e h t a M DEMO
s c i t a m ARY D N SECO ATION C EDU
nez Jimé a r e l ero J. Co telu Alb s z aña I. Ga ra C e l o R. C
12 M ON
THS LICEN CE
INCLUDED
DIGITAL PROJECT
1
GLO BAL
THINKERS
Conntoewnledtgse in the course Basic k
Practice makes perfect! 1. 2. 3. 4. 5. 6.
Make sure you understand the problem statement Draw Plan well Representing data using an outline Proceed systematically Test
Problems CHALLENGES THAT LEAVE THEIR MARK
Natural numbers and integers ......................................................10
1 Natural numbers
........................................................................................................................................... 68
1. Positive and negative numbers 2. The set of integers 3. Addition and subtraction with integers 4. Addition and subtraction with brackets 5. Multiplication and division with integers 6. Combined operations 7. Powers and roots of in Exercises and problems Maths workshop Self-assessment
Volleyball equipment. .............................................................................................................. 90 CHALLENGES THAT LEAVE THEIR MARK
..............................................................................................................12
1. Numeral systems 2. Large numbers 3. Rounding natural numbers 4. Basic operations with natural numbers 5. Expressions with combined operations Exercises and problems Maths workshop Self-assessment
2 Powers and roots
........................................................................................................ 32
1. Powers 2. Powers of base 10: Uses 3. Operating with powers 4. Square roots Exercises and problems Maths workshop Self-assessment
3 Divisibility
.................................................................................................................................... 48
1. The relation of divisibility 2. Multiples and divisors of a number 3. Prime and composite numbers 4. Decomposing a number into its prime factors 5. Lowest common multiple 6. Greatest common divisor Exercises and problems Maths workshop Self-assessment
2
4 Integers
Decimal numbers and fractions................................................... 92
5 Decimals
......................................................................................................................................... 94
1. The structure of decimal numbers 2. Addition, subtraction and multiplication with decimals 3. Dividing decimals 4. Square roots and decimal numbers Exercises and problems Maths workshop Self-assessment
6 The metric decimal system
..................................................................... 112
1. Magnitudes and measurements 2. The metric decimal system 3. Units of measurement for fundamental magnitudes 4. Conversion of units 5. Complex and simple amounts 6. Measuring surface areas Exercises and problems Maths workshop Self-assessment
7 Fractions
...................................................................................................................................... 130
1. What are fractions? 2. The relationship between fractions and decimals 3. Equivalent fractions 4. Problems with fractions Exercises and problems Maths workshop Self-assessment
18 Operating with fractions
......................................................................... 144
1. Reducing to a common denominator 2. Adding and subtracting fractions 3. Multiplying and dividing fractions 4. Combined operations 5. Problems with fractions Exercises and problems Maths workshop Self-assessment
19 Proportionality and percentages
........................................... 160
1. Proportionality between magnitudes 2. Direct proportionality problems 3. Inverse proportionality problems 4. Percentages 5. Percentage increases and decreases Exercises and problems Maths workshop Self-assessment
How much water do taps waste?. ........................................................................178 CHALLENGES THAT LEAVE THEIR MARK
12 Geometric shapes
................................................................................................. 224
1. Polygons and other plane shapes 2. Symmetries in plane shapes 3. Triangles 4. Quadrilaterals 5. Regular polygons and circles 6. Pythagorean theorem 7. Applications of the Pythagorean theorem 8. Geometric shapes 9. Polyhedra 10. Solids of revolution Exercises and problems Maths workshop Self-assessment
13 Areas and perimeters
.................................................................................248
1. Measuring quadrilaterals 2. Measuring triangles 3. Measuring polygons 4. Measuring circles 5. The Pythagorean theorem for calculating areas Exercises and problems Maths workshop Self-assessment
Geometry............................................................................................ 180
Garden shed. ..................................................................................................................................... 266
10 Algebra
CHALLENGES THAT LEAVE THEIR MARK .........................................................................................................................................182
1. Letters instead of numbers 2. Algebraic expressions 3. Equations 4. First methods for solving equations 5. Solving first-degree equations with one unknown 6. Solving problems through equ Exercises and problems Maths workshop Self-assessment
11 Lines and angles
..................................................................................................... 206
1. Basic elements of geometry 2. Two important lines 3. Angles 4. Angle measures 5. Operating with angle measures 6. Angular relationships 7. Angles in polygons 8. Angles in a circumference Exercises and problems Maths workshop Self-assessment
Processing information. ................................................................268
14 Graphs of functions
...........................................................................................270
1. Cartesian coordinates 2. Points that provide information 3. Points that are related 4. Interpreting graphs 5. Linear functions. Equation and representation Exercises and problems Maths workshop Self-assessment
15 Statistics and probability
......................................................................286
1. Statistical analysis process 2. Frequency and frequency tables 3. Statistical graphs 4. Statistical parameters 5. Random events. Probability Exercises and problems Maths workshop Self-assessment
Analysing a set of different types of randomly mixed elements. ................................................................................................................................................ 306
Annex Glossary. .................................................................................................................................................. 308
3
5
S R E B M U N L A M I C E D
Language Bank 1
How would you count if the number zero didn’t exist? Think about it and discuss with a partner.
Most ancient civilisations used decimal-based numeral systems, meaning they represented quantities using ten different digits. There’s no doubt that the use of the decimal base comes from counting the fingers on our hands. This explains the origin of the word 'digit' which comes from the Latin 'digitus', meaning finger. In Babylon, however, they used a base 60 (sexagesimal) numeral system. Complicated, right? Well, surprisingly this system was maintained for many centuries in some cultures —such as Arabic— and it has survived to the present day to measure time. Why do you think time is measured this way? Well, because the system for measuring time was established in Babylon. In the 7th century in India, a positional notation (the value of each digit depends on the place it occupies) was added to the decimal base. In all previous numbering systems each digit had a value independent of its place. The invention of zero was a very important step and helped reach this breakthrough, this is because zero was used to indicate places where there is no quantity. This decimal-positional numeral system was used in Europe for a long time only to designate integers (the Babylonian sexagesimal system was still used for the parts of the unit)! It was in the 16th century that the decimal system was also extended to quantify parts of units, incorporating new orders (tenths, hundredths, thousandths...), with the same structure as the integer orders.
94
Use what you have learnt to solve the problem Carmela has a tendency to put on weight, so the dietician has advised her to lower her sugar intake. The dietician suggested replacing soft drinks (which contain a lot of sugar) with natural fruit juices. In the local greengrocer's, they sell freshly squeezed orange juice in two different sizes:
fruit juice
€ 2.35
€ 1.10
50 cL
20 cL
1 How many litres is each bottle? 2 How many small bottles are equal to one large bottle? 3 How much does a litre of juice cost in each size? Carmela buys one large bottle and fills up three glasses.
4 How much juice should she pour in each glass? Write the result both in centilitres, rounding to the nearest tenth, and in litres, roundina to the nearest thousandth.
5 How much does each glass cost? Write the result in cents. Now you’re ready to do your own research.
6 Ask yourself the same questions but now using another product in the supermarket, gazpacho for example.
7
Better still, make a budget for a birthday party. State the number of guests, make a shopping list, check prices, etc.
95
1
THE STRUCTURE OF DECIMAL NUMBERS Place value of decimal units We use the place value of decimals to express amounts smaller than the unit.
1 TENTH
• If we divide a unit into ten equal parts, each part is one tenth. 5.3
5
5.4
5.5
6
1 UNIT 1 0.1 = — 1 0.1 = — 10 10
1 HUNDREDTH 10 TENTHS
5.3 → Five units and three tenths • If we divide one tenth into ten equal parts, each part is one hundredth. 5.3
5 1 0.01 = — 1 0.01 = — 100 100
1 THOUSANDTH 100 HUNDREDTHS
5.36
5.5
5.36 → Five units and thirty-six hundredths • If we divide one hundredth into ten equal parts, each part is one thousandth. 5.36
1 000 THOUSANDTHS
5.4
5.365
5.37
1 0.001 = — 1 0.001 = — 1000 1000
5.365 → Five units and three hundred and sixty-five thousandths • In the decimal numeral system, a unit of any place value is divided into ten
units of the place value immediately below.
1 U = 10 t =100 h = 1 000 th = …
€2.34 ↓
tenths hundredths
2
thousandths ten thousandths 0.3
units
hundred thousandths millionths
ten 0.04
…
Two ones and thirty-four hundredths
T
U.
t
h
th t th h th m
1
3.
0
5
7
…
4
Thirteen ones and five hundred and seventy-four ten thousandths • To read a decimal number:
— First, read the integer in units. anayaeducacion.es Practise reading decimal numbers.
96
— Then, read the decimal place expressed in the place value of the units of the decimal figure to the right.
UNIT
5
Order of decimal numbers Decimal numbers are ordered on the number line.
Remember
5.36
The zeros to the right of a decimal number do not change the value of the number. U.
t
h
2.
5
2.
5
0
2.
5
0
5.365
5.37
–1.7 < –0.5 < 0.4 < 1.7 < 2.5
th
We can also compare decimal numbers without a number line by looking at the digits and their position: • To compare two decimal numbers, U. t h th we compare the integer part. 5. 3 7 5 For example: 6. 1 0 0 5.375 < 6.1 → because 5 U < 6 U • If the integer part is the same, we make the decimal places of the numbers the same by adding zeros to the right. Then, we compare the decimal places. For example: U. t h th 3.25 3.4 3. 2 5 ↓ ↓ 3. 4 0 3.25 < 3.40 → because 25 h < 40 h
0
2.5 = 2.50 = 2.500
Let's practise! 1 Write the following numbers in figures:
a) Eight tenths.
b) Two hundredths.
c) Three thousandths.
d) Thirteen thousandths.
2 Write the following numbers in words:
3
U.
