YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 1
Chapter 1 Understanding Whole Numbers
Lesson 1 Whole Numbers through Millions
The number system we use is the Hindu-Arabic system that developed in India. In this system, using ten digits: 0,1,2,3,4,5,6,7,8, and 9, we can represent any number we like, no matter how big or how small the number is. This is made possible by the place value property of the system. The value of the digit in a number depends on its position. Starting with the tens place, each place has a value of 10 times the place value to its right. Space is used to separate each group of three digits called period. To read numbers in figures, start from the left and read the period name of the digits except the unit or ones period. A symbol for nothing or zero had to be invented. Zero “holds the place” for a particular value, when no other digits goes in that position. For example, the number “100” in words means one hundred, no tens and no ones. The digits are grouped by threes starting from right. Each group is called period. The first group is called ones period and has ones, tens and hundreds places. The second group is the thousands period and has thousands, ten thousands, and hundred thousand places. The third group of three digits is the Millions period. It has the Millions, ten Millions, hundred Millions places. The fourth group of three digits is the Billions period. It has the Billions, ten Billions and hundred Billions places.
0
0
0
Ones
1 0 0 0 0 10 Hundred Millions = 1 000 000 000
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Ones Period
Tens
Thousands Period
Hundred Thousand s Ten Thousand s Thousand s Hundreds
Millions
Hundred Millions
Ten Millions
Millions Period
Ten Billions Billions
Billions Period Hundred Billions
The place value chart for 1 million is given below,
0
0
Page 2
Let us place the number in the place value chart
Hundred Thousand s Ten Thousand s Thousand s Hundreds
Tens
Ones
Ones Period
Millions
Thousands Period
Ten Millions
3
Millions Period Hundred Millions
Ten Billions Billions
Hundred Billions
Billions Period
7
5
1
8
2
5
6
9
4
In going from left to right, The value of digit 3 is 3 000 000 000; The value of digit 7 is 700 000 000; The value of digit 5 is 50 000 000; The value of digit 1 is 1 000 000; The value of digit 8 is 800 000 The value of digit 6 is 60 000 The value of digit 9 is 9000 The value of digit 4 is 400 The value of digit 2 is 20 The value of digit 5 is 5
In expanded form: 3 000 000 000 + 700 000 000 + 50 000 000 + 1 000 000 + 800 000 + 60 000 + 9 000 + 400 + 20 + 5 In words: Three billion, seven hundred fifty one million, eight hundred sixty nine thousand, four hundred twenty five In standard form: 3 751 869 425
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 3
Practice Exercise 1.1 A. Read each number 1. 803 574 357 874 2. 586 403 856 372 3. 574 350 596 795 4. 397 492 402 544 5. 960 504 640 240 B. Write each number in standard form 1. Seven billion, nine hundred fifty eight million, seven thousand, one hundred twelve 2. Four hundred sixteen billion, eight hundred one million, ninety 3. One hundred million, sixty five million, eight hundred fifty thousand, twelve 4. Forty billion, one hundred five million, seventy thousand twenty four, six hundred ten 5. Five billion, five million, five thousand, five C. Write each number in words 1. 960 504 640 240 2. 397 492 402 544 3. 803 574 357 874 4. 586 403 856 372 5. 574 350 596 795
D. Write in expanded form 1. 574 350 596 795 2. 586 403 856 372 3. 960 504 640 240 4. 803 574 357 874 5. 397 492 402 544
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 4
E. Fill in the blanks In the number 408 000 112 536 1. The digit 8 is in the _____ place. 2. The value of the digit 4 is _____. 3. The hundred thousands place is the digit _____. 4. The digit 5 is in the _____ place. 5. The ten millions place has the digit _____
Lesson 2 Whole Numbers through Billions Let us count in hundred millions. What number comes after 900 million? 100 000 000 one hundred million 200 000 000 two hundred million 300 000 000
three hundred million
400 000 000
four hundred million
500 000 000
five hundred million
600 000 000
six hundred million
700 000 000
seven hundred million
800 000 000
eight hundred million
900 000 000 1 000 000 000
nine hundred million one billion
The place value chart for 1 billion is given below,
0
0
0
0
Ones
0
Tens
Millions
0
Ones Period Hundreds
Ten Millions
1
Thousands Period Hundred Thousand s Ten Thousand s Thousand s
Hundred Millions
Millions Period
Billions
Ten Billions
Hundred Billions
Billions Period
0
0
0
10 Hundred Millions = 1 000 000 000 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 5
After the millions period, the next group of three digits is the billions period. It has the three place values – billions, ten billions, and hundred billions. Write in standard and expanded and expanded form: two billion, five hundred three million, one thousand, seventy-five.
Ten Thousands
Thousands
Hundreds
Tens
Ones
Ones Period
Hundred Thousands
0
Thousands Period
Millions
Hundred Millions 5
Ten Millions
Billions
Millions Period
2
Ten Billions
Hundred Billions
Billions Period
3
0
0
1
0
7
5
To complete the value chart, we again put zeros as place holders. In standard form: 2 503 001 075 In expanded form: 2 000 000 000 + 500 000 000 + 3 000 000 + 1 000 + 70 + 5 Write in words and expanded form: 6 035 010 700
Ten Thousands
Thousands
Hundreds
Tens
Ones
3
Ones Period
Hundred Thousands
0
Thousands Period
Millions
6
Ten Millions
Hundred Millions
Millions Period
Billions
Ten Billions
Hundred Billions
Billions Period
5
0
1
0
7
0
0
In words: six billion, thirty-five million, ten thousand, seven hundred In expanded form: 6 000 000 000 + 30 000 000 + 5 000 000 + 10 000 + 700
In 7 105 028 000
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 6
Ten Thousands
Thousands
Hundreds
Tens
Ones
Ones Period
Hundred Thousands
0
Thousands Period
Millions
Hundred Millions 1
Ten Millions
Billions
Millions Period
7
Ten Billions
Hundred Billions
Billions Period
5
0
2
8
0
0
0
The billions period has the digit 7; 7 is in the billions place with a value of 7 000 000 000, The millions period has the digits 105; 1 is in the hundred millions place with a value of 100 000 000, 5 is in the millions place with a value of 5 000 000, The thousands period has the digits 028; 2 is in the ten thousands place with a value of 20 000, 8 is in the thousands place with a value of 8 000 The ones period has zero digits.
In expanded form: 7 000 000 000 + 100 000 000 + 5 000 000 + 20 000 + 8 000 In words: seven billion, one hundred five million, twenty-eight thousand Remember this: o
Putting the number in the place value chart will help us in writing the expanded and standard forms.
Ten Billions
What comes after 9 billion? 1 000 000 000
one billion
2 000 000 000
two billion
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 7
3 000 000 000
three billion
4 000 000 000
four billion
5 000 000 000
five billion
6 000 000 000
six billion
7 000 000 000
seven billion
8 000 000 000
eight billion
9 000 000 000
nine billion
10 000 000 000
te billion
The place value chart for 10 billion is given below.
Thousands
Hundreds
Tens
Ones
0
Ten Thousands
0
Ones Period
Hundred Thousands
0
Thousands Period
Millions
Hundred Millions
1
Ten Millions
Billions
Millions Period
Ten Billions
Hundred Billions
Billions Period
0
0
0
0
0
0
0
Write in standard form: twenty-eight billion, eighty million, eight thousand, eight hundred. In expanded form: 20 000 000 000 + 8 000 000 000 + 80 000 000 + 8 000 + 800 In standard form: 28 080 008 800 Remember this: The place value chart clearly gives the position of each given digits. Adding zeros as place values holders will help us write the standard and expanded form
Write in expanded form and in words: 30 405 178 000 The digit 3 is in the ten billions place with a value of 30 000 000 000 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 8
4 is in the hundred millions place with a value of 400 000 000 5 is in the millions place with a value of 5 000 000 1 is in the hundred thousands place with a value of 100 000 7 is in the ten thousands place with a value of 70 000 8 is in the thousands place with a value of 8 000
In expanded form: 30 000 000 000 + 400 000 000 + 5 000 000 + 100 000 + 70 000 + 8 000 In words: thirty billion, four hundred five million, one hundred seventy-eight thousand.
Hundred Billions What number comes after 99 999 999 999? We add 1 and get 100 000 000 000 or one hundred billion.
Hundreds
Tens
Ones
0
Thousands
0
Ten Thousands
0
Ones Period
Hundred Thousands
Hundred Millions
0
Thousands Period
Millions
Billions
1
Ten Millions
Ten Billions
Millions Period
Hundred Billions
Billions Period
0
0
0
0
0
0
0
Write in words, then in expanded form: 620 050 004 300
Hundreds
Tens
Ones
5
Thousands
0
Ten Thousands
0
Ones Period
Hundred Thousands
Hundred Millions
2
Thousands Period
Millions
Billions
6
Ten Millions
Ten Billions
Millions Period
Hundred Billions
Billions Period
0
0
0
4
3
0
0
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 9
In words: six hundred twenty billion, fifty million, four thousand, three hundred In expanded form: 600 000 000 000 + 20 000 000 000 + 50 000 000 + 4 000 + 300
In the number four hundred billion, one hundred five million, seventy thousand,
Hundreds
Tens
Ones
0
Thousands
1
Ten Thousands
0
Ones Period
Hundred Thousands
Hundred Millions
0
Thousands Period
Millions
Billions
4
Ten Millions
Ten Billions
Millions Period
Hundred Billions
Billions Period
5
0
7
0
0
0
0
The standard form is 4000 105 070 000 There are eight zeros that act as place holders. The digit 1 is in the hundred millions place The digit 4 is in the hundred billions place The value of digit 5 is 5 000 000 The ten thousands place is the digit 7.
Answer this: Fill in the blanks. In the number 536 002 408 000 1. 2. 3. 4. 5. 6.
The digit 3 is in the __________place. The value of digit 6 has the digitis __________. The millions place has the digit __________. The digit 5 is in the _________ place. The value of the digit 8 is __________. The hundred thousands place is the digit _________. In the number nine hundred two billion, fifty-six million, thirty seven, 7. The value of the digit 2 is __________ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 10
8. The digit 9 is in the __________ place. 9. In the hundred millions place is the digit __________. 10. The number 0s that act as place holder is __________. Practice: A. Give the place value and the value of the indicated digit in the number 230 080 900 007. Place Value 1. 2. 3. 4. 5.
The digit 8 The digit 2 The digit 7 The digit 3 The digit 9
__________________ __________________ __________________ __________________ __________________
Value _________________ _________________ _________________ _________________ _________________
B. Give the place value and the value of the digit in the number one hundred two billion, three hundred forty million, eight thousand Place Value
Value
6. The digit 2
__________________
_________________
7. The digit 4
__________________
_________________
8. The digit 1
__________________
_________________
9. The digit 8
__________________
_________________
10. The digit 3
__________________
_________________
C. Write each number in standard in standard form. 1. Fifty-one billion, three million, seven hundred thousand _______________ 2. One hundred eighteen billion, sixty-seven million, twelve thousand _______________ 3. Seven billion, nine hundred three million, fifteen thousand _______________ 4. Three hundred nine billion, ninety-eight thousand _______________ 5. Four hundred sixteen billion, five million, seventy _______________ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 11
D. Write each number in words 6. 403 700 060 000 _________________________________________ _______________________________________________________ 7. 580 058 508 000_________________________________________ ______________________________________________________ 8. 162 000 340 009 _________________________________________ _______________________________________________________ 9. 396 020 400 050 _________________________________________ _______________________________________________________ 10. 701 017 071 000 _________________________________________ _______________________________________________________
Lesson 3 Regrouping Numbers
In order to write a number in standard form, there should only be 1 digit in each place value position. If there were two-digit numbers in given place value then we regroup the numbers. Regroup the following numbers and write in write in standard form: a) 36 ones, b) 24 hundreds, c) 15 ten thousand, 23 tens.
In regrouping the numbers, we add to the next higher place value. a. 36 ones is 3 tens 6 ones = 30 + 6 = 36 b. 24 hundreds is 2 thousands 4 hundreds = 2 000 + 400 = 2 400 c. 15 ten thousands, 23 tens is 1 hundred thousand, 5 ten thousands, 2 hundreds, 3 tens = 100 000 + 50 000 + 200 + 30 = 150 230
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 12
Regroup in order to write in standard form.
thousands hundreds tens ones 5 6 12 8 12 tens is 1 hundred, 2 tens Add the hundreds
thousands hundreds tens ones 5 7 2 8 In standard form: 5 728
Regroup in order to write in standard form.
thousands hundreds tens ones 3 12 3 26 26 ones is 2 tens, 6 ones Add the tens.
thousands hundreds tens ones 3 12 5 6 12 hundreds is 1 thousand, 2 hundreds Add the thousands
thousands hundreds tens ones 4 2 5 6 In standard form: 4 256 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 13
5
14
ones
28
tens
thousands
13
hundreds
Ten thousands
2
Hundred thousands
millions
Regroup in order to write the number in number in standard form.
