miles did he drive?

b) Alex are deposits three checks; one for $36.98, another for $17.27, and a third for $260. How much is his total deposit?

c) Mariaâ&#x20AC;&#x2122;s car gets 42.1 mpg on the highway, how many gallons of fuel will she use traveling 340 highway miles? (Round answer to tenths of a gallon.)

d) Kevin needs to cut 5 equal strips of metal from a 14 meter strip. How long will each of the small strips be?

Percents

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Two

Appendix

e) Roger purchases a TV and put $40 down, then pay $32.60 a month for 8 months. What is the total cost of the TV?

Section 2.7 Decimal Applications

Decimals

g) If the total precipitation (rainfall and snow) for the year in Key West, Florida is expected to be 37.9 inches, and it has already rained 26.82 inches, how many more inches of precipitation are likely?

Whole Numbers

f) If the revenues from an extra quarter cent sales tax amount to $148,136.45 and are divided equally among 7 different departments, how much will each department receive? (Round to the nearest cent).

Fractions Percents

i) What is the area of Nashâ&#x20AC;&#x2122;s room? (A = L x W )

Ratio and Proportion

h) Nashâ&#x20AC;&#x2122;s room measures 9 by 12.5 feet. What is the perimeter of his room? (P = 2L + 2W)

Appendix

Whole Numbers

Chapter Two

Section 2.8 Fraction Preview Relationship between decimals and fractions. Decimals and fractions both represent parts of a whole number.

Decimals

Both of these numbers below are read, “One and twenty-three hundredths. Whole Number 1.23 A mixed fraction is the sum of a whole number followed by a proper fraction.

Ratio and Proportion

Fractions

Whole number

Percents

Decimal part

Fraction part 23 100

For every fraction there is an equivalent decimal. But there may be many fractions that share the same equivalent decimal. For example, the decimal

is equal to

5/10 , 50/100 , and 1/2

How to change decimals to fractions Notice that the number of digits to the right of the decimal is the number 7 7 7 of zeros placed after the one below. .7 = 10 , .07 = 100 , .007 = 1000 Example One: 3.23 is read “three and twenty-three hundredths.” These 23 words can also be expressed as the fraction 3 100 Example Two: 0.045 is read “forty-five thousandths” and is written

Appendix

Example Three: .45 is read “forty-five hundredths” and is written in 45 fractional form as 100 .

45 1000

Section 2.8 Fraction Preview

a) 0.25 =

b) 0.5 =

c) 0.333 =

d) 1.75 =

e) 45.4 =

f) 100.125 =

Example One: Change the following fraction to a decimal by dividing or on the calculator 1 ÷ 2

the Bo

To pG oes in

“Top in the box” is a catchy way to remember that the numerator goes into the “box” of the division problem. In our example, the 1 is the top that goes into the box.

.5 2 )1.0

Preview B: Convert the fractions to decimal numbers. 1

1 2

1 4

3 4

Percents

Ratio and Proportion

Example Two: 4 2 can be changed to a decimal by keeping the 4 as our whole number and then dividing the 2 into the 1 as in example one, to get 0.5. The final answer is 4.5, “ four and five tenths.”

Fractions

Decimals

Changing fractions to decimals

Whole Numbers

Preview A: Convert the following decimals to fractions.

Appendix

Whole Numbers

Chapter Two

Answers to Practice Problems Practice 1 a) 29

b) 0.81

Ratio and Proportion

Fractions

Decimals

g) 0.045

h) 0.4321

Practice 2 a) 9

c)29.81 d) 9034.7

b) 8

i) 120.5

c) 7

d) 6

e) 3

e) 1.004 j) 0.017

f) 0

f) 100.062 k) 1.7

g) 9

h) 4 i) 7

j) 1

Practice 3 a) 0

b) 0

c) 0

d) 0

e) 1

f) 0

g) 0

h) 1

i) 0

j) 1

k) 1

l) 1

m) 0

n) 1

o) 1

p) 0

q) 0

r) 1

s) 0

t) 1

Practice 4

a) Ty Cobb b) Rogers Hornsby c) Joe Jackson d) Pete Browning e) Ed Delahanty f) Tris Speaker g) Ted Williams h) Billy Hamilton i) Dan Brouthers j) Babe Ruth

0.366 (.36636) 0.359 (.35850) 0.356 (.35575) 0.349 (.34892) 0.346 (.34590) 0.345 (.34468) 0.344 (.34441) 0.344 (.34429) 0.342 (.34213) 0.342 (.34207)

a) 0.133

b) 0.009

c) 0.020

d) 5.123

e) 6.667

f) 40.619

g) 2.000

h) 0.13

i) 0.01

j) 0.33

k) 5.57

l) 48.00

m) 1.3

n) 0.7

o) 8.0

p) 100.0

q) 12

r) 10

s) 1

t) 12

u) 500

v) 1000

Appendix

Percents

Practice 5

Answers to the Decimals Practice Problems

< 8.2 < 8.7 b) 0.001 < 0.01 < 0.0101 < 0.1 < 0.15 c) 93.909 < 93.999 < 93.99901 < 94 < d) 16 < 16.3 < 16.38 < 16.8 < 16.83 a) 7.8

8.014

3.48989

3.4899

3.48999

94.0001

3.49

Decimals

e) 3.489

7.96

Whole Numbers

Practice 6

Practice 7 a) 6.21 > 2.62 > 2.612 > 2.061 > .66 b) 140.1 > 14.11 > 14.1 > 14.01 > 1.401

Fractions

c) 0.00701 > 0 .007 > 0.0067 > 0.00618 > 0.006 d) 1.1 > 1 > 0.1 > 0.019 > 0.01 e) 5.91 > 5.1 > 5.09 > 5.01 > 5

a) 14.1

b) 29.77

c) 316.90356

d) 9.88767

e) 206.48

f) 1233.8766

g) 116.916

h) 8414.791

i) 430.8

j) 15

k) 87.001

l) 241.776

m) 34.818

n) 219.03

o) 7.4

p) 20.03

q) 5000.0186

r) 0.009

a) 0.0035368 b) 1.44

c) 68.929

d) 0.1608 e) 0.000001

f) 7.8144

g) 533.2114 h) 9.009

i) 0.001

k) 205.2

l) 0.8964

n) 3.96

m) 5.344

a) 2.3

b) 20

f) 0.005 g) 0.04

c) 2000

d) 200

h) 6.25

i) 2000

e) 6.25

Appendix

Practice 10

j) 51.88536

Percents

Practice 9

Ratio and Proportion

Practice 8

Decimals

Whole Numbers

Chapter Two

Practice 11 a) 0.0454…≈ .05

b) 0.222 ≈ 0.22

c) 283.333…≈ 283.33

d) 0.706 ≈ 0.71

e) 1.444 ≈ 1.44

f) 227.555 ≈ 227.56

g) 4.6

h) 58.22

i) 2.20930 ≈ 2.21

Practice 12 a) 76.2 miles

b) $314.25

c) 8.07 ≈ 8.1 gal. d) 2.8 meters

e) $300.80

f) $21,162.35

g) 11.08 inches

h) 43 feet

i) 112.5 square feet

Fractions

Preview A a)

25 100

Preview B a)

0.5

1.5

75 100

0.25

3.25

Appendix

Percents

Ratio and Proportion

5 10

4 10

0.75

45.75

333 1000

100

125 1000

Chapter

Fractions Table of Contents

Section 3.1 Reading Fractions.....................................................89 Practice 1: Write as a fraction. . . . . . . . . . . . . . . . . 90 Practice 2: Write each part of your day as a fraction. . . . . . . . . . 91

Section 3.2 Complementary Fractions.............................................92 Practice 3: Write the fraction complements. . . . . . . . . . . . . 93

Section 3.3 Estimating Fractions..................................................94 Practice 4: Estimate these results.. .

Section 3.4 Equivalent Fractions..................................................96 Practice 5: Build up the fraction to the given denominator. . . . . . . . 98 Practice 6: Reduce these fractions to their lowest terms.. . . . . . . . 99

Section 3.5 Comparing Fractions.................................................101 Practice 8: Compare fractions using the symbols > or < or = . . . . . 102 Practice 9: Order these fractions from smallest to largest. . . . . . . . 104 Practice 10: Order these fractions from largest to smallest.. . . . . . 104

Section 3.6 Multiplying Fractions................................................105 Practice 11: Multiply these fractions. Reduce if possible. . . . . . . Practice 12: Change these mixed numbers to improper fractions: . . . Practice 13: Simplify improper fractions to mixed numbers.. . . . . Practice 14: Reduce improper fractions and simplify to mixed numbers. . Practice 15: Multiply mixed numbers. . . . . . . . . . . . .

. 107 . . 109 . . 110 . . 111 . . 112

Section 3.7 Dividing Fractions....................................................113 Practice 16: Divide these fractions. Cancel and or simplify. . . Practice 17: Divide the following mixed numbers. . . . . .

. .

114 115

Section 3.8 Fraction Word Problems x รท........................................116 Practice 18: Fraction Word Problems. . . . . . . . . . . . . .

119

Section 3.9 Addition and Subtraction...........................................121 Practice 19: Write the LCD for the following sets of fractions. . . . . . 123 Practice 20: Add or subtract the following fractions. . . . . . . . . . 125 Practice 21: Add or subtract the following mixed numbers.. . . . . . . 129

Section 3.10 Fraction Word Problems + -......................................130 Practice 22: Fraction Applications; add or subtract.. . . . . . . . .

134

Section 3.11 Ratio & Proportion Preview........................................137 Answers to Practice Problems ....................................................139

Copyright ÂŠ2009 Central New Mexico Community College. Permission is granted to copy, distribute and/or modify this document, under the terms of the GNU, Free Documentation License. Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is available on http://www.gnu.org/copyleft/fdl.html

The following MALL activities are available for class check out: Fraction Kit2 Dividing the Gold Fraction Trading I Fraction Trading II Pizza Problem (warm up) Pizza Problem

REFC

Reduce and Estimate Fractions Card



These techniques will help you find common factors that can be divided into both the numerator and denominator. Reducing is an important last step in getting the final answer.

You may cut this paper on the dotted line to create a Reduce and Estimate Fractions Card (REFC ). One side provides techniques that will help you to reduce a fraction and the other side is a handy reference for estimating fractions.

Divide by 2 if:

The top AND the bottom numbers are EVEN numbers 2

Like: 4 , 26

ON the facing page

, 44

Divided by 3 if:



The sum of the top numbers can be divided by 3 AND the sum of the bottom numbers

is a colorful chart of fractions. Look at it for guidance throughout the chapter.

Mastering fractions is far more than just memorizing rules. To really understand fractions you will need to learn how they interact with each other. We have created a set of hands on materials that compliment each section in this chapter. They are designed to help you see how the rules work. If you understand the “why and hows” you wont forget the rules. The Fraction Kit project begins with creating 10 different sized fractions from colorful strips of paper. Use a baggie or large envelope to store the your work. Start as soon as possible.

You are ready to begin:



It is time to build your Fraction Kit

can be divided by 3

Like:

561 762

5 + 6 + 1 = 12 7 + 6 + 2 = 15

12 and 15 can be divided evenly by 3

Divide by 5 if:

The top AND bottom numbers end in 0 or 5

Like:

5 15

60 75

255 460

Divide by 10 if: The top AND bottom numbers end in 0.

Like:

20 40

140 260

320 440

Fraction Kit Divide by 25 if:

The top AND the bottom numbers end in 25 or 50 or 75 or 00.

Like:

225 400

140 260

320 440

REFC

Estimating Fractions

Reduce and Estimate Fractions Card

A fraction smaller than 1/2 estimates to 0. A fraction is equal to or larger than 1/2 estimates to 1. Below are list of fractions and their estimates:

These techniques will help you find common factors that can be divided into both the numerator and denominator. Reducing is an important last step in getting the final answer.

≈0

13 16

≈1

≈0

≈1

≈0

≈1

14 16

≈1

≈0

≈1

15 16

≈0

≈1

≈0

≈1

≈0

≈1

7 ≈1 10

8 ≈1 10

9 ≈1 10

10 ≈1 10

4 ≈1 8

7 ≈1 8

8 ≈1 8

1 ≈0 6

2 ≈0 6

3 ≈1 6

1 ≈0 5

2 ≈0 5

3 ≈1 5

1 ≈0 4

2 ≈1 4

3 ≈1 4

1 ≈0 3

2 ≈1 3

3 ≈1 3

1 ≈1 2

2 ≈1 2

≈1

≈0

12 16

≈1

≈0

≈1

5 ≈1 10

12 12

≈1

3 12

≈ 1+1=2

≈ 3+0=3

The top AND the bottom numbers are EVEN numbers 14

The sum of the top numbers can be divided by 3 AND the sum of the bottom numbers can be divided by 3

Like: 5 ≈1 8

6 ≈1 8

, 44

Divided by 3 if:

6 ≈1 10

561 762

5 + 6 + 1 = 12 7 + 6 + 2 = 15

12 and 15 can be divided evenly by 3

Divide by 5 if:

The top AND bottom numbers end in 0 or 5 4 ≈1 6

5 ≈1 6

4 4 ≈1 ≈1 4 4

6 ≈1 6

5 ≈1 5

Like:

≈ 2+1=3

5 15

60 75

255 460

Divide by 10 if: The top AND bottom numbers end in 0.

Like:

When estimating mixed numbers add the whole number to the estimated number. 1

Divide by 2 if:

≈0

Like: 4 , 26

4 ≈0 10

3 ≈0 10

≈0

4 ≈0 12

2 ≈0 10

≈0

≈1

1 ≈0 10

≈0

20 40

140 260

320 440

Divide by 25 if:

The top AND the bottom numbers end in 25 or 50 or 75 or 00.

Like:

225 400

140 260

320 440

Section 3.1 Reading Fractions

Decimals

We count one apple , two apples, three apples etc.

Whole Numbers

Section 3.1 Reading Fractions

When we count bags of chips, what is the whole?

a bag or a chip?

It all depends on the wording. As you study fractions you will need to locate both the whole and the parts of the whole.

Fractions

Now, to count baskets of apples. We would count one basket, two baskets, three baskets etc. In this situation our “whole” changes from apples to baskets of apples.

Your Turn: a dollar or a penny?

If you count dollars what is the whole?

a dollar or a penny?

If you count staircases, what is the whole?

a step or a staircase?

If you count steps on a staircase what is the whole? a step or a staircase? If you count pizza pies, what is the whole?

a slice or the pie?

If you count slices of pizza, what is the whole ?

a slice or a pie?

36 apples +

Percents

Let’s return to the baskets, if there are 36 apples in one basket, but I take one apple away, then:

Ratio and Proportion

If you count pennies, what is the whole?

35 apples = 107 apples

1 basket + 1 basket + Almost one whole basket or 2

35 36

baskets of apples.

Appendix

If we want to count the baskets of apples as well as the apples inside the baskets we would use fractions.

Decimals

Whole Numbers

Chapter Three

A fraction is written with one number over another and is separated by a ‘divide’ line called a fraction bar. 1

Look at: 2

and 4

These two fractions represent parts of one whole. The denominator (bottom number) refers to number of equal parts in the whole (one ) and the numerator (top number) tells us how many of the parts of the whole are present. 1 2 says, one of two equal parts.” Read as “one half” 3 4

says, three of four equal parts.”

Read as

“three fourths”

Example One: A class of 20 students has 6 people absent. Those six absentees are part of the whole class of 20 people therefore, six Fractions Ratio and Proportion

6 represent the fraction of people absent. 20

twentieths or

Example Two: One whole “Goodbar” candy breaks up into 8 equal small sections. If someone ate 5 of those sections, that person ate five eights or 5 of the “Goodbar.” 8

Your Turn: What fraction represents the people NOT absent.

Practice 1: Write as a fraction. a) Two of five equal parts

Percents

b) One of four equal parts c) Eleven of twelve equal parts d) Three of five equal parts e) Twenty of fifty equal parts

Appendix

f) It is 25 miles to Grandma’s house. We have already driven 11 miles. What fraction of the way have we driven?

Section 3.1 Reading Fractions Whole Numbers

g) Joey, cuts a whole pizza into twelve slices. Seven slices were eaten. What fraction of the pizza was eaten?

Practice 2: Write each part of your day as a fraction. There are 24 hours in one day. So if you sleep 8 hours of equivalents. For this exercise use your own typical day.

Fractions to be expressed

Decimal Factoid Numerator ÷ Denominator = Decimal

1/4 =1 ÷4 = .25 1/2 =1 ÷2 = .50 3/4 =3 ÷4 = .75

Fraction

Fractions

your whole day, that fraction is 24 . Include the decimal

Decimals

h) There are 24 students in Maria’s class. Eight have passed the fractions test. What fraction of the students in her class have passed the fractions test?

Decimal Equivalent

The number of hours you spent at school. The number of hours you spent eating yesterday. The number of hours you studied outside of class. The number of hours you talked on the phone. The number of hours that you watched television. The number of hours that you took care of children. The number of hours spent commuting. The number of hours that you spent at friends homes or that somewhere visiting friends.

Percents

The number of hours that you spent at your workplace.

Ratio and Proportion

The number of hours you slept.

The amount of time that you spent doing domestic chores (cleaning, laundry, cooking, etc.). Anything else?