6 Look at the table
a) 1.2
b) 12.56
c) 5.184
d) 1.06
e) 5.004
f ) 2.018
Write the following numbers in figures: a) Eleven units and fifteen hundredths. b) Eight units and eight hundredths.
t
and answer.
h
th t th h th m
4
0
2
0
0
3
0
0
0
a) How many hundredths are there in 40 thousandths? b) How many hundredths are 200 ten thousandths? c) How many millionths are there in 3 thousandths? 7 Write down the value of each letter: 3
A
B
4
D
C
c) One unit and three hundred and eleven thousandths. d) Five unit and fourteen thousandths.
6.2
N
M
P
Q
6.4
4 Write the following numbers in words:
a) 0.0007
b) 0.0042
c) 0.0583
d) 0.00008
e) 0.00046
f ) 0.00853
g) 0.000001
h) 0.000055
i) 0.000856
5 Write the following numbers in figures:
1.56
X
Y
Z
1.57
T
8 Order the following amounts from lowest to highest:
a) Fifteen ten thousandths.
a) 5.83
5.51
5.09
5.511
5.47
b) One hundred and eighty-three hundred thousandths.
b) 0.1
0.09
0.099
0.12
0.029
c) Fifty-eight millionths.
c) 0.5
– 0.8
– 0.2
1.03
–1.1 97
1
THE STRUCTURE OF DECIMAL NUMBERS
There are more decimals between two decimals • Let’s choose any two decimals, for example 2.3 and 2.6. It is clear that there are other decimals between them: 2.3 < 2.4 < 2.5 < 2.6 • Now, let’s find a decimal number between 2.3 and 2.4. There is one tenth of difference between these two numbers. This tenth can be divided into ten hundredths. 5.36
5.365
5.37
If we add any of these hundredths to 2.3, we get a decimal between 2.3 and 2.4. 2.3 = 2.30 < 2.32 < 2.35 < 2.38 < 2.40 = 2.4 We can continue this process indefinitely. We can do the same with any other pair of numbers. eas
id Consolidating
1 In your notebook, write a decimal number between each pair of numbers.
It must be different from the ones in the example on the left. a) 2 <
< 2.1
b) 2.1 <
< 2.11
c) 4.9 <
<5
d) 4.99 <
<5
Example
a) 2.00 < 2.05 < 2.10 b) 2.100 < 2.105 < 2.110 c) 4.90 < 4.95 < 5 d) 4.990 < 4.995 < 5
Let's practise! 9 Copy in your notebook and write a number between
each pair of numbers: a) 7 < < 8 c) 2.6 < < 2.8 e) 0.4 < < 0.5
b) 0.3 < < 0.5 d) 1.25 < < 1.27 f ) 3.42 < < 3.43
13 Lola has a weighing scales in the bathroom. It
measures to the nearest tenth of a kilo. If the weight does not match with an exact number of tenths, the result blinks between the tenth before and after it. How much does she weigh if the result blinks between 53.6 kg and 53.7 kg?
10 Write a decimal number between each pair of numbers:
a) 0.5 and 0.6 d) 0 and 0.1 g) 0.9 and 1 11
b) 1.5 and 1.6 e) 3 and 3.1 h) 2.9 and 3
c) 1.35 and 1.36 f ) 3.2 and 3.21 i) 2.99 and 3
Write a decimal number that is at an equal distance from both numbers. a) 4 and 5 b) 1.8 and 1.9 c) 2.04 and 2.05
12 Add three numbers between 2.7 and 2.8. They must
be regularly spaced.
98
14 The 100 metre sprint is taking place at an international
athletics event. Two judges measure the winning runner’s time, but they get slightly different results: • Judge A → 9 seconds and 92 hundredths
2.7
2.8
• Judge B → 9 seconds and 93 hundredths
2.700
2.800
What time would you give the winner?
UNIT
5
Rounding Sometimes, we have numbers that contain too many decimal places. In those cases, it is necessary to replace them with more manageable numbers that have an approximate value. Example
The bank calculated the interest on my two bank accounts: A → €18.2733 B → €35.3682 However, it really paid me: A → €18.27
B → €35.37
Why was I paid a different amount to the amount calculated? The smallest unit of money is one cent. Therefore, they had to round the results with many decimal places to the nearest cent. • For account A, the amount 18.2733 is closer to 18.27 than it is to 18.28. Therefore, it is rounded to 27 cents (the number in the hundredths column does not change).
Take note Numbers are often rounded in commercial and banking transactions. In general, numbers that are rounded down are compensated by numbers that are rounded up.
18.27
18.2733
18.28
• For account B, the amount 35.3682 is closer to 35.37 than it is to 35.36. Therefore, it is rounded to 37 cents (we add one to the hundredths figure). 35.36
35.3682
35.37
As you can see, we used the value nearest to the full cent. To round a number to a specific place value: • We delete all the figures to the right of that place value. • If the first figure that we delete is greater than or equal to five, we add one to
the previous figure. If it is not, it does not change.
Let's practise! 15 Round these numbers to the nearest tenths:
a) 6.27 d) 0.094
b) 3.84 e) 0.341
c) 2.99 f ) 0.856
18 Approximate the weight of each box to the nearest
gram. A gram is one thousandth of a kilo.
16 Round these numbers to the nearest hundredths:
a) 0.574 d) 3.0051
b) 1.278 e) 8.0417
c) 5.099 f ) 2.998
17 Approximate the capacity of a bottle to the nearest
decilitres.
4L 4 : 3 = 1.3333…
5.000 kg
19 Copy and complete in your notebook:
! The value 3.5777… = 3.57 has been rounded to 3.6. ! 3.57 3.6 3.5 The rounding error is less than five… 99
2
ADDITION, SUBTRACTION AND MULTIPLICATION WITH DECIMALS You already know how to add up, subtract and multiply decimal numbers. Therefore, we can quickly revise these concepts and add negative numbers.
Addition and subtraction Problem solved
To add up or subtract decimals: • We write the figures in the correct column, making sure the decimal points coincide. • We add up (or subtract) ones and ones, tenths and tenths…
anayaeducacion.es Practise adding up and subtracting decimal numbers.
A farmer poured two pitchers of milk into a cooling tank. One pitcher contained 12.35 litres and the other one 7.65 litres. Then, he removed two buckets to make cheese. One bucket contained 8.9 litres and the other one contained 5.45 litres. How many litres are left in the tank? added
removed
12.35 8.9 + 7.65 + 5.45 20.00 14.35 ?L 12.35 L remaining 7.65 L 8.9 L 5.45 L 20.00 – 14.35 (12.35 + 7.65) – (8.9 + 5.45) = 20 – 14.35 = 5.65 5.65 Answer: There are 5.65 litres of milk left in the tank.
Multiplication Problem solved
To multiply decimals: • We multiply them in the same way that we multiply integers. • We put the decimal point in the product. We leave as many decimal figures as the total number of decimal places in the factors.
anayaeducacion.es Practise multiplying decimal numbers.
pies
photoco
ge .04 per pa ............... €0 ... ... 10 a to 1 5 per p ge ...... €0.02 ... ... ... 0 10 11 to per page 0 ... €0.019 10 n a th More
It costs 2.50 to park a car for one hour. How much does it cost to park it for three and a quarter hours (3.25 h)? 3. × 1 6 6 5 8. 1
2 5 ← 2 decimal places 2. 5 ← 1 decimal place 2 5 ⏐ ↓ 0 2 5 ← 2 + 1 = 3 decimal places
Answer: We pay €8.125 for three and a quarter hours, which can be rounded to €8.13.
Multiplying by 10, 100, 1 000… Remember that to multiply a decimal number by 10, 100, 1 000… you simply have to move the decimal point one, two, three… places to the right. Exemple
Look at the prices of the advert on the left, let’s calculate: • The cost of 10 photocopies → 0.04 · 10 = €0.40 • The cost of 100 photocopies → 0.025 · 100 = €2.50 • The cost of 1 000 photocopies → 0.019 · 1 000 = €19.00
100
UNIT
5
eas
id Consolidating
1 Solve without a calculator:
a) 1 – 0.4 d) 0.75 – 0.5
Help
b) 1.5 – 0.6 e) 1.25 – 0.75
c) 2.1 – 0.2 f ) 2 – 1.25
1 Imagine it in a number line: 0.8
2 Take note, copy and complete in your notebook:
0
a) 1.5 – 1 = 0.5 → 1 – 1.5 = … b) 1 – 0.75 = 0.25 → 0.75 – 1 = … c) 2.2 – 0.4 = 1.8 → 0.4 – 2.2 = …
1
0.8 – 0.5 = 0.3 2 0.5 – 0.3 = 0.2 → 0.3 – 0.5 = –0.2
3 Calculate:
a) (–0.3) · 4 c) (–0.1) · 0.4
0.5
3 Use the rule of signs:
b) 0.8 · (–2) d) (–0.2) · (–0.3)
• (–0.5) · 3 = –1.5 • (–0.3) · (–0.4) = +0.12
Let's practise! 1 Solve without a calculator:
a) 0.8 + 0.4 d) 1 – 0.3
8 Calculate the following without a calculator:
b) 1.2 + 1.8 e) 2.4 – 0.6
c) 3.25 + 1.75 f ) 2.5 – 0.75
2 Remember the rules for calculating using positive and
negative numbers. Solve without a calculator: a) 0.4 – 0.6 b) 0.9 – 1.6 c) 0.25 – 1 d) 1.2 – 1.5 e) 0.5 – 0.75 f ) 2 – 1.95
e) 4.03 · 2.7
f ) 5.14 · 0.08
b) 3.5 – 0.2 · (2.6 – 1.8)
c) (5.2 – 6.8) · (3.6 – 4.1) d) (1.5 – 2.25) · (3.6 – 2.8) 10 True or false?
a) If we multiply a number by 0.8, its value increases. b) The result of multiplying a number by 1.1 is greater than the original number. c) To multiply by 100, we move the decimal point two places to the right. d) Moving the decimal point one place to the left is the same as multiplying by ten.