7
2
14 hundreds is 1 thousand, 4 hundreds
6
4
ones
8
tens
thousands
15
hundreds
Ten thousands
2
Hundred thousands
millions
28 ten thousands is 2 hundred thousands, 8 ten thousands
7
2
6
4
ones
8
tens
thousands
5
hundreds
Ten thousands
3
Hundred thousands
millions
15 hundred thousands is 1 million, 5 hundred thousands
7
2
In standard form: 3 586 472 Regroup and write in standard form. 3 billions, 15 hundred millions, 13 ten thousands, 26 tens
Starting from the right, we have the following regrouping: 26 tens is 2 hundreds, 6 tens; YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 14
13 ten thousands is 1 hundred thousand 3 ten thousands 15 hundred millions is 1 billion; 5 hundred millions
Adding the corresponding digits in each place values, the regrouped number is 4 billions, 5 hundred millions, 1 hundred thousand, 3 ten thousands, 2 hundreds, 6 tens. In standard form: 4 500 130 260
Test: A. Regroup and write in standard form 1) 14 tens 2) 29 hundreds 3) 17 thousands 15 ones 4) 36 ten thousands 5) 15 ten millions 23 hundred thousand thousands, 14 tens
Lesson 4 Comparing and Ordering
To compare the numbers with the same number of digits, we align the digits by place value. From the left check the digit in each place value until they are different. Compare the values of the digits to find the greater number.
Compare: 365 182 and 370 265 6>7 Therefore, 365 182 < 370 265
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 15
Compare: 2 562 104 and 2 536 205 Aligning the place values, we have the following 6>3
Therefore, 2 562 104 > 2 536 205
Test Compare the pairs of numbers. Write < or > in the blank. 1) 2) 3) 4) 5) 6) 7) 8) 9)
345 _____ 543 1 753 _____ 1 573 48 162 ____ 4 816 625 402 _____ 623 835 703 195 _____ 698 302 5 864 _____ 6 423 7 562 _____ 7 395 216 365 _____ 843 643 468 132 _____ 468 152
Ordering Numbers
Arrange these numbers in order from the greatest to the least: 62 895, 65 102, 63 470 The ten thousand digits are the same. The thousands digit in order: 5 > 3 > 2 The numbers in order: 65 102, 63 470, 62 895
Arrange these numbers in order from the greatest to the least: 345 702, 348 205, 362 367
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 16
6 > 4, so 362 367 is the highest number Comparing the two other numbers, we go to the thousands place. 345 702 348 205 8 > 5 so 348 205 > 345 702 The number in order from the greatest to the least: 362 367, 348 205, 345 702
Are the numbers arranged in ascending order ( from the least to the greatest or descending order ( from greatest to least) ? 7 350,
7 500, 73 000
First, we arrange the numbers by columns. 7 350
for the hundreds digits, 3 < 5
7 500 73 000
the greatest number
Therefore, the numbers are arranged in ascending order or from the least to greatest. Test: A. Arrange these numbers in order from the greatest to the least. 1) 2) 3) 4) 5)
5 173, 3 715, 7 153 ______________________________ 4 250, 2 450, 4 520 ______________________________ 7 165, 7 615, 7561 ______________________________ 123 045, 120 345, 125 304 _______________________ 45 186, 45 816, 45 681 ___________________________
B. Listed below are the elevations of five mountains in the Philippines. Arrange the mountains in order from the highest to the lowest.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 17
Mountain
Elevation
Mt. Pulag, Benguet
2 922
Mt. Apo, Davao
2 954
Mt. Banahaw, Quezon 2 158 Mt. Kanlaon, Negros
2 435
Mt. Mayon, Albay
2 463
Lesson 5 Rounding Off Numbers
To the nearest Ten and the nearest Hundred
To round off a number to the nearest ten, locate the tens digit and look at the ones digit. Number 236 is closer to 240 therefore we round up o 240.
236 is closer to 200 than to 300 so we round down 236 to 200. Example: The school is 550 m from the church. About how many meters away is the school from the church? 550 is halfway between 500 and 600. We round up to 600. The school is about 600 m from the church.
Remember this: Round up if the ones digit is 5 or greater. Round down if the ones digit is less than 5 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 18
Replace the ones digit by zero To round off a number to the nearest hundred, locate the hundreds Locate the hundreds digit and look at the tens digit. Round up if the tens is 5 or greater Round down if the tens is less than 5. Replace the tens and ones digits by 0s.
Practice: Round off each number to the nearest ten: 1) 2) 3) 4) 5)
42 48 45 94 86
_____ _____ _____ _____ _____
Round off to the nearest hundred: 1) 2) 3) 4) 5)
351 _____ 358 _____ 256 _____ 637 _____ 5 961 _____
To the Nearest Thousand To round off a number, we follow these steps: Step 1.
Locate the rounding place.
Step 2.
Look at the digit to the right. If the digit is 5 or greater, round up (or add 1). If the digit is less than 5, round down (or it remains the same).
Step 3.
Replace each digit to the right of the rounding place with 0s.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 19
Example Round the number 428 531 a. to the nearest thousand;
the digit to the right of 8 is 5. We round up. Add 1 to the thousands digit. Replace the digits to the right with 0s
428 531 rounded up to 429 000
b. to the nearest ten thousand; the digit to the right of 2 is 8. Since 8 > 5,we round up. Add 1 to the ten thousands digit. Replace the digits to the right with 0s. 428 531 rounded up to 430 000
c. to the nearest hundred thousand the digit to the right of 4 is 2. Since 2 < 5 , we roun down. The hundred thousands digit remains the same. Replace the digits to the right with 0s. Test: A. Round off to the nearest thousand 1) 2) 3) 4) 5)
4 275 __________ 4 806 __________ 6 522 __________ 20 268 _________ 49 702 _________
B. Round off to the nearest ten thousand 6) 7) 8) 9)
72 958 _________ 45 028 _________ 19 340 _________ 835 641 ________
C. Round off to the nearest hundred thousand 1) 125 600 __________ 2) 370 915 __________ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 20
3) 856 048 __________ 4) 639 256 __________ 5) 2 953 423 ________
To the Nearest Million and the Nearest Billion Round off 4 513 269 923 (a)to the nearest million the digit to the right of 3 is 2. Since 2 < 5, we round down 4 513 269 923 round down to 4 513 000 000 (b) to the nearest billion. The digit to the right of 4 is 5, so we round up. 4 513 269 923 round up to 5 000 000 000
Test: Given the number 4 653 721, round off to nearest 1) Thousand _______________ 2) Hundred thousand _______________ 3) Million _______________ Given the number 27 475 296 531, round off to the nearest 1) Ten Million 2) Billion
_______________ _______________
Lesson 6 Roman Numerals The ancient Romans used letters to represent numbers. They are called Roman Numerals. I stands for 1
C stands for 100
V stands for 5
D stands for 500
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 21
X stands for 10
M stands for 1 000
L stands for 50 Here are some guidelines in finding the value of Roman Numerals. 1. Add to get the values of the symbols that are repeated. The symbols I,X,C, and M repeated up to three times only. Each of V,L, and D are never repeated. III 1+1+1 =3
XX 10 + 10 = 20
CCC 100+100+100 = 300
MM 1 000 + 1 000 = 2 000
2. Add if the symbols decrease in the value from left to right. VI LX CL 5+1 50+10 100+50 =6 =60 =150 DCCLXIII
MMVIII
500+100+100+50+10+3 =763
1 000+1 000+5+3 =2 008
3. Subtract if a symbol is lesser value comes before a symbol of greater value IV = 4 IX = 9 XL = 40 CDIX = 409
Practice Exercises: Find the value of each Roman numeral.
1. LXXXII 2. CCCLX 3. MMCCLV 4. XXXIX 5. CDXLVI
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 22
Chapter 2 Understanding Addition and Subtraction of Whole Numbers
Lesson 1 Understanding Addition of Large Numbers
Remember this When we want to put together sets of objects, we add. The symbol for Addition is the plus sign, +. In the addition sentence, the numbers to be added are the addends. The result of adding is the sum. Properties of addition Here are the properties of addition that can help in finding the sum. Identity or Zero Property of Addition. The sum of zero and a number is the number itself. 0+4=4 Commutative or Order property of Addition. When the order of the addends is changed, the sum remains the same. 4+5=5+4=9 Associative or Grouping Property of Addition. When the grouping of three or more addends is changed, the sum remains the same. (4 + 5) + 6 = 9 + 6 = 15 4 + (5 + 6) = 4 + 11 = 15 We use these properties to find the shortcuts in adding more than two addends. We may change the order and the grouping of the addends so that we can add first the friendly numbers. Friendly numbers are those numbers whose sums can easily be obtained mentally. Friendly numbers that add up to multiples of 10 are called compatible numbers of 10. Can you name friendly numbers of ten?
We may also group the friendly numbers of multiples of 10. Example: 3 + 4 + 7 = (3 + 7) + 4 = 10 + 4 = 14 1 + 4 + 16 + 9 = (4 + 16) + ( 1+ 9) = 20 + 10 = 30
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 23
Test: Fill in the missing number. Name the property of addition used. Write I for Identity Property, C for Commutative Property, and A for Associative Property 1) 23 + _____ = 23 2) 14 + _____ = 5 + 14 3) 0 + ______ = 9 4) 3 + ( 5 + 2 )= (3 + _____) + 2 5) 6 + 3 = 3 + _____
Adding Large Numbers
Hundreds
Tens
Ones
4
Thousands
1
Ten Thousands
3
Hundred Thousands
Millions
The first step is to align the place values of the numbers â&#x20AC;&#x201C; the ones, the tens, the hundredths, the thousandths, the ten thousands, the hundred thousands and the millions. Then we add by column starting from the right, the ones place.
1 1 3 2 7
2 2 2 2 8
3 1 1 2 7
3 3 0 3 9
0 1 3 3 7
1 2 2 1 6
Take note that the sum of the digits in each column is less than 10. There is no need to regroup in any of the place values.
The following are the steps in adding or subtracting whole numbers 1. Write the numbers in a column. Align digits having the same place value. 2. Start adding or subtracting the ones digits of the numbers. Regroup when necessary. 3. Continue the process up to the digits in the highest place. Regroup when necessary. Example: Addition 5,683,492 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 24
+ 9,462,857 15,146,349 Subtraction 986,422 75,510 910,912 Practice: Find each sum a) 894 + 16,372 b) 5,604,723 + 896,071 Find each difference a) 5,648 – 4,215 b) 7,614 – 978
Lesson 2 Subtraction of whole number In subtracting numbers, subtract the digits having the same place value starting from the ones place up to the highest place value. You may use estimation to check whether the answer is reasonable or not. Addition is the reverse process of subtraction. In problems involving subtraction, addition may be used to check if the difference is correct. Example: 847 387 – 415 026 847 387 – 415 026 432 361 Practice: a) 5 685 – 3 421 b) 72,458 – 10 232 c) 83,713 – 33 201 d) 697 542 – 432 420 e) 764 238 – 224 125
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 25
Lesson 3 Exponential form 2 x 2 x 2 x 2 x 2 x 2 = 26 = 64 2 is the base and 6 is the exponent The exponent tells how many times the base is multiplied by itself to get the product. In 26 = 64, the product, 64, is called the power of 2. The expression 2 6 is read as “two raised to the sixth power”. Write the number 152 in standard form is 15 x 15 = 225 When 2 is used as the exponent, you may say that the base is squared, instead of saying “raised to the second power”. Thus 152 is read as “fifteen squared”. 73 = 7 x 7 x 7 = 49 x 7 = 343 When 3 is used as an exponent, you may say that the base is cubed, instead of saying “raised to the third power”. Thus, 73 is read as “seven cubed” 41 = 4 any number raised to the first power is the number itself. 90 = 1 any number, except 0, raised to the zeroth power is 1. Write the number using exponents 5,000 = 5 x 1000 and 1000 = 103 Then 5000 = 5 x 103 600,000 = 6 x 100,000 and 100,000 is 105 Then 600,000 = 6 x 105 Practice: Write the number in standard form 1. 2. 3. 4. 5.
25 53 751 107 7 x 105
Write the missing exponent YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 26
6. 49 = 7 7. 64 = 4 8. 81 = 3 9. 125 = 5 10. 32 = 2 Write each number in exponential form 11. 10,000 12. 800 13. 50,000,000 14. 1,000,000 15. 700,000 Lesson 4 Multiplying by multiples of 10, 100 and 1000 Multiplying by 10 10 x 1 = 10 (1 ten) 10 x 2 = 20 (2 tens) 10 x 35 = 350 10 x 72 =720 10 x 40 400 10 x 50 = 500 Can you find any pattern or rule when multiplying by 10?