Appendix

The total hours in your “whole” day?

Whole Numbers

Chapter Three

Section 3.2 Complementary Fractions Fractions express how many parts are in the whole and how many parts to count. The fraction can also tell us how many parts have not been counted. We call this the complement. Together the fraction plus the complement make 1 .

Decimals

The fraction ,

3 4

, means 3 of 4 equal parts. That means 1 of the 4 was not

counted as it is somehow different from the original 3.

5 8

means 5 of 8 equal parts. Through subtraction ( 8-5 ) we can figure

out, then, that 3 parts have not been counted. This is expressed as the

Fractions

5 complement 3 . Together 8 8

and

3 8

make

8 8

which equals

Complementary Situations

Ratio and Proportion

Example One: Itâ&#x20AC;&#x2122;s 8 miles to town. Marty has driven 5 miles. What fractional part does he have left to drive? The 8 miles is our whole (1) way to town and 8 - 5 is 3. So Marty has 3 miles or

3 8

Percents

5 8

of the way to go before he gets to town.

3 8

8 8

(One is the whole way to town).

Example Two: Martyâ&#x20AC;&#x2122;s car uses 16 gallons of gas a week. By Tuesday he has used 7 gallons. How many gallons does he have left to use for the rest of the week? What fractional part is that?

Appendix

16 7 9 = 16 16 16 of a tank left.

Section 3.2 Complementary Fractions

Lunch Paper

Cake

Total

Whole Numbers

Your Turn: Marty drops his daughter Maria off at the college. He gives her 10 dollars. She spends 4 dollars for lunch, 1 dollar on a newspaper and gives a 5 dollars towards a birthday cake for Tyler. Complete the chart below.

Dollar amount

Decimals

Fractional Part of the total

5 10

Decimal Equivalent

Practice 3: Write the fraction complements.

b) After class, Maria collects donations for Tyler’s birthday cake. Maria gives 5 dollars, Joey donates 4 dollars, Rhonda gives 3 and then gets the rest from from Tyler’s boss. The cake costs 20 dollars. Complete the chart below.

Dollars

Rhonda

Fractional part of the total. Decimal Equivalent.

Tyler’s Boss All together

20 20

.15

Appendix

d) Tyler offers to drive Maria home from college. She lives at her Dad’s, which is 8 miles away. After 3 miles, they stop for a run in the Bosque. What fraction of the way will they have left to drive?

Percents

c) The cake had 12 slices. 5 were eaten. What fraction of the cake was left?

Ratio and Proportion

Maria Joey

Fractions

a) There are 20 people in Maria’s class. 11 are women. What fractional part of the class are men?

Whole Numbers

Chapter Three

Helpful Tools “The Reducing and Estimating Fractions Card”

REFC and

Decimals

Section 3.3 Estimating Fractions

Use these

The Color Fraction Bar Graphic

Marty is building a doghouse for Rimmie, his Jack Russell terrier, and is estimating the amount of carpet he’ll need to staple onto the plywood frame. Between the floor, walls, roof, and trim, he figures he’ll need a carpet roll that is at least 7 1/2 feet long. He gets to the carpet store and looks in the bargain bin, where he finds what seems like enough carpet. It is 8 1/3 feet long, costing $2.00 per feet. He wants to quickly make sure if 1) it’s enough carpet and 2) he has enough cash on him to buy it or if he needs to put it on his debit card. Is there enough carpet? Marty needs to round off the 7 1/2 to the nearest whole number. 7 1/2

rounds

to ___ ?

[1/2 is closer to 1 than 0, so the answer is 8 (7 + 1)].

How much will it cost? Here he rounds off 8 1/3 to __

Ratio and Proportion

Fractions

[1/3 is less than half, which rounds to 0, so the answer is 8.]

At 2 dollars a foot he budgets at least $16 (2x8) so will use his debit card instead of the cash in his pocket. The purpose of this section is to get you accustomed to rounding off fractional numbers. The colored fraction chart may come handy here. If the fraction is smaller than 1/2, then round the fraction to zero. For example: since 1/4 is smaller than 1/2 it is rounded to zero. If the fraction is larger than or equal to 1/2 then it rounds off to 1. For example: 3/4 is larger than 1/2 we round to 1. Quick Check : Round each fraction to 0 or 1. Answers

Percents

1 8

b) ≈ e) ≈

5 10 ≈

2 3 3 8 3 4

c) ≈ f) ≈ i) ≈

Appendix

1 3

1 4 5 8 2 5

a) ≈

≈

Cover the answers on the right as you work the problems.

b) 0

f) 0

0 g)

≈

i) 1

Section 3.3 Estimating Fractions Whole Numbers

In this next set of estimates we use mixed numbers. Mixed numbers 7 represent amounts between the whole numbers. For example, 2 10 is between the whole numbers 2 and 3. Quick Check: Round each fraction to its nearest whole number.

5 10

≈

2 3

3 8

e) ≈ h) ≈

c) ≈

4 f)

≈

5 8

3 4

1 4

2 5

≈

Cover the answers on the right as you work the problems.

2 d)

3 g)

≈

4 f)

5 h)

2 i) 3

In the following examples we first round each mixed number to the nearest whole number and then perform the given operation such as: addition, subtraction, multiplication or division.

Fractions

1 8

Decimals

1 3

Answers b) c)

Example One: Estimate the answer. First round, then add or subtract.

+ 2 9/10 ≈ 2 + 3

= 5

b) 5 1/3 - 2 1/2 ≈ 5 - 3 = 2 c) 5 2/3 - 2 1/2 ≈ 6 - 3 = 3 Example Two: Estimate the answer. First round, then multiply or divide a) 3 2/3 x 4 1/2 ≈ 4 x 5 = 20 b) 1 2/3 ÷ 1 1/2 ≈ 2 ÷ 2 = 1

Percents

c) 5 1/3 x 2 1/2 ≈ 5 x 3 = 15 d) 6 1/3 ÷ 2 1/2 ≈ 6 ÷ 3 = 2

Practice 4: Estimate these results. b) 5 3/4 + 1 2/3 ≈

c) 4 7/8 - 3 1/3 ≈

d) 8 2/3 x 2 1/2 ≈

e) 4 1/8 - 2 2/3 ≈

f) 8 3/16 ÷ 3 2/3 ≈

Appendix

a) 1 5/6 + 1 2/3 ≈

Ratio and Proportion

a) 1 13/16

Whole Numbers

Chapter Three

Section 3.4 Equivalent Fractions Just as you can cut 3 pizzas in any number of ways and still have 3 pizzas, circles can also be partitioned into a set of equivalent fractions without changing the quantity. They are called fraction forms of one.

Decimals

4 4

Fractions

=1=

Take a look at ways we use 1 without changing the value of the number.

5 x 1 = 5

9x1=9

1 x 16 = 16

รท 1 = 5

9 รท 1 = 9

16 รท 1 = 16

Ratio and Proportion

=8 = 8

= 5

= 9

= 16

We can multiply by a fraction form of 1 to get a second fraction with the same value as the first. 1 4

2 2

2 8

In the visual representation below, each fourth is cut by a factor of two.

1 4

and also equals

2 8

Appendix

Percents

Building Up Fractions This section will show you the mathematical steps needed to produce equivalent fractions. We do this by multiplying both the numerator and denominator of the fraction by the same number. 2 3

4 4

8 12

2 and 3 were both multiplied by 4 so, 2

8 12

3 5

2 2

6 10

3 and 5 were both multiplied by 2 so, 3

6 10

Section 3.4 Equivalent Fractions

8 8

8 16

1 and 2 were both multiplied by 8 so, 1

8 16

6 12

1 = 1 x 4 2 2 4

1 = 1 x 5 5 2 2

5 10

1 = 1 x 3 3 2 2

4 8

3 6

All of the above fractions are equivalent.

LOOK!

1 = 2

6 12

= 4 8

= 5 10

= 3 6

Example One: Build up 8 for a denominator of 16 Ask yourself, 8 times what equals 16. Answer 2. = 10 16

Example Two: Build 3 4 3 4

3 x 3 4 3

Both were multiplied by the factor 2

for a denominator of 12

Percents

5 x 2 8 2

Ratio and Proportion

Note: All of these fractions have the same decimal of 0.5

5 8

Fractions

1 = 1 x 6 6 2 2

Decimals

1 Here are examples of building up the fraction with several version of 2 one.

Whole Numbers

1 2

9 12

Both were multiplied by the factor of 3

Appendix

The trick is picking the right factor to multiply.

Whole Numbers

Chapter Three

Example Three: Here is one beyond the color fraction bars. 1 Build 3 up to have a denominator of 48. Ask yourself, 3 times what equals 48. Since the factors for 48 are not easily evident, you can divide 48 by 3. The answer is 16. 1 x 16 3 16

A factor means the number being multiplied. The answer is called the product.

= 16 48

Decimals

Example: 17 x 3 = 51 17 and 3 are factors. The 51 is the product.

Practice 5: Build up the fraction to the given denominator.

Ratio and Proportion

Fractions

1 2

2 8

5 8

b) x

8 e)

16 h)

2 3

3 4

1 4

c) 5 6

f) 2 5

i) 1 4

Percents

Reducing Fractions Multiplying factor versions of “1” to a fraction is called building up a fraction. Dividing common factors in groups of “1” is called reducing a fraction. Reducing is the opposite of building up , so instead of multiplying, we divide. A fraction is fully reduced when it is in its lowest terms. 5 Example One: Reduce 10 to its lowest terms.

Appendix

5 10

÷ 5

1 2

Note: Review the REFC card at the begining of this chapter for hints on how to pick factor forms of one.

Section 3.4 Equivalent Fractions Whole Numbers

6 Example Two: Reduce 8 to its lowest terms. 6 8

2 2

1/8

2/8

1/4

4/8

5/8

2/4

6/8

7/8

3/4

8/8

4/4

Example Three: Reduce 24 to its lowest terms. One good strategy is to check and see if you can divide by the prime numbers: 2, 3, 5 or 10. You might need to divide twice before you reach its lowest terms.

3 12

3 3

and now continue,

and now you’re done.

Ratio and Proportion

30 10 ÷ 120 10 =

Practice 6: Reduce these fractions to their lowest terms. a)

6 12

2 4

4 8

10 16 2 6 9

14 16 5 10 2

16 80 12 10 So when reducing, remember an equivalent fraction can be obtained by dividing both the numerator and denominator by the same number. Be sure to keep dividing until you can’t find another factor.

Appendix

Percents

6 8

Fractions

3/8

Decimals

The gray bar on these number lines illustrate equivalent fractions. Same length just broken up differently.

Whole Numbers

Chapter Three

Example One: Reduce 18 to its lowest terms. (notice that this fraction is not on your color fraction chart 12 18

รท

Example Two: Reduce

Decimals

8 2

รท

2 to its lowest terms.

Note: Fractions with denominators of 1 can also be written as a whole number.

1 2 Fractions

Practice 7: Reduce or Build up the fraction to the given denominator. a)

Ratio and Proportion

Note: We used the factor 6 instead of a 2 and then a 3. This shortcut will save you

2 3

3 4

h) 14 7

10 = 3 30

1 5

Appendix

100

u) 7 9

17 51

t) 40 = 50

2 5

r) 1 4

q) 7 9

10 15 l)

6 8 n)

5 = 6 30

k) 7 = 6 42 20 = 11 22

i) 1 8

f) 12 = 2 24

5 6

Percents

3 12

Section 3.5 Comparing Fractions

Fraction sizes are most easily compared when they have the same denominator. Let’s see how it is done.

Whole Numbers

Section 3.5 Comparing Fractions

Example One: 1 or 4

Visually it is easy to see that the 1/2 is larger. Here is the mathematical method. Since 2 goes into 4 evenly, build up the half to become fourths. 1 2

2 2

= 2 4 Fractions

Now we can compare apples with apples, 2 4

1 4

Since 8 goes into 16 evenly, build up the ths to become 16ths

7/8

The following symbols are used to compare quantity.

> means “greater than” i.e. 8 > 5

14 16

< means “less than”

i.e. 5 < 8 = means “equal to” 9 = 9 or nine is equal to nine

14 16

Note: the tip points to the smaller number.

Looks like they’re equal. 14 16

7 8

101

Appendix

This is your answer:

What does this symbol > mean?

Percents

Ratio and Proportion

Example Two: Which is larger 16 or 8

Decimals

1 Which is larger 2

Whole Numbers

Chapter Three

Practice 8: Compare fractions using the symbols > or < or = a)

Decimals

3 4

10 16

9 16

5 8

2 5

7 8

3 10

15 16

2 3

1 2

3 4

2 3

Ratio and Proportion

Fractions

Placing Fractions in Order Which is more, a dime, a nickel or a quarter ?

10 ¢

25 ¢

5 ¢

Your answer should be:

25¢

10 ¢

5 ¢

Now look at these coins as values in fractional form:

25 100

10 100

5 100

Percents

The denominator of 100 represents cents.

Comparing the Simple fraction A simple fraction has a numerator of 1.

Appendix

Rewrite the following fractions from smallest to largest:

1 1 1 3 ,5 ,2

here our problem’s visual

102

☞

Section 3.5 Comparing Fractions

1 5

1 3

1 2

Whole Numbers

When fractions have one in their numerator, the larger the denominator, the smaller the fraction. Therefore, the order for these fractions is:

Your Turn: Recreate the visuals from smallest to largest.

Decimals

Comparing fractions using common denominators Example: Rewrite the fractions from largest to smallest. 5 8

3 4

Build up the fractions so that they all share the same common denominator. In this case 16 will be our common denominator.

1 1

5 8

2 2

3 4

4 4

9 16

10 16

12 16

Percents

Now we can order the fractions confidently, from largest to smallest. 12 10 9 16 16 16

Ratio and Proportion

9 16

Fractions

9 16

Returning to our original fractions, here is the correct order.

3 4

5 8

9 16

.75

.63

103

.56

Appendix

And below we see the decimal equivalents. (Rounded to the nearest hundredth place). Placing fractions into decimal equivalents is another way to compare size.

Practice 9: Order these fractions from smallest to largest. a)

Practice 10: Order these fractions from largest to smallest. a)

Appendix

Percents

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Three

104

Section 3.6 Multiplying Fractions

To multiply a fraction simply multiply the numerators together and then multiply the denominators together. Finally reduce if possible. 21 (7)(3) 7 3 = X = 40 (8)(5) 8 5

Example Two:

(1)(2) 2 2 1 2 1 = (2)(3) = = X now reduce 6 3 2 6 3

now reduce

60 80

30 2 รท 2 = 40 2 2

15 and again 20

รท

15 = 20 5 5

3 4

Although repeated reducing works, there are better and faster methods worth learning. Mathematicians are basically lazy people. And if there are faster ways to get the answer, we use them. Next we will show you how we use factoring to make it easier to reduce a number.

What do you get?

Percents

Your Turn: Multiply these numbers (2) (3) (2) (5)

Ratio and Proportion

รท

30 reduce again 40

Fractions

Example Three:

15 60 4 X 16 = 80 5

Decimals

Example One:

Whole Numbers

Section 3.6 Multiplying Fractions

Multiply these numbers (2) (2) (2) (2) (5) What do you get? Appendix

What is this fraction? (2) (3) (2) (5) (2) (2) (2) (2) (5)

105

Whole Numbers

Chapter Three

Using what you have learned about â&#x20AC;&#x153;fraction forms of oneâ&#x20AC;? , we are going to eliminate the pairs of one until we have arrived at our fully reduced fraction. 60

(2) (3) (2) (5) (2) (2) (2) (2) (5)

80 60 80

Decimals

60 80 60 80

(3) (2) (2)

Fractions Ratio and Proportion

Why can we cancel forms of one? Any number multiplied by one is still that number.

5 5 =

Your Turn 7 x 1 x 1 x 1 =

Using what you have learned about factoring, we will now show you a method called canceling. We cancel when the numerators and denominators can be divided evenly by the same number. 1

2 1 x = 2 3 3

the twos canceled 3

Example One: Multiply the following 22

6 =

First cancel by dividing by 3, then cancel by dividing by 11. Finally multiply. Here the numbers are partially factored and then canceled.

Percents

11 3(2)

3 x 11(2)

1 4

Here is what it looks like to divide in your head and cancel as you go: 3 22

Appendix

In each case the number 7 does not change.

Your Turn

7 x 1 = 7 and 7 x 2 = 14 = 7 2 1 1 2

Fraction Forms of One 2 8 10 98 same 2 8 10 98 same

11 6

1 4

In math the answer is not considered complete until fully reduced.

106

Section 3.6 Multiplying Fractions Whole Numbers

Example Two: Multiply a whole number times a fraction.

3 glasses one third full = one whole glass

1 3

And here is the procedure, 3 is written as the equivalent fraction 1 Then multiply as usual. = 3 3

3 x 1 1 3

To make a whole number a fraction, just put a one under it.

18 =

18 1

8 =

Fractions

Practice 11: Multiply these fractions. Reduce if possible. a)

d) 15

1 x 4 = 5 2

e) x 1 = 5

1 x 12 = 3

h) 4

3 x = 16 4

1 x 6 = 16 2

6x 1 = 3

2 16

4 6

= Percents

Ratio and Proportion

x 2 = 2 3

Decimals

Visually this means, this means three thirds.