5 Copy in your notebook and add the missing decimal
b) 3.8 · 12 → 456 d) 11.7 · 0.45 → 5265
6 Multiply:
b) 35.29 · 10 e) 6.24 · 100
c) 4.7 · 1 000 f ) 0.475 · (–10)
7 Multiply:
a) (–2) · 0.7 c) –0.6 · (–3) e) (–0.2) · (–0.8)
d) 3.70 · 1.20
a) 8.3 + 0.5 · (3 – 4.2)
a) 17.28 – 12.54 – 4.665 b) 17.28 – (12.54 – 4.665) c) 12.4 – 18.365 + 7.62 d) 12.4 – (18.365 + 7.62)
a) 3.26 · 100 d) 9.48 · 1 000
c) 27.5 · 10.4
• 5.6 – 2.1 · (0.5 – 1.2) = 5.6 – 2.1 · (–0.7) = = 5.6 + 1.47 = 7.07
4 Solve in your notebook:
point to each product: a) 2.7 · 1.5 → 405 c) 0.3 · 0.02 → 0006
b) 2.6 · 5.8
9 Follow the example to solve the following operations:
3 Add three more terms to the following series:
a) 0.25 - 0.50 - 0.75 - … b) 8.25 - 8.2 - 8.15 - 8.1 - …
a) 3.25 · 16
b) (–0.5) · 4 d) 0.2 · (–10) f ) (–4) · (–0.25)
11
We cut a section measuring 0.97 m off a 2 m bar. How long is the remaining bar?
12 It took Jon Dalton twenty-two seconds and three
tenths of a second to complete the 200-metre sprint. It took Bobi García twenty-three seconds and fourteen tenths of a second to complete the same sprint. How much faster was Jon than Bobi?
13 The ironmonger sells white cable for €0.80 per metre
and black cable, which is thicker, for €2.25 per metre. How much do we pay for 3.5 m of white cable and 2.25 m of black cable? 101
3
DIVIDING DECIMALS Now you will learn more about dividing decimal numbers. Let’s begin with divisions that have an integer divisor.
Integer divisor. Approximation of the quotient We will now see how we can get the decimals of the quotient to reach the approximate that we want. Problems solved
To find the decimal quotient: • When we bring down the tenths decimal place of the dividend, we add the decimal point to the quotient and continue the division. • If there are not enough decimal places in the dividend, we add zeros until we reach the desired approximation.
1. We divide a 15-litre barrel of oil into four identical bottles. How many litres do we put in each bottle? 15 3
4 3
1 5. 0 3 0 2
4 3.7
→ The integer quotient gives a remainder of 3 ones. → We can transform the three ones of the remainder into 30 tenths if we put the decimal point in the quotient. Then, divide this by 4. There is a remainder of 2 tenths.
1 5. 0 4 3 0 3.75 → We can continue the division by transforming the 2 tenths into 20 hundredths. 20 0 Answer: We put 3.75 litres in each bottle.
2. Emily buys a cheese that weighs one kilo and seven hundred and twenty-five grams. She shares the cheese with her two sisters. How much cheese does each of them receive? 1. 725
anayaeducacion.es Practise dividing decimal numbers.
3 0
→ 1. 725 2
3 0.5
→ 1. 725 22 15 0
3 0.575
Answer: Each sister receives 0.575 kg of cheese (575 grams).
Dividing by 10, 100, 1 000… Remember that to divide a number by 10, 100, 1 000… you simply have to move the decimal point one, two, three… places to the left. Example
Considering the weight of a pack of 500 sheets, calculate: • The weight of 100 sheets → 2 331 : 5 = 466.2 grams • The weight of 10 sheets → 466.2 : 10 = 46.62 grams • The weight of 1 sheet → 466.2 : 100 = 4.662 grams 2 331 grams
102
To divide a decimal number by a unit followed by zeros, we move the decimal point to the left the same number of places as there are zeros after the unit.
UNIT
5
Division with decimal divisors Until now, we have not seen divisions with decimals in the divisor. We can use a property you already know to help us solve this type of problem. ➜ a key property of division
Compare the following examples: Examples
Remember When we multiply the dividend and the divisor by the same number, the remainder is multiplied by this number. · 10
13 01
2 6
130 010 · 10
20 6
• If we pack 15 kilos of plums into 3 boxes, we put 5 kilos in each box. 15 3 0 5
• If we pack 150 kilos of plums into 30 boxes, we put 5 kilos in each box. 150 30 00 5
When we multiply the number of kilos (the dividend) and the number of boxes (the divisor) by 10, the result stays the same. Property of division: When we multiply the dividend and the divisor by the same number, the quotient stays the same. ➜ how to remove decimal places from the divisor
When the divisor is a decimal number, we can use the property mentioned above to re-write the division. This way the result and the divisor are integers. Examples
A waiter fills a bottle with 0.6 litres of milk. Approximately, he adds 0.04 litres of milk to each cup of coffee. How many cups of coffee can he make with one bottle? 0.6
0.04
60 20 0
4 15
· 100
⎯→ We multiply the dividend and the divisor by 100. According to the property of division, the quotient · 100 does not change. ⎯→ The divisor is now an integer. Now, we can do the division. Answer: He can make 60 : 4 = 15 cups of coffee.
When there are decimals in the divisor: • We multiply the dividend and the divisor by the unit followed by as many
zeros as there are decimal places.
• This changes the division to transform the divisor into and integer.
However, the quotient does not change.
Problems solved
Find the quotient of the following divisions: 21 : 16.8 · 10
· 10
210 0420 0840 000
168 1.25
· 10 000
0.3 : 0.0025 3000 50 00
· 10 000
25 120
103
3
DIVIDING DECIMALS eas
id Consolidating
1 Calculate by rounding the quotient to the nearest tenths:
Exemples
a) 10 : 3
b) 16 : 9
c) 25 : 7
d) 9.2 : 8
e) 15.9 : 12
f ) 45.52 : 17
1 Take note:
18 40 50 1
2 Calculate the quotient to two decimal places:
a) 3 : 4
b) 3 : 7
c) 30 : 8
d) 2 : 9
e) 6 : 11
f ) 5 : 26
3 To divide 7.158 : 0.03, we multiply
3 Copy and complete each division in your notebook:
18 : 2.4
· 10
… … …
· 10
24 7.5
by 100.
2.7 : 0.075
· 1 000
2700 … …
7 rounded to 2.57 ⎯⎯⎯⎯→ 2.6
· 1 000
… …
· 100
7.158 : 0.03
· 100
715.8 : 3
Let's practise! 1 Divide without a calculator:
a) 1 : 2 d) 1 : 4 g) 1.2 : 2
b) 5 : 2 e) 2 : 4 h) 1.2 : 3
7 Calculate:
c) 7 : 2 f) 5 : 4 i) 1.2 : 4
b) 53 : 4 e) 6.2 : 5
c) 35 : 8 f ) 12.5 : 4
b) 8 : 100 e) 5.7 : 100
c) 2 : 1 000 f ) 2.8 : 1 000
3 Divide:
a) 5 : 10 d) 3.6 : 10
b) 0.5 : 4 e) 0.08 : 2
c) 0.3 : 9 f ) 0.02 : 5
c) 0.8 : 1.25
d) 2 : 5.4
e) 3.2 : 8.36
f ) 3.654 : 6.3
Express the capacity of one bottle in litres.
9 A road maintenance company agrees to mark a 15 km
stretch of new motorway in eight days. Approximately, how many kilometres do they mark each day?
10 How many rows of 0.2 m × 0.2 m × 0.2 m boxes can
we paile up in a container that is 1.85 m high? How much space is there between the top box and the roof of the container?
4 Calculate to three decimal places where possible:
a) 0.9 : 5 d) 1.2 : 7
b) 0.7 : 1.4
8 Three bottles of soft drink contain one litre in total.
2 Calculate to two decimal places where possible:
a) 28 : 5 d) 47 : 3
a) 0.4 : 0.84
11
A fruit seller sells melons for €1.25/kg. How much does a melon that costs €4.40 weigh?
5 Copy and complete in your notebook:
a) 8 : 0.9 = … : 9 c) 2 : 1.37 = … : 137
b) 15 : 0.35 = … : 35 d) 7 : 0.009 = … : 9
6 Transform each division into an equivalent without
decimals in the divisor and calculate the quotient: a) 32 : 0.8 b) 6 : 0.7 c) 1.82 : 0.7 d) 18 : 0.24 e) 0.72 : 0.06 f ) 1.52 : 0.24 g) 7 : 0.05 h) 0.2 : 0.025 i) 11.1 : 0.444
104
12 A vaccine requires 0.25 millilitres (0.00025 litres) of
active ingredient per dose. How many doses do we get from one litre of active ingredient?
4
SQUARE ROOTS AND DECIMAL NUMBERS You already know the concept of square root. We apply it in the same way to decimals. a = b ↔ b2 = a
0.81 = 0.9 ↔ (0.9)2 = 0.81
However, most numbers do not have an exact square root. In these cases, we must find the approximate root. 2 " 2 2 = 4 < 7.5 7.5 = * 3 " 3 2 = 9 > 7.5
2.7 " 2.7 2 = 7.29 < 7.5 7.5 = * 2.8 " 2.8 2 = 7.84 > 7.5
7.5 = 2. …
7.5 = 2.7…
Finding the square root on a calculator It takes a long time to find approximate square roots by trial and error. It is easier and faster to use a calculator. 7.5 → 7 . 5 $ → {“…|«°\‘“|} Usually, we do not need all the decimal places on the calculator. We round the answer to a certain number of decimal places. 2.7 " Rounded to the tenths 7.5 = ) 2.74 " Rounded to the hundreths
Calculating with pen and paper Think about the algorithm you learnt in unit 2 to calculate the square root of natural numbers. You can use the same method with decimal numbers. Remember that we separate the figures into groups of two, to the right and left of the decimal point. Example
√ 7 . 50 2 –4 3
√ 7 . 50 2.7 –4 47 · 7 = 329 3 50 –3 29 21
√ 7 . 50 –4 3 50 –3 29 21 –16 4
00
2.73 47 · 7 = 329 543 · 3 = 1 629
00 29 71
We can continue this process by adding pairs of zeros to the radicand until we reach the approximation that we want. Let's practise! 1 Solve without a calculator:
2 Round to the nearest tenths and hundredths:
a) 0.01
b) 0.09
c) 0.25
a) 58
b) 7.2
c) 0.5
d) 0.64
e) 0.0001
f ) 0.0049
d) 14
e) 8.5
f ) 0.03 105
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S M E L B O R P D N EXERCISE A The decimal numeral system 1 Write the following numbers in words: a) 13.4 b) 0.23 c) 0.145 d) 0.0017 e) 0.0006 f ) 0.000148 2
Write the following numbers in figures: a) Eight ones and six tenths. b) Three hundredths. c) Two ones and fifty-three thousandths. d) Two hundred and thirteen hundred thousandths. e) One hundred and eighty millionths.