To multiply a number by 10, we affix one zero after the number. Hence, we also have the following: 3 529 x 10 = 35 290 10 x 893 104 = 8 931 040
Let us now multiply the multiples of 10. Recall that a multiple of a number is the product of that number and a whole number: 0,1,2,3,4,â&#x20AC;Ś The multiples of 10 are 0, 10,20,30,40,50â&#x20AC;Ś YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 27
Observe how the product is obtained. 20 x 1 = 20 20 x 2 = 40 20 x 3 = 60 4 x 50 = 200 5 x 50 = 250 8 x 50m= 400 15 x 30 = 450 22 x 30 = 660 53 x 30 = 1590 7 x 20 = 140 70 x 20 = 1 400 700 x 20 = 14 000 Remember this To multiply a whole number by a multiple of 10: 1. Multiply the nonzero digits; 2. Count the number of zeros in the factors; and 3. Affix the same number of zeros in the product. Thus we have the following: 110 231 x 30 = 3 306 930 495 123 x 20 = 9 902 460 732 000 x 40 = 29 280 000
Multiplying by 100 To multiply a number by 100, affix two zeros to that number. Thus, we also have the following: 126 753 x 100 = 12 675 300
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 28
100 x 5 862 000 = 586 200 000 The multiples of 100 are 200,300,400â&#x20AC;Ś. When multiplying with multiples of 100. We have the following: 300 x 1 = 300 300 x 2 = 600 4 x 200 = 800 40 x 200 = 8 000 400 x 200 = 80 000 Remember this To multiply a whole number by 100: 1. Multiply the nonzero digits; 2. Count the number of zeros in the factors; and 3. Affix the same number of zeros in the product Thus we have the following: 121 234 x 200 = 24 246 8000
Multiplying by 1 000 1000 x 1 = 1000 1000 x 2 = 2000 1000 x 35 = 35 000 1 000 x 56 = 56 000 1 000 x 70 = 70 000 1 000 x 80 = 80 000 Remember this To multiply a whole number by a multiple of 1 000 1. Multiply the nonzero digits 2. Count the number of zeros in the factors; and 3. Affix the same number of zeros in the product. Test YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 29
Find the product 1. 20 x 9 = _____ 2. 20 x 90 = _____ 3. 20 x 703 = _____ 4. 20 x 558 = _____ 5. 396 x 20= _____ 6. 2 539 145 x 30 = _____ 7. 536 x 400 = _____ 8. 1 342 000 x 500 = _____ 9. 159 034 x 600 = _____ 10. 895 000 x 2 000 = _____
Lesson 5 Multiplying with zeros in the multiplier
Find the product: 42 856 x 205 42 856 X 205 214 280 000 000 8 571 2___ 8 785 480 A shorter way is to omit the second partial product since there are zero tens in 205. 42 856 X 205 214 280 8 571 2___ 8 785 480 Find the product: 35 100 246 x 300 35 100 246 x 300 10 530 073 800 There are 0 ones and 0 tens so we omit the partial products for the ones and the tens of the multiplier.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 30
Remember this in writing the partial products be sure to align the correct place values.
Lesson 6 Estimating Products
To estimate products, 1. Round the factors to the highest place value; 2. Multiply the nonzero digits in the factors, then add the number of zeros in the factors Estimate the product 729 x 79 = 700 x 80 = 56000 9894 x 62 = 10000 x 60 = 600000 Practice: 1. 2. 3. 4. 5.
921 x 42 768 x 53 293 x 48 653 x 48 6204 x 543
Lesson 7 Solving word problems Kathy ordered 165 packages of table napkins for her restaurant. Each package holds 1 000 napkins. How many napkins did she order?
What is asked? Total number of napkins Kathy ordered What are given? 165 packages of table napkins, 1 000 napkins in each package Operation to be used: Multiplication YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 31
Number Sentence: 165 x 1 000 = 165 000 Complete Answer: Kathy ordered a total of 165 000 napkins
Test Solve each problem. 1. Norman’s bakery sells 435 cans of broas each week. (a) How many cans of broas are sold in 16 weeks? Sharon’s bakery sells 516 cans of broas for each week. (b) how much more cans of broas does Sharon’s bakery sell in 16 weeks? What is asked? What are given? Operations to be used: Number sentence: Complete Answer: 2. In a chicken farm, a farmer collected 1 625 eggs everyday. How many eggs are collected in 45 days? What is asked? What are given? Operations to be used: Number sentence: Complete Answer:
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 32
3. Mrs. Campos bought 12 boxes of drawing paper. Each box has 250 sheets of paper in it. (a) How many sheets of paper are there in all? (b) If Mrs. Campos needs 2 165 sheets of drawing paper for her classes, How many sheets of paper are left? What is asked? What are given? Operations to be used: Number sentence: Complete Answer: 4. Andyâ&#x20AC;&#x2122;s allowance is P120 each day while his sister Beth receives P 75 each day. (a) what is their total allowance in 20 school days? (b)how much more did Andy receive? What is asked? What are given? Operations to be used: Number sentence: Complete Answer: 5. There are 40 pupils who joined the educational tour. The budget for each pupil is P 200 for food, P120 for transportation, and P 50 for the museum fee. What is the total budget for the 40 pupils? What is asked? What are given? Operations to be used: Number sentence: Complete Answer:
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 33
Chapter 4 Understanding Division of large numbers
Lesson 1 Division Features of division 1. 2. 3. 4.
Any nonzero number divided by one is the number itself. A number divided by itself is one. Zero divided by any number, except zero is zero. Division by zero has no meaning
Division by using the long method 24 24 – 4 = 20 – 4 = 16 – 4 = 12 – 4 = 8 – 4 = 4 – 4 = 0 Since we are able to subtract 4 ix times, the answer of 24 divided by 4 is 6 The shortcut method of dividing 6 -24 0 To check division, we multiply 6 x 4 = 24 Example 8 x 4 = 32, then 32 9 x 5 = 45, then 45 Practice: 1. 24
2
2. 14
7
3. 18
3
4. 16
2
5. 25
5
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 34
Lesson 2 Dividing whole Numbers by 10, 100, 1000 Basic facts 5 x 10 = 50 then 50 ÷ 10 = 5 25 x 10 = 250 then 250 ÷ 10 = 25 4 x 100 = 400 then 400 ÷ 100 = 4 43 x 1000 = 43 000 then 43 000 ÷ 1 000 = 43 Remember this: When dividing by 10, 100, or 1 000, simply remove the same number of zeros in the dividend as in the divisor to obtain the quotient. Test Find the quotient 1. 270 ÷ 10 = 2. 540 ÷ 10 = 3. 4 200 ÷ 100 = 4. 2 800 ÷ 100 = 5. 3 500 ÷ 10 = Solve each problem 1. When the number is divided by 1 000, the quotient is 350. What is the number? 2. What is the quotient when 3 x 105 is divided by 1 000? 3. What number must be multiplied by 72 to obtain 7 200? 4. A total of 2 700 bottles were placed equally in boxes. If 27 boxes were used, how many bottles were there in each box? 5. The participants in a sportfest were divided into groups with 10 members in each group. If there were 450 participants, how many groups were formed? Lesson 3 Dividing multiples of 10, 100, 1000 Basic fact: 5÷1=5 50 ÷ 10 = 5 500 ÷ 100 = 5 12 ÷ 4 = 3 120 ÷ 4 = 30 1 200 ÷4 = 300 12 00 ÷ 4 = 3 000 What conclusion can we make? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 35
There is at least the same number of zeros in the quotient as in the dividend when the divisor is a one-digit number. 1 800 ÷ 3 = 600 450 000 ÷ 5 = 90 000 When both dividend and divisor have zeros, we cancel out first the same number of zeros then we divide. Test Find the quotient 1. 160 ÷ 20 = 2. 200 ÷ 40 = 3. 450 ÷ 90 = 4. 3 200 ÷ 400 = 5. 2 700 ÷ 300 =
Lesson 4 Dividing large numbers by One – and Two- digit Divisors By one-digit Divisors An important step in division is to decide where to place the first digit of the first quotient. First we count the number of digits in the divisor. Next, draw a line after the same number of digits in the dividend. If the number formed from the digits of the dividend to the left of this line is larger than the divisor, the first digit of the quotient is right above the last digit of this number. Otherwise, move the line one digit to the right. The first digit of the quotient will be right before this line. Example: Divide: 462 ÷ 3 Decide where to begin 3
462
place the first digit on the quotient The quotient will have 3 digits
3 , 4 so there are enough 3’s Divide the hundreds 1 3 462 3
3 x 12 = 3 < 4 multiply: 3 x 1 = 3
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 36
16
subtract: 4 – 3 = 1 compare: 1<3 Bring down: 6
Divide the tens
3
1 462 3 16 15 12
3 x 5 = 15 < 16 multiply: 3 x 5 = 15 subtract and compare: 16-5 = 1, 1<3 bring down 2
Divide the ones
3
1 462 3 16 15 12 12 0
3 x 4 = 12 multiply: 3 x 4 = 12 subtract and compare: 12 – 12 = 0
There is no remainder or the remainder is zero To check: 154 x 3 = 462 ( the dividend) To check the answer, multiply the quotient and the divisor. The answer must be the dividend. Divide: 526 ÷ 6
87R4 6 526 48 46 42 4 remainder
6 x 8 = 48 < 52 6 x 7 = 42 < 46
To check: (87 x 6) = 522 + 4 = 526 (the dividend) In checking, add the remainder to the product of the quotient and the divisor.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 37
Test 1. 4 935
2. 6 7 365
3. 3 15 238
4. 7
506 435
5. 8 36 018
Remember this: Steps in Dividing Whole numbers
Decide where to place the first digit of the quotient Estimate the partial quotient Divide Multiply Subtract and compare Bring down Repeat the steps up to one’s digit Check
Lesson 5 Estimating quotients Mrs. Cruz baked 5 635 cookies for a field trip. The cookies are placed in 71 tin cans. About how many cookies are in each can? To estimate quotients we follow these steps: Round off the divisor to its greatest place value position Round off the dividend to a multiple of the divisor Use basic division facts involving zeros to divide YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 38
80 70 5 600
71 5 735 Round 71 to 70 Round off 5 635 to 5 600 70 x ? = 5 600 Find 5 600 ÷ 70 using basic division facts involving zeros There about 80 cookies in each tin can. Example:
340 89 520
round off
About 300 300 90 000
300 90 000
think : 300 x _____ = 90 000 3 x 300 = 900
Test Estimate the quotient 1. 782 23 568 2. 394
364 512
3. 415 25 038
Lesson 7 Solving Word Problems
The share of four farmers in making copra from coconuts is ₱ 6 400. If they worked 5 days, how much does each farmer earn in one day? What is asked? The earnings of each farmer in one day What are given? 4 farmers sharing ₱ 6 400, 5 days Operations to be used: Division Number Sentence: Share of each farmer:
6 400
Amount earned in one day:
4 = 1 600 1 600 ÷ 5 = 320
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 39
Complete Answer: Each farmer earns ₱ 320 a day.
Practice: Solve the following problems 1. Two hundred students traveled by bus in an educational tour. Five buses which can seat 42 students each were used. If four buses were filled with an equal number of students, how many were in the fifth bus? 2. A bookstore got ₱12 375 for selling science books to 15 students. How much did each student pay for the books? Test yourself: Choose the best answer. Circle the letter of your choice. 1) In the division sentence 56 divided by 8 equals 7, 8 is the a. dividend b. divisor c. quotient 2) In the division sentence 45 divided by 5 equals 9, the dividend is a. 5 b. 9 c. 45 3) In the division sentence 40 ÷ 8 = 5, 5 is the a. divisor b. quotient c. dividend 4) In the division sentence 28 ÷ 7 = 4, the dividend is a. 4 b. 7 c. 28 5) What will divide by 9 to obtain the quotient 7? a. 56 b. 63 c. 54 6) If 70 is divided by 7, the quotient is a. 9 b. 10
c. 28
7) If 6 is divided by zero, the quotient is a. 6 b. zero
c. no answer
8) If zero is divided by 15, the quotient is a. zero b. 15 c. no answer
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 40
9) If 15 is divided by 1, the quotient is a. 1 b. 15
c. no answer
10) If 12 is divided by twelve, the quotient is a. 12 b. 1 c. no answer 11) a. 1
In 5 1 324, the first digit of the quotient is above b. 2 c. 3
12) 1 000 รท 25 is a. 400 b. 4
c. 40
13) 35 000 รท 700 is a. 500 b. 50
c. 5 000
14) 8 000 รท 100 is a. 10 b. 80
c. 8
15) 1 345 275 725 is equal to a. 205 b. 215
c. 250
16) If 840 is divided by 35, the quotient is a. 23 b. 24 c. 25 17) 2 352 is divided by 112 equals a. 21 b.22 c. 23 18) The quotient of dividing 20 910 by 205 is a. 100 b. 101 c. 102 19) If 820 410 is dividend by 4002, the quotient is a. 204 b. 205 c. 206 20) The quotient when 277 380 is divided by 1 340 is a. 205 b. 206 c. 207
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 41
Chapter 5 Understanding Decimals Lesson 1 Understanding Tenths of a Number
A number with one or more places to the right of the ones place is called a decimal. The dot that separates the whole number and the decimal part is called the decimal point. 0.25, 0.4, 0.125 are examples of decimal number When a whole is divide into ten equal parts, each part is one tenth or whole.
of the
may be written in decimal form 0.1.