3 x 10 = 3 5 Appendix

107

Whole Numbers

Chapter Three

Mixed Numbers and Improper Fractions A mixed number includes a whole number and a fraction such as: 1 2 1 4 3 2

An improper fraction is has a numerator that is larger than the denominator.

Decimals

35 3

Your Turn: What decimal numbers are associated with these fractions? 1

Fractions

Below is a visual illustration of the mixed number 2 4

Your Turn: How many

1 4

ths are grayed out on the numberline?

Percents

Ratio and Proportion

Converting Mixed Numbers into Improper Fractions In order to multiply or divide mixed numbers, you will need to first convert them to â&#x20AC;&#x153;improperâ&#x20AC;? fractions. 1 2 Example: Change these mixed number to their improper fraction. 2 4 , 35 Here is the procedure, a) Multiply the denominator by the whole number. b) Then add the numerator - this amount will be your new numerator. c) The new denominator remains the same. 1

2 4 =

Appendix

3 5 =

108

4x2+1 4

3x5+2 5

8+1 4

15 + 2 5

= 4

= 5

Section 3.6 Multiplying Fractions

7 58

2 93

Whole Numbers

Practice 12: Change these mixed numbers to improper fractions:

Decimals

1 13 2

12 5

11 3 Fractions

Converting Improper Fractions into Mixed Numbers

Work Space 37 5 8 = 48

4 8) 37 -32 5

Percents

Work Space

Ratio and Proportion

a) To convert an improper fraction back to a mixed number divide the denominator (bottom number) into the numerator (top number). b) Your answer is the whole number, the remainder is the numerator. c) And the denominator stays the same.

4 4) 17 -16 1

17 1 4 = 44

Appendix

109

Practice 13: Simplify improper fractions to mixed numbers. a)

27 8

25 2

13 5

17 3

93 8

27 16

93 16

35 12

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Three

Reducing your fraction can take place before or after you simplify, but your final answer must always be fully reduced.

Example: Simplify these improper fractions to mixed numbers and reduce if possible. 26 2 1 8 = 3 8 now reduce to 3 4

Work Space 3

8)26

Percents

- 24 2

Same problem but this time we reduced first. 2 รท 2

13 = 4

1 = 34

Work Space 3

4)13

12 1

Appendix

26 8

110

Section 3.6 Multiplying Fractions

65 10

40 6

22 4

22 8

Whole Numbers

Practice 14: Reduce improper fractions and simplify to mixed numbers.

Decimals

32 12

8 3

25 2

Fractions

30 9

You must first change the mixed number to an improper fraction, before you can multiply. Multiply the numerators and then multiply the denominators. Finally reduce if possible. 1

Example One: Multiply the following mixed numbers: 2 4 x 1 7

Percents

First: Convert the mixed number to an improper fraction.

1 4

2x4+1 4

8+1 4

= 4

1 7

1x7+1 7

7+1 7

= 7

8 7

72 72 = 28 28 and reduce

รท 4 = 7

111

and simplify

4 7

Appendix

Then: Multiply straight across. 9 4

Ratio and Proportion

Multiplying Mixed Numbers

Whole Numbers

Chapter Three

Work Space

OR cancel first then multiply straight across. 9 41

8 7

18 7

7)18

4 7

14 4

12 Example Two: Reduce this fraction to its lowest terms 16 .

Decimals

Here is another method. We know there are several groups of “one” in this fraction, so we can just divide them all out. First: Identify the primes 12 16

(2) (2) (3) = 3 (2) (2) (2) (2) 4

12 = 16

(2) (2) (3) = 3 (2) (2) (2) (2) 4

12 16

4 4

3 4

After we cancel, our fraction is fully reduced.

Practice 15: Multiply mixed numbers.

Convert these mixed numbers to improper fractions, multiply and then simplify your answers.

1 12

3 x 14

Appendix

Percents

Ratio and Proportion

Fractions

Now we just cross them out which is also called canceling.

112

x 55

1 43

7 x 18

Section 3.7 Dividing Fractions

1 x 28

1 34

1 x4

x 42

Whole Numbers

d) 1 2

Decimals

3 7x 18

4 25

x 5

x 32 Fractions Ratio and Proportion

Section 3.7 Dividing Fractions When dividing fractions, flip the divisor and multiply.

And here is a whole number’s reciprocal,

4/1 is the reciprocal of 1/4

Percents

The divisor is the second fraction found to the right of the ÷ symbol. Flip means the reciprocal. i.e. 5/6 is the reciprocal of 6/5

Example One: Divide the following fractions.

3 5

= 1 2

x 5 3

=5 6

Appendix

Here is a tip to remember the rule for dividing fractions. Visualize the divsion sign as a pancake flipper.

113

Whole Numbers

Chapter Three

Example Two: Divide the following fractions: 1 ÷ 4

1 2 ÷ 4 1

1 1 x 4 2

1 8

Example Three: Divide the following fractions: 5 1 5 ÷ = 8 4 8 2

Decimals

4 x 1

5 2

1 2

Example Four: Divide the following fractions: 1

8 ÷ 16

8 1 16 = 128 ÷ = 8 x 1 16 1 1

2 3

5 6

b) =

6 ÷ 2 =

7 8

9 10

÷2 =

Appendix

Percents

Ratio and Proportion

Fractions

Practice 16: Divide these fractions. Cancel and or simplify.

114

2 5

7 8

÷6

1 4

1 2

c) 3

f) =

i) =

1 4

1 2

÷4 =

3 5

÷ 10 =

Section 3.7 Dividing Fractions

When dividing mixed numbers, first change the mixed or whole numbers to improper fractions, then invert the fraction on the right of the ÷ symbol. Now you will be ready to multiply and simplify. Here problems are worked out vertically.

1 2 5 2

÷1 ÷

1 3

1 2

÷6

6 1

15 2

4 3

Decimals

Example Two:

Example One:

Fractions

15 1 x 2 6

5 3 x 2 4

15 1 x 2 62

15 8

Whole Numbers

Dividing Mixed Number Fractions

5 4 1 4

Ratio and Proportion

Practice 17: Divide the following mixed numbers. Cancel and simplify when possible.

1 1 8

3 3

2 3

1 5 16

1 1 8

Percents

1 4 8

Appendix

115

d) 3

g) 8

3 4

5 6

4 6 5

2 1 3

5 8 8

7 1 10

1 1 2

6 3

2 5

1 4

3 2

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Three

Section 3.8 Fraction Word Problems x ÷ Refer to your hints for solving problems found in the appendix.

1 13 3 13 1 13 ÷3= ÷ = x = 2 2 1 2 3 6 =

1 6 pounds

The word “Per” means each. In math it often indicates you need to divide.

Appendix

Percents

Example One: If 3 boxes of candy weigh 6 1/2 pounds altogether. Find the weight per box.

116

Section 3.8 Word Problems for Multiplication and Division of Fractions

12 x 3

= 42

x 2

inches

Whole Numbers

Example Two: If one “2 by 4” board is precisely 3 1/2 inches wide, find the total width of twelve boards, the “2 by 4”s are laid side by side.

Multiply because all 12 boards are the same width.

inches thick.

1000

Decimals

Example Three: A piece of paper is

How many sheets of paper will it take to make a stack 1 inch high?

1÷

4 1000

1 1

1000 4

1000

4) 1000

= 250 sheets

Fractions

We are looking to see how many sheets stack to 1 inch high. This is a division problem.

Example Four: Joey can bike 30 laps around a track in 1 hour How many laps could he make in 4 1/2 hours?

1 42

9 x 30 = 2 1

30 x = 135 laps! Yikes 1

1 10 ÷ 2 2

10 = 1

5 ÷ 2

10 = 1

2 x 5

20 = 5

= 4 books

Appendix

The number of books that take up the space in the book bag means to divide.

117

Percents

Example Five: Jennifer’s book bag is 10 inches wide. How many books will the bag hold if a student places his 2 1/2 inch wide books inside the bag ?

Ratio and Proportion

Think, if Joey can bike 30 laps in one hour, this implies that he can also run 60 laps in 2 hours, 90 laps in 3 hours, etc. This repeated addition means to multiply.

Whole Numbers

Chapter Three

Example Six: A seamstress uses 1 1/3 yard of material to make one New Mexico State Flag, how many yards of material will be needed to make 9 flags? 1 9x13

9 = 1

4 x3

36 = 3 = 12 yards

If 1 1/3 yards of material makes one flag, then 1 1/3 + 1 1/3 yards of material makes two flags.

Decimals

Repeated addition means multiply. Example Seven: A full can of diet soda weighs about 3/4 of a pound. You buy a 12-pack of diet soda. What is the total weight of the cans of soda?

Percents

Ratio and Proportion

Fractions

12 x

12 3 x 1 4

36 = 9 pounds 4

Note: If one can of soda weighs 3/4 of a pound, then two cans will weigh 1 1/2 pounds, three cans will weigh 2 1/4 pound, etc. This is multiplication (if you are not sure make a chart). Example Eight: You have to read 1/3 of the pages in the Fractions Chapter in the Math 550 book to catch up to your classmate. This is 14 pages. How many pages are in the entire chapter? 1 14 รท 3

14 = 1

3 x 1

42 = 1 = 42 pages

Since 1/3 of the chapter is 14 pages long, 2/3 would be 28 pages and 3/3 would be 42 pages. This means you need to divide (order counts)!

Example Nine: Candice is going to serve 1/8 of an apple pie to every guest at a birthday party. She expects 24 guests to show up. How many apple pies will she need to feed all of her guests? 24 x

Appendix

3 4

1 8

24 1

1 8

24 = 3 pies 8

Each guest will get 1/8 of a whole apple pie. This means that one pie will feed 8 guests. You need to multiply.

118

Section 3.8 Word Problems for Multiplication and Division of Fractions

Fraction Word Problems

Reduce and simplify your answers if possible. a) A stack of boards is 21 inches high. Each board is 13/4 inches thick. How many boards are there?

Whole Numbers

Practice 18:

Decimals

in 21/2 inches of threads?

Ratio and Proportion

c) Craig has a bolt that has 161/2 turns per inch. How many turns would be

Fractions

b) A satellite makes 4 orbits around the earth in one day. How many orbits would it make in 61/2 days?

Percents

13 d) If a bookshelf is 28 1/8 inches long, how many 2 16 will it hold?

inch thick books Appendix

119

e) Sandra needs to make 16 costumes for the school play. Each costume requires 2 1/4 yards of material. How many yards of material will she need?

f) The Coffee Pub has cans of coffee that weigh 3 1/4 pounds each. The Pub has 8 cans of coffee left. What is the total weight of 8 cans?

g) Candice baked 9 pies that weigh a total of 20 1/4 pounds. How much does each pie weigh?

h) Kevin has read 3/4 of a book, which is 390 pages. How many pages are in the entire book?

i) DJ Gabe is going to serve 1/3 of a whole pizza to each guest at his party. If he expects 24 guests, how many Pizzaâ&#x20AC;&#x2122;s will he need?

Appendix

Percents

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Three

120

Section 3.9 Addition and Subtraction of Fractions

5 8

2 8

7 8

3 4

2 4

5 4

Example One: Add 2 quarters and 3 dimes. Does that make 5 cents? 5 quarters? 5 dimes? Or 5 dollars? None of the above! 5 coins makes the most sense because that would be a common unit. But 5 coins only reveal the quantity of coins. And that is not much information, what we really want to kow is how much money it makes. So to add 2 quarters and 3 dimes we must first convert them to the common language of cents. Here is what these coins look like in fractional form.

3 Three dimes are equal to three tenths of a dollars. So 3 tenths = 10 Here is 2 fourths plus 3 tenths in mathematical form; +

3 10

First we need to convert the fractions so that they have the same denominator. Since we count in pennies, we are going to use 100 as the denominator. 2 4

25 x 25

50 = 100

or 50 pennies

3 10

10 10

30 100

or 30 pennies

30 + 100

80 = 100

121

Appendix

Now we can add the like units. 50 pennies + 30 pennies = 80 pennies. Or in fractional form it looks like this: 50 100

Percents

2 4

Ratio and Proportion

2 Two quarters are equal to two fourths of a dollar. So 2 fourths = 4

Fractions

But, we can not add fractions that have different denominators, the unlike denominators have little in common. In order to add fractions with unlike denominators we must build up one or both of the denominators until they share the same number.

Decimals

Adding Fractions is simple when the denominators are the same. Same or like denominators are also called common denominators. This is because it is as if the fractions speak the same language. i.e the language of 4ths, or 5ths or 8ths etc. Here we add only the numerator and keep the denominator the same. For example:

Whole Numbers

Section 3.9 Addition and Subtraction

Whole Numbers

Chapter Three

Common Denominators When adding and subtracting fractions, we first need to build a common denominator (take another look at section 3.3). 1 Example Two: Add 2

1 + 4

a) Start with the largest denominator.

(4 is the largest)

Decimals

b) See if you can build up the smaller denominator so that it will match the larger. In our example, 4 is the largest denominator and 2 can be multiplied by a 2 to get a 4 therefore, 4 is the Common Denominator. 1 Example Three: Add 2

1 + 3

Ratio and Proportion

Fractions

Letâ&#x20AC;&#x2122;s find the common denominator. a) Start with the largest denominator.

b) See if the fraction with the smaller denominator can be built up evenly to obtain the larger denominator. There is no whole number that can be multiplied by a 2 to become 3, so the two cannot easily build to a 3. d) If the smaller denominator can not become the larger through multiplication, multiply the two together to obtain a common denominator. 3x2=6 Here, 6 is a Common Denominator. And in this case it is the Lowest Common Denominator (LCD) . 3 Example Four: Add 4

2 + 3

a) 4 is the largest denominator. b) 3 can not be multiplied by a factor to get 4. SO c) Multiply 4 x 3 = 12. 12 is the Common denominator and also the LCD.

Appendix

Percents

(3 is the largest)

122

Section 3.9 Addition and Subtraction of Fractions Whole Numbers

Practice 19: Write the LCD for the following sets of fractions. a)

1 2

1 3

2 4

2 3

5 8

d) 1 2

2 3

1 12

2 3

5 6 Decimals

1 2

g) 1 2

1 2

1 4

1 5

h) 7 10

5 10

5 6

1/6 4/15

Percents

Example One: Find the LCD for + 6 x 15 = 90 90 is a common denominator. But is it the lowest? Letâ&#x20AC;&#x2122;s check by writing out the multiples for 6 and 15.

Ratio and Proportion

Note: So far we have seen methods to get a common denominator but it may not always get a lowest common denominator or LCD. Here is a method to yield the LCD.

Fractions

1 16

This way we can be sure 30 is the LCD. And 30 is easier to manage than 90! Appendix

123

Whole Numbers

Chapter Three

3/4 5/6

Example Two: Find the LCD for + 4 x 6 = 24 24 is a common denominator. But is it the lowest? Letâ&#x20AC;&#x2122;s check by writing out the multiples for 4 and 6.

Decimals

This way we can be sure 12 is the LCD. Although we can still see that 24 works as well.

Step by step for adding or subtracting fractions. Example One: Add the following fractions.

Ratio and Proportion

Fractions

3 5

+ 5

4 = 5

If the denominators are the same, simply add the numerators and keep the denominator.

Step by step for adding or subtracting fractions with different decnominators. a) Rewrite the problem vertically. b) Find the LCD. c) Change to equivalent fractions (by building up). d) Add or subtract the numerators (denominators stay the same).

1 2

1 4

Percents

Example Two:

1 2 1

+ 4

2 2 1 1

2 4 1 4 3

Appendix

124

Section 3.9 Addition and Subtraction of Fractions

5 8

- 1

5 8

3 3

15 24

1 3

8 8

8 24

Whole Numbers

Example Three:

Example Four:

2 3

+ 4 = +

2 3

4 4

8 12

3 4

3 3

9 12

Practice 20: Add or subtract the following fractions.

5 = 1 12

Fractions

17 12

Decimals

7 24

Simplify and reduce when possible.

9 16

1 4

d) =

3 5

3 4

+ 10 =

1 2

Percents

+ 4

Ratio and Proportion

3 8

Appendix

125

Whole Numbers

Chapter Three

2 3

1 2

Ratio and Proportion

Fractions

Decimals

+ 16

2 4

3 6

3 4

1 3

Adding and Subtracting Mixed Numbers Note: Donâ&#x20AC;&#x2122;t forget to add or subtract the whole numbers as well!. Example Two

Example One 1

68 + 5=

1 2 + 23 = Rewrite Vertically

Rewrite vertically

= 16

1 + 23

2 = 26

68 + 5

11 8

Appendix

Percents

7 16

126

Section 3.9 Addition and Subtraction of Fractions

+ 5

Example Four 6

39 -12

Rewrite Vertically

2 53

10 = 5 15

3 9

= 3 18

3 5

9 15

1 - 12

9 = 1 18

19 4 4 5 15 = 5 + 1 15 = 6 15

3 2 18

Decimals

Whole Numbers

Example Three

1 =26 Fractions

Borrowing and Subtraction 2.67 - .88

Take a look $2.67 - .88

Example Five: Subtract the following: =

rewrite vertically

0 15 8

0 +8

8 = 14 + 8 7 13 8

Appendix

7 - 13 8

8 = 14 + 8

Percents

15 - 13 8

Note: We always borrow a “one” that is written in fraction form 6/6, 7/7, 3/3 etc. And the fraction form of one we choose is dependent on the denominator.