3
Express the following numbers as decimals: a) 6 tenths. b) 27 ones. c) 200 hundredths. d) 800 thousandths.
4
Copy and complete in your notebook: a) 8 U = 80 t = … h = … th b) … U = … t = 30 h = … th c) … U = … t = … h = 1 700 th
5
Write the following numbers in figures: a) Half a one. b) Half a tenth. c) Half a hundredth. d) One quarter of a one.
Order. Representation. Rounding 6 Order from lowest to highest. ! a) 1.4 1.390 1.39 1.399 b) – 0.6 0.9 – 0.8 2.07
a
b
6
m r
n
2.3 5.28
c
6.5
s
o 5.29
d
t
u
q
106
11
Solve without a calculator: a) How much do we add to 4.7 to make 5? b) How much do we add to 1.95 to make 2? c) How much do we add to 7.999 to make 8?
12
Solve the following calculations: a) 13.04 + 6.528 b) 2.75 + 6.028 + 0.157 c) 4.32 + 0.185 – 1.03 d) 6 – 2.48 – 1.263
13
Solve the following expressions: a) 5 – (0.8 + 0.6) b) 2.7 – (1.6 – 0.85) c) (3.21 + 2.4) – (2.8 – 1.75) d) (5.2 – 3.17) – (0.48 + 0.6)
Multiplication and division 14
Multiply and divide without a calculator: a) 0.12 · 10 b) 0.12 : 10 c) 0.002 · 100 d) 0.002 : 100 e) 0.125 · 1 000 f ) 0.125 : 1 000
15
Multiply: a) 0.6 · 0.4 c) 1.3 · 0.08 e) 2.65 · 1.24
Calculate to two decimal places where possible: a) 0.8 : 0.3 b) 1.9 : 0.04 c) 5.27 : 3.2 d) 0.024 : 0.015 e) 2.385 : 6.9 f ) 4.6 : 0.123
17
Copy and complete in your notebook: a) 72 : … = 7.2 b) 3.8 : … = 0.038 c) … : 1 000 = 0.05 d) … : 100 = 2.3
18
Which numbers divide the range 2-3 into four equal sections?
Problem solved
Round each of these numbers to the nearest one, tenth and hundredth. a) 2.499 b) 1.992 c) 0.999
b) 0.03 · 0.005 d) 15 · 0.007 f ) 0.25 · 0.16
16
v
Approximate 3.70965 to the nearest… Ones → 4 Thenths → 3.7 Hundredths → 3.71 Thousandths → 3.710 9
Addition and subtraction
e
p 2.4
Write a decimal number between: a) 0.5 and 0.6 b) 1.1 and 1.2 c) 0.24 and 0.25 d) 6.16 and 6.17 e) 1 and 1.1 f ) 3 and 3.01
Calculations
Associate a number to each letter:
7
8
1.41 –1.03
10
2
3
UNIT
19
Write down the numbers that divide the range 0.7-0.8 into five equal parts. 0.7
20
21
22
23
24
25
27
Divide. What do you notice? a) 3 : 0.5 b) 5 : 0.5 d) 0.4 : 0.5 e) 0.7 : 0.5
b) 0.62 : 0.1 – 4.3 – 12 · 0.1 c) 15 · 0.5 + 0.5 : 0.2 – 9.8
c) 22 · 0.5 f ) 4.2 · 0.5 c) 11 : 0.5 f ) 2.1 : 0.5
Calculate, look at the results and answer: a) 200 · 0.1 30 · 0.1 8 · 0.1 What happens to a number when we multiply it by 0.1? b) 7 : 0.1 35 : 0.1 0.5 : 0.1 What happens to a number when we divide it by 0.1?
d) 5.5 · 0.2 + 1.1 + 0.66 : 0.6 28
Problem solved
3.25 · 2.4 – 1.5 · (2.1 – 3.9) = 7.8 – 1.5 · (–1.8) = = 7.8 + 2.7 = 10.5 3.25 × 2.4 1300 650 7.800 29
3.9 – 2.1 1.8
1.5 × 1.8 120 15 2.70
Calculate:
Give examples, do some research and complete these sentences in your notebook: a) Multiplying by 0.2 is the same as dividing by… b) Dividing by 0.2 is the same as multiplying by…
b) 0.36 – 1.3 · (0.18 + 0.02)
Multiply without a calculator: a) 18 · 0.1 b) 15 · 0.01 d) 5 · 0.2 e) 200 · 0.02 g) 20 · 0.5 h) 20 · 0.05
f ) – (3.5 · 1.2) : 2.1 + (0.865 – 3)
Divide without a calculator: a) 7 : 0.1 b) 9 : 0.01 d) 2 : 0.2 e) 6 : 0.02 g) 1 : 0.5 h) 1 : 0.05
c) 400 · 0.001 f ) 3 000 · 0.002 i) 2 000 · 0.005
c) 2.5 – 1.25 · (2.57 – 0.97) d) 6.5 · 0.2 – 0.4 : (2.705 – 3.105) e) 12 : 6.4 – 2 · (1 : 8) g) (–5.33 + 1.79) · 3 – (8.75 : 0.5) 30
4.8 + 2.6 · 0.5 – 18 · 0.1
a) 2.755 – 0.5 · (1.69 – 0.38) b) 2.3 · (6.07 – 3.77) – 0.45 Square roots 31
4.8 + 1.3 – 1.8 3 + 1.3 4.3 4.8 + 2.6 · 0.5 – 18 · 0.1 = 4.8 + 1.3 – 1.8 = = 3 + 1.3 = 4.3
Look at the example and solve with a calculator: • 1.42 – 2.4 · (2.15 – 1.6) ⇒ ⇒ 2.15 - 1.6 = * 2.4 µ 1.42 ≤ Ñ ⇒ {∫∫≠…‘} 1.42 – 2.4 · (2.15 – 1.6) = 0.1
c) 8 : 0.001 f ) 10 : 0.002 i) 1 : 0.005
Problem solved
7.8 + 2.7 10.5
a) 1.9 + 2 · (1.3 – 2.2)
Combined operations 26
Calculate using mental arithmetic: a) 5.6 – 0.8 : 0.5 + 6.2 · 0.5
0.8
Multiply. What do you notice? a) 6 · 0.5 b) 10 · 0.5 d) 0.8 · 0.5 e) 1.4 · 0.5
5
32
Solve without a calculator: a) 0.04
b) 0.16
c) 0.36
d) 0.0009
e) 0.0025
f ) 0.0081
Solve these square roots with a calculator. Round the results to the nearest hundredths. a) 13
b) 217
c) 2 829
d) 42
e) 230
f ) 1425 107
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S M E L B O R P D N EXERCISE A Problem solving 33
37
Problem solved
If the body and lid weigh 120 grams, how much does each filled jar weigh?
Solve a similar problem with simpler data.
George moves forward 67 cm on each step. There is a distance of 1 km and 340 m from home to school. How many steps does he need to take to go from home to school?
We put 15 kilos of honey into 25 jars.
Answer first: We put 10 kilos of honey into 20 jars. If the body and lid weigh 0.2 kilos, how much does each jar weigh? 38
Four cups weigh the same as five glasses. Each cup weighs 0.115 kg. How much does each glass weigh? Answer first: Four cups weigh the same as five glasses. Each cup weighs 100 grams. How much does each glass weigh?
• Here is a similar problem with simpler data: George moves forward 0.5 on each step. Between home and school there is a distance of 400 m. How many steps does he take to go from home to school? We divide the distance between the length of a step: 400 : 0.5 = 800 steps Answer: George takes 800 steps to go from home to school. • Using the same method, solve the original problem. 34
A fish weighing one and a quarter kilos cost €15.75. How much is a fish weighing 1.4 kilos? Answer first: A fish weighing one and a half kilos cost €15. How much is a fish weighing two kilos?
108
40
This term, Rachel took three maths exams. Her marks were 5.5, 7 and 2.40. What is her average mark?
41
Mr. Orondo went on a diet after seeing his dietician. The table below shows his weight on the first day of the past six months: 1st
2nd
3rd
4th
5th
6th
91.38
90.16
88.815
87.801
86.9
86.15
a) In which month did he lose the most weight? b) How much weight did he lose in total? 42
A box contains 80 teabags which weigh 3.125 g each. How many grams of tea are there in the box?
43
We pay €0.50 to enter a public car park, plus €0.012 per minute. How much did we pay if we parked for one hour and thirteen minutes?
We use a 2.8-litre bottle to fill four 45-centilitre glasses. How much water is left in the bottle? Answer first: We use a 2-litre bottle of water to fill four 0.2-centilitre glasses. How much water is left in the bottle?
36
Marta bought three croissants at the bakery for €4.05. The customer who came in after her bought four croissants and paid with a €10 note. How much change did he get?
Target 6.1. Water is a scarce resource. Imaine that you spend 5 minutes in the shower every day. The tap pours 0.4 L per second. How many times did you shower if you used approximately 7.2 hectolitres of water? What could you do to use only half the water you use now? Answer first: If the pump was broken and it was losing 4 litres of water per second, how long would it take to lose 8 000 litres of water?