Just like in whole numbers, the decimals may also be written in the place value chart. To the right of the ones place is the tenths place with a decimal point in between. 0.5 read as “five tenths” 1.4 read as “ one and four tenths or one point four” When writing a decimal number in words, the word and stands for the decimal point. = 0.5 = 0.2 = 1.4 Practice: A. Write the decimal form a) b) c) 1 d) 3
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 42
e) 10 B. Write the fraction form a. b. c. d. e. f.
0.4 0.8 0.3 2.6 7.5
Lesson 2 Understanding the hundredths
The decimal square above is divided into 100 equal parts. Each part is onehundredth, or
of the whole. In decimals, we write
as 0.01.
We read 0.01 as one hundredth or zero point zero one. On the decimal place value chart, the hundredths digit is next to the tenths digit. ones . tenths hundredths 0 . 0 1 Zero ones
zero tenths
The digit 1 is in the hundredths place. Example Write the shaded part as a fraction and as a decimal. Then write the number in words. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 43
As a fraction: As a decimal: 0.04 In words: four hundredths We read: four hundredths or zero point zero four
On the place value chart: ones . tenths hundredths 0 . 0 4 The digit 4 is in the hundredths place. Example: Write the shaded part as a mixed number, as a decimal, and in words. Then write the number in the place value chart.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 44
= 1 and
1
= 0.08
= 1.08
In words: one and eight hundreds We read: one and eight hundredths or one point zero eight ones . tenths hundredths 1 . 0 8 The digit 1 is in the ones place. The digit 0 is in the tenths place. The digit 8 is in the hundredths place.
Test A. Write the decimal and fraction or mixed number for each shaded part of the figure. Then write in words.
1.
_____ = _____ ____________
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 45
2.
_____ = _____ ____________ 3.
_____ = _____ ____________
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 46
B. Shade the figure for each given decimal. 4.
= _____
5.
C. For each value chart, write the number in words, as a decimal, and as a fraction or mixed number.
6. tens ones . tenths hundredths 0 . 9 0 ________________ _____ = _____
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 47
7. tens ones . tenths hundredths 0 . 7 5 ____________ _____ = _____ 8. tens ones . tenths hundredths 1 3 . 0 5 ____________ _____= _____ D. Write each of the following fractions in decimals, in words and in the place value chart 9. tens ones . tenths hundredths
= _____
_______________
10. tens ones . tenths hundredths
1
= _____
_______________
Test A. Write each fraction or mixed number as a decimal. 1.
= _____
2.
= _____
3.
= _____
4.
= _____
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 48
5.
= _____
B. Write each decimal as a fraction and in words 1. 0.06 = _____ _______________ 2. 0.60 = _____ _______________ 3. 0.07 = _____ _______________ 4. 7.06 = _____ _______________ 5. 6.07 = _____ _______________ C. Give the place value of the underlined digit. 1. 0.25 2. 1.60 3. 3.72 4. 46.95 5. 29.56 D. Write as a fraction and as a decimal 1. Seven tenths _____ = _____ 2. Seven hundredths _____ = _____ 3. One and four tenths _____ = _____ 4. Six and four hundredths _____ = _____ 5. Twenty six and forty hundredths _____ = _____
Lesson 3 Understanding through the thousandths There are 1,000 meters in 1 kilometer. The girl scouts hiked 375 meters to the camp site. What part of the kilometer did they hike? 375 In fractions, we write :
1 000
In decimals, we write: 0.375 In words, we write: three hundred seventy-five thousandths We read the decimal as: three hundred seventy-five thousandths or zero point three hundred seventy-five. In the place value chart, we add one decimal place to the right of the hundredths place.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 49
ones . tenths hundredths thousandths 0 . 3 7 5 The digit 3 is in the tenths place. The digit 7 is in the hundredths place. The digit 5 is in the thousandths place. Example 1: This time the denominator is 1 000, there will be three decimal places.
a.
= 0.005
five thousandths
b.
= 0.050
fifty thousandths
c.
= 0.500
five hundred thousandths
Example 2: Write as a fraction and then as a decimal a. Four thousandths =
as a fraction 0.004 as a decimal
b. Fourteen thousandths =
as a fraction 0.14 a decimal
c. three hundred four thousandths =
as a fraction 0.304 as a decimal
Example 3: Write the mixed number 42
in a place value chart. Then write it in decimal
form and in words. tens ones . tenths hundredths Thousandths 4 2 . 1 3 8 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 50
In decimals: 42.138 In words: forty-two and one hundred thirty-eight thousandths The digit 4 is in the tens place. The digit 2 is in the ones place. The digit 1 is in the tenths place. The digit 3 is in the hundredths place. The digit 8 is in the thousandths place. Test
A. Write each fraction as decimal. 1.
= _____
2.
= _____
3.
= _____
4.
= _____
5.
= _____
B. Write each expression as a decimal and as a fraction. 6. Twelve thousandths : _____ = _____ 7. Fifty-nine thousandths: _____ = _____ 8. Sixty-five thousandths: _____ = _____ 9. Eight hundred three thousandths: _____ = _____ 10. Seven hundred twenty-six thousandths: _____ = _____ Lesson 4 Changing Common Fractions to Decimal Form
Equivalent fractions are fractions that describe the same quantity. To change to equivalent fractions of higher terms, we multiply both numerator and denominator by a common factor. Thus, when changing commons to decimal
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 51
form we ask the question: what factor will we multiply the denominator to make it 10, or 100, or 1 000?
Study these examples:
=
=
=
= 0.50
=
So and
= 0.50
are equivalent fractions since they describe the
same quantity =
=
=
= 0.25
=
= 0.25
Likewise, and =
=
=
= 0.06 =
=
=
are equivalent fractions.
= 0.068 = 0.625
Practice: Change the fractions into decimal form by finding their equivalent fractions with denominators of 10, 100, or 1000.
=
=
= 0.75
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 52
=
=
=
=
=
=
=
=
=
=
=
= =
= =
=
=
Lesson 5 Comparing and Ordering Decimals
Comparing decimals For fractions with the same denominator, the larger the numerator the greater is the fraction. Thus,
since 35 is < 45. 0.35 < 0.45 0.45 > 0.35
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 53
To compare decimals that are less than one whole, we can also use the place value position in the same way as we do for whole number. 1. First align the decimal points of the numbers to be compared. 2. Then starting from the tenths to the right, locate the largest place value with digits that are not the same. 3. In this place value, the decimal which has the greater digit is the greater decimal. 4. When changed to fraction with the same denominator, the greater numerator is the greater fraction. Example: compare 21.6 and 21.4 4 and 6 are not equal 4 < 6, so 21.4 < 21.6 Equivalent decimals are decimals that represent the same quantity. This implies that for decimal numbers, adding a zero to the last digit does not change the value of the decimal. Then we have the following: 0.2 = 0.20
and
0.80 = 0.8
Compare 7.01 and 7.1 In comparing the two numbers, we may affix a zero so that the two will have the same number of decimal places. 7.01
7.10
Since 0 < 1, then 7.01 < 7.10
Compare 3.052 and 3.025 5 > 2, so 3.052 > 3.025
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 54
Compare 43.06 and 34.068 We may affix zero so the number of digits are the same. Since 0 < 8, then 34.060 < 34.068 When the whole number parts of two decimals are not the same, the decimal having the higher whole number part is larger.
Exercises: Write <, =, > in the blanks 1.
0.58 _____ 0.50
2. 23.15 _____ 23.15 3. 6.025 _____ 6.205 4. 41.5 _____ 41.50 5. 98.324 _____ 98.423
Ordering Decimals
Arrange these numbers from the least to the greatest. 1.23, 1.32, 1.25 To arrange decimals in order, we compare two decimals at a time. Be sure to align the decimal points of the number of decimal places for the numbers. 1.23, 1.32, 1.25 Since 3 > 2, then 1.32 is the greatest number.
1.23 1.25 Since 3 < 5, then 1.23 < 1.25 The numbers arranged from the least to the greatest: YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 55
Example: order the decimals from least to greatest: 0.32, 0.08, 0.3, 0.2
Aligning the decimal points, we have the following: 0.32 0.08 0.3 0.2 Since 0 < 2 < 3, then 0.08 is the least number.
Exercises: Order each set of numbers from the least to the greatest. 1. 8.56, 38.65, 83.05 _____, ____, _____ 2. 35.13, 35.31, 35.01, 35.30, 35.03 _____, _____, ______, ______, ______ 3. 78.0, 78.20, 78.02, 78.22 _____, _____, _____, _____ 4. 22.75, 22.7, 22.5, 2.85, 20.750 _____, _____, _____, _____ 5. 17.105, 17.015, 17.150, 17.051, 17.510 _____, _____, _____, _____, _____
Lesson 6 Rounding Decimals
On the number line, 3.7 is nearer to 4 than to 3 so we round off to 4. ₱ 47.25 is between ₱47.00 and ₱ 48.00. It is nearer to ₱ 47.00 than to ₱ 48.00. We round off ₱ 47.25 to the nearest peso, ₱ 47.00 We often round off to the nearest whole number, yet we can also round off to any place value position. Round off 6.54 to the nearest tenths. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 56
6.54
6.5
The digit to the right is 4 so we round down. Round off 27.369 to the nearest hundredths. 27.369
27.37
9 > 5 so we round up by adding 1 to the hundredths digit.
Exercise: Round off each decimal to the nearest whole number. 1. 45.75 = ______ 2. 29.49 = ______ 3. 100.5 = ______ 4. 125.68 = ______ 5. 312.05 = ______
Lesson 7 Addition and Subtraction of Decimals
When adding decimals, the tenths should be aligned with the tenths, hundredths with the hundredths, and so on. To ensure that the digits of the decimals are aligned according to their place values, the decimal points must be aligned first. For example, find the sum: 45.21 + 3.34 Tens Ones 4 5 . + 3 . sum 4 8 .
Tenths 2 3 5
hundredths 1 4 5
To obtain the sum, we add the digits in each column or place value starting from the right YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 57
45.21 + 3.34 = 48.55
We add 0s as place holders so that the addends will have the same number of decimal places. Study the following examples. 1. 0.6
0.6
1
1.0
5.1
5.1 6.7
2. 1.135 + 2 + 3.06 + 0.4 = 1.135 2.000 3.060 0.400 6.595 Exercises: 1) 3.04
2) 32.14
+1.25
+ 7.3
3) 42.025 15.43 + 2.5
Find the sum: a) b) c) d) e)
0.43 + 2.34 = __________ 1.33 + 1.45 = __________ 3.63 + 1.27 + 4.1 = _________ 51.45 + 0.34 + 7.2 = __________ 40 + 10.08 + 5.3 = __________
Adding of Decimal Numbers with Regrouping
Find the sum: 2.47 + 0.56 First, align the decimal points. Then we add the decimals in the same way as when adding whole numbers starting from the right. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 58
Add the hundredths, tenths, then the ones. 2.17 0.56 2.73 Find the sum: 5.63 + 8.78 14.41 Exercise: find the sum 1. 0.48 + 0.35 = 2. 0.37 + 0.26 + 0.5 = 3. 9.53 + 5.47 = 4. 12.68 + 17.33 + 5.7 = 5. 21.71 + 3.9 + 12.89 =
Subtracting Decimal Numbers
The procedure for subtracting decimals is the same as for whole numbers. This time it is important that the decimal points as well as the digits of the same place value are aligned. Then subtract from right to left. Find the difference: 5.67 â&#x20AC;&#x201C; 3.42 Subtract the hundredths, tenths, then the ones. 5.67 3.42 2.25 Exercise: a) 0.46 â&#x20AC;&#x201C; 0.32 = YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 59
b) 12.75 – 10.25 = c) 26.47 – 12 .05 = d) 3.561 – 1.32 = e) 10.75 – 6.45 =
Subtracting decimal Number with Regrouping As long as decimal points are aligned, subtracting decimal numbers can be carried out as if subtracting whole numbers. Thus, regrouping may proceed in the same manner. Affix zeros to the decimals when necessary.
Find the difference 9.2 – 5.48 8 11 1 10 9.20 + 5.48 3.72 To check if the answer is correct, add the difference and the subtrahend to get the minuend. 5.48 + 3.72 = 15.00 Exercises: a) b) c) d) e)
0.804 – 0.275 = 8.026 – 4.108 = 20.5 – 9.66 = 100 – 57.235 = 18 – 7.5 =
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 60
Lesson 8 Solving Word Problems Here are some suggestions in solving word problems. a) Read the problem carefully. b) Find out what the given conditions are. c) Determine what is being asked. d) Decide what operations to use. e) Write the number sentence. f) State the answer. Greg has five 1-peso coins and three 25-centavo coins. Jon has two 1-peso coins and two 25-centavo coins. How much more money does Greg have? What is asked? How much more money has Greg? What are given? Greg, 5 1-peso coins and 3 25-centavo coins; Jon, 2 1-peso coins and 2 25-centavo coins Operations to be used: Addition and subtraction Number sentences: Greg’s total amount ₱1 x 5 = ₱5 ₱0.25 x 3 = ₱ 0.75 ₱5.00 + ₱ 0.75 = ₱5.75 Jon’s total amount: ₱1 x 2 = ₱2 ₱0.25 x 2 = ₱.50 ₱2 + ₱.50 = ₱2.50 Difference in the amount: ₱5.75 - ₱2.50 = ₱3.25 Greg has ₱3.25 more than Jon. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 61
Practice: 1) Pencil is 19.8 cm long. A ballpen is 13.2 cm long. By how many centimeters is the ballpen shorter than the pencil? 2) A house lizard is 10.50 cm. a beetle is 4.75 cm long. How many centimeters is the house lizard longer than the beetle? 3) David and her sister Anna were given ₱ 1 000 each as Christmas gift. With the money, David bought a blue shirt for ₱379.95, a belt for ₱290.50 and a yellow shirt for ₱315.75. Anna bought a blouse for ₱499.95, and a pair of shoes for ₱495.75. Between the two, who received more change and by how much?