Ratio and Proportion

Sometimes subtraction requires borrowing. In the case above the ones column has insufficient funds and must “borrow” from the tens column. Subtracting with mixed numbers often presents the same dilemma. In our new case the fraction has insufficient funds and must borrow from the whole number.

1 8

127

Whole Numbers

Chapter Three

Example Six: Subtract the following. 1

rewrite vertically 1 56

6 =4 +6

1 + 6

7 =46

5 -26

2 2 6

5 -2 6 Decimals

- 26

1 =23

1 83

- 6 10

rewrite vertically 1 8 3 7 - 6 10

10 8 30

21 = - 6 30

30 10 = 7 30 + 30

40 = 7 30

21 - 6 30

Appendix

Percents

Ratio and Proportion

Fractions

Example Seven: Subtract the following. Notice that this one is not on the color fraction chart but that the rules are the same.

128

19 1 30

Section 3.9 Addition and Subtraction of Fractions

Simplify and reduce when possible.

a) 4

+ 8 10

+ 2 10

Whole Numbers

Practice 21: Add or subtract the following mixed numbers.

= Decimals

16 6

1 12

+ 12

f) 2

- 3

10 4

- 1=

h) 1

- 23

Percents

+33

Ratio and Proportion

Fractions

6 - 48

Appendix

129

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Three

21 + 17 16

-47

129 3

71 100

- 98 15 =

- 25 =

Section 3.10 Fraction Word Problems + Example One: If brand X can of beans contains 15 1/2 ounces and brand Y contains 12 3/4 ounces, how many more ounces is in the brand X can? 1 15 2

2 = 15 4

6 = 14 4

3 3 - 12 4 = 12 4

3 = 12 4

Means to Subtract

Percents

Your answer is Brand x contains 2 4

more than Brand Y. Means to Add

Appendix

Example Two: Find the total snowfall for this year if it snowed November, 2 1/3 inches in December and 1 3/4 inches in January. 1 2

6 12

130

1/2 inch in

Section 3.10 Fraction Word Problems + -

4 = 2 12

3 + 14

9 = 1 12

Whole Numbers

1 23

7 = 4 12

7 Answer to total rainfall is 4 12

Example Three: A doctor recommends her patient take 12 ounces of a certain bottle of medicine in one day. She takes 2 1/2 ounces in the morning, 1 1/4 ounces in the afternoon and 3 1/8 ounces in the evening. How many more ounces does the patient have to take before bedtime to use up the entire bottle of medicine?

1 14

2 = 1 8

1 38

1 = 38

This number reflects how much she drank. The next step takes how much she drank and subtracts it from the amount left in the bottle.

7 - 68

7 = 68 1 58

Percents

Next subtract this total of meds taken from the original amount in the bottle. 8 12 = 11 8

Ratio and Proportion

7 68

Fractions

This is a two step problem. First total all the ounces she drank in one day. 1 4 22 = 28

Decimals

19 3 12

ounces of medicine

131

Appendix

Example Four: Sandra is sewing together fabric to make winter wear for her four dogs. She had 45 inches of material. She will use 9 1/3 inches of material for her oldest dog, 12 1/12 inches for her biggest dog, and 5 5/6 inches for her smallest dog. How much material will she have left for her favorite dog? This is a two step problem. First, add the parts together. Hint: adding will tell you how much total material was used. 4 1 9 = 9 12 3

Whole Numbers

Chapter Three

1 1 12 12 = 12 12

5 + 5 6

10 = 5 12 15 26 12

12 3 3 3 1 = 26 + 12 + 12 = 26 + 1+ 12 = 27 12 = 27 4

Ratio and Proportion

Fractions

Decimals

Next we will subtract the material she used from how much material she started with to find out how much is left. Hint: “Have Left, “ means to subtract.

4 = 44 4

1 27 4

1 = 27 4 3 17 4

inches of material left for her favorite dog.

Example Five: A car travels 150 1/10 mile one day, 125 1/5 miles the second day, and 65 7/10 miles the third day. How far has the car traveled in those three days? Hint: “How far” implies total number of miles. Therefore add the numbers. 1 1 150 10 = 150 10

1 125 5

2 = 125 10

Percents

7 7 + 65 10 = 65 10 10 340 10 = 341 Example Six: A garden has irregular sides: 14 1/2 feet, 13 3/4 feet, 12 1/8 feet. Find the fourth side if the distance around the garden (its perimeter) is 50 1/2 feet?

Appendix

This is a two step problem. First we find out how long it is around the three sides we have been given.

1 14 2

4 = 14 8

132

Section 3.10 Fraction Word Problems + -

13 4 +

12 8

= 13 8

= 12 8

39 8

= 39 + 8

+ 8

= 39 + 1 + 8

= 40 8

This number reflects the distance of the three sides. Now just subtract that from the whole perimeter 1 4 50 2 = 50 8 3 3 - 40 8 = 40 8

1 10 8

feet on the fourth side

Fractions

Decimals

Whole Numbers

Example Seven: You drink 6 1/4 ounce of soda from a can that has 11 1/2 ounces in it. How much is left in the can? How much is left means to subtract. 1 11 2

2 = 11 4

1 - 6 4

1 =6 4 1 5 4

Ratio and Proportion

ounces Percents

Hint: The â&#x20AC;&#x153;remaining pieceâ&#x20AC;? means to subtract.

1 64

1 = 64

5 =54

133

Appendix

Example Eight: If I cut 4 1/2 inches off a 6 1/4 inch long sandwich, how long is the remaining piece of the sandwich? (It was actually a 6 inch sub, but my friend Gabe took out his ruler and measured it to be just a bit longer.)

Whole Numbers

Chapter Three

1 -42

2 = 44

2 =44 3 14

inches

Example Nine: Find the total width of all the books in your book bag if they are 1 1/2 inches wide, 7/8 inches wide and 2 13/16 inches wide?

Decimals

Hint â&#x20AC;&#x153;Totalâ&#x20AC;? means to add the numbers together.

1 2

7 8

Ratio and Proportion

Fractions

2 16

8 = 1 16 =

14 16

13 = 2 16 35 3 16

3 3 = 3 + 2 16 = 5 16

inches wide

Practice 22: Fraction Applications; add or subtract. a) Three boards are laid side by side. Find the total width of 3 boards that are 1 3/4 inches wide, 7/8 inches wide, and 1 1/2 inches wide.

Appendix

Percents

b) A 7.15 H tire is 6 5/8 inches wide and a 7.15 C tire is 4 3/4 inches wide. What is the difference in their widths?

c) A patient is given 1 1/2 teaspoons of medicine in the morning and 2 1/4 teaspoons at night. How many total teaspoons does the patient receive daily?

d) If 3 1/2 feet are cut off a board that is 12 1/4 feet long, how long is the remaining part of the board? (Draw your own picture).

134

Section 3.10 Fraction Word Problems + -

Percents

i) Sandra wants to make five banners for the parade. She has 75 feet of material. The length needed to create four of the banners are: 12 1/3 ft, 16 1/6 ft, 11 3/4 ft, and 14 1/2 ft. How much material is left for the fifth banner?

Ratio and Proportion

h) Three sides of parking lot are measured to the following lengths: 108 1/4 feet, 162 3/8 feet, and 143 1/2 feet. If the distance around the lot is 518 15/16 feet, find the fourth side.

Fractions

g) I set a goal to drink 64 ounces of water a day. If I drink 10 1/3 ounces in the morning, 15 1/2 ounces at noon, and 20 5/6 ounces at dinner, how many more ounces of water do I have to drink to reach my goal for the day?

Decimals

f) If 3 1/2 ounce of cough syrup is used from a 9 1/4 ounce bottle, how much is left?

Whole Numbers

e) A runner jogs 1 1/2 miles east, 2 1/4 miles south, and 1 3/16 miles west. How far has she jogged?

Appendix

135

The following practice exercises are a mixture of fractions not found on the color fraction chart in your book. The same rules apply. Use your REFC card to help you with some of the divisablity rules.

Practice 23 A mix of challenging fraction problems a)

3 147 5

+ 7 24 =

3 - 39 4

Decimals

Whole Numbers

Chapter Three

X 53

17 8

Ratio and Proportion

Fractions

10 8

f) 9 10

20 10 รท 2 4

- 25 =

h) 7 8

- 28 =

Appendix

Percents

+ 12 4

136

รท 24

+56

Section 3.11 Ratio and Proportion Preview

“All fractions are ratios, but not all ratios are fractions.” Another true statement because like Chihuahuas fractions are a very small breed. Fractions can only show the relationship between a part and a whole (numerator and denominator). Ratios can go on and relate parts to parts, wholes to wholes, or two totally different quantities that measure different things, called rates.

Fractions

A Ratio can mean three different things, 1) A fractional quantity that tells us how much (3 1/2 cookies). 2) A comparison of two like units ( red flowers to orange flowers). 3) A comparison of two unlike units called a Rate (miles to gallons).

Decimals

“All Chihuahuas are dogs, but not all dogs are Chihuahuas.” Makes sense doesn’t it? That’s a true statement because Chihuahuas make up only a small part of the dog group which includes a lot of other breeds of dogs.

Whole Numbers

Section 3.11 Ratio & Proportion Preview

One important ratio is called a “per unit rate” which means the denominator is one. “Per” is a good clue that you are looking at a rate.

55 miles or 55 miles per hour is rate of speed. 1 hour

Your Turn: What is the abbreviation for miles per hour ?

miles per gallon,

revolutions per minute, inch

Percents

Your Turn: Can you think of the abbreviations for the following common automobile terms?

Ratio and Proportion

$10.40 1 hour or $10.40 per hour is a rate of pay.

pounds per square Appendix

137

A proportion is a statement that two ratios or two rates are equal. Creating these statements make it easier to answer problems like the ones given below. The problems can be answered in other ways as well. Go ahead and give it a try. Or you might want to revisit them later after you have mastered the proportion techniques found in Chapter 4. Your Turn: If I earn $12.40 per hour, how much can I make in a 40 hour week?

Your Turn: A car traveled 270 miles in 4.5 hours. How fast was it going?

Your Turn: Joannaâ&#x20AC;&#x2122;s recipe calls for 3 cups of sugar and makes 3 dozen cookies. How many cups of sugar does she need to make 6 dozen cookies?

Your Turn: A # 4 cheeseburger meal costs $4.89. How much will three # 4 cheeseburger meals cost?

Appendix

Percents

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Three

Learning how to solve proportions will be a great asset. So stay tuned to your next chapter.

138

Answers to the Fractions Practice Problems

Answers to Practice Problems Practice 1 a) 2 5

b) 1 4

c) 11 12

d) 3 5

e) 20 50

f) 11 25

g) 7 12

h) 8 24

Practice 2

Everyoneâ&#x20AC;&#x2122;s answer will be a little different but here is a sample School 9/24 .40 Sleeping 7/24 .29 Eating 1/24 .04 Homework 2/24 .08 Phone Time 1/24 .04 TV 1/24 .04 Childcare 0/24 0 Work 0/24 0 Getting Ready 1/24 .04 With Friends 0/24 0 Domestic Chores 1/24 .04 Driving 1/24 .04 Total Hours 24/24 1.0

Practice 3

a)9/20 are men b) Dollars

Maria Joey

Rhonda

Tylerâ&#x20AC;&#x2122;s Boss All together

Fractional part of the total. Decimal Equivalent.

.25

.15

1.00

c) 7/12 of the cake left to eat. d) 5/8 of the trip to go.

Practice 4 a) 4

b) 8

c) 2

d) 27

139

e) 1

f) 2

Whole Numbers

Chapter Three

Practice 5 a)

1 x4 = 4 2 4 8

2x2 3 2

Decimals

Fractions

2x2 = 4 5 2 10

Practice 6 3

1 2

= 3 12

8 1

3 4

Appendix

140

7 8

Percents

1 x3 4 3

2 1

1 x4 = 4 4 4 16

4 Ratio and Proportion

5 x 2 = 10 8 2 16

5 x 2 = 10 6 2 12

3x3 = 9 4 3 12

= 4 6

2 x2 = 4 8 2 16

1 2

1 5

Answers to the Fractions Practice Problems

Practice 7 a)

1 2

8 = 16

3 4

9 = 12

g) 14 7 j)

b) 2 3 e)

1 8

6 8 n)

20 = 10 11 22

p) 7 9 s)

Practice 8 a)

7 9

5 = 1 6 30

a) 4

3 8

7 16

b) 8

141

2 3

2 5

8 20

1 5

5 25

5 20

17 51

35 45

3 12

Practice 9 1

10 15 l)

3 4

15 18

10 = 1 3 30 1 4

4 5

40 = 50

14 18

3 24

7 = 1 6 42

5 6

12 = 1 2 24

2 1

8 = 12

15 16

1 3 1 4

Whole Numbers

Chapter Three

Practice 10 a)

7 8

Decimals Fractions

1 3

3 16 3 16

Practice 12 a)

47 8

27 2

Practice 13

1 4

b) e)

5 12

1 3

1 6

2 5

3 16

h) 2

1 12 2

b) 29 3

31 4

5 2 62 5

9 8

35 3

3 3 8

1 12 2

3 2 5

5 11 8

2 5 3

11 1 16

Appendix

Percents

1 2

Practice 11 a)

Ratio and Proportion

142

13 5 16

11 2 12

Answers to the Fractions Practice Problems

Practice 14 a)

6 2 e)

1 3 3

Practice 15 a)

1 1 16

5 9 8

Practice 16 a)

4 5

7 16

2 3

2 2 3

12 5 e)

4 1 5

8 8

13 16

1 2

2 2 3

2 8 d)

2 5

3 2

143

1 19 2

2 4 3

2 3 2

3 2 4

1 12 2

Ratio and Proportion

Fractions

Decimals

Whole Numbers

Chapter Three

Practice 17 a)

e) 4

2 3 3

3 1 8

7 1 8 3 5 10

3 5 4

Practice 18

Percents

1 5 3 4 5

a) 12 boards b) 26 orbits

c) 411/4 turns d) 10 books

e) 36 yards

g) 21/4 lbs

f) 26 lbs

Practice 19

h) 520 pages

a) 6

b) 12

c) 8

d) 6

e) 16

f) 4

g) 10

h) 12

Practice 20 a)

5 8

1 1 6

Practice 21

Appendix

1 4 2

7 10 5 8

5 16

9 16 10

1 1 4 9 38 16

1 6 2

1 3 8 11 30 15

144

3 17 4

5 2 6 5 7

i) 8 pizzas

1 4

5 12

5 14 12

1 1 8

37 71 50

Answers to the Fractions Practice Problems

Practice 22 a)

1 i) 20 4 feet

c) 54

4 d)

1 8 inches

3 8 4 feet

1 17 3 ounces

Practice 23 1 16 2

17 35 24 5 8

7 1 8 inches

15 4 16 miles 13 104 16 feet

17 107 20

37 50

3 3 4 teaspoons

3 5 4 ounces

1 9 5

1 2 9

145

Chapter

Ratios and Proportions Table of Contents

Section 4.1 Ratios with Like Units......................................................................................149 Practice 1: Express the following comparisons as ratios, then reduce. . . . . 150 Practice 2: Express the ratios in fraction form, then reduce.. . . . . . . 151 Practice 3: Express the following comparisons as ratios, then reduce.. . . . 151

Section 4.2 Ratios with Unlike Units .................................................................................152 Practice 4: Convert to like units, set up a ratio, then simplify. . .

153

Section 4.3 Rates.................................................................................................................154 Practice 5: Express rates in fraction form, then reduce.. . . . . . . . 155

Section 4.4 Proportions......................................................................................................156 Practice 6: Are these proportions true? Yes or No. . . . . . . . . . Practice 7: Solve these proportions for the given variable. . . . . . . .

157 161

Section 4.5 Ratio and Proportion Word Problems............................................................162 Practice 8: Create a Proportion with a variable and solve. . . . . . . .

164

Section 4.6 Percent Preview...............................................................................................166 Answers to Practice Problem..............................................................................................167

The following MALL activities are available for class to check out: Drawing to Scale Shadows Recipe Conversion

147

148

Section 4.1 Ratio with Like Units

In this section we are going to explore ratios. Fractions represent only one type of ratio. They exclusively represent a part to the whole ratio. Here is an example of a part to a whole fraction; the fraction representing the ratio of weekend days (2) to the whole week (7) is

Tues

Wed

Thurs

Fri

Sat

Sun

Decimals

Mon

Whole Numbers

Section 4.1 Ratios with Like Units

Ratios can also be used to compare numbers in a variety of other ways, such as part to part or whole to part. Example One: What is the ratio of the work week to weekend?

Example Two: What is the ratio of weekend to the work week ?

Fractions

5 to 2 or 5 : 2 or 2

Example Three: What is the ratio of the whole week to the workweek? 7 to 5 also written as

A ratio statement can be written four ways:

17 hours awake to 7 hours asleep 17 to 7

17 : 7

17 7

Appendix

17/7 is not a fixed quantity so it should not be reduced to 2 and 3/17th. The fraction bar is kept to represent the relationship between two numbers.