35
39
44
Rosie and James go to the supermarket and buy: — Five litres of milk at €1.05 per litre. — A 0.92-kg bag of cod at €13.25 per kg. — A packet of biscuits costing €2.85. — A quarter of a kilo of ham for €38.40 per kg. How much did they pay?
UNIT
45
Ana, Anwar, Anthony and Mary each eat a sandwich at a sandwich shop. Anna, Anwar and Anthony have the same sandwich, but Mary has a ham sandwich, which costs €1.80 more. They paid €14.60 in total. How much did Mary’s sandwich cost?
46
A dairy company sells yoghurts for €1.20 each. One third of this price is for the packaging. Half of it is for the production and marketing costs, and for profit. The rest is for the contents. How much does the contents cost?
47
We supply a tank with a capacity of 19.36 cubic metres with a well connected to a pump that provides 4.3 litres of water per second. How long does it take to fill the tank if the pump starts working when the tank is empty?
52
How many shelves that are 0.8 m long and 0.25 m wide can a carpenter make by cutting a 2.40 m × 1.75 m board?
53
A vine-grower harvested 42 tonnes of grapes. One in every five kilos is table grape. The remaining are for making wine. If he needs 1.25 kilos of grapes to make one litre of wine, how much wine can he make this time?
54
A wine seller buys a batch of 30 000 litres of wine for €72 000. He puts the wine into 75-centilitre bottles. He pays €14 for one hundred empty bottles and €10 for one thousand corks. He makes a profit of €54 000. How much does he charge for each bottle of wine?
‘+’ problems 55
Remember that one cubic metre is the same as 1 000 litres. 48
A baker has a basket that weighs 8.5 kg when empty. When it is full of loaves of bread that weigh 250 grams, it weighs 18.750 kg. How many loaves of bread are in the basket?
49
A car moves forwards 2.68 metres every time its wheels turn. How many times do they turn in the 620 km journey from Madrid to Barcelona? (Round your answer to the nearest hundreds).
50
A farmer wants to surround his farm (shown in the diagram below) with a wire fence that costs €12.99 per 5-metre roll. How much will the wire cost in total? 9.85 m
19.95 m 28.2 m
51
Martina has two mobile phone contracts with two different companies: A and B. Phone company A charges 30 cents to connect a call and 20 cents per minute. Company B does not have a connection charge, but charges 25 cents per minute.
Briefly explain which phone Martina should use depending on the time she thinks she will need for her call. 56
The tables below show the number of shots and baskets two players made and scored in their last five games: 1st
2nd
3rd
4th
5th
shots
4
3
4
2
5
baskets
2
3
3
2
4
1st
2nd
3rd
4th
5th
shots
5
7
3
8
7
baskets
2
5
2
7
5
player a
5.75 m
The stationary shop sells pens for €1.65 and markers for €2.40. How many pens can I buy if I buy two markers and do not want to spend more than €10? How much money do I have left?
5
player b
Which player do you think is most likely to score a basket? Justify your answer. 109
P O H S K R O W S H T A M READ AND LEARN Different types of decimals Decimals can be classified into three different groups based on the figures to the right of the decimal point: ➜ Terminating decimals
They have a specific number of decimal places. For example, if four siblings share €25 of pocket money between them, each sibling gets 25 : 4 = €6.25. The number 6.25 is a terminating decimal. ➜ Recurring decimals
The decimal places go on forever and never stop, but they repeat cyclically. For example, if # a hiker walks 111 metres in 99 steps, they move forwards 1.121212... = 1. 12 metres per step. # The number 1. 12 is a recurring decimal. ➜ Decimals that are neither terminating nor recurring
The decimal places go on forever and they never stop, but they do not repeat cyclically. For example, we can make one up: 0.123456789101112131415… • What are the next three figures?
INVESTIGATE 1:9
0.11111…
2:9
0.22222…
3:9
)
a) Use a calculator to complete some rows of this table: 0.1
b) Now, divide some numbers in this series by 9: 1 - 10 - 19 - 28 - 37 - … • What do these numbers have in common? • What do the quotients have in common? c) Do the same for the numbers in the series below: 2 - 11 - 20 - 29 - 38 - … 3 - 12 - 21 - 30 - 39 - … 4 - 13 - 22 - 31 - 40 - … • What do you notice? • Which numbers do you have to divide to obtain 4.555…?
110
UNIT
5
PRACTICE MAKES PERFECT! Calculations... and a bit of creativity • How tall is the pedestal?
• Three motorcyclists, Robert Red, Walter White and Gemma Grey are getting ready to go for a ride. — Robert says: “Have you noticed that one of our motorcycles is red, one is white and the other is grey, but none of the colours match the owner’s surname?” — The owner of the white motorcycle says: “I hadn’t noticed, but yes, you are right!” What colour is each person’s motorcycle?
35 cm
29 cm
• Obtain the number 10 by multiplying: a) Three different numbers. b) Three decimal numbers. c) Three numbers with one decimal place.
? × ? × ? = 10
SELF-ASSESSMENT
anayaeducacion.es Answer key.
1 Write the following numbers in figures:
7 Calculate:
a) Twenty-eight thousandths. b) Two ones and seven hundredths. c) One hundred and thirty-two ten thousandths. d) Nine millionths.
< 4.6
b) 0.1 <
< 0.11
! b) 5.6
6 What is the number each letter points to?
a
Commitment
b
c
b) 4.2 – 0.2 · (5 – 0.6)
c) (4.2 – 0.2) · 5 – 0.6
d) 4.2 – (0.2 · 5 – 0.6)
a) 7 : 13
b) 54.5 : 12
c) 8.34 : 15.25
11
Michael works at a shop wrapping presents. He is paid eighty cents for every present he wraps. Yesterday, he wrapped 30 presents. How much did he earn?
12 To give Rosie a present, we need €33 between 10
2.9
2.8
a) 4.2 – 0.2 · 5 – 0.6
pay for a melon that weighs 2.800 kilos?
5 Round to the nearest tenths and hundredths.
a) 2.726
d) 2.6 : 100
10 One kilo of melon is €1.75. How much do you
4 Complete with a decimal in your notebook.
a) 4.5 <
c) 6.8 · 100
9 Calculate to two decimal places:
3 Order the following numbers from lowest to highest
and write them on a number line. 2.07 - 2.27 - 2.71 - 2.7 - 2.17
b) 2.8 · 3.75
8 Calculate:
2 Think and answer.
a) How many thousandths make up one tenth? b) How many millionths are there in one thousandth?
a) 2.8 – 3.75 + 1.245
d
friends. To give my mother a present, we need €10 between her 3 children. Which present is the most expensive for me?
Watch the video for target 15.1. Think of something you can do to achieve that goal. Make a commitment to put your idea into practice.
111
6
FRACTIONS
Language Bank 1
Why do you think it took three thousand years for unit fractions to reach Europe? Discuss.
In ancient Egypt, during the 18th century BCE, they handled fractions in a very unusual way: they only recognised fractions with a numerator of 1 (unit fractions)! 1 2
1 3
…
1 7
…
1 10
…
1 100
There was a need to measure and divide, fractions were used to meet this need. For example, in the Rhind Papyrus (16th century BCE), the earliest surviving mathematical account, they struggled to divide four loaves of bread into seven. How did they do it? Instead of writing the division as
4 , they put: 7 1 1 + 2 14
They wrote all the fractions as the sum of different unit fractions! This process made handling fractions a very complicated task, so they had to use long and complex tables. Strangely enough, not only did the Greeks copy this method of dealing with fractions, but the method also reached Europe in the 13th century —3 000 years later— where it was combined with the use of ordinary fractions. 112
3
10
100
1 3
1 10
1 100
Use what you have learnt to solve the problem Let's go back to the Egyptian problem on the previous page. The Egyptian way of presenting the solution was probably due to the way they distributed the numbers: • The loaves are split in half. Each of the seven people takes half a loaf and half a loaf is left over. • The remaining half loaf is divided into seven pieces and each person takes one piece.
1 What fraction of bread does each of these last parts represent?
2 Using a sum of fractions, indicate the two pieces of bread that each of the seven people involved in the division gets.
+
We find it simpler to divide each loaf into seven parts and give each person four of those parts.
3 What fraction of bread does each person get?
4 Using fractions, show the series of equalities you see in the graph and check that —in both ways of dividing out the bread— each person gets the same amount. →
+
=
=
=
1 1 7 1 + = + =… 2 14 14 14
5 Think of some other similar distributions, solve them by the two methods above and compare the results.
113
1
MAGNITUDES AND MEASUREMENTS We use the properties and characteristics of objects to collect and transmit information about them. material: colour:
1.6 m
shape:
It weighs 6 units.
Metallic grey
Cylindrical
weight:
3.2 m
Stainless steel
483 kg
capacity:
6.43 m3
We can measure and quantify some of these properties with numbers. We call them magnitudes. Examples of magnitudes: mass, length, surface area, temperature, voltage, sound intensity, power of an engine…
How do we measure magnitudes? Measuring a magnitude means comparing one amount to another predetermined and fixed amount called a unit of measurement.
OK, but… Which units?
amount to be measured
unit of
the capacity of the jug is less
measurement
than
5 glasses
We can measure magnitudes with different units. The unit we use must be recognised and accepted by everyone if we want the measurement to be significant. It must be conventional and standardised.
Let's practise! 1 True or false?
a) The kilometre is a magnitude.
3 Use the following units to express the weight of the box:
a) A green cube.
b) A red cube.
b) Palms are a unit of length. c) A computer’s memory capacity is a magnitude. d) A tape measure is a unit of measurement. e) Weighing scales are a type of measuring instrument. f ) The decibel is a unit used to measure the intensity of sound. 2
Colour and shape are qualities, not magnitudes. Why? 114
4 Which magnitudes do these units measure?
a) Second
b) Bit
c) Degrees Celsius
d) Gram
e) Volt
f ) Square metre
2
THE METRIC DECIMAL SYSTEM Throughout history, each region, country and cultural group adopted its own different units of measurement. I would understand it better in pounds.
How many arrobas is that?
A quintal.