Test Circle the letter of the best answer. 1. The decimal form of the fraction is a. 0.25
b. 0.025
c. 0.35
2. The decimal number 0.75 is equivalent to the fraction a.
b.
c.
3. In the decimal 3.475, the tenths digit is a. 5
b. 4
c. 7
4. Twenty-four hundredths is written in decimals as a. 0.24
b. 0.024
c. 2 400
5. Which of the following is not equivalent to the other two decimals? a. 8.050
b. 8.05
c. 8.50
6. One and two-fifths when written as a decimal is a. 1.25
b. 1.20
c. 1.40
7. The sum of 3.42 + 4.64 is a. 13.40
b. 13.36
c. 13.30
8. The difference between 12.4 and 8.36 is YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 62
a. 4.21
b. 4.16
c. 4.04
9. Two whole chickens weigh 1.44 kg and 1.54 kg. The sum rounded off to the nearest kg is a. 3
b. 2
c. 4
10. The difference between 2.369 and 5 is? a. 2.369
b. 2.631
c. 2.361
11. In the decimals 3.025, 3.05, 3.2, the smallest is a. 3.025
b. 3.05
c. 3.2
12. The fraction 4/9 is written in decimals as a. 0.45
b. 0.44
c. 0.43
13. The difference between 20.4 and 16.35 is a. 4.06
b. 4.05
c. 4.1
14. A piece of pork weighs 1.45 kg and a cut of beef weighs 1.6 kg. The total weight is best written as a. 3.2 kg 15. 4
b. 3.05 kg
c. 3.1
is written decimals as a. 4.48
b. 4.24
c. 4.048
16. Mrs. Uy had five mangoes weighed. The weight is 1.5 kg when two mangoes are lifted, the weighing scale reads 1.1 kg the weight of two lifted mangoes is a. 0.3
b. 0.4
c. 0.5
17. The decimal number 0.375 when expressed as a fraction is a.
b.
c.
18. Two 500-peso bills, three 100-peso bills, and three 25-centavo coins can be written as a. ₱1 200.75
b. ₱2 300.75
c. ₱1 300.75
19. The mixed number 15 is written in decimals as a. 15.75
b. 15.3
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
c. 15.375 Page 63
20. The difference 21 – 8 written as a decimal a. 13.4
b. 12.4
c. 14.4
Chapter 5 Understanding fractions
Lesson 1 The meaning of fractions Fraction as a Part of the Region Mother divided a pizza into six equal slices. Ana ate two slices of pizza. What part of the pizza did Ana eat? We can use the fraction to tell what part of the pizza Ana eat. Ana ate of the pizza. Read as “two sixths”. A fraction is a part of a whole. It is written using two whole numbers separated by a bar. The number above the bar is the numerator. It tells the number of parts considered. The number below the bar is the denominator. It tells the total number of equal parts the whole region is divided into. Let’s check understanding Match the fraction in words seven twenty-fourts nine tenths two thirds three fifths one half
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 64
Fraction as a part of a group There are five boys and seven girls in the room. (a) What part of the pupils are boys? (b) What part of the pupils are girls? The part of boys is The part of girls is
There are three red roses and six white roses in the vase. (a) What part of the flowers are red roses? (b) What part are white roses? The part of red roses is The part of white roses is
Fractional part of a number We can use division to find a fractional part of a number. Edgar collected 15 postcards. He wants to give
of the postcards to his friends.
How many postcards will he give away? What is of 15? 15 postcards in all is to be divided into 5 equal parts 15 รท 5 = 3 Thus, Edgar will give away 3 of his postcards. What is of 12? 12 in all is to be divided in 3 equal groups We divide 12 by 3 = 4 of 12 = 4
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 65
Remember this: To find the fractional part of a number, divide the number by the denominator of the fraction to get the number of equal groups. Then multiply the quotient by the numerator of the fraction. Luz has 20 t-shirts.
of them are pink, her favorite color. How many t-shirts are
pink? What is of 20? 20 t-shirts in all is to be divided in 5 equal groups we divide 20 by 5: 20 ÷ 5 = 4 There are 4 t-shirts in each group. There are 2 groups of pink t-shirts so we multiply by 2. 2x4=8 Therefore, of 20 is 8. Luz has 8 pink t-shirts.
Which is less, of 12 or of 10? of 12
of 10
12 ÷ 3 = 4
10 ÷ 2 = 5
Therefore, of 12 is 4.
Therefore, of 10 is 5
Since 4 < 5, then of 12 <
of 10.
Let’s check understanding A. Find each fractional part 1.
of 9
2.
of 25
3.
of 24
4.
of 18
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 66
5.
of 24
B. Compare and write <, =, or > in the blanks. 1.
of 8 _____ of 12
2.
of 20 _____
3.
of 30 ______ of 15
4.
of 42 _____ of 40
5.
of 25
of 32 _____ of 27
Let’s think and apply Answer the questions correctly. 1. A bakery is selling 12 cakes.
of the cakes are chocolate cakes. How
many chocolate cakes are there? 2. Mark has 18 friends in the class.
of them are girls. How many of Mark’s
friends are girls?
3. There are 20 cars in the parking lot.
of the cars are blue. How many blue
cars are there in the parking lot? 4. There are 48 marbles in the box.
of the marbles are blue and the rest are
red. How many red marbles are there in the box?
5. In a class of 36 students,
of the students failed in the examination. How
many passed the examination?
Fractions representing division We may use a fraction to represent a division sentence. 6 ÷ 2 = 3 may be written as = 3 24 ÷ 4 = 6 may be written as
=6
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 67
12
= 4 may be written as
=4
Remember this A division sentence may be expressed as a fraction where the dividend is he numerator and the divisor is the denominator. Likewise, a fraction is an indicated division. The numerator is the dividend and the denominator is the divisor
Terms used for fractions
1. The fractions
are all proper fractions.
In a proper fraction, the numerator is less than the denominator. 2. The fractions ⅓, ⅕, ⅙, ⅛ are all unit fraction. A unit fraction is a proper fraction whose numerator is 1 3. The fractions ⅔, ⅖, ⅜, ⅘ are all common fractions A common fraction is a proper fraction whose numerator is not 1. 4. The fractions , ,
,
,
are all improper fractions.
In an improper fraction, the numerator is equal to or greater than the denominator. 5. The fractions
,
,
,
, are similar fractions, the denominators are the
same. 6. The fractions ,
are all dissimilar fractions.
In the dissimilar fractions, the denominators are not the same. Improper fractions Improper fractions are equal to or greater than one. They may be expressed as whole numbers or mixed numbers. In the mixed number, the whole number part is the quotient. The remainder is the numerator and the divisor is the denominator of the fractional part. = =
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 68
To change mixed number to improper fraction, we multiply the denominator of the fractional part by the whole number. Then add the product to the numerator. This becomes the numerator of the improper fraction. The denominator of the fractional part is the denominator of the mixed number. Change
to an improper fraction
Step 1. Multiply the denominator by the whole number. 4 x 3 = 12 Step 2. Add the product to the numerator. This becomes the numerator of the improper fraction. (3 x 4) + 2 = 12 + 2 = 14 Step 3. The denominator of the fractional part is the denominator of the improper fraction. Denominator : 3 Thus,
Practice: Change the improper fraction to mixed numbers.
1. 2. 3. 4. 5. Change the mixed numbers to improper fractions 1. 2.
3. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 69
4. 5.
Lesson 2 Factors multiples, and equivalent fractions Counting numbers that are multiplied in order to obtain a given number are called factors of the number. 4 x 6 = 24
4 and 6 are factors of 24
8 x 3 = 24
8 and 3 are factors of 24
2 x 3 x 4 = 24 2, 3, and 4 are factors of 24 Let us now find the factors of 24. Start with the counting numbers 1,2,3, â&#x20AC;Ś 1 x 24 = 24 2 x 12 = 24 3 x 8 = 24 4 x 6 = 24 The factors of 24 are: 1,2,3,4,6,8,12 and 24 Let us have another example: The factors of 16: 1,2,4,8,16 1 x 16 = 16 2 x 8 = 16 4 x 4 = 16 Two or more numbers may have factors that are common to each other. The largest of these factors is called the greatest common factor of the GCF of the numbers. To find the GCF of two or more numbers, we follow these steps: YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 70
List the factors of each number. List the factors that are common to all numbers Find the greatest common factors(GCF). Factors of 24: 1,2,3,4,6,8,12 and 24 Factors of 16: 1,2,4,8, and 16 Common factors:1,2,4,and 8 GCF of 24 and 16: 8
Remember this: The greatest common factor or GCF of tow or more numbers is the largest among their common factors. When a counting number has exactly two factors: itself and 1, it is called a prime number. Number 7 is a prime number. The numbers 2 and 3 are also prime numbers. Can you name other prime numbers less than 20? Practice: List all the factors of each number in each pair or set of numbers. Circle the common factors. Then state the GCF. 1. 4: _____ 8: _____
GCF of 4 and 8 : _____
2. 9: _____ 15: _____
GCF of 9 and 15 : _____
3. 16: _____ 21: _____
GCF of 16 and 21: _____
4. 20: _____ 32: _____ 48: _____
GCF of 20,32,48: _____
5. 12: _____ 18: _____ 24: _____
GCF of 12, 18, and 24: _____
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 71
Equivalent fractions Fractions that name the same part of a whole, a region, or a set are called equivalent fractions. To find fractions that are equivalent to a given fraction, we multiply or divide both numerator and denominator by the same number.
=
=
=
Thus, = =
The fractions =
and
and
=
=
=
are equivalent fractions
are equivalent fractions
=
The fractions
and are also equivalent fractions.
Practice Give three fractions equivalent to each given fraction. 1. = 2. = 3. = 4. = 5. =
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 72
Reducing a fraction to lower terms A fraction is in its lowest term if the common factor of the numerator and denominator is 1. To reduce factor to its lowest terms, divide both numerator and denominator by their greatest common factor or GCF. Example: Reduce
to lowest terms
Factors of 16: 1,2,4,8,16 Factors of 20: 1,2,4,5,10,20 GCF of 16 and 20 :4
=
=
To check Factors of 4: 1,2,4 Factors of 5: 1,5 The common factor is 1. Thus is in lowest term. Practice: Find the GCF and reduce to lowest terms
1. 2. 3. 4. 5. Least common multiple Let us skip count by 2â&#x20AC;&#x2122;s: 2,4,6,8,10, â&#x20AC;Ś
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 73
Let us skip count by 5’s: 5,10,15,20, 25,… Let us skip count by 10’s: 10,20, 30, 40, 50, … The numbers in each list are multiples of the first number. Observe that the multiples can be obtained by multiplying the first number by 1,2,3,and so on. Given a pair or set of number, the smallest among the multiples that are common to them is their least multiple to or LCM. To find the least common multiple of two numbers: List the multiples of each number. Determine the multiples common to the numbers. Find the least among the multiples. Remember this The least common multiple or LCM of two or more numbers is the smallest among the common multiples of the numbers. Example : Find the least common multiple of 2 and 3 Multiples of 2: 2,4,6,8,10,12,14,16,18,20,… Multiples of 3: 3,6,9,12,15,1,8,21,24,… The common multiples of 2 and 3: 6,12,18 The least common multiple is 6. Practice: Find the least common multiple (LCM) of these numbers 1. 2 and 5 2. 6 and 12 3. 5 and 6 4. 12 and 20 5. 8 and 7 6. 6 and 7 7. 8,6 and 3 8. 2,3 and 8 9. 4,5 and 6 10. 2, 3 and 4
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 74
Lesson 3 Comparing and ordering fractions
Is greater than ? The answer is yes.
In comparing unit fractions, the greater the denominator, the smaller is the value of the fraction. Thus
>
>
and so on
The fraction that is arranged from least to greatest is: , , , , YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 75
Practice: 1.
_____
2.
_____
3.
_____
4.
_____
5.
_____
Similar fractions
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 76
We
see
that
<
Remember this In comparing similar fractions, the greater the numerator, the greater the value of the fraction.
>
Thus,
>
Since the numerators are arrange as 11 > 9 > 5 > 3 > 1. Arranging the fractions in order from highest to lowest , we have:
,
,
Practice: Arrange the fractions from the greatest to the least
1. , , 2. , ,
3.