149

Percents

Nutshell

7 : 5 or 5

Ratio and Proportion

2 to 5 or 2 : 5 or 5

Whole Numbers

Chapter Four

Example Four: Aliceâ&#x20AC;&#x2122;s income is $1800/month. If she spends $600/month on rent, the ratio of total income to rent can be expressed as:

1800 : 600

1800

and reduced to 3:1 or 1

600

Decimals

This means, of every $3 she earns, $1 is spent on rent. Be sure to watch the order in which numbers fit into a ratio by placing the first number mentioned in the numerator and the second number in the denominator. Reduce the ratio when possible. If you get a whole number keep the denominator as a one.

Ratio and Proportion

Fractions

Your Turn: If Alice spends $ 600 on rent then she spends $ 1200 on everything else. What is the ratio Alice spends on rent to what she spends on everything else? Notice that this is part to part ratio.

Example Five: Joanna is snack mom for her daughterâ&#x20AC;&#x2122;s class, so she is baking cookies. The recipe calls for 4.5 cups flour and 2 cups sugar. What is the ratio of flour to sugar? 4.5 : 2

4.5 2

Note: This is a ratio and not a fraction

Practice 1: Express the following comparisons as ratios, then reduce. The Wagner Farmland Experience has 14 Chickens, 8 Goats, and 6 Sheep.

Percents

a) What is the ratio of Chickens to Goats? b) What is the ratio of Chickens to Sheep?

Appendix

c) What is the ratio of Sheep to Goats?

d) What is ratio of Sheep to all the animals (including sheep) ?

150

Section 4.1 Ratio with Like Units

a) 3 : 12

b) 25

: 5

d) 12 : 3

e) 42 to 4

c) 100 to 10

Whole Numbers

Practice 2: Express the ratios in fraction form, then reduce.

f) 7 : 30

Decimals

Practice 3: Express the following comparisons as ratios, then reduce.

Fractions

a) What is the ratio of Penguins to all of the players? b) What is the ratio of Redwings to all of the players? One Redwing player trips a Penguin, and is put in the penalty box for 5 minutes. c) While he is in the penalty box, what is the ratio of Redwings to Penguins?

Percents

Another Redwing makes a penalty and goes into the box, but so does a Penguin. For the following few minutes, there will be two Redwings and one Penguin off the floor.

Ratio and Proportion

A Hockey Story. The Detroit Red Wings are playing the Pittsburgh Penguins, and each team has 6 members.

d) What is the ratio of Redwings on the floor to Redwings in the Penalty Box?

151

Appendix

e) Now what is the ratio of Penguins on the floor to Penguins in the Penalty Box?

Decimals

Whole Numbers

Chapter Four

Section 4.2 Ratios with Unlike Units Converting to Like Units

Ratios should be written in the same units of measure. This makes comparisons easier and more accurate. For instance, the comparison of 1 day to 1 year is less clear than 1 day to 365 days. So the aim is to always convert units to the same measure before creating the ratio. Example One: Convert to like units and then set up a ratio in fractional form for: “3 hours to 72 minutes” Since these units (hours and minutes) are not alike, you must convert one unit to the other so that you have minutes to minutes or hours to hours. It is often easier to convert to the smaller unit of measure. In this case we will convert hours to minutes.

Fractions

Since 1 hour is 60 minutes, 3 hours is equal to 180 minutes (3 x 60). Now replace the 3 hours with 180 minutes in the ratio; then reduce and simplify. 3 hours 72 minutes

180 minutes 72 minutes

5 minutes

2 minutes

5 2

Percents

Ratio and Proportion

The label “minutes” can cancel leaving: 5 to 2 Summary for converting to like units: 1. State the ratio:

3 hours to 72 minutes

2. Change units to match:

180 minutes to 60 minutes

3. Place the ratio into fractional form then reduce:

180 minutes

72 minutes

Example Two: Compare 2 quarters to 3 pennies. To keep the same value, substitute 50 cents for the 2 quarters 2 quarters 3 pennnies

50 pennies 3 pennies

Example Three: Compare 1 quarter to 1 dollar? 1 quarter

Appendix

1 dollar

152

1 quarter

= 4 quarters

1 4

Section 4.2 Ratio Unlike Units

Remember, the first item mentioned goes on top: 1 dollar 1 quarter

4 quarters 1 quarter

4 1 Some commonly used Unit Equivalents:

Example Five: Compare 4 yards to 3 feet.

3 feet

12 feet 3 feet

4 feet 1 feet

1 hr = 60 min 3 ft = 1 yd 12 in = 1 ft 365 days = 1 yr

Decimals

Since 1 yard is 3 feet then 4 yards is 12 feet (3 x 4). 4 yards

Whole Numbers

Example Four: How would you write the ratio of “a dollar to a quarter?”

a) 5¢ to $2

Fractions

Practice 4: Convert to like units, set up a ratio, then simplify.

Ratio and Proportion

b) 12 feet to 2 yards

c) 30 minutes to 2 hours

Percents

d) 5 days to 1 year

e) 1 dime to 1 quarter (convert both to pennies) Appendix

153

Decimals

Whole Numbers

Chapter Four

Section 4.3 Rates

Comparing different measures. Rates are a type of ratio that compares totally differing types of quantities; such as miles and hours. These are called rates because they cannot be converted to a common unit. Rates are simplified through division. For clarity, both labels of measurement remain. Example One: Express the following rate and reduce to lowest terms. 80¢ for 2 lbs. of bananas.

Money (Cents = ¢) and Weight (lbs.) measure two different quantities 80 ¢

now divide

2)80

The rate is 40 cents per pound.

Notice that we don’t say “per one pound.” Instead, “per pound,” implies “per one pound.” And sometimes you won’t even hear “per,” just “40 cents a pound” The word “per” represents the fraction bar and is often abbreviated with the letter p. On the following page you will see some more examples of rates such as: mpg’s, rpm’s, mph or in phrases with the word “per” like “dollars per hour” or “ feet per second.” We might say milk is “three dollars a gallon” but we don’t write prices like that. Instead we write “per” or use the fraction bar: $3.00 / gal Example Two: Express the following rate and reduce to lowest terms. It took 8 gallons of gas to drive 200 miles. 200 miles: 8 gallons 200 miles 8 gallons

25 8)200

OR 25 miles per gallon (mpg)

Appendix

Percents

Ratio and Proportion

Fractions

2 lbs

154

Even though gallons was mentioned first in the example, it’s a usual custom to say “miles per gallon” or mpg’s.

Section 4.3 Rates

200 miles : 240 minutes When we compare distance to time, we usually say “miles per hour” (mph), not “miles per minute,” so we will convert the time to hours.

Whole Numbers

Example Three: Express the following rate as miles per hour.

Since 1 hour is 60 minutes, divide 240 minutes by 60 and get 4 hours.

240 min

200 miles 4 hours

now divide

4)200

The speed is 50 mph (read “50 miles per hour.”)

Practice 5: Express rates in fraction form, then reduce.

Fractions

So order and custom both have to be considered when you write rates. Choose the order that will make the most sense. For instance, in example two, we used miles per gallon, but using the opposite order is a way to joke around about a real gas- guzzling vehicle: “So, how many gallons does that thing use per mile?” Ha-ha!

Decimals

200 miles

a) Apples are $4.00 for 5 lbs. How much do apples cost per pound?

Percents

c) Sloane drove 150 miles in two and a half hours. What is the speed in miles per hour ?

Ratio and Proportion

b) Marty, a trucker travels 630 miles in 3 days. How many miles did he average in a single day?

d)Tyler drives 84 miles on 2 gallons of gas. What is his gas mileage (miles per gallon) ? Appendix

155

Decimals

Whole Numbers

Chapter Four

Section 4.4 Proportions Definition: A proportion is a mathematical sentence stating that two ratios or rates are equal. 3 men 2 women

9 men 6 women

$ 1.50 3 lbs

Ratios have like units and Rates have unlike units. Proportions compare 2 rates or 2 ratios. If they are equal then the proportion is said to be true.

= $ 4.50 9 lbs

Proportions can also be written using colons. 3 : 2 :: 9:6

1.5 : 3 :: 4.5 : 9

Be careful to notice the meaning of all those dots as the last proportion because two of them are decimal points.

Percents

Ratio and Proportion

Fractions

Example One: Are these two ratios in proportion to one another? 2 3

2 x 3 = 6 9 3 3

6 9

Yes!

In this case the fraction on the right is built up by a factor of 3, a whole number. Sometimes fraction are built up by 3.5 , a decimal value. 2 2 3.5 7 7 x = = 8 28 8 3.5 28 These are not so easy to decode so we use other methods of testing for equality. The Cross Products Rule: When a proportion is true, the cross products are equal. Cross products are the products of the two diagonal factors in a proportion. Written in colon form the 2 and 28 are called the extremes and the 8 and 7 are called the means. means

2 (28) = 56

2 : 8 :: 7 : 28 or

Appendix

extremes

156

7 28

8 (7) = 56

Section 4.4 Proportions

5 pounds

2 dollars

7 pounds

5 lbs : $2.00 :: 7lbs : $2.80

2.8 dollars

Whole Numbers

Example Two: Are these two rates equal? 5 pounds cost $2 so 7 pounds cost $2.80

This is read “5 pounds is to $2.00 as 7 pounds is to $2.80 .”

5 ÷ 2 = 2.5

and

7 ÷ 2.8 = 2.5.

Decimals

Another way to find out if the proportions are equal is to compare decimal versions of the same number.

When fractions share the same decimal, they are equal. So yes they are in proportion!

5 ( 2.8) = 14

and

2 ( 7) = 14.

Yes!

Fractions

Let’s double check. Are their cross products equal?

Practice 6: Are these proportions true? Yes or No. 1 3 = 3 5

d) 10 5

12 6

1.5 b) 4.5 = 3 9

2.25 9 = 2 8

12 e) 6 = 11 5

2 1 = 16 8

Percents

Your Turn: Create a proportion that is true using the following numbers: 4 , 6, 8, 12

Ratio and Proportion

Your Turn: Create a proportion that is true. Use you own numbers. Appendix

157

Decimals

Whole Numbers

Chapter Four

Solving Proportion Problems Example One: Solve to find the missing number. 3 2

9 ?

We know that in proportions, the cross products are equal. Think of cross products as mates. 2 and 9 are mates and 3’s mate is missing. To solve for the missing number let X mark the spot. Here is what we do to find the lonely mate.

3 2

9 X

3(X) = 2 (9) Cross products are equal.

2 (9)

Fractions

3(x)

3(X) = 18

Multiply the mates. Three ’s mate is missing.

X = 18 ÷ 3

Eighteen divided by the lonely mate.

X=6

Your missing mate is the number six.

Here it is again:

Ratio and Proportion

Multiply the mates 2 x 9 = 18 (the mates) then take that answer and divide by the lonely mate (3) 18 ÷ 3; X = 6 Plug your answer back into the equation.

And get the proportion:

3 2

9 6

Now check your proportion by checking the cross products:. 2 (9) = 18 and 3 (6) = 18

Appendix

Percents

Calculations are correct!!

158

Section 4.4 Proportions

Whole Numbers

Example Two:

Fractions

1 inch 50 miles

6.5 inches X miles

then solve it using the Lonely Mate Formula as follows.

It is 325 miles from Flagstaff to Albuquerque. Example Three: Solve for X to create a proportion that is true. 8

x 24

Plug your answer back into the equation.

And get the proportion:

3 8

9 24

Percents

Multiply the mates 3 x 24 = 72 then take that answer and divide by the lonely mate 72 รท 8; X = 9

Ratio and Proportion

Multiply the mates 50 x 6.5 = 325 then take that answer and divide by the lonely mate 325 รท 1;

Decimals

A map of Arizona and New Mexico shows Flagstaff 6.5 inches apart from Albuquerque. The legend also notes that each inch on the map is equivalent to 50 miles. From this we can calculate the distance from Flagstaff to Albuqerque. We set up the proportion as 1 inch is to 50 miles as 6.5 inches is to X miles. Here the letter X represents the answer to our question.

Now check your proportion by checking the cross products:. 3 (24) = 72 and 8 (9) = 72

Calculations are correct!! Appendix

159

Decimals

Whole Numbers

Chapter Four

Example Four: Solve for X to create a proportion. This example uses fractions instead of whole numbers but the method is the same. Find the mates and then divide by the lonely mate.

2/5 6

4/5

= x

Multiply the mates 6 x

24 2 รท 5 ; X = 12 then take that answer and divide by the lonely mate 5

Now check with the cross products. 6

Fractions

Multiply 1

4 5

12 1

24 5

Example Five: Solve for X to complete the proportion. x : 150 :: 5 : 25 Write this in fraction form first x

Ratio and Proportion

4 24 5 = 5

150 = 25

150(5) = x(25)

Multiply the mates 150 (5) = 750 then take that answer and divide by the lonely mate 750 รท 5; X = 30 Plug your answer back into the equation.

And get the proportion:

150 = 25

30 (25) = 750 and 150(5) = 750 Calculations are correct!!

Appendix

Percents

Now check your proportion by checking the cross products:.

160

Section 4.4 Proportions

x = 50

1/6 : x :: 9 : 63

5 12

80 x

Decimals

3 10

Whole Numbers

Practice 7: Solve these proportions for the given variable.

.75 x 1 = 20

Fractions

x 1.8

5 1.5 x = 3.6

Percents

x: 81 :: 27: 2.43

24 3 = 56 x

Ratio and Proportion

1 3 =

Appendix

161

3 miles 15 miles 10 miles = x miles

1500 copies = 3 hour

x copies 1 hours

14 dollars 1 hour

x dollars 8 hours

30 miles 1 gallon

x miles 7gallons

Fractions

Decimals

Whole Numbers

Chapter Four

Percents

Ratio and Proportion

Section 4.5 Ratio and Proportion Word Problems Proportions can be used to solve practical problems in which one situation can be compared to another. Missing information can be uncovered by using the cross product, then division rule (also called the â&#x20AC;&#x153;lonely mateâ&#x20AC;? formula). Example One: Candice wants to give a party for 60 people. She has a punch recipe that makes 2 gallons of punch and serves 15 people. How many gallons of punch should she make for her party? 1. Set up a ratio from the recipe: gallons of punch to the number of people. 2 gallons 15 people

Appendix

2. Set up a ratio of gallons (X) to the people coming to the party. (place the gallons on top). x gallons 60 people

162

The units (words) of each rate need to line up. The tops need to be the same unit and the bottoms need to match as well.

Section 4.5 Ratio and Proportion Word Problems

2 gallons 15 people

x gallons 60 people

4. Solve using the “lonely mate” formula.

Whole Numbers

3. Set up a proportion.

2 X 60 = 120 and then 120 ÷ 15 = 8 Decimals

This means she should make 8 gallons of punch! Does the answer seem correct to you? 8 gallons is a lot of punch, but 60 is a lot of people, too

1. Set up a ratio between the amount of money he makes and the dozen envelopes he stuffs (one dozen = 12).

Fractions

Example Two: Travis stuffs envelopes for extra money. He makes 25 cents for every dozen he stuffs. How many envelopes will he have to stuff to make $10.00?

$.25

12 envelopes

$ 10 x envelopes

3. Set up a proportion between the two ratios. $.25

$ 10 x envelopes

4. Multiply by the mates and divide by the lonely mate. 10 x 12 = 120 and then 120 ÷ .25 = 480

Percents

12 envelopes

Ratio and Proportion

2. Set up a ratio between the amount of money he will make and the number of envelopes he will have to stuff.

This means that he will have to stuff 480 envelopes to make $10.00 !

Appendix

163

Decimals

Whole Numbers

Chapter Four

Example Three: A three-pound can of coffee costs $5.67. How much does a pound of coffee cost? 3 pounds $5.67

1 pounds x

5.67 x 1 = 5.67 then 5.67 รท 3 pounds = 1.89 ; The answer is $1.89 Example Four: A bank robber stands frozen in front of the police. His money bag is 12 inches tall and casts a 14 inch shadow. His own shadow is 7 feet long. How tall is the bank robber? These two figures are in proportion to each other in the following way. x feet tall

7 x 12 = 84 then 84

12 inches tall 14 inches shadow

รท 14 = 6 feet with arms up!

Practice 8: Create a Proportion with a variable and solve. a) Alice will need to track the weight of premature infants when she graduates. For practice she is asked to calculate the following. A premature infant is gaining 1.5 ounces in one day. At that rate, how many days will it take for him to gain 9 ounces?

Percents

Ratio and Proportion

Fractions

7 foot shadow

Appendix

b) Johnny works at the copy center as part of his work study. He knows that the copier can duplicate 2400 copies in one hour. How many copies can it make per minute?

164

Section 4.5 Ratio and Proportion Word Problems

Decimals

d) Marty buys Rimmie a five-kilogram sack of Beefy-Bones dog food weighs 11 lbs. How many pounds would a 1-kilogram sack weigh?

Whole Numbers

c) Sandra is working part-time as a seamstress. The company will pay her $.50 for every 4 pockets she sews on. How many pockets would she have to sew on to earn $60.00?