WHEAT
This made it difficult for different societies to communicate with each other. Therefore, it was necessary to create a system of measurement that all countries recognised and used. At the end of the 18th century (1792), the French Academy of Sciences proposed the metric decimal system.
quadrant
The metric decimal system (MDS) is a set of units of measurement for the most basic magnitudes. It has a structure: • The fundamental units are linked to each other. earth’s meridian = 40 000 km
1L
magnitude
fundamental unit
length
→
metre
→ One ten-millionth of a quadrant of the Earth’s meridian.
capacity
→
litre
→ The capacity of a cube measuring one decimetre each side.
mass
→
gram
→ The mass of one centimetre cubed of water.
1 kg
• Each unit also has a set of multiples and submultiples, all related by powers
of base 10, which use these prefixes: multiples
1 dm
kilo hecto 1 000 U 100 U
submultiples deca 10 U
←
unit 1U
→
deci 0.1 U
centi mili 0.01 U 0.001U
Let's practise! 1
Research. The arroba is an ancient unit of mass that people used in Spain. Unfortunately, its value was different in different regions. a) Find out how many kilos an arroba weighed in Castilla and in Aragón. b) Describe some of the problems these differences caused.
2 Name:
a) The multiples of the metre. b) The multiples of the gram. c) The submultiples of the litre. d) The submultiples of the gram. 3 Remember that one quadrant of the Earth’s meridian is
one quarter of the Earth:
a) How many metres is one quadrant of the meridian? b) How many metres is the whole meridian? 115
3
UNITS OF MEASUREMENT FOR FUNDAMENTAL MAGNITUDES Measures of length As you already know, the metre is the base unit of length in the metric decimal system. Here are its multiples and submultiples of the metre: 10
10
10
10
10
10
km
hm
dam
m
dm
cm
mm
1 000 m
100 m
10 m
1
0.1 m
0.01 m
0.001 m
Ten units of any position make one unit of the following position. This is why we say that units of length go up and down in tens. We must choose an appropriate unit when we measure lengths. For example: — To express the thickness of this book: millimetres (14 mm) or centimetres (1.4 cm), but not metres (0.014 m). — To express the distance from Oviedo to Sevilla: kilometres (665 km), but not centimetres (66 500 000 cm). ➜ very small units of length
Diatom algae under an optical microscope.
Scientific and technological advances allowed us to discover a microscopic world. As a result, we need units smaller than the millimetre. Here are some examples: • The micrometre → 1 µm = 0.001 mm (one thousandth of a millimetre) We use it to measure microorganisms (microbes, bacteria…). • The nanometre → 1 nm = 0.000001 mm (one millionth of a millimetre) • The angstrom → 1 Å = 0.0000001 mm It is used to measure atomic distances. ➜ very large units of length
Sombrero Galaxy in the Virgo constellation.
We use enormous units to measure large lengths, like the distance between stars and planets: • The astronomical unit → 1 AU ≈ 150 million kilometres → It is the approximate distance from the Earth to the Sun. • The light year → 1 light year ≈ 9.5 billones kilometres → It is the distance light travels in one year. It is used to measure the distance between galaxies.
Let's practise!
True or false?
1
2 Which unit would you use to measure these lengths?
a) The distance from the Earth to the Sun is 1 AU.
a) The width of a road.
b) The distance from Mars to the Sun is greater than one light year.
b) The length of a river.
c) We measure the radius of an atom in angstroms.
d) The diameter of a screw.
d) Ten thousand micrometres make one millimetre.
e) The diameter of the solar system.
116
c) The thickness of a wooden board.
UNIT
6
Measures of capacity In the metric decimal system, the litre is the base unit to measure capacity. One litre is the capacity of a cube with a one-decimetre edge. Here are the multiples and submultiples of the litre:
Remember
10 1 dm
1L
1 litre → 1 dm3 1 kL = 1 000 litres → 1 m3 1 mL = 0.001 litres → 1 cm3
10
10
10
10
10
kL
hL
daL
L
dL
cL
mL
1 000 L
100 L
10 L
1
0.1 L
0.01 L
0.001 L
Each unit of capacity in the metric decimal system equals ten units of the position immediately below it. In other words, units of capacity go up and down in tens. Exemples
— A barrel has a capacity of 2.5 hectolitres, or 250 litres. — A can of soft drink has a capacity of 33 centilitres. Traditional units celemín
(Castilla) → 4.625 L
bushel → 12 celemines
Measures of mass In the metric decimal system, the gram is the base unit to measure mass. One gram is the mass of water that fits in a cube with a one-centimetre edge. Since grams are very small, we measure everyday objects in kilograms. The gram has multiples and submultiples that go up and down in tens. 10
Celemín is an ancient unit for measuring capacities.
10
10
10
10
10
kg
hg
dag
g
dg
cg
mg
1 000 g
100 g
10 g
1
0.1 g
0.01 g
0.001 g
We can measure very large masses using two multiples of the kilogram: • A metric quintal (q) → 1 q = 100 kg • A metric ton (t) → 1 t = 1 000 kg Exemples
Mass and weight To help you distinguish between weight and mass, think: An object has the same mass (amount of matter) on the Earth and on the Moon. But on the Moon, it weighs much less. Why?
— A tablet for the flu contains 15 milligrams of active ingredient. — A fish weighs 1.6 kilograms. — The lorry transports 3.4 tons. Note: In everyday life, we often talk about mass and weight as the same thing (1 kg mass = 1 kg weight). However, this is not possible in science because they are different magnitudes. When you see the word “weight”, you should assume it is everyday language. In scientific language, we always say “mass”.
Let's practise! 3 True or false?
a) A 25-litre barrel has a mass of 25 kilos. b) Ten centilitres make one millilitre. c) Ten decagrams make one hectogram. d) One kilo of oil weighs less than one kilo of water. e) One kilo of oil takes up more space than one kilo of water.
4 Which unit would you use to measure the following?
a) The capacity of a shampoo bottle. b) The mass of a bag of oranges. c) The amount of water in a reservoir. d) The annual mussel production in Galicia. e) The amount of saffron added to a paella. 117
CONVERSION OF UNITS
4
We can use a table of multiples and submultiples to convert units of length, capacity and mass. For this type of conversion, we must move the decimal point. Examples
Observe
km
To convert the unit, multiply or divide by the unit followed by zeros.
3.
5
0
two steps
km - hm - dam - m
m - dm - cm
m
dm
cm
mm
→ 3 500 m
0.
27.4 cm →
three steps
(. 1 000)
3.5 km →
hm dam
0.
2
7.
→ 0.274 m
4
3.5 km → 3.5 · 1 000 = 3 500 m
(: 100)
27.4 cm → 27.4 : 100 = 0.274 m
eas
id Consolidating
1 Copy and complete the table with the capacity of the can in the units
given.
Example
A can weighs 375 g.
33 cL
L
dL
cL
mL
0.
3
3
0
kg
hg
dag
g
0
3
7
5
0.375 kg = 3.75 hg = 37.5 dag = 375 g
… L = … dL = … cL = … mL
Let's practise! 1 The kangaroo’s height appears in the table. Express it: m
a) 1.4 g
dm cm mm
1
a) In metres. c) In centimetres.
2
b) In decimetres. d) In millimetres.
3 Express the following amounts in litres:
c) 74.86 hL f ) 3 800 mL
4 Convert to hectometres:
a) 6 km 118
b) 0.54 km
c) 5 dg
d) 62 cg
a) 3 kg = ... g
b) 420 g = ... kg
c) 1.4 hg = ... dag
d) 28.7 dg = ... g
e) 39 dg = ... mg
f ) 470 mg = ... cg
7 Express the elephant’s weight in kilos, grams and tons.
a) 0.2 kg → 0.2 · 1 000 = … g b) 5.3 hg → 5.3 · … = … g c) 3.7 dg → 3.7 : 10 = … g d) 280 cg → 280 : … = … g b) 42.6 dL e) 1.46 daL
b) 0.6 g
6 Copy and complete in your notebook:
7
2 Copy and complete in your notebook:
a) 2.75 kL d) 350 cL
5 Convert to milligrams:
c) 80 dam
d) 28 m
t
q
4
6
kg hg dag g
0
0
0
0
0
Which units are the most suitable to express the elephant’s weight? 8 Copy and complete in your notebook:
a) 4 q = ... kg
b) 280 kg = ... q
c) 3.7 t = ... kg
d) 9 700 kg = ... t
anayaeducacion.es Practise converting units of length, capacity and mass.
5
COMPLEX AND SIMPLE AMOUNTS When a measurement is expressed in more than one unit, we say it is complex. When it is expressed with just one unit, we say it is simple. complex
simple
simple
2 m 5 dm
2.5 m
250 cm
The example below shows how to convert complex amounts into simple amounts: a) Express the capacity of the tank in litres. b) Convert the capacity of the fabric softener bottle into a complex measurement. kL
5 kL 8 hL 7 daL
5 kL 8 hL 7 daL → 0.639 L →
hL daL
5
8
7
L
dL
cL
mL
0 0.
6
3
9
→ 5 870 L → 6 dL 3 cL 9 mL
Operations with complex amounts 0.639 L
We can also use the table of multiples and submultiples of the main unit to make calculations that contain complex amounts. The following activities will show you how.
eas
id Consolidating
1 Complete and solve:
kL
a) A petrol tanker that transports 3 kL 5 hL 2 daL of petrol delivered 9 hL 7 dal 5 L. How many litres remain? b) Each bottle of pills contains 3 g 2 dg 4 cg of active ingredient. How many grams of active ingredient do we need to make 75 bottles?
–
3
hL daL
5
2
hg dag g
3 × +
L
(3 kL 5 hL 2 daL ) – (9 hL 7 daL 5 L ) = … L Answer: There are ... litres of petrol remaining.
0
dg cg
2 7
4 5
3.24 g · 75 = … g Answer: We need ... grams of active ingredient.