,
,
4. , , 5.
, ,
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 77
Dissimilar fractions The fractions , ,
and
are dissimilar fractions. They do not have the same
denominators. To compare dissimilar fractions, we rename the fractions so that they become similar fractions. Then we can apply the rule for comparing similar fractions. To rename a fraction means to change it to an equivalent fraction. To rename fraction to an equivalent similar fraction we follow these steps. Find the least common multiple of the denominators. We call this number the least common denominator or LCD of the fractions. Divide the LCD by the denominator of each given fraction and multiply by the numerator. This is the numerator of the equivalent fraction. The LCD is the denominator. Let us have some examples. Example 1 Compare the fractions
and
Multiples of 5: 5,10,15,20,… Multiples of 4: 4,8,12,16,20,… LCM: 20 LCD of and is 20. Changing and to similar fractions
= =
( (
)
= )
= =
=
Comparing the fractions Since 8 < 15, then
<
Therefore, < YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 78
To compare dissimilar fractions, rename the fractions to similar fractions by finding first the least common denominator or LCD. For similar fractions, the greater the numerator, the greater is the value of the fraction. Example: Compare the fractions
Multiples of 5: 5,10,15,20,25,30,â&#x20AC;Ś Multiples of 6,12,18,24,30,â&#x20AC;Ś LCM: 30 LCD of = =
is 30.
(
)
(
)
=
=
=
=
Since 24 > 20, then Therefore, > For fractions with the same numerators, the greater the denominator the lower is the value of the fraction. Practice Find the LCD between each pair of fractions. Next, change to similar fractions using the LCD. Then compare the fractions.
1.
a. LCD: _____ b. = _____; = _____ c.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
_____
Page 79
2.
a. LCD: _____ b.
= _____; = _____
c. 3.
_____
a. LCD: _____ b. = _____; c.
4.
= _____
_____
a. LCD: _____ b.
= _____;
c.
_____
= _____
Lesson 4 Addition of fractions Adding similar fractions To add similar fractions, simply add the numerators to obtain the numerator of the sum. The denominator of the sum is the common denominator of the fractions. Example:
1. + =
=
Reduce to lowest terms. We change improper fraction to a mixed number.
+ =
=1 =1
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 80
+
=
=2
Practice: Add the similar fractions
1. + 2. + 3. + 4. + 5.
+
Adding dissimilar fractions To add dissimilar fractions, rename first the fractions to equivalent similar fractions. Then apply the rules for adding similar fractions. Example: Find the sum:
+
Multiples of 3: 3,6,â&#x20AC;Ś LCD of and is 6. =
(
)
=
=
+ = + = or We always reduce the sum to its lowest terms. The GCF of 3 and 6 is 3 so we divide both numerator and denominator by 3 to get .
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 81
Cross multiplication method In cross multiplication method, the numerator of the product is the sum of the cross products ( numerator x denominator). The denominator of the product is the product of the denominators of the addends. That is,
+
=
(
) (
)
Example: Use cross multiplication method to find the sum:
+
(
) (
)
=
=
+
=1
Adding improper fractions To find the sum of improper fractions we still follow the rules governing proper fractions - whether similar or dissimilar. Since
and
are dissimilar fractions, we
have to change them to similar fractions first before adding the numerator. LCM for 2 and 3: 6 LCD for
and
is 6
= = + = +
=
=3
Practice: Add the fractions 1. +
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 82
2. + 3. + 4. 5.
Adding mixed numbers When adding mixed numbers, we first get the sum of the whole numbers. Then take the sum of the fractional part. Then combine the two sums.
Example: find the sums 1. 2 + 4 = 2 + 4 + = 6 2. 6 + 5 = 6 + 5 + + = 11 + +
3. 3 + = 3 + + = 3 +
(
) (
11 + = 11 + 1 = 12
)
=3+
=3
Practice: Find the sum 1. 6 + 2 = 2. 3 + 8 = 3. 7 + 4. + 2 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 83
5. 3 + 1
Lesson 5 Subtraction of fractions
We subtract similar fractions by finding the difference of the numerators and leaving the denominator the same.
Example:
Practice: Find the difference of each pair of fractions. 1.
=
2. 3. 4.
=
5.
=
When the fractions are dissimilar, we follow the same rule in addition. Change the fractions to similar fractions before getting the difference between the numerators. Example: Find the difference: YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 84
LCM of 5 and 3: 15 = = So,
=
=
Using the cross multiplication method (4 x 3) – ( 5 x 2)
=
To subtract mixed numbers, subtract the fractions first. Then, subtract the whole numbers. Example: Subtract: 4 - 1 = Subtract the fractions: - = Subtract the whole numbers: 4 – 1 = 3 4 -1 = 3 =3
Subtracting fractions with regrouping When then fractional part of the minuend is less than that of the subtrahend, there is a need to regroup. Example: 6
=5
=5
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 85
To subtract a mixed number from a whole number, regroup the whole number and express it as a similar fraction to the other fraction. Example: regroup: 8 = 7 + = 7
Practice: Find the difference:
1.
=
2. 3. 4. 5.
Lesson 6 Solving word problems The following is a guide when solving word problems 1. 2. 3. 4. 5. 6.
Read the problem carefully. Know what is given. Determine what is asked. Decide what operations to be used. Write the number sentence State the answer.
Example: Mr. Reyes gave
of his money to his son Jess, and
to his daughter Cora. What
part of his money was given to the two children? What was part left to himself? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 86
What are asked? 1. Part of the money given to Jess and to Cora. 2. Part of the money left to Mr. Reyes himself What are given? of Mr. Reyes money given to Jess, is given to Cora. Operations to be used: Addition and subtraction Number sentences:
1.
=
=
2. Answer: Part of the money given to the children is . of the money is left to Mr. Reyes himself.
Practice: 1. A container for drinking water was
full in the morning. Then it became
full in the evening. a. What part of the container of water was consumed during the day? What is asked? What are given? Operation to be used: Number sentence:
Complete answer: b. If the capacity of the container of water is 60 liters (L), how many liters were consumed on that day? What is asked? What are given? Operation to be used: Number sentence:
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 87
Complete answer:
2. Mr. Salgado sold
of the mango fruits produced in his farm to Mr. Tan,
to
Mr. Chang and the rest to Mrs. Rosal. What part of the mangoes was sold to Mrs. Rosal?
What is asked? What are given? Operation to be used: Number sentence:
Complete answer:
Lesson 7 Multiplication of fractions
In multiplying fractions, cancel out common factors between numerator and a denominator before finding the product. When a factor is the whole number, write as a fraction with denominator 1. When a factor is a mixed number, change it first to improper fraction.
Observe that 2 is a common factor in both numerator and denominator. Since
=
1, then we can already â&#x20AC;&#x153;cancel outâ&#x20AC;? this number before we obtain the product. In multiplying fractions using cancellation, determine first which numerators and denominators have common factors. Divide them by their GCF. Then find the product of the fractions. This process is also called cancelling out common factors. Example:
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 88
Multiply Practice: 1. 2. 3. 4. 5.
Summary
A fraction is a part of a whole. A proper fraction is a fraction whose numerator is less than the denominator. A unit fraction is a proper fraction with numerator equal to 1. A common fraction is a proper fraction with numerator greater than 1. An improper fraction is a fraction whose numerator is equal to or greater than the denominator. A mixed number is composed of a whole number and a fraction. Equivalent fraction are different fractions that tell the same amount. The greatest common factor of GCF is the largest among the common factors of two or more numbers. The least common multiple or LCM is the smallest among the common multiples of whole numbers. To change improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number part of the mixed YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 89
number. The remainder is the numerator and the divisor is the denominatoror the fraction part. To change a mixed number to an improper fraction, multiply the denominator by the whole number and add the product to the numerator to get the numerator of the improper fraction; retain the denominator. Similar fractions have the same denominator. Dissimilar fractions have different denominators. Among unit fractions, the greater the denominator, the less is the value of the fract6ion. Dissimilar fractions may be compared by first changing them into similar fractions. To add similar fractions, add the numerators to obtain the numerator of the sum. Then copy the common denominator. To subtract similar fractions, take the difference of the numerators of the minuend and the subtrahend to obtain the numerator of the difference. Then copy the common denominator. To add or subtract dissimilar fractions, change them first to similar fractions. Then apply the rules for adding or subtracting similar fractions. The least common denominator ( LCD ) of two or more fractions is the least common multiple of the denominators of the fractions. To reduce to lowest term, divide the numerator and the denominator and the denominator by their GCF. To multiply two fractions, multiply the numerators to obtain the numerator of the product. Then get the product of the denominators. Test yourself Circle the letter of the best answer.
1. Which of the following is a unit fraction? a. b.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 90
c. 2. Which of the following is not a common fraction? a. b. c. 3. Which of the following is the proper fraction? a. b. 1 c. 4. Which of the following of the improper fraction? a. b. c. 5. Which improper fraction is equivalent to
?
a. 6. Which mixed number is equivalent to
?
7. The sum of and 1 is 8. The sum of and is 9. The difference between and is equal to 10.
The product of and is
11.
The product of and 4 is
12.
The difference: 2
13.
The sum of 1 and 2 is
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
is
Page 91
14.
The product of 2 and 3
15.
The difference between 3 and 2 is
Chapter 7 Understanding Plane Figures Lesson 1 Lines and angles A point is an exact location in space. It is represented by a dot. A capital letter is used to name a point. A line is a set of all points in a straight path that extends in both directions. The arrowheads indicate that the line goes on and on in both directions. We use any two points on the line to name it.
A
B
A line segment is a part of a line. It is a straight path that has two endpoints. It does not go on forever. We use the endpoints to name a line segment.
A
B
A ray is also a part of a line. It is a straight path with one endpoint. This means that a ray continues on and on in one direction. It is named by its endpoint and another point on the line. E
F
A plane is a set of points that form a flat surface. It extends in all directions. We use the points to name a plane. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 92
Lines that meet at one point are called intersecting lines.
When intersecting lines form square corners, they are perpendicular lines. The symbol for perpendicular is .
When lines do not meet, they are parallel lines. The symbol for parallel is â&#x20AC;&#x2013;.
Angles on the plane
When two rays meet at their endpoints, an angle is formed. The two rays are called the sides of the angle. The common endpoint is called the vertex of the angle.
An angle may also be named by the three points that make it; with the vertex at the middle a number may also denote an angle.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 93
Angle measures The unit of measure of an angle is the degree ( ). The instrument used for measuring an angle is called protractor. To use the protractor for measuring an angle, let one side of the angle , ray in the figure coincide with the horizontal line of the protractor that indicates 0 0 or 1800. Let the vertex coincide with the protractor: read the measure of the angle where OA passes through the scale.
Two angles having the same measure are said to be congruent angles.
When the two side of an angle are perpendicular to each other, the angle formed is called a right angle. The measure of a right angle is 90 0 is called an acute angle.
An angle with measure greater than 900 but less than 1800 is called an obtuse angle.
Lesson 2 Identifying Plane figures
A polygon is any closed figure in a plane whose sides are line segments that do not cross at points other than the endpoints. When two sides of a polygon have common endpoint, they are adjacent sides. The common endpoint of adjacent sides of a polygon is called a vertex. Two adjacent sides form an angle. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 94
A polygon with three sides is a triangle. It has three angles.
A polygon with four sides is a quadrilateral. It has four angles.
A polygon with five sides is a pentagon. It has five angles.
A polygon with six sides is a hexagon. It has six angles.
A polygon with seven sides is a heptagon. It has seven angles.
A polygon with eight sides is an octagon. It has eight angles.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 95
A polygon with nine sides is a nonagon. It has nine angles.
A polygon with ten sides is a decagon. It has ten angles.
When the sides of a polygon are congruent, it is called a regular polygon. Congruent sides are sides that have the same length. We indicate congruent sides by the number of ticks on the congruent sides. The angles of a regular polygon are also congruent. Example:
A regular triangle
3 congruent sides
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 96
A regular quadrilateral
4 congruent sides
A regular pentagon
5 congruent sides
A regular Hexagon
6 congruent sides
Quadrilaterals A quadrilateral is a polygon with four sides. Quadrilaterals have special names. When a quadrilateral has exactly one pair of parallel sides, it is a trapezoid.
trapezoid When a quadrilateral has two pairs of parallel sides it is a parallelogram. The parallel sides are also congruent.
A parallelogram that has four congruent sides is a rhombus.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 97
A parallelogram whose sides form four right angles is a rectangle. Adjacent sides in a rectangle are perpendicular.
A parallelogram that has four right angles is a square. Adjacent sides in a square are perpendicular.
Polygons and circles are called plane figures.
Test: A. How many triangles can you see in the figure? Name all of them. K
L J
N
M
B. Based on the properties of quadrilaterals, write T if the statement is true and F if it is false. 1. A trapezoid is a parallelogram. 2. A rhombus is a parallelogram. 3. A rectangle is a rhombus. 4. A square is a rectangle. 5. A square is a rhombus.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 98
Lesson 3 Kinds of triangle Triangles are classified by the measures of their angles and by the number of sides they have that are congruent. When classified according to measures, a triangle may be acute, right or obtuse. When classified according to the number of equal sides, a triangle may be scalene, isosceles or equilateral.