Fractions Percents

f) Candice finds a recipe that makes 6 servings of Creamy Pasta. She would like to make only 3 servings. If the recipe normally asks for 3/4 cup of milk. How much milk will she need for 3 servings?

Ratio and Proportion

e) In preparation to work at the bedside, Alice is given the following homework assignment. A patient is receiving 1 liter of IV fluids every 8 hours. At that rate, how much will he receive in 3 hours? ( 1 liter equals 1000ml).

Appendix

165

Fractions

Decimals

Whole Numbers

Chapter Four

Section 4.6 Percent Preview You just finished a unit on Ratios and Proportions of all kinds. Your next unit is going to deal in depth with a very special ratio called “per cent”. Percents are actually another way to handle fraction and decimal ideas. Oh no, not another fraction/decimal idea! Who do we have to thank for that? Well believe it or not…the Romans. Weird looking letter and numbers are not the only thing we got from the Romans. If you saw “Gladiator,” you heard the military rank of Centurion used. A Roman Centurion was sort of a Lieutenant in charge of 100 soldiers (100..cent… centurion..century…get it) ? Well among his many other duties the centurion was responsible for reporting to his commander after a battle the count of how many men were still standing in his group of 100 (century). The centurion also was required to slit the throats of those who could not stand... but that’s another story. So if he found 75 of his guys alive on the battle field, he could report “75 per century” survival (or LXXV per C). 100 turned out to be a real handy reference number, especially when money systems developed using place value and base 10 so percents stuck around. Percents are actually ratios, that always refer to 100 as the whole, so 75% is 75 per 100. That little symbol (%) is a shorthand form of 100.

20% =

20 100

= 2

= 1 5

And here is the same percent in decimal form 20% = 20 ÷ 100 = 0.2 Yes, that’s right, three ways to show the same idea of parts and wholes. So just what should the tip be on a bill that comes to, $28.00? You probably have a tip calculator on your phone, but here’s how you figure it out using a proportion:

20 x = 100 28

Percents

Ratio and Proportion

If you go out to eat, you should leave a tip for the server, it’s just polite—to say nothing of that’s how they make their money. 20% tip means 20 cents on every dollar spent on the bill. Look at that as Fraction:

To solve that proportion: 20 x 28.00 ÷ 100. That’s $5.60 so be cool and give

Appendix

the server six dollars

166

Answers to Practice Problem Ratios & Proportions

Practice 1:

a) 14 : 8 reduced 7 : 4 b) 14 : 6 c) 6 : 8 reduced

3 : 4 d) 6 : 28

Practice 2: a)

3 1 = 12 4

reduced 7 : 3

25 5 = 1 5

reduced 3 : 14

Practice 3:

a) 6 : 12 reduced 1 : 2 b) 6 : 12 c) 5 : 6

Practice 4: a).

1 40

Practice 5: a)

80 Cents 1 Pound

Practice 6: a) no

b) yes

d) 4 : 2

2 1

210 miles 1 day

c) yes

Practice 7:

a) x = 15 c) x = 1.17 or 7/6 e) x = 0.6 g) x = 900 i) 50 miles k) 500 copies

d) yes

12 4 = 1 3

42 21 = 2 4

7 30

reduced 1 : 2 reduced 2 : 1

1 4

e) no

1 73

60 miles 1 hour

e) 5:1

2 5

42 miles 1 gallon

f) yes

b) x = 192 d) x = 15 f) x = 7 h) x = 12 j) $ 112 l) 210 miles

Practice 8: a) x = 6 days

d) x = 2.2 lbs. or 2

100 10 = 10 1

b) x = 40 copies/minute c) x = 480 pockets

1/5

lbs

e) x = 0.375 L or

167

3/8 L

f.) x =

0.375 cup or

cup

Chapter

Percents Table of Contents

Section 5.1 The Percent Ratio.......................................................................................... 171 Practice 1: Convert these fractions into percents. . . . . . . . . . . 171 Practice 2: Find the complement.. . . . . . . . . . . . . . . 173

Section 5.2 Pieces in the % Puzzle.................................................................................. 174 Practice 3: Identify the percent, part, and base. . . . . . . . . . Practice 4: Including the unknown, identify the “%” , “P” and “B”. . . . .

174 175

Section 5.3 The Percent Proportion................................................................................ 176 Practice 5: Translate the following into a percent proportion.. . . . . . 177 Practice 6: Solve each Percent Problem. . . . . . . . . . . . . . 179 Practice 7: Solve each percent problem for the unknown. . . . . . . . 180 Practice 8: Solve each percent problem for the unknown. . . . . . . . . 181

Section 5.4 Percent Word Problems................................................................................ 182 Practice 9: Percent Word Problems . . . . . . . . . . . . . .

184

Section 5.5 Equivalent Forms.......................................................................................... 186 Practice 10: Complete the Decimal - Fraction Chart. . . . . . . . . Practice 11: Complete the Fraction - Decimal - Percent Chart. . . . . . Practice 12: Complete the Fraction - Decimal - Percent Chart. . . . . .

186 187 188

Section 5.6 Simple Interest.............................................................................................. 189 Practice 13: Solve each of these interest problems:. . . . . . . . .

190

Answers to Practice Exercises.......................................................................................... 191 Copyright ©2009 Central New Mexico Community College. Permission is granted to copy, distribute and/or modify this document, under the terms of the GNU, Free Documentation License. Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is available on http://www.gnu.org/ copyleft/fdl.html

The following MALL activities are available for class check out: Payroll Buying a Car Ordering Supplies Books n Bux

169

170

Section 5.1 Percent Ratio

A Percent is a special ratio that compares a number to 100 and uses the % sign. “Per” and “cent” make up the word percent. Per means to divide and Cent means 100. So together we can see it means divided by 100 or as compared to 100.

Whole Numbers

Section 5.1 The Percent Ratio

Your Turn: A penny is also called a _________ and 100 years is also called a ____________.

Example One: Convert these percents into fractions

a) 25%

b) 50%

c) 67.5%

All we need to do is remove the % sign and replace it with a 25 a) 100

50 b) 100

100

67.5 c) 100

Your Turn: Convert 17% to a fraction.

Fractions

Decimals

We use percents because it helps us get a handle on both the very large and very small numbers.

Example Two: Convert these fractions into percents. 40

a) 100

100

b) 100

All we need to do is remove the

a) 40%

c) 100

100

and replace it with a % sign.

b) 100%

c) 75 %

Ratio and Proportion

Your Turn: Convert 1/100 to a percent.

a) 3/100

b) 45/100

c) 50/100

d) 100/100

e) 217/100

f) .1/100

Percents

Practice 1: Convert these fractions into percents.

Convert these percents into fractions. h) 11%

i) 98%

j) 43% k) 200% l) .5% Converting a fraction that has a denominator of 100 isn’t too complicated.

171

Appendix

g) 75%

Decimals

Whole Numbers

Chapter Five

But Fractions with different denominators, are a little trickier. Fractions, decimals and percents ( % ) are used to represent parts smaller than one. So if it sounds to you like there are three ways to show amounts that are less than one… you’re totally right. There are! Why is that? Well, sometimes it’s just a matter of style, like these 3 ways of printing the letter “a”:

A A

Sometimes it’s a matter of custom for particular uses or professions. In Cooking, carpentry and sewing, we frequently use fractions.

Ratio and Proportion

Fractions

Take 1 3/4 cup sugar, 2 1/2 cups flour, 1/2 tsp. salt, and 2/3 cup shortening, mix that all up, then part it out in little pieces on a cookie sheet. Bake it and that’s exactly what you get…cookies! Money information usually comes in decimal form. For example, if your bank balance is $8.45 and you deposit your check for $207.45, your balance goes up to $215.90 (at least until you start paying bills.) Percents are common in business and politics due to the really huge numbers involved. It is easier to understand the larger amounts if we can related them to something smaller. Like 100. Think the national debt, our population, or the cost of health care. Let’s look at an example having to do with pay checks. So what exactly does it mean when 24% of your paycheck is taken out for Uncle Sam the tax man? It means that 24 of every 100 dollars you make goes to taxes. 24% of a pay check of $200 dollars is only 48 dollars but 24% of your boss’s check of $2000 is going to be a lot more, like 480 dollars!

When a Pharmacy Tech. says that peroxide solution is 3%, this means there is 3 ounces of peroxide for every 100 ounces of solution. The rest (97 oz) is water. When we refer to ratios, the units can vary a lot (ounces, cc’s, gallons, etc.).

Appendix

Percents

In Pharmacy and Chemistry we also use percents, but sometimes for the opposite reason. The amounts they deal with are often very small.

172

Section 5.1 Percent Ratio

Here are three forms of complements. Notice that the fractions and decimals add to one, but percents add to 100. 3 7 10 10 + 10 = 10

.3 + .7 = 1.00

30% + 70%= 100%

Whole Numbers

Complementary Percents

100% is one whole.

Prefer Brand X

Don’t like Brand X

40%

ALL the People

Decimals

Example One: When we say 40% of people prefer Brand X that also implies that 60% of people don’t like Brand X.

100% 100% - 40% = 60% Fractions

Example Two: Complements don’t only come in pairs. There can be more than two percentages adding to 100. For instance, if 55% of voters voted for one candidate while 30% voted for another, we still have 15% not accounted for. Perhaps they did not vote at all! Voters for Voters for Undecided ALL the One Two Voters 30%

100%

Ratio and Proportion

55%

100% - 55% - 30% = 15% Example Three: What is the complement of 15% ? 100% - 15% = 85%

Practice 2: Find the complement. b) 78%

c) 65.5%

d) 25%

e) 16 %

f) 88.9%

g) 5%

h) 44%

Percents

a) 35%

i) If 20% of a class failed an exam what percent passed?

173

Appendix

j) If 75% of car buyers prefer the color white and 15% like black, what percent (%) like other colors? Hint: make a chart.

Whole Numbers

Chapter Five

Section 5.2 Pieces in the % Puzzle Every percent problem has three possible unknowns, or variables: the percent, the part, or the base. In order to solve any percent problem, you must be able to identify these variables. .

Decimals

In the problems below, the percent will have a percent sign (%). The base always follows the word “of.” The base means the “whole” amount. The part usually will be at the beginning of the sentence (in front of “is” or “=”). Sometimes the part will follow the is or equal (=) at the end of the sentence. The part means the amount that is examined in relationship to the whole.

Percents

Ratio and Proportion

Fractions

Look at the following examples. All three variables are known and identified as percent, base, and part. Example One: a)

21 is 70% of 30

25% of 200 is 50

70 is the percent

25 is the percent

30 is the base

200 is the base

21 is the part

50 is the part

6 is 50% of 12

15 = 33.3% of 45

6 is the part

15 is the part

50% is the percent

33.3% is the percent

12 is the base

45 is the base

Example Two: Identify the part ( P ) base ( B ) or percent ( % ) P % base 170 is 25% of 680

Appendix

Practice 3: Identify the percent, part, and base Write percent (%) over the percent, a “P” over the part, and a “B” over the base. a) 8 is 40% of 20 b) 25% of 8 = 2

174

Section 5.2 Pieces in the % Puzzle

e) 5 is 1% of 500

f) 16% of 300 = 48

g) 20 is 50% of 40

h) 1/2 % of 250 = 1 1/4

i) 66 2/3 % of 3 is 2

j) 1 is 33 1/3 % of 3

(P) b) 35

(%) 75%

(B) of _____

(Base is unknown)

Fractions

Example One: One of the three variables (P, B, or %) is the unknown in these percent problems. Identify the percent, part and base in each problem by writing “%” over the percent, a “P” over the part, and a “B” over the base. (%) (B) (P) a) 30% of 500 is ______ (Part is unknown)

Decimals

d) 75% of 100 is 75

Whole Numbers

c) 15 = 50% of 30

Practice 4: Including the unknown, identify the “%” , “P” and “B”

c) 43 is what percent of 483?

d) 1.6 is 8% of what?

e) 39.7% of what is 8.1?

f) 40 = ______% of 40?

g) ______ % of 803 is 1?

h) 1/2 % of 567.375 is what?

i) 48 = 16% of _____?

j) What percent of 30 is 20?

Percents

b) What is 87.5% of 8?

Ratio and Proportion

a) 7% of 78 is ______?

Appendix

175

Whole Numbers

Chapter Five

Section 5.3 The Percent Proportion Percents provide a comparison of numbers to each other. In a percent problem, a part is compared to the base while a percent is compared to 100.

Decimals

Let’s examine the following percent problem

21 is 70% of 30

70% implies the ratio

70/100

The part 21 is compared to base 30 in the ratio

21/30

Keep in mind that these two ratios are equal to one another.

Fractions

Whenever one ratio is equal to another ratio, the equation is called a proportion. All percent problems can be set up as proportions. There are other ways to solve percent problems. We are going to review this one method as it seems to be an easy one. Example One: Translate the following into a percent proportion.

70 % of 30 is 21

Ratio and Proportion

70 100

21 30

is a proportion

In proportions, since the two ratios are equal, if you cross-multiply you will get the same answer. 70 x 30 = 2100

and

100 x 21 = 2100

Note: If you don’t get the same answer, it means that the ratios are not equal nor in proportion to each other.

Percents

Example Two: Translate the following into a percent proportion

6 is 50% of 12 = 50 100

6 12

and let’s check with cross multiplication

Appendix

50 x 12 = 600

176

100 x 6 = 600 looks good!

Section 5.3 The Percent Proportion

a) 10 is 25% of 40

b) 50% of 200 is 100

Whole Numbers

Practice 5: Translate the following into a percent proportion.

Solving percent problems for an unknown

Use this formula. % 100

Decimals

Cross multiplying is an easy way to solve all percent problems when one of the three numbers is missing. p B

2) To find the unknown, cross multiply the numbers in opposite corners.

Fractions

1) Set up percent problems by placing the numbers in ratios; replace the unknown value with the letter it represents.

3) Then divide by the remaining number (lonely mate).

use this formula

% 100

p B

1) Set up percent problems by placing the numbers in ratios, replace the unknown value with the letter it represents. 6 P = 100 20

6 x 20 = 120

Percents

2) Multiply the numbers in opposite corners (there is only one way you can cross multiply).

Ratio and Proportion

Example One: What is 6% of 20?

3) Divide by the remaining number (lonely mate). 120 รท 100 = 1.2 1.2

100)120.0

Appendix

Answer: 1.2 is 6% of 20

177

Whole Numbers

Chapter Five

Example Two:

What % of 50 = 7?

Use this formula: % 100

p B

Decimals

1) Set up percent problems by placing the numbers in ratios; replace the unknown value with the letter it represents. In this case it is the % sign. % 100

7 50

2) Multiply the numbers in opposite corners 7 x 100 = 700

Fractions

4) Divide by the remaining number (lonely mate): 700 รท 50= 14 14

50)700

Ratio and Proportion

Answer: 7 is 14% of 50

Example Three:

4 is 25% of what?

1) Set up percent problems by placing the numbers in ratios and replace the unknown with the letter it represents. 25 100

4 B

Percents

2) Cross multiply the opposite corners: 4 x 100 = 400 3) Divide by the remaining number (lonely mate): 400 รท 25= 16 16

25)400

Appendix

Answer: 4 is 25% of 16

178

Section 5.3 The Percent Proportion

Round to the hundredths place if necessary.

e) What % of 20 is 40?

f) What percent of 156 is 78?

g) What is 80% of 40?

h) What is 75% of 80?

i) What is 10% of 50?

j) What is 100% of 38?

Percents

d) What % of 60 is 12?

Ratio and Proportion

c) 67 is 100% of what?

Fractions

b) 5 is 20% of what?

Decimals

a) 3 is 50% of what?

Whole Numbers

Practice 6: Solve each Percent Problem.

Appendix

179

Practice 7: Solve each percent problem for the unknown. b) 57 is 30% of what?

c) 5 is 31.25% of what?

d) What percent of 109 is 23?

e) What percent of 76 is 11.4?

f) What percent of 42 is 3.57?

g) What is 25.5% of 12?

h) What is 30% of 72?

i) What is .5% of 45?

j) What is 6.5% of 28.6?

Appendix

Percents

Ratio and Proportion

a) 94 is 80% of what?

Decimals

Round answers to the hundredths place, if necessary.

Fractions

Whole Numbers

Chapter Five

180

Section 5.3 The Percent Proportion

Round decimal answers to the nearest hundredth, if necessary. Reduce fractional answers to lowest terms. b) 20.5 = 13% of what?

c) 66 % of 300 = what?

d) What percent of 16.7 is 4.3?

e) 34 % of 103 is what?

f) 66.3 is what% of 56?

g) 16 is what percent of 38.1?

h) 172 is 35.83% of what?

i) .25% of 44 is what?

j) 25 is 8.5 % of what?

Decimals

a) 5 is 33 % of what?

Whole Numbers

Practice 8: Solve each percent problem for the unknown.

Fractions Ratio and Proportion

Percents Appendix

181

Whole Numbers

Chapter Five

Section 5.4 Percent Word Problems Here are several tips that will help you solve percent word problems. 1) Make sure you understand the question that is asked.

Decimals

2) Sort out the information to make a basic percent problem, such as “30% of what is 17?” 3) Keep in mind that sometimes, you will have to subtract or add some of the numbers in order to find the complement. 4) The base will always be the original number, price, or total. And is found right after the word “of.”