Let's practise! 1 Express in metres:
3 Fred buys a chicken that weighs 2 kg 200 g and a rabbit
a) 6 km 4 hm 8 dam c) 5 m 4 dm 7 cm
b) 5 hm 3 m 6 dm d) 3 dam 7 cm 1 mm
2 Express as complex amounts:
a) 3.68 kL
b) 7.42 dL
c) 22.36 hL
d) 365 cL
e) 2 364 L
f ) 2 408 mL
that weighs 0.760 kg. How much do they weigh together?
4 Marta went to the supermarket to buy five two-litre
bottles of olive oil. However, each bottle contained 20 cL extra. How much oil does Marta have in the five bottles? 119
6
MEASURING SURFACE AREAS In the past, measuring surface area was particularly important for measuring farmland. Since land was irregular in shape, people invented strange methods for measuring surface area. Look at the following example: ➜ the sowing method
The unit of surface area was the land that could be sown with one unit of capacity of the seed. Another unit: the yugada The yugada → The amount of land that a pair of oxen can plough in a day.
For example: One fanega of land → The amount of land that one fanega of grain covers.
1 fanega = 12 celemines = 55.5 litres These units of measurement were not very accurate. People interpreted them in different ways and gave them different values. They were not strictly mathematical. However, the units you will study now are.
Exact and invariable units to measure surface area We measure surface area by measuring the space contained within a square (square unit). Then, we calculate the surface area by finding how many square units the shape contains. Exemples
We can use different units to measure surface area: red u2 blue
→ square unit → 1 u2
a
b
u2 Sa = 15 u2
d
Sc = 23 u2
Sd = 22.5 u2
We usually define square units using the corresponding linear units. traditional units
units of the mds
1 foot
1 inch
The yellow shape measures: — 20 red u2 — 5 blue u2
Sb = 7.5 u2
c
↑
↑
square inch 120
↑
square foot
1 centimetre
Remember
square centimetre
UNIT
6
Approximate measurement of an irregular surface area It is easy to measure the surface area of a polygon using square units. However, when the surface is irregular, we use estimation. Look at the following example: Exemples
Sblue = 48.5 u2 Sgreen = 28 u2
The surface area inside the red line is greater than the surface area inside the green polygon, and it is smaller than the surface area inside the blue polygon. We can use this information to estimate that the surface area inside the red line is 38 square units. This value is between 28 and 48.5 units: 28 u2 < Sred < 48.5 u2 → Sred ≈ 38 u2
Let's practise! 1 One bushel of wheat seeds weighs 47 kg.
a) How many kilos of wheat do we need to sow a field that measures 10 bushels? b) How many bushels of land can we sow with 1 000 kg of wheat?
5 How many square inches are there in a square with a
five-inches side? How many square feet are there in a rectangle that is three feet tall and four feet long? 5 inches
2 How long does it take for three pairs of oxen to plough
3 feet
a field with a surface area of 48 yugadas?
3 We know that a tractor can plough the field from the
previous question in two days. How many pairs of oxen do we need to replace the tractor?
4 feet
6 Calculate the surface area of the square, rhombus and
rectangle in square centimetres:
4 Calculate the surface area of the following shapes in
square units:
a
5 cm
b c
d
7 Calculate the surface area of
the blue polygon and the green polygon. Then, estimate the surface area of the circle.
1 u2
121
6
MEASURING SURFACE AREAS
Surface area units in the metric decimal system The square metre is the main unit for measuring surface area. It also has multiples and submultiples.
Agricultural units We use them to measure fields (agro = field).
100
100
100
km2 hm2 dam2 1 000 000 m2 10 000 m2 100 m2
• Hectare (ha)
1 ha = 10 000 m2 = 1 hm2
ha
• Are (a)
a
100
m2 1
100
100
dm2 cm2 mm2 0.01m2 0.0001 m2 0.000001 m2
ca
The diagram below shows the relationship between these units. It shows how a square metre can be broken down into square decimetres:
1 a = 100 m2 = 1 dam2 • Centiare (ca)
1 dm 1 dm
1 ca = 1 m2
1m
• One square metre is divided into 10 rows. Each row is 10 square decimetres. Therefore: 1 m2 = 10 × 10 dm2 = 100 dm2
1m
• It happens the same with every unit in relation to the following unit. Surface area units increase and decrease in hundreds.
Conversion of units The surface area of Tenerife is 2 034 km2 = 203 400 ha.
We use a table to convert surface areas from one unit to another. However, we must remember that each time we move up or down a unit, we must move the decimal point two places. In other words, each conversion is the same as multiplying or dividing by 100. Practise doing this in the activity below.
eas
id Consolidating
1 Copy and complete. First, convert the units shown directly in the table. Then,
multiply or divide by the unit followed by zeros. km2 hm2 dam2 m2 ha a ca
6.2 dm2 → 2 47 200 m → 252 800 m2 →
dm2 cm2 mm2
6 2 0 0 0 → … cm2 → 62 000 mm2 4 7 2 0 0 0 2 5 2 8 0 0
6.2 dm2 . 100 … cm2 . 100 … mm2 47 200 m2 : 100 … dam2 : 100 … hm2 252 800 m2 : 10 000 … ha : 100 … km2
122
→ 472 dam2 → … hm2 → … ha → … km2
UNIT
6
Operations with complex amounts The table we use to convert units can also help us with operations that contain complex surface area amounts. Complete the activities below to practise this method. eas
id Consolidating
2 Complete and solve in your notebook:
a) We cover the floor of a stadium in artificial grass. It has a surface area of 1.02 ha. Inside, there is a football field with a surface area of 73 dam2 53 m2 50 dm2. What is the surface area of grass outside the field? hm2 dam2 m2
1.02 ha – 7 353.5 m2 = … m2
dm2
1 0 2 0 0 0 0 –
Answer: The surface area outside the football field is … m2.
b) A sports centre renovates its six 200-m2 tennis courts by replacing the floor with artificial grass. They buy rolls of grass measuring 10 m2 75 dm2. Are 125 rolls enough to complete the renovation? hm2 dam2 m2
Surface covered by 125 rolls:
dm2
(10 m2 75 dm2) · 125 = … m2
× 1 2 5
Surface area of the 6 tennis courts: … m2 Answer: …
+
Let's practise! 8 Give the most suitable unit to express:
12 Express in square metres:
a) 5 km2 48 hm2 25 dam2 b) 6 dam2 58 m2 46 dm2 c) 5 m2 4 dm2 7 cm2
a) The surface area of Portugal. b) The surface area of a reservoir. c) The surface area of a house. d) The surface area of a piece of paper.
13 Write as complex amounts:
a) 587.24 hm2 b) 587 209.5 m2 c) 7 042.674 dm2
9 Express in square metres:
a) 0.006 km2 d) 70 dm2
b) 5.2 hm2 e) 12 800 cm2
c) 38 dam2 f ) 8 530 000 mm2
10 Express in square centimetres:
a) 0.06 dam2 11
b) 5.2 m2
c) 0.47 dm2
d) 8 mm2
Copy and complete in your notebook: b) 825 hm2 = ... km2 a) 5.1 km2 = ... hm2 c) 0.03 hm2 = ... m2 d) 53 000 m2 = ... dam2
14 Calculate:
a) (6 dam2 52 m2 27 cm2) – 142.384 m2 b) 5 246.9 cm2 + (18 dm2 13 cm2 27 mm2) c) (15 hm2 14 dam2 25 m2) · 4 15 A 17.56 hm2 farm has 13.45 ha of dry land planted
with cereal and 11 850 m2 of irrigated land. The rest is fallow land. How much of the surface area of the farm is fallow land?
anayaeducacion.es Practise converting surface area units.
123
folio. for your port from this unit se resources oo ch to r be Remem
S M E L B O R P D N EXERCISE A Magnitudes and units 1
5
True or false? a) The radius of the Moon is measured in astronomical units. b) The radius of a cell is expressed in micrometres. c) We measure the amount of air in a room in square metres.
6
Convert to grams: a) 1.37 kg
b) 0.7 kg
c) 0.57 hg
d) 1.8 dag
e) 0.63 dag
f ) 5 dg
g) 18.9 dg
h) 480 cg
i) 2 500 mg
Express the weight of the loaf of bread in the picture in kilograms and then, in milligrams.
d) We express the mass of a train in tons. e) The amount of petrol a lorry carries can be expressed in litres and kilos. note: When the answer is false, write the correct sentences. 2
320 g
7
a) 15 000 kg 8
d) 1 kg
Copy and complete in your notebook:
c) 7 hg = ... dag = ... g = ... dg d) 42 g = ... dag = ... cg = ... mg
c) A spoonful of syrup. d) The amount of petrol in an oil tanker.
9
Express in centilitres:
e) The weight of a cat.
a) 0.15 hL
b) 0.86 daL
c) 0.7 L
f ) The amount of corn harvested from a farm.
d) 1.3 L
e) 26 dL
f ) 580 mL
g) The fabric of a tent.
10
h) The surface area of a field. 27 m
6.8 m2
6.7 t
8 mL
95 hL
80 cm
3.4 ha
2 500 g
Conversion of units Use the example to help you complete these sentences:
Express the capacity of the bottle in decilitres. Express the capacity of the glass as a fraction of a litre.
3 L 4 11
25 cL
Copy and complete in your notebook: a) 4.52 kL = ... hL
b) 0.57 hL = ... daL
• To convert from kilometres to metres, we multiply by one thousand.
c) 15 daL = ... L
d) 0.6 L = ... cL
e) 850 mL = ... dL
f ) 1 200 cL = ... L
a) To convert from decalitres to decilitres… .
g) 2 000 mL = ... dL
h) 380 daL = ... kL
b) To convert from milligrams to grams… .
12
c) To convert from decametres to hectometres…
b) 8 dam 5 m 7 cm
Copy and complete in your notebook: b) 2 380 m = ... km = ... hm = ... cm
Express in metres: a) 3 km 8 hm 5 dam c) 1 m 4 dm 6 cm 7 mm
a) 2.7 hm = ... km = ... dam = ... dm
124
c) 400 kg
b) 0.005 kg = ... g = ... mg = ... dag
b) The height of a building.