When all the angles of a triangle have measures less than 900, it is an acute angle. When all the angles in a triangle are acute, it is an acute triangle. B
A
C
When one angle of a triangle measures 900 , it is a right triangle.
When one angle of a triangle measures greater than 900 , it is a obtuse triangle.
The sum of the angles of any triangle is 1800. When all the sides of a triangle are congruent or are equal in length, it is called an equilateral triangle.
Equilateral triangle An equilateral triangle is also equiangular triangle. Each angle in an equilateral triangle measures 600. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 99
Isosceles triangle
The base angles of an isosceles triangle are congruent. When the third angle of an isosceles triangle measures 900, then the triangle is an isosceles right triangle.
Isosceles right triangle In a scalene triangle, the shortest side is opposite the smallest angle; the longest side is opposite the largest angle.
Scalene triangle
Test A. Match the name of the triangle in column B with its description in column A. write the letter of your answer in the blank. A _____ 1. One angle measures 900.
B a. scalene triangle
_____ 2. All sides are congruent.
b. equilateral triangle
_____ 3. Two sides are congruent.
c. obtuse triangle
_____ 4. All angles are acute. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
d. isosceles triangle Page 100
_____ 5. No two sides are congruent.
e. right triangle
_____ 6. One angle is obtuse.
F. acute triangle
B. Write A if the statement is always true, B if the statement is sometimes true, and C if the statement is not true at all. _____ 1. An isosceles triangle is equiangular. _____ 2. One angle of a right triangle measures 900. _____ 3. An equilateral triangle is a scalene triangle. _____ 4. An isosceles triangle is an acute triangle. _____ 5. The sum of the angles in a triangle is 1800. _____ 6. A scalene triangle is an obtuse triangle. _____ 7. The base angles of an isosceles triangle are obtuse angles. _____ 8. The angles of an right triangle other than the right angle are both acute. _____ 9. A triangle has two obtuse angles. _____ 10. A scalene triangle has two congruent angles.
Lesson 4 Quadrilaterals
A quadrilateral is a polygon with four sides. Figure MATH is a quadrilateral. The four sides are ̅̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅̅ vertices: M, A, T, and H. It has four angles: M
̅̅̅̅̅ it has four .
A
H T YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 101
There are two pairs of opposite sides: ̅̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅̅ are opposite sides. ̅̅̅̅̅ sides.
̅̅̅̅̅ have a common vertex, M. ̅̅̅̅̅
̅̅̅̅̅ are opposite sides; ̅̅̅̅̅ are adjacent
Can you name other pairs of adjacent sides? If exactly one pair of opposite sides of a quadrilateral are parallel, it is a trapezoid. In quadrilateral LOVE, sides ̅̅̅̅̅ Quadrilateral LOVE is a trapezoid. L
V
̅̅̅̅̅ are parallel. That is ̅̅̅̅̅ ̅̅̅̅̅.
O
E
If the two nonparallel sides of a trapezoid are congruent, it is called an isosceles trapezoid.
When a quadrilateral has two pairs of parallel sides it is a parallelogram. The parallel sides are also congruent.
If the angles of a parallelogram are right angles, the parallelogram is a rectangle.
A square is a rectangle with all sides congruent. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 102
If all the sides of a parallelogram are congruent, it is a rhombus. Figure DART is a rhombus because the opposite sides are parallel and all the sides are ̅̅̅̅̅ congruent. That is, ̅̅̅̅̅ ̅̅̅̅̅ need not to be congruent. D
T
̅̅̅̅̅. In a rhombus, the four angles A
R
A parallelogram whose sides form four right angles is a rectangle. Adjacent sides in a rectangle are perpendicular.
A parallelogram that has four right angles is a square. Adjacent sides in a square are perpendicular.
If the angles in a rhombus are all right angles, then the rhombus is a square. A square may also be defined as a rhombus in which all the angles are right angles. Polygons and circles are called plane figures. Capital letters are used to name plane figures.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 103
Lesson 5 Construction of geometric figures
The ruler is an instrument used in drawing a straight line. It is also used in measuring length. The rule has a scale that measures length in centimeters ( cm ) , and a scale that measures in (inches) in. it can measure up to the tenth of a centimeter, or the eight and sixteenth of an inch. Note that 1 inch is approximately equal to 2.54 cm. Practice: A. Measure the length of each line segment to the nearest tenth of a centimeter. 1. _____ 2. _____ 3. _____ 4. _____ 5. _____
Lesson 6 Drawing plane figures To draw an equilateral triangle Step 1. Draw a line segment. Label the endpoints as A and B. A
B
Step 2. Place the pivot of the compass at A and the marker at B. Use the marker of the compass to draw a large arc through B.
A YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
B Page 104
Step 3. Transfer the pivot point at B and with the same sitting, draw an arc to intersect the first arc. Label the point of intersection C.
A
B
Step 4. Draw the line segments AC and BC.
A
B
Answer the following. 1. Using your ruler, measure the length of the three sides.
̅̅̅̅̅ = _____ cm ̅̅̅̅̅ = _____ cm ̅̅̅̅̅ = _____ cm 2. Is the figure an equilateral triangle? To draw a triangle of any length To draw a triangle ABC where AB = 5 cm, AC = 4 cm, BC = 3 cm
Step 1. Using a ruler, draw a line segment AB with length 5 cm Step 2. Set the opening of the compass at 4 cm place the pivot at A and draw an arc above the line segment.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 105
Step 3. Set the opening of the compass at 3 cm. place the pivot at B and draw an arc intersecting the first arc. Label the point of intersection C. Step 4. Draw line segments AC and BC.
Answer the following: 1. Measure the length of the sides of the triangle.
̅̅̅̅̅ = _____ cm ̅̅̅̅̅ = _____ cm ̅̅̅̅̅ = _____ cm 2. What kind of triangle is
ABC? _____
To draw a regular hexagon Step 1. Use the compass to draw a circle. Place a point on the line of the circle. Label it A. Step 2. Place the pivot of the compass at point A and with the same setting, draw a small arc that intersects the circle. Label the point of intersection B. Step 3. Place the pivot at B and again with the same setting, draw another small arc to intersect a circle. Label the point of intersection C. continue the process until you go back to point A. Label the points of the intersections D, E, F. Step 4. Use a ruler to draw line segments that will join the consecutive points. Answer the following:
1. Measure the length of the sides of the figure: ̅̅̅̅̅ = _____ cm ̅̅̅̅̅ = _____ cm ̅̅̅̅̅ = _____ cm ̅̅̅̅̅ = _____ cm ̅̅̅̅̅ = _____ cm YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 106
̅̅̅̅̅ = _____ cm 2. What kind of polygon is the figure? _____ 3. Is this a regular polygon? _____
Chapter 8 Understanding Perimeter, Area and Volume Lesson 1 Perimeters of plane figures Mr. Dizon wants to put a fence around his rose garden which is in the form of a triangle. The sides measures 10 m, 8 m and 6 m. How many meters of fencing material does he need? To find the answer, we need to know the distance around the garden. The distance around a plane figure is called the perimeter. To find the perimeter, add the length of all the sides. 10 m + 8 m + 6 m = 24 m Mr. Dizon needs 24 m of fencing material for his rose garden.
The perimeter of a polygon is the sum of all the sides. Find the perimeter 5 cm 4 cm
6 cm
8 cm
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 107
Perimeter = 4 + 5 + 6 + 8 = 23 cm Did you know that the word perimeter comes from the Greek word perimetros. Peri means “around”, metros means “to measure”.
Practice Find the perimeter. 12 cm 4 cm
4 cm 12 cm
6 cm 4 cm
11 cm
15 cm
8 cm
8 cm
8 cm
8 cm
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 108
Perimeter of a triangle The perimeter of a triangle is the sum of the lengths of the sides. If a, b, and c represent the three sides of the triangle, then Perimeter = a + b + c b a c
In triangle ABC
B 6 cm
A
6 cm
5 cm
C
Perimeter = 6 + 6 + 5 = 17 cm If the triangle is equilateral, then the three sides are equal lengths. If s represents the length of one side, then Perimeter = 3 x s s
s
s
In equilateral triangle MAT, the length of one side is 7 cm. Perimeter = 3 x 7 = 21 cm 7 cm
7 cm 7 cm
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 109
Perimeter of a quadrilateral In any quadrilateral, the perimeter is equal to sum of the four sides. In quadrilateral ABCD, Perimeter = a + b + c + d b = 12 cm a = 6 cm
c = 7 cm
d = 10 cm Perimeter = 6 + 12 + 7 + 10 = 35 cm
Perimeter of a square The four sides of a square is represented by s, then the perimeter of the square is Perimeter is 4 x s
s = 3 cm s
s
s Find the perimeter of a square if one side is 3 cm. Perimeter = 4 x 3 = 12 cm
Perimeter of a rectangle In a rectangle, the two opposite sides are equal. We call the longer side the length and the shorter side the width. If l represents the length and w the width, then the perimeter of the rectangle is Perimeter = l + w + l + w or Perimeter = ( 2 x l ) + ( 2 x w ) What is the perimeter of a rectangle whose length is 5 cm and width is 2 cm? Since l = 5 cm and w = 2 cm YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 110
Perimeter = ( 2 x 5) + ( 2 x 2) = 10 + 4 = 14 cm
Another example: Two squares are placed side by side to form a rectangle. Of one of the side of the square is 5 cm, find the perimeter of the rectangle formed. What is given? A rectangle with l= 5 + 5 = 10 cm and w = 5 cm What is asked? The perimeter of a rectangle, P What is the formula to be used? Perimeter = ( 2 x L) + ( 2 x w ) Solution: Perimeter = (2 x 10) + (2 x 5) = 20 + 10 = 30 cm Answer: the perimeter of the rectangle formed is 30 cm.
Perimeter of a rhombus The four sides of a rhombus are congruent. If s represents one side of a rhombus, then Perimeter = s + s + s + s =4xs
Example: A residential lot is in the form of a rhombus whose side is 15 m. Martha wants to build a fence around the lot. How long is the fence needed to enclose the lot? Perimeter = 4 x s = 4 x 15 m = 60 m
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 111
Perimeter of regular polygons A regular polygon is a polygon that has congruent sides. Thus, the perimeter of a regular pentagon is Perimeter = s + s + s + s + s =5xs The perimeter of the regular hexagon is Perimeter = 6 x s The perimeter of a regular n-gon, or a polygon that has n sides with s as the length of a side is Perimeter = n x s
Example: Find the perimeter of a regular heptagon whose side is 12 cm. A regular heptagon has seven congruent sides. Thus, Perimeter = 7 x 12 cm = 84 cm Another example: The perimeter of a regular dodecagon is 240 cm. what is the length of one side? A regular dodecagon has 12 congruent sides. Perimeter = 12 s 240 = 12 s s = 240 รท 12 = 20 cm One side of the regular dodecagon is 20 cm. Another example: The perimeter of a regular polygon is 90 cm. one side of the polygon is 6 cm. How many sides does the polygon have? Let n represent the number of congruent sides Perimeter = n x s
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 112
90 = n x 6 n = 90 รท 6 = 15 The polygon has 15 congruent sides. Test Answer the following: 1. One side of a regular hexagon is 3 cm. a hexagon has six sides. What is the perimeter? 2. The length of a rectangle is 15 cm and the width is 10 cm. what is the perimeter? 3. One side of an equilateral triangle is 6 m. find its perimeter. 4. A nonagon has nine sides. If each side of a regular nonagon is 10 cm, what is its perimeter? 5. Marie wants to put a lace around a square handkerchief whose side is 40 cm. what is the length of the lace that she needs to sew around the handkerchief? Solve the following problems. 1. Five squares, 4 cm on a side are arranged as shown in the figure. Find the perimeter of the figure formed.
2 cm
A
E
B
D
H
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
C
Page 113
2. Four equilateral triangles are arranged to form a big triangle. One side of the small equilateral triangle is 2 cm. find (a) the perimeter of
triangle
ABC, (b) the perimeter of quadrilateral BEDC, and (c) the perimeter of quadrilateral EDCH. 3. The figure consists of four equilateral triangles and a square. One side of the equilateral triangle is 3 cm. find (a) the perimeter of figure ABCDEFGH, the perimeter of figure ABDFH, and (c) the perimeter of figure BCDFGH.
A H
B
G
C F
D
E 4. The figure consists of a rectangle ABCD, and two pairs of equilateral triangles. With the given lengths of the rectangle, find (a) the perimeter of EBFCGDHA, (b) the perimeter of ABCGD, and (c) the perimeter of ABFCDH.
E
A
4 cm
2 cm
B 2 cm
D
4 cm
C
G YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 114
5. The width of the rectangle is 10 cm. the length is 2 times the width. Find the perimeter. 6. The width of a rectangular residential lot is 20 m by 30 mm.It was divided into two identical rectangles as shown below. Find the perimeter of one of those rectangles.