Percents

Ratio and Proportion

Fractions

Example One: Kevin won 80% of the games he pitched. If he pitched 35 ball games, how many games did he win? 80% of 35 is what? 80 100

P 35

28 100 g2800

80 x 35 = 2800 He won 28 games.

The complementary information says he also lost some games. (35 - 28 = 7) He lost 7 games. This wasn’t the question posed in the problem. But it might be! So read carefully! Example Two: Craig, an electrician, worked 7 months out of the year. What percent of the year did he work? Round your answer to the nearest hundredth. Factoid: 12 months = 1 year

What percent of 12 is 7? % 100

Appendix

7 12

7 x 100 = 700 and next we divide Craig worked 58.33% of the year.

182

58.33 12 g 700.00

Section 5.4 Percent Word Problems

Example Three: There are 28 students in a class. Sixteen of those students are men. What percent of the class are women? (Round to the nearest tenth). %

Women

Total People

100%

So we ask ourselves, 12 is what % of 28? %

12 28

Multiply 100 x 12 = 1200

now divide

42.86 28)1200.00

Fractions

100

To figure out the number of women in the class we subtracted 16 from 28.

Decimals

Men

Whole Numbers

Sometimes the information needed to solve a percent word problem is not stated directly. So you will need to sort out the numbers given in the problem. Organizing all the information into a table (box) format can help you see what numbers you have and what you need. You can often complete the chart using logic and addition or subtraction.

Next round to the nearest tenth.

Example Four: Travis took a math test and got 35 correct along with 10 incorrect answers. What was the percentage of correct answers? (Round to the nearest hundredth). Correct Answers

Incorrect Answers

Total Answers

100%

100

35 45

First 35 x 100 = 3500 then

Percents

35 is what % of 45?

The total will be our base. And to figure out the total we added the parts.

Ratio and Proportion

The women are 42.9% of the class are women.

77.777 45 g3500.000

Notice how the complements complete the table.

183

Appendix

So 77.78% of the answers were correct.

Practice 9: Percent Word Problems Set up a basic percent proportion. Sometimes you will have to do extra steps to solve the problem. Follow rounding directions. a) Alice earned a grade of 80% on a math test that had 20 problems. How many problems on this test did she answer correctly? (Round to the nearest whole number).

Decimals

Whole Numbers

Chapter Five

c) A metal bar weighs 8.15 ounces and 93% of the bar is silver. How many ounces of silver are in the bar? (Round to the nearest thousandth).

d) Peggy put $580 into a savings account for one year. The interest earned on the account was $37.70 for the year. What is the interest rate (%) on this savings account? (Round to the tenth of a percent).

e) Alice answered 46 problems on another test correctly and received a grade of 92%. How many problems were on the test, if all the problems were worth the same number of points? (Round to the nearest whole number).

Appendix

Percents

Ratio and Proportion

Fractions

b) There are 36 carpenters in Craigâ&#x20AC;&#x2122;s crew. On a certain day, 29 were present. What percent showed up for work? (Round to the nearest tenth).

184

Section 5.4 Percent Word Problems

Decimals

g) Joanna bought a handy cordless electric drill at 85% of the regular price. She paid $32.89 for the drill. What was the regular price? (Round to the nearest cent).

Whole Numbers

f) Gabe found a wrecked Trans-Am that he could fix. He bought the car for 65% of the original price of $17,200. What did he pay for the car? (Round to nearest dollar).

h) Craig’s crew is made up of 9 men and several are women. 75 % of the crew are men. How many people are on the crew? Fractions

i) Alex earns $12,800 a year. The income tax withheld for the year was $1,920.00. What percent of his earnings were taken out for taxes?

185

Appendix

l) Tyler’s favorite softball team, the Royals played 75 games and won 55 of them. What percent of the games did they lose? (Round to the nearest tenth).

Percents

k) There are 32 students in Candice’s culinary arts class. Nine of those students are women. What percent are men? (Round to the nearest tenth).

Ratio and Proportion

j) At a sale, Kevin bought a shirt for $15. This price was 80% of their original price. What was the original price? (Round to the nearest cent).

Decimals

Whole Numbers

Chapter Five

Section 5.5 Equivalent Forms You might remember from the start of the unit that fractions, decimals and percents are all ways to show parts (amounts that are between whole numbers.) Because they all do the same thing we are able to change one form into another. In section 2.8, you changed fractions into decimals and then back again. Letâ&#x20AC;&#x2122;s review. Change a fraction to a decimal: divide the top number by the bottom number:

Fractions

Work Space .5 2)1.0 - 10

= .5

Ratio and Proportion

Percents

.2 5)1.0 - 10 0

85 = 8.2 0

Change decimals to fractions: Determine the place value of the decimal number, then construct a denominator matching the place value. The tenths place value with have a 10 denominator, a hundredths place value will have a 100 denominator etc. For example:

Appendix

Work Space

seven tenths = .7 = 10

fifteen hundredths = .15 =

Practice 10: Complete the Decimal number

Decimal - Fraction Chart

Fraction with denominator of 100

.6 .2

95/100 .01 .001

186

5 15 Ăˇ 5 100

Fully reduced Fraction

1/4 1/10

3 20

Section 5.5 Equivalent Forms

.27 = 27.% = 27%

1.40 = 140.%= 140%

7.00 = 700.% = 700%

1.4 x 100 = 140%

.25 x 100

.0023 x 100

Practice 11: Complete the Fraction - Decimal - Percent Chart

Decimals

Or you can convert a decimal on a calculator by multiplying by 100

Whole Numbers

Changing decimals to percents: Move the decimal point two places to right and if needed add zeros as place holders. Note that once the decimal point is moved to the right of the ones place, it can be dropped. As shown below.

Rounded to he nearest hundredth. Decimal number

Percent

.22

250%

41/50

80/100

Ratio and Proportion

1/100

Reduced Fraction Fractions

15%

Fraction with denominator of 100

.75

Changing Percents to Decimals: move the decimal point two places to the left (if needed add zeros as placeholders).

30% รท 100 = 0.30

.57%

187

รท 100 = 0.0057

Appendix

30% = .30 00.57% =.0057 250% = 2.5 Or you can convert a percent to a decimal on a calculator by dividing by 100. As shown below.

Percents

1/7

Whole Numbers

Chapter Five

Sometimes there are fractional percents. They are still percents. In these cases, just change the fraction to a decimal first and move the point second. Example One: Convert 300 thousandth.

4/7%

to a decimal and round to the nearest

4 Ăˇ 7 = . 5714286 300 4/7 % = 300.5714286 % = 3.00571428 â&#x2030;&#x2C6; 3.006

Rounded to the nearest hundredth. Decimal

Percent

Fraction with denominator of 100

Reduced Fraction

1/20

10%

Fractions

Decimals

Practice 12: Complete the Fraction - Decimal - Percent Chart

1/8

Ratio and Proportion

.2 1/3 .4

50%

Percents

75/100 83% 1.00

Appendix

600/100 20%

188

Section 5.6 Simple Interest

One of the everyday uses of percents include interest. It is the way we earn money on our savings and the way we spend money through acquiring debt. It is the way mortgages are figured, monthly payments are determined on car loans. And of course it is the way credit card companies make so much money.

This is the formula to express simple interest. I (interest) = P (principal) x R (rate) x T (time) I=PxRxT

Decimals

There are many kinds of interests but for this course we are only going to look at simple interest.

Whole Numbers

Section 5.6 Simple Interest

I = PRT

Fractions

The Interest ( I ) is the dollar amount earned or owed on a lump sum. The Principal (P) is the amount borrowed or deposited called the lump sum. The interest Rate (R) is how many dollars per hundred dollars (%) you will earn or owe. Hint: When using Rates be sure to translate the % to its decimal form. The Time (T) is assumed to be one year unless otherwise stated. If the time is in months, it can be found using the following ratio:

Example One: Jennifer borrowed $500.00 from the bank. She must repay the loan in 2 years. The rate for the loan is 6%. How much interest does she owe? First 6% is converted to .06 decimal equivalent.

Ratio and Proportion

months given 12

I = 500 X .06 X 2

This means she must not only pay back the 500 dollars but she must also pay 60 dollars to bank for the convenience of having the money before she saved it. Normally you start paying a loan back as soon as you borrow it. So what would her two year monthly payments be? (Sometimes loans are deferred and that means the repayment doesnâ&#x20AC;&#x2122;t begin until a certain date such as the end of school).

189

Appendix

Well she owes 500 dollars plus 60 dollars interest. That is a total of $560. There are 24 months in 2 years. So if we divide that into monthly installments, we will have : 560 Ăˇ 24 = $23.33 every month for two years.

Percents

I = $60.00

Whole Numbers

Chapter Five

Example Two: Candice puts $4500 in on an investment that earns 8.5% over 18 months? How much money will she earn? Again the rate is converted from 8.5% to the decimal .085 I=PxRxT I = 4500 X .085 X 18

= $573.75 is the interest earned.

Decimals

12 How much will be in the bank after 18 months? Well she will still have the $4500 but now she will also have earned $573.75. So altogether she now has 4500 + 573.75 = $5073.75

a) Travis gets a student loan from the New Mexico Educational Assistance Foundation to pay for his educational expenses this year. Find the interest on the loan if he borrowed $2000.00 at 8% for a year.

b) Gabe is starting a small business in Albuquerque. He borrow $10,000 from the bank at a 9% rate for 5 years. Find the total interest he will pay on this loan.

c) Elena is tired at the end of the term and decide to borrow $500.00 to go on a trip to Reno. She goes to the bank and borrows the money at 11% for 2 years. How much interest will she pay?

Percents

Ratio and Proportion

Fractions

Practice 13: Solve each of these interest problems:

Appendix

d) What is the interest on a loan of $2500 that is borrowed at 9% for 7 months? How much would it cost to repay the loan?

e) Do you understand what interest means? Circle one

190

YES! NO!

Answers to the Percents Problems

Practice 1: a) 3%

Whole Numbers

Answers to Practice Exercises Chapter 5 Percents

b) 45%

c) 50%

d) 100%

g) 75/100 or reduced to 3/4

h) 11/100

i) 98/100 reduced to 49/50

j) 43/100

k) 200/100 reduced to 2

l) .5/100 = 5/1000 = fully reduced to 1/200

a) 65%

b) 22%

c) 34.5%

d) 75%

e) 84%

f) 11.1%

g) 95%

h) 56%

i) 80% passed

f) .1%

Decimals

Practice 2:

e) 217%

j) 10% of car buyers like other colors

Practice 3: b) % B P

f) % B P

g) P % B

c) P % B

Fractions

a) P % B

d) % B P e) P % B

h) % B P i) % B P

j) P % B

Practice 4: b) P % B

c) P % B

d) P % B

f) P % B

g) % B P

h) % B P i) P % B

e) % B P j) % B P

Even though we did not ask you to calculate the answers, if you must know, here they are! a) 5.46

b) 7

c) 8.90

d) 20

e) 3.22

h) 2.836875 ≈ 2.84

i) 300

j) 66 2/3 %

f) 100% g) 0.1245% ≈ .12%

Ratio and Proportion

a) % B P

Practice 5: 25 100

= 40

50 100

100

= 200

Percents

Practice 6: b) 25

c) 67

d) 20%

e) 200%

f) 50%

g) 32

h) 60

i) 5

j) 38

a) 117.5 f) 8.5%

b) 190 g) 3.06

c) 16 h) 21.6

Practice 7:

d) 21.10% i) 0.23 ≈ (.225)

191

e) 15% j) 1.86 ≈ (1.859)

Appendix

a) 6

Decimals

Whole Numbers

Chapter Five

Practice 8: a) 15.15

b) 157.69

c) 198

d) 25.75%

e) 35.02

f) 118.39%

g) 41.99%

h) 480.04

i) 0.11

j) 294.12

Practice 9:

a) 16 problems e) 50 problems i) 15% taxes for the year k) 71.9%

Ratio and Proportion

Fractions

Percents

d) 6.5% h) 12 people in the crew

Practice 10: Complete the Decimal number

Appendix

b) 80.6% c) 7.580 oz f) $11,180 g) $38.69 j) $18.75 dollars for the shirt l) 26.7% games lost

Decimal - Fraction Chart Fraction with Fully reduced denominator of Fraction 100

60/100

3/5

20/100

1/5

.25

25/100

1/4

.95

95/100

19/20

10/100

1/10

.01

1/100

.001

.1/100

1/1000

Practice 11: Fraction - Decimal - Percent Chart Decimal number

Percent

Fraction with denominator of 100

Reduced Fraction

.15

15%

15/100

3/20

.22

22%

22/100

11/50

.01

1/100

.82

82%

82/100

41/50 x

2.50

250%

250/100

2 1/2 or 5/2

80%

80/100

4/5

.75

75%

75/100

3/4

192

Answers to the Percents Problems

.14

Percent

Fraction with denominator of 100

14%

14/100

Reduced Fraction

1/7

Practice 12: Reduced Fraction

.05

5/100

1/20

.10

10%

10/100

1/10

.125

12.5%

12.5/100

1/8

20%

20/100

1/5

.33

33.3%

33.3/100

1/3

40%

40/100

2/5

50%

50/100

1/2

.75

75%

75/100

3/4

.83

83%

83/100

1.00

100%

100/100

6.00

600%

600/100

20%

20/100

1/5

Ratio and Proportion

Fraction with denominator of 100

Fractions

Percent

Decimals

Decimal

Whole Numbers

Decimal number

Practice 13: Percents

a) $160.00 b) $4500.00 c) $110.00 d) $131.25 e) Yes? Good Job! No? Ask for help!

Appendix

193

Chapter

Appendix Table of Contents

Glossary .................................................................................................................... 196 Web sites................................................................................................................... 199 Helpful Hints for Word Problems............................................................................. 200 Zip Code Map of Albuquerque................................................................................ 201 Visualizing Fractions, Decimals and Percents........................................................ 202 Solving Fraction Problems Using a Calculator....................................................... 208 Practice B1: Simplify the following fractions .. . . . . . . . . . . Practice B2: Convert to mixed numbers to improper fractions . . . . . . . Practice B3: Multiply fractions using a calculator.. . . . . . . . . . Practice B4: Divide fractions using a calculator.. . . . . . . . . . . Practice B5: Add fractions using a calculator.. . . . . . . . . . . Practice B6: Subtract fractions using a calculator. . . . . . . . . . .

209 209 210 211 212 213

195

Whole Numbers

Glossary

Glossary Addition: operation of combining numbers to find the sum or total. Approximate: close in value but not exact

Decimals

Approximation Sign - is a squiggly equal sign â&#x2030;&#x2C6; . And this means that the answer has been rounded to a specific place value. An equal sign = means that the answer is exact. Associative Property of Addition: when we add three numbers, we can group them in any way and still get the same answer. i.e 3 + (7 +2) = 12 (3 +7) + 2 = 12

Fractions

Associative Property of Multiplication: when we multiply three numbers, we can group them in any way and still get the same answer Average: the sum of a set of numbers divided by the number of numbers. Also called the mean. Communicative property of Addition: the order in which we add numbers does not change the answer.

Percents

Ratio and Proportion

Communicative property of Multiplication: the order in which we multiply numbers does not change the answer. Composite Number: a natural number that is not prime. Which means it contains a factor other than 1. Consecutive Number: Numbers which follow each other in order, without gaps, from smallest to largest. 12, 13, 14 and 15 are consecutive numbers. Decimal fractions: a fraction with 10, 100, 1000 (any multiple of 10) in the denominator Decimal number: the numbers in the base 10 number system, having one or more digits to the right of a decimal point. Denominator: the bottom part of a fraction. Difference: the result of subtracting two numbers.

Appendix

Digit: these ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are digits. Example: the number 365 has three digits: 3, 6, and 5. Division: the process of splitting a product into two of its factors. Equation: a mathematical statement that says that two expressions have the same value; any number sentence with an =.

196

Even number: a number that is divisible by 2. Expanded notation: a number that is written as a sum of the powers of 10 i.e. 743 = 700 + 40 + 3

Factor: one of two or more numbers that are multiplied together. The answer is the product. In the equation 4 x 5 = 20, 4 and 5 are factors, 20 is the Product.

Greatest Common Factor: (GCF) the largest number that divides two or more numbers evenly. The greatest common factor of two numbers (i.e. 5, 10) is the largest number that can be divided by both without a remainder (5). (Handy for reducing fractions)

Improper fraction: a fraction with a numerator that is greater than or equal to the denominator. Inequality: a mathematical expression which shows that two quantities are not equal.

Ratio and Proportion

Identity property of 0: adding 0 to a number does not change that number. In that it keeps the numbers â&#x20AC;&#x153;identity.â&#x20AC;?

Fractions

Fraction: a number used to name a part of a group or a whole. The number below the bar is the denominator, and the number above the bar is the numerator.

Decimals

Exponent: a short hand way to indicate repeated multiplication of the same number. The small number in the upper right hand corner indicates how many 3 times that factor should be repeated. In the following number 6 Six (the base) must be multiplied by itself 3 (the exponent) times or 6 x 6 x 6.