7. One side of a regular octagon is 10 cm. what is the perimeter? 8. The perimeter of a regular decagon is 70 cm. what is the length of one side of the decagon? 9. The perimeter of a regular polygon is 48 cm. each side of the polygon is 6 cm. How many sides does the polygon have? What kind of polygon is it? 10. The perimeter of a rectangle is 80 m. if the length of the rectangle is 30 m, what is the width of the rectangle?
Lesson 2 Areas of parallelograms and triangles The area of a figure is the number of unit squares needed to cover the figure.
1 square unit To find the area of a figure we find the number of unit squares that cover it.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 115
How many unit squares are there in the figure? Counting the unit squares, there are 10 of them. Then, rectangle has an area of 10 unit squares. If one side of a unit square is 1 cm long, then the area of figure is 10 square centimeters or 10 cm2. Test Find the area of each figure by counting the numbers of unit squares.
_____
_____
The area of rectangle and square
To find the area of a rectangle, multiply the length by the width. Area = length x width =lxw Where l is the length and w is the width Example: In square ABCD, there are 4 rows with 4 units square in each row. Hence, the area is 4 x 4 = 16 unit squares.
To find the area of a square, multiply one side by the other side. Since the lengths of the side of a square are equal, then the area is
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 116
Area = side x side = s x s = s2 Find the area of square LMNO Since s = 5 cm the area is Area = s x s = 5 cm x 5 cm = 25 cm2
Area of a parallelogram To find the area of a parallelogram we need to know its base and its height or altitude. Any side of the parallelogram can be its base. The height or altitude is the length of the line segment perpendicular from the base to the opposite vertex. A=bxh Example: A large table library is in the form of a rectangle. The length is 4 m, the width is 2 m. what is area of the table top? Length (l) = 4 m; width, (w) = 2 m Area = l x w = 4 m x 2 m = 8 m2 Example 2: The area of a rectangle is 50 m2. If the width is 5 m, what is the length of the rectangle? Area = length x width 50 m2 = l x 5 m L=
= 10 m
The length of the rectangle is 10 m.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 117
Area of a triangle Area of a triangle = x (area of rectangle) = x (l x w) = x (base x height of triangle) = x (b x h) Area = base x height Remember this The area of a triangle is
x b x h where b is the base and h is the
height. In a right triangle, the base and the height are lengths of sides of the triangle. In an acute triangle, the line segment that gives the height is inside the triangle. In an obtuse triangle, if the base is one side of the obtuse angle then the line segment that gives the height is outside the triangle. Test Find the area of the following triangles. 30 cm
50 cm
40 cm 1. The base is _____ 2. The height is _____ 3. The area is _____
50 cm
25 cm
35 cm
4. The base is _____ 5. The height is _____ 6. The area is _____ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 118
5 dm
5 dm 4 dm 6 dm
7. The base is _____ 8. The height is _____ 9. The area is _____ 10. The perimeter _____ Test Answer the following questions 1. The length of a rectangle is 4 m and the width is 2 m. what is its area? 2. A square handkerchief is 30 cm on one side. What is the area of the handkerchief? 3. A classroom is 8 meters long and 5 meters wide. What is the area of the classroom? 4. A rectangular vegetable garden has an area 16 m 2. One side is 8 m. What is the perimeter of the vegetable garden? 5. The perimeter of a square picture frame is 100 cm. what is its area? 6. Mr. Pablo wants to put a large painting on the wall. It is 3 m long and 4 m high. What is the area that will be covered by the painting? 7. The mango farm of Mr. Reyes is 30 meters by 45 meters. What is the area of the farm? 8. A rectangle has an area of 36 cm2. What is the width of the rectangle if the length is 9 cm? 9. Mr. Torres is painting the wall of his house. If the wall is 5 m high and 6 m wide. What is the area of the wall? 10. The floor of a house has dimensions 10 m by 12 m. The house is sitting on a square lot that is 15 m on one side. What is the area of the yard surrounding the house? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 119
Lesson 3 Understanding solid figures Solid figures are made up of plane figures put together. Their boundaries are plane surfaces. Some solid figures like cube, rectangular prism, and triangular pyramid have straight edges. These are called polyhedra ( singular: polyhedron). In a polyhedron, the flat surface are called faces. Each face is a polygon in shape. Two faces meet or intersect at an edge. Three or more edges meet at a vertex. In a rectangular prism, there are six faces. Each face is a rectangle. There are eight vertices and twelve straight edges. The two identical and parallel faces on top and bottom are called bases. In a rectangular prism, the two bases are in the form of a rectangle.
A cube is a special type of a rectangular prism. In a cube, all the edges have the same length. All the faces of a cube are squares. The two bases are also squares.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 120
In a triangular pyramid, each face is a triangle. It has four faces, four vertices and six edges. In a triangular pyramid, there is only one base. It is triangular in shape. Opposite the base is the vertex where the other three faces intersect.
There are also solid figures that do not have straight edges. A right circular cylinder or simple cylinder, is a solid figure. It has two circular bases. The lateral surface is a curved surface that connects the two parallel bases. A right circular cylinder has no vertices or straight edges.
A right circular cone or simple cone, has a circular base. The other end is a vertex. The lateral surface is a curved surface that connects the vertex and the circular base. A cone has no straight edges.
A sphere is also a solid figure. All points on the sphere are of the same distance from a point called the center. A sphere is curved all around, no vertices and bo straight edges. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 121
Test
A. Write on the blank the kind of solid figure that is being described. 1. A __________ has six square faces. 2. A __________ has a circular base and a vertex connected by a curved surface. 3. All points in a __________ sphere are of the same distance from the center. It has no straight edges or vertices. 4. A __________has six faces in the shape of a rectangle, eight vertices, and twelve edges. 5. A __________ has four faces, four vertices, and six straight edges. B. Identify the solid figure representing these objects. 6. Television set __________ 7. Ice cream cone __________ 8. Pipe __________ 9. Volleyball __________ 10.
Book __________
11.
Globe __________
12.
Funnel __________
13.
Classroom __________
14.
Marble __________
15.
Die __________
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 122
Lesson 4 Volume of a rectangular prism
The volume of an object is the amount of space it occupies. The bigger the object, the more space it occupies, and therefore, the greater its volume. To find the volume of a solid figure, we count the number of cubes in the solid. A unit cube has one unit length at all edges. Volume = length x width x height V=Lwxh Recall that, in the metric system, a unit length may be measured in metre (m), in decimeter (dm) , in centimeter (cm), or in millimeter (mm). Thus for a unit cube that is 1 cm on one edge, the volume is 1 cubic centimeter or 1 cm. Example for the rectangular prism we find the volume as V=lxwxh If L = 5 cm , w = 3 cm , h = 2 cm
V= 5 cm x 3 cm x 2 cm V = 30 cm
Test Find the volume of rectangular prism 1. L = 6 cm W = 3 cm H = 5 cm 2. L = 4 cm W = 3 cm H = 2 cm
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 123
3. L = 5 cm W = 4 cm H = 10 cm 4. L = 4 cm W = 3cm H = 2 cm Test yourself Choose the letter of correct answer. 1. The perimeter of a triangle is 30 cm. two sides are 10 and 8 cm long the third side has length of _____ cm. 2. The perimeter of an isosceles triangle is 32 cm. one of the congruent sides is 10 cm long . the third side has length of _____ cm. 3. One side of a square is 8 cm long. The perimeter of the square is _____ cm. 4. The width of a rectangle is 5 cm. the length is twice the width. The perimeter is _____ cm. 5. One side of a regular octagon is 6 cm long. Its perimeter is _____ cm. 6. A rectangle has width 6 cm. if the length is twice the width, then the area is _____ cm2. 7. The length of a rectangle 4 cm longer than the width. If the length is 12 cm, the area is _____ cm2. 8. The altitude to the base of a parallelogram is 7 cm. if the area is 63 cm2, then the base is _____ cm. 9. The area of a square is 49 cm2. Therefore, one side has length _____ cm. 10. The length of a rectangle is 14 cm. if the width is half as long, then the area of the triangle formed by two of its sides and a diagonal is _____ cm2.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 124
Chapter 9 Understanding the Bar Graph
Lesson 1 Reading and interpreting bar graph
The information gathered in the survey is called the data. The data from the table may also be presented in a bar graph. A bar graph presents data using bars or rectangular blocks of different lengths to represent the frequency or amount o0f the data. There are several things we should look into when reading a bar graph: the title of the graph, the horizontal and vertical axes including the labels and scale, and the bars. A bar graph may also have horizontal bars. The scale is set at the horizontal axis while the label is at the vertical axis. Example: The bar graph below shows the favorite subjects of some grade 4 students.
Science Computer Filipino English Math 0
2
4
6
8
10
12
Let us answer the following questions. 1. What subject does the students like the most? Filipino 2. What subject does the students like the least? Computer 3. How many students chose Math as their favorite subject? 9
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 125
4. How many more students preferred Filipino than Science? 11 â&#x20AC;&#x201C; 8 = 3. There are 3 more students who prefer Filipino than Science. 5. How many students were asked about their favorite subject? 8 + 5 + 11 + 7 + 9 = 40 students
Test A survey was conducted in a neighborhood about kind of pets at home. Below is the result. Answer the questions that follow. 16 14 12 10 8 6 4 2 0 Dog
Cat
Bird
Fish
1. What kind of pet is most popular? _____ 2. What kind of pet is least popular? _____ 3. How many more cats than birds are there in a survey? _____ 4. How many less birds than dogs are there? _____ 5. How many more dogs than cats are there? _____
Lesson 2 Constructing a bar graph
A bar graph has the following parts.
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 126
Title of the graph. The title should be clear and contain important information about the data being presented. Axes and their labels. There are two axes in each graph. The axes labels should clearly give the information presented on each axis. Onje axis represents data groups the other represents the amount or frequency of data groups. Axes scale. A scale on the bar graph tells what each unit length represents. The scale is divided into equal intervals. Bar. Rectangular blocks that represent the amount or frequency of the data. A bar graph may be drawn vertically or horizontally. The table below presents the number of hours that Elena spends in studying her lessons at home. Days of the week Number of study Hours Monday 2 Tuesday 1.5 Wednesday 2.5 Thursday 3 Friday 1
Let us present the data in a horizontal bar graph. The scale will have 0.5 increments and will be located on the horizontal axis. The days of the week will on the vertical side.
Days of the week
Friday Thursday Wednesday Tuesday Monday 0
0.5
1
1.5
2
2.5
3
3.5
Number of Study Hours
1. On what day did Elena study the longest time? Thursday YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 127
2. On what day did Elena study the shortest time? Friday 3. How many hours did Elena study on Tuesday? 1.5 hours 4. How many hours did Elena spend studying for the whole five days? 2 + 1.5 + 2.5 + 3 + 1 = 10 hours 5. What is the average number of hours that Elena spent studying for thr five days? 10 รท 5 = 2 hours
Test The preference of students on ice cream flavor is given below. Ice cream flavor Avocado Mango Vanilla Chocolate Strawberry
Number of students 5 15 12 18 9
Draw a bar graph for the given data. What is the title of the bar graph? What is the vertical label? What is the horizontal label?
Summary Data is information gather for a survey Data gathered at a particular point in time is best represented by a bar graph. It is more attractive than just presenting the data in a table. A bar graph presented must have the following components: a title, axes and their labels, axes scale, bars. A bar graph is interpreted just like the one presented in the table.
Test yourself Choose the letter of the best answer. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 128
A. The enrollment in five elementary schools in a town is given below. Answer the question that follows
School E School D School C School B School A 0
School School A School B School C School D School E
200
400
600
800
1000
Number of pupils 700 800 950 650 400
1. The enrolment of school D is nearest to a. 600
b. 620
c. 660
d. 690
2. The combined enrollment of schools A and E is a. 1000
b. 1100
c. 1200
d. 1300
3. The number of pupils in school A is less than that of School C by around a. 200
b. 250
c. 300
d. 400
4. The enrollment in school B is how many more than the enrollment in school E? a. 300
b. 350
c. 400
d. 450
5. How many more pupils are enrolled in school C compared to school D? a. 300
b. 350
c. 400
d. 450
B. Given in the table and bar graph below are the different ways by which pupils go to a certain school. Answer the questions that follow. Means of going to school Number of pupils Jeep or Bus
70
School Bus
80
Private car
95
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
Page 129
walking
65
walking Private Car school Bus Jeep or Bus 0
20
40
60
80
100
6. What is the total number of pupils in the school? a. 300
b. 310
c. 320
d. 330
7. How many more pupils take the school bus than those who walk? a. 13
b. 14
c. 15
d. 16
8. How many less pupils walk than those who take a jeep or bus? a. 4
b. 5
c. 6
d. 7
9. How many more pupils take the school bus compared to those that take a jeep or bus? a. 7
b. 8
c. 9
d. 10
10. The combined number of pupils who take the private car and walk compared with the combined number of pupils who take the jeep and the school bus is a. 8
b. 9
c. 10
YOUNG JI INTERNATIONAL SCHOOL/COLLEGE
d. 11
Page 130