Whole Numbers

Equivalent fractions: two fractions that are equal to the quantity or value.

Least Common Denominator: the smallest multiple of the denominators of two or more fractions.

Lowest terms: The numerator and denominator of a fraction that have had all common factors but 1 factored out and canceled.

Percents

Like fractions: fractions that have the same denominator.

Mixed number: an amount written as a whole number plus a proper fraction.

Multiplication property of zero: when you multiply any number times zero, the result is zero.

197

Appendix

Multiple: a multiple of a number is the product of that number and any other whole number. i.e. 4,6,8 are multiples of 2

Whole Numbers

Glossary

Multiplication: the operation of repeated addition. i.e. 5 x 3 = 5 + 5 + 5 Multiplicative identity is one: any number multiplied by 1 gives you the same number. Number Line: A line on which every point represents a real number.

Decimals

Numerator: the top of a fraction. Odd number: a whole number that is not divisible by 2 (not even). Operation: addition, subtraction, multiplication, and division are the basic

Fractions

arithmetic operations. +,

-,x,รท

Percent: a fraction, or ratio, in which the denominator is assumed to be 100. The symbol % is used for percent. i.e 14% means 14/100. Perimeter: the sum of the lengths of the sides of a polygon (square , triangle, rectangle etc).

Ratio and Proportion

Power: a number that indicates the operation of repeated multiplication also called the exponent Prime Number: a number whose only factors are itself and 1. Product: the name for the number that results in two factors being multiplied. In the equation 4 x 5 = 20, 4 and 5 are factors, 20 is the Product. Proper fraction: a fraction whose numerator is less than its denominator. Its value is less than one.

Percents

Proportion: a statement that two ratios or two rates are equal. Quotient: the answer to a division problem. Rate: a ratio that compares different kinds of units.

Appendix

Ratio: the comparison of two quantities. Reciprocal: a ratio or fraction that has been flipped. The reciprocal of Reduce: fractions in their smallest number terms

198

2/3 is 3/2

Simplest Form: a fraction that is fully simplified, reduce or in lowest terms has only the number one left as a common factor.

Whole Numbers

Repeating Decimal: a decimal in which the digits endlessly repeat a pattern. Also called unending, infinite, repeating, or periodic decimal. i.e. .333 or .3838

Subtraction: the result of subtracting two numbers.

Terminating Decimal:when converting a fraction to a decimal form, the resulting decimal has no remainder. Also called a finite decimal, i.e. .375 or .2

Decimals

Sum: the result of adding two or more numbers, also called the total.

Whole Numbers: the set of numbers that includes zero and all of the numbers we can get by counting, adding and multiplying

Web sites

Fractions

Zero: the one and only place holder. Also means none!

http://www.math550.com

Visual Fractions http://www.visualfractions.com/index.htm

Ratio and Proportion

Mrs. Moreanoâ&#x20AC;&#x2122;s quiz site http://www.quia.com/pages/cnmmath.html

National Library of Virtual Manipulatives http://nlvm.usu.edu/ Percents

Mathworksheet Generator http://themathworksheetsite.com/ Prime Number Practice http://www.mathgoodies.com/factors/prime_factors.html

Appendix

199

Whole Numbers

Glossary

Helpful Hints for Word Problems The best thing to do with a word problem is to read it carefully and then try to imagine yourself in the situation. This way you take advantage or your everyday practical problem solving skills. These skills will help in your real life in math class as well.

Decimals

When possible draw a sketch to help you understand a problem. Rewriting the problem with similar numbers can be help you understand what operations are needed. Of course once you figure out what operations are needed you will apply your method to the original numbers. Look for words in the story that give hints as to what to do. Some times the answer words for the different operations are there.

Sum or Total – the answer from adding numbers Difference – the answer from subtracting numbers Product – the answer from multiplying numbers Quotient – the answer from dividing numbers

It’s also a good idea to estimate the answer of the problem before you begin to work it out. That way if your result is way off from what you expected, you can rethink and recheck. Estimating cuts down on the possibility of making a lot of silly errors.

Just practical everyday logic can help. Suppose you had a problem about the price of a loaf of bread and you came out with the answer that the bread cost $110 a loaf. That alone should be enough to let you know you better check your work or try again. Sometimes you will need to do more than one operation to solve a problem. For instance, in a bank balance you will have to add up all the deposits along with the beginning balance and then subtract out all the checks and bank charges. Some problems give you more information than you need, so just read carefully and decide what you need and what you don’t.

Appendix

Percents

Ratio and Proportion

Fractions

200

Decimals

Fractions Ratio and Proportion

Percents

Appendix

201

Whole Numbers

Zip Code Map of Albuquerque

Visualizing Fractions, Decimals and Percents Illustrate the following decimal, fraction, or percent using the 100 square grids found on the facing page.

a) 35%

b) 20%

c) 1/5

d) 1/2

e) 1%

f) 1/2%

g) 100%

h) 0.1

i) 27/100

j) 0.155

k) .25

l) 1/8

Based on the drawings that you have made, insert > , < or = to complete each of the following statements. 1)

35%

1/5

27/100

100%

11)

.25

20%

2) 20%

1/5

0.155

4) 1/2%

1/2

1/2%

6) 1/2%

0.1

20%

0.155

0.1

10) 0.155

1/5

27/100

12) 1/8

20%

Based on the drawing you have made, fill in the missing forms of the given values.

Fraction (reduced)

13) 14)

1/2

Decimal

Percent

15)

16)

17)

0.155

18)

.25

19) 20)

1/8

20%

100%

27%

Graphing Fractions, Decimals and Percents a) 35%

b) 20%

c) 1

d) 1

e) 1%

f)1 %

e) 100%

j) 0.155

h) 0.1

k) 0.25

i) 100

l) 8

Solving Decimal Problems using a calculator Try this with your model: Round numbers to the tenths place.

Fractions

Decimals

Whole Numbers

Glossary

Casio 260 Solar Press buttons:

MODE

Texas Instruments

Press buttons

MODE

7 Ratio and Proportion

Casio MS 115

1 Display reads 0.0

2nd

5 times Screen will show: Fix SCI Norm 1 2 3 Press

FIX (

-TI

1 Display reads 0.0

for Fix Screen will show: Fix 0 ~ 9 rounding to the tenths is 1 decimal place so Press

Enter the number and press the = sign. a) 2.93 = ________

b) 3.777 = ________

c) 4.333 = ________

d) 8.993 = ________

e) 28.31 = ________

f) 12.98 = ________

g) 124.19= ________

h) 139.73= _______

i) 17.08 = ________

b) 3.8

c) 4.3

d) 9.0

e) 28.3

f) 13.0

g) 124.2

204

Answers to Practice A-1

h) 139.7

i) 17.1

Appendix

Practice A-1: Round using a calculator to the tenths place.

a) 2.9

Percents

Other models: Ask if your calculator can round, some do and some do not.

Solving Decimal Problems using a Calculator

Try this with your model: Round numbers to the hundredths place value using a calculator. Casio 260 Solar Press buttons:

Texas Instruments

Press buttons

MODE

7 2 Display reads 0.00

2nd

5 times Screen will show: Fix SCI Norm 1 2 3 Press

2 Display reads 0.00

for Fix Screen will show: Fix 0 ~ 9 rounding to the tenths is 2 decimal place so Press

Practice A-2: Round using a calculator to the hundredths place.

d) 7.5552 = _____e) 111.773 = _____f) 17.456 = ____

Percents

Enter the number and press the = sign. The number in bold will be rounded to the hundredths place. a) 69.111 = _____b) 44.9732 = _____c) 3.4479 = ____

Ratio and Proportion

FIX (

-TI Fractions

MODE

Casio MS 115

Decimals

Step 1 Locate the number in the tenths place. Step 2 Look at the next number to the right. a. If this number is less than 5, drop it and all digits to its right. Example: 3.541 = 3.5 b. If this number is 5 or more, drop it and all digits to its right but then add 1 to the digit in the tenths place. Example: 3.571 = 3.6

Whole Numbers

Do you remember the rule for rounding to the nearest tenths place value without the calculator?

g) 36.149 = _____h) 87.9999 = _____i) 2.555 = ____

Appendix

a) 69.11 i) 2.56

b) 44.97

c) 3.45

Answers to Practice A-2

d) 7.56

e) 111.77

205

f) 17.46

g) 36.15

h) 88.00

Do you remember the rule for rounding to the nearest hundredths place value without the calculator? Step 1 Find the number in the hundredths place. Step 2 Look at the next number to the right. a. If this number is less than 5, drop it and all digits to its right. Example: 3.1541 = 3.15 b.If this number is 5 or more, drop it and all digits to its right but then add 1 to the digit in the tenths place Example: 3.1571 = 3.16

Try this with your model: Round numbers to the thousandths place value using a calculator. Casio 260 Solar MODE 7 3

2nd

5 times Screen will show: Fix SCI Norm 1 2 3 Press

FIX (

-TI

3 Display reads 0.000

for Fix Screen will show: Fix 0 ~ 9 rounding to the tenths is 3 decimal place so Press

Practice A-3: Round using a calculator to the thousandths place.

Enter the number and press the = sign. The number in bold will be rounded to the thousandths place. a. 67.7888 = _____ b. 5.22222 = _____ c. 8.9999 = ___ d. 39.1111 = _____ e. 1.99917 = _____ f. 7.0006 = ___ g. 21.7685 = _____ h. 42.4321 = _____ i. 9.2287 = ___

e) 1.999

f) 7.001

Ratio and Proportion

Press buttons

d) 39.111

Percents

Press buttons

MODE

Display reads 0.000

Appendix

Texas Instruments

206

c) 9.000 i) 9.229

Fractions

Press buttons:

Casio MS 115

Answers to Practice A-3 a) 67.789 b) 5.222 g) 21.769 h) 42.432

Decimals

Whole Numbers

Glossary

Solving Decimal Problems using a Calculator

-TI

Whole Numbers

Display reads 0.000

Decimals

Try this with your model: Round numbers to the whole number place value using a calculator. Casio 260 Solar

Casio MS 115

Texas Instruments

Press buttons

Press buttons:

MODE

2nd

5 times Screen will show: Fix SCI Norm 1 2 3 Press

0 Display reads 0

FIX (

for Fix Screen will show: Fix 0 ~ 9 rounding to the tenths is o decimal place so Press

Practice A-4: Round using a calculator to the nearest whole

Answers to Practice. A-4 a) 34 b) 678 c) 8 d) 845

e) 45

f) 25

g) 6

h) 631

i) 14

Finally letâ&#x20AC;&#x2122;s reset your calculator back to the floating decimal Casio 260 Solar Press buttons:

7 9

Texas Instruments

Press buttons

MODE 5 times Screen will show: Fix SCI Norm 1 2 3 Press

2nd FIX (

-TI

Percents

MODE

Casio MS 115

Ratio and Proportion

number. Enter the number and press the = sign. The number in bold will be rounded to the nearest whole numbers. a) 34.1122 = ____ b) 677.88 = _____ c) 7.9999 = _____ d) 844.77 = ____ e) 45.322 = _____ f) 24.937 = _____ g) 6.3142 = ____ h) 631.40 = _____ g) 14.100 = _____

Fractions

3 1

207

Appendix

for Norm Press

Fractions

Decimals

Whole Numbers

Glossary

Solving Fraction Problems Using a Calculator

UAs you have learned in the fraction chapter, simple fractions can be easy to add, subtract, multiply and divide.

Unfortunately, this is not always the case. In this section, we present a way to operate on the more difficult fractions through the use of a calculator. Remember, calculators are tools. You should always think before writing anything down. Try solving the fraction on a piece of paper. Then, use the calculator to check to see if your work is right.

Casio 260 Solar

Texas Instruments Texas Instruments TI 30X II

TI-34II

You will need to Locate the following

a Ratio and Proportion

Casio MS 115

b c

a key

b c

key

Shift Key

b c

key

/ key.

key

Simp Unit Key 2nd key.

Key

D

Example One: Simplify 12/18 a

for fraction bar and 18 and = sign

Percents

Press 12

/ key and 18.

Simp Press key then =. Repeat until the lowest reduced form is reached.

Your answer is 2/3

Appendix

Example Two: Simplify 7/8

Press 7

b c

Press 7 for fraction bar and 8 and = sign

Your answer is 7/8

208

Press

/ key and 8.

Simp

key then =

Solving Fraction Problems using a Calculator

Whole Numbers

Practice B1: Simplify the following fractions .

14 30

h) =

Decimals

Answers to Practice B1

f) 1 7

d) 14 e) 4

c) 5 8

b) 1 5

2 a) 3

g) 24 15 h) 25

Converting Mixed Numbers to Improper Fractions Casio 260 Solar

an improper fraction

Casio MS 115

3 a

Now press Shift

3/4 to

b c

Texas Instruments 30 X II

Press 1

and 4.

key and a

b c

Texas Instruments

TI-34II c

Now press

key and and equals.

2nd

ď&#x20AC;´D

nd Now press 2

/ 4

Press 1 Unit 3

Ratio and Proportion

Press 1

Fractions

Example One: Convert

b c

Your answer is

Practice B2: Convert to mixed numbers to improper fractions .

3 4 =

7 2 = 2

6 5

= Appendix

Answers to Practice B2

e) 4

f) 5

209

d) 2

9 5

g) 4

h) 5

8 4 =

c) 6

4 6

2 3 =

b) 3

5 3 =

17 a) 3

Percents

Multiply fractions using a calculator Example One: Multiply proper fractions Casio 260 Solar Press Find

Casio MS 115 Press

key.

Casio 260 Solar

Press 1

b c

Find

TI-34II

key.

Find

Casio MS 115

5/8

Texas Instruments Texas Instruments 30 X II

Press

1/2

Press Find Unit Key Find / key.

b c

Texas Instruments

TI-34II

/ 2x

Press 1

/ 8

. Your answer is 5

Example Two: Multiply mixed fractions 2

Casio 260 Solar

Press 2

b c

1 a

Casio MS 115

b c

1/3

x 7

1/2

Texas Instruments Texas Instruments 30 X II

TI-34II

1 a

Press 2 Unit 1 7 Unit 1 /

/ 3x 2

The answer is 17 1/2

Practice B3: Multiply fractions using a calculator.

x42

4 8

x27

6 3

e) 30

x27

x 5

Answers to Practice B1

5 6

1 7

f) 10

3 5

x 8

d) 8

1 3

x 8

b) 24

1 a) 12

Whole Numbers

Divide fractions using a calculator 2

Example One: Divide proper fractions 21

Casio 260 Solar

Press 2

Texas Instruments Texas Instruments

30 X II

TI-34II 2

7 =

/ 21 ÷ 2

press =

The answer is 1/3 1

Example Two: Divide mixed fractions 4 5

Press 4

Casio MS 115

1 a

b c

5÷

÷ 23

Texas Instruments Texas Instruments

30 X II b

1 a

TI-34II

4 Unit 1 Unit 1 /

/ 5 ÷2 3

Fractions

Casio 260 Solar

7 Decimals

Casio MS 115

÷ 7

press = Ratio and Proportion

The answer is 1

4/5

Practice B4: Divide fractions using a calculator.

÷23

7 7

÷42

e) 1 63

f) 3 25

4 5

9 5

÷ 3

÷22

Percents

÷ 9

d) 2 35

Answers to Practice B1

12 ÷ 4

b) 1 c) 1 3

5 a) 9

Appendix

211

Example One: Add proper fractions

Casio 260 Solar

Press 1

Casio MS 115

+ 8

Texas Instruments 30 X II

3+ 7

b c

Texas Instruments

TI-34II

1 Unit 1 Unit 1 /

/ 3 +7 2

5 6

Practice B5: Add fractions using a calculator.

7 7

+24

+ 2 15

+ 3

4 5

+ 62

Answers to Practice B5

3 5

+ 3

+ 15

9 28

Percents

The answer is 8

10 10 f)

Appendix

+ 7 2

Texas Instruments 30 X II

5 15

Ratio and Proportion

Press 1

Casio MS 115

2+ 3 /

7/8

Fractions

8 =

Example Two: Add mixed fractions 1 3

TI-34II

The answer is

Casio 260 Solar

Texas Instruments

Decimals

Add fractions using a calculator

Whole Numbers

Glossary

212

Solving Fraction Problems using a Calculator

Whole Numbers

Subtract fractions using a calculator 3

Example One: Subtract proper fractions

Casio 260 Solar

4 - 1

b c

Texas Instruments 30 X II

Texas Instruments

TI-34II 3

8 =

/ 4- 1 /

8 Decimals

Press 3

Casio MS 115

- 8

press = 5

The answer is

Example Two: Subtract mixed fractions 9 3

Press 9

Casio MS 115

1 a

b c

3 - 7

Texas Instruments 30 X II

1 a

b c

Texas Instruments

TI-34II

9 Unit 1 Unit 1 / press =

/ 3 -7 2

Ratio and Proportion

The answer is 1

Fractions

Casio 260 Solar

- 7 2

5 6

Practice B6: Subtract fractions using a calculator.

7 5

- 38

11 15 d)

Answers to Practice B6â&#x20AC;&#x2122;

213

Appendix

4 16 - 2 4

- 2 15

- 3

Percents

1 16 e)

13 5

- 2

7 1

- 4

4 40 f)