4
b) 8 200 kg
a) 5.4 t = ... kg = ... hg = ... dag
Link each statement to the correct measurement: a) A stride.
3
Express these masses in tons:
13
Express in grams:
c) 47 m = ... dam = ... dm = ... hm
a) 4 kg 5 hg 2 dag 3 g
b) 9 hg 8 dag 5 g 4 dg
d) 382 cm = ... m = ... dm = ... mm
c) 6 dag 8 g 6 dg 8 cg
d) 7 dg 6 mg
UNIT
14 15
Write the following as complex amounts: a) 4.225 kg b) 38.7 g c) 1 230 cg d) 4 623 mg
b
34.2 dL
3.24 L
16
Surface area units 22
Express the contents of each container as a complex amount: a
c
23 18 cL
Convert to litres: a) 8 kL 6 hL 3 L c) 1 daL 9 L 6 dL 3 cL
24
b) 5 hL 2 daL 7 L 2 dL d) 4 L 2 dL 5 cL 7 mL
Operations with complex amounts 17
Calculate the total length of this track in metres:
25
2 km 700 m
3 842 m
26
25 hm 7 dam 8 m
18
19
Calculate and express your result in the units stated: a) 27.46 dam + 436.9 dm → m b) 0.83 hm + 9.4 dam + 3 500 cm → m c) 0.092 km + 3.06 dam + 300 mm → cm d) 0.000624 km – 0.38 m → cm
27
28
How much does the box of biscuits weigh?
20
21
?
Calculate and express as complex amounts: a) 57.28 g + 462 cg b) 0.147 t – 83.28 kg c) 0.472 kg · 15 d) 324.83 hg : 11 Calculate and express your result in litres: a) 0.05 kL + 1.2 hL + 4.7 daL b) 42 dL + 320 cL + 2 600 mL c) 7.8 daL – 52.4 L
Consider, show and explain the difference between half a square metre and the surface area of a square with a side that is half a metre long. Copy and complete in your notebook: a) 1 km2 = ... m2 c) 1 hm2 = ... m2
b) 1 m2 = ... dm2 d) 1 m2 = ... cm2
e) 1 dam2 = ... m2
f ) 1 m2 = ... mm2
Copy and complete in your notebook: a) 4 km2 = ... dam2 c) 0.005 dam2 = ... dm2
b) 54.7 hm2 = ... m2 d) 0.7 dm2 = ... mm2
e) 5 400 m2 = ... hm2
f ) 174 cm2 = ... dm2
Convert these amounts to square decimetres: a) 0.146 dam2
b) 1.4 m2
c) 0.36 m2
d) 1 800 cm2
e) 544 cm2
f ) 65 000 mm2
Express in hectares: a) 572 800 a
b) 50 700 m2
c) 25.87 hm2
d) 6.42 km2
Express as complex amounts: a) 248 750 dam2
b) 67 425 m2
c) 83 545 cm2
d) 2 745 600 mm2
Look at this farm. Calculate the total surface area. hm22 55 dam dam22 9 m 22hm m22
220 g 0.53 kg
6
3,25 3.25 ha ha
29
Calculate and express in square metres: a) 0.000375 km2 + 2 500 dm2 b) 0.045 hm2 – 29.5 m2 c) 520 mm2 · 1 500 d) 6.96 hm2 : 24
30
Calculate and express as complex amounts: a) 725.93 m2 – 0.985 dam2 b) 0.03592 km2 + 27.14 ha + 3 000 a c) 467 108.25 dam2 : 30 d) (15 hm2 16 dam2 38 m2 ) · 30 125
folio. for your port from this unit se resources oo ch to r be Remem
S M E L B O R P D N EXERCISE A Problem solving 31
35
Problem solved
A liquid medicine comes in 3-centilitre bottles. Each dose of the medicine is a 0.3-millilitre drop. How many doses are in each bottle?
Explain the solutions and write the unit after each result.
A local council divides its 1.8 hectare farm into smaller 7.5 dam2 sections of land. They sell the sections as family gardens. They cost €3 000 each. How much money does the council make from the sales?
• Surface area of the farm: 1.8 ha = 1.8 hm2 = 18 000 m2 • Surface area of one section: 7.5 dam2 = 750 m2 • Number of sections: 18 000 : 750 = 24 sections • Profit from selling the sections: 24 · 3 000 = €72 000 32
Copy and complete by writing the units: a) Each pill contains 20 mg of active ingredient. How much active ingredient do we need to make 100 000 pills? • 20 · 100 000 = 2 000 000 … • 2 000 000 … = 2 000 … = 2 … b) On average, an athlete moves forwards 65 cm with each step. How many steps does he take to complete a 10 km race? • 10 km = 10 000 … • 65 cm = 0.65 … • N.o of steps: 10 000 : 0.65 = 15 384.61 … . He takes approximately 15 385 steps.
33
How many 20-cL glasses can you fill with a bottle that contains two and a half litres of water?
34
A can of olive oil filled ten 75-cL bottles of oil. How many litres of oil were removed from the can?
126
3 cL
36
A farmer has a field of 1.4 ha. He planted 15 plots of beetroot. Each plot has a surface area of 2 dam2. How many square metres does he have left for other crops?
37
We filled a jug with 15 cans of oil. Each can contained 4 L 6 dL 4 c. How many litres are in the jug?
38
A spoonful of rice weighs 22 dg. Each spoonful contains 66 grains of rice. How many grains are there in one kilo of rice?
39
We need 200 g of red paint to paint one square metre of wood. We have a two-kilo can of paint. Is it enough to paint a wooden cube with sides that measure one metre?
40
We need 140 wood boards that measure 80 cm × 20 cm each to cover the floor of the living room. a) How many square centimetres does each board cover? b) What is the surface area of the room?
41
Problem solved
We know that one litre of water weighs 1 kg. If a lorry is carrying 240 hectolitres of water, how much does its load weigh in tons?
• 1 hectolitre (100 litres) weighs 100 kg. • 240 hectolitres weigh 240 · 100 = 24 000 kg. • 24 000 kg are 24 000 : 1 000 = 24 tons. 42
How much weight can carry a water tank with a capacity of 25 hL 5 daL?
UNIT
43
A water tanker weighs 17.7 tons when it is empty. It weighs 25.2 tons when it is full. How many litres of water does it carry when it is full?
44
How much water is there in the bottle on the right of the scale? 46 cL
25 g
2 L 5 dL
45
46
51
A food company mixes 1.3 tons of roasted coffee with 800 kilos of natural coffee. Then, it packs the resulting mixture into 200-gram bags. If each box contains ten bags and it sells each box for €35, how much profit does it make by selling the entire batch?
52
A hiking club organised an orientation route. It marked a 40.56 ha rectangle of land on the map. Then, the club divided it into squares using a coordinates system, as shown in this diagram:
?L
A town receives its water supply from a reservoir. The reservoir contains 1.4 million cubic metres of water. The town uses 5.6 million litres of water a day. How many days does the water from the reservoir last? note: One cubic metre is equal to 1 000 litres. How many square units does the yellow shape contain?
a) How long is the side of each square? b) Which is the surface area of the marked land? ‘+’ problems 53
47
6
Calculate the surface area of these shapes in square centimetres:
A building has a rectangular garden. In it, there is a central area with plants that measures 42 m × 24 m and a path around it. The path is two metres wide and it is paved with stones that are 25 cm long. How many stones were needed to pave the path? 2m 42 m 24 m
1 cm 2
28 m
b
a
46 m
48
Imagine that you cut the path and put all four sides in a straight line:
How many square centimetres does this shape contain? 1 cm2
49
One litre of oil weighs 918 grams. Does one ton of oil fit in a 10-hectolitre container?
50
The diameter of one grain of pollen is 25 micrometres. How many grains of pollen do you need to make a one metre line?
46 m
24 m
46 m 140 m
54
24 m
2m
A gardener wants to fertilise a grass field. The fertiliser he uses is concentrated and he must dilute 10 mL of fertiliser per litre of water. a) If each bottle contains 2 litres of the solution, how many litres of water does he need to dilute each bottle? Once he has diluted it, 5 litres the solution cover 100 m2 of grass. b) How many bottles does he need to fertilise a field that measures one hectare? 127
P O H S K R O W S H T A M READ AND LEARN Stories and measurements ➜ There are many stories about the seven-league boots. These boots were magical and made it possible to walk very long distances. Remember what you learned about traditional measurements. How many kilometres are seven leagues?
Measurements and sayings ➜ You might know the expression ‘meterse en camisa de once varas’. • How many metres are eleven yards? • What does the saying mean?
➜ Do you know the book Twenty Thousand Leagues Under
the Sea by Jules Verne? How long was that journey in kilometres?
➜ Another common expression is ‘an ounce of common
sense is worth a pound of theory’. Do some research and explain what it means.
INVESTIGATE Calculating distances by measuring time ➜ If you count the number of seconds that pass between
hearing thunder and seeing the lightning, you can find out how far away the lightning struck. Light travels so quickly (300 000 km/s) that we see lightning almost at the exact moment it is formed. However, it takes us longer to hear thunder, because sound travels more slowly (331 m/s). If thunder takes 15 s, this means it travelled 15 · 331 = 4 965 m. In other words, the storm is approximately 5 km away.
128
➜ Submarines use a similar but more sophisticated
technique called sonar. It makes it possible to avoid underwater terrain and to find other vessels. Sonars send ultrasound waves through the water. When the waves hit obstacles, they are reflected back to the submarine. The time it takes the waves to return tells the sonar how far away things are. This is shown on a screen.
UNIT
6
PRACTICE MAKES PERFECT! Think and organise • A road maintenance company installed numbered signs to mark kilometres and striped posts to mark hectometres.
• A supermarket sells one-litre, two-litre and five-litre bottles of water. You want to buy 8 litres. How many different ways can you do this?
1L
How many signs and posts did it need to cover the 10 to 20-kilometres stretch (including the 10 and 20-kilometre points)?
2L
• We are near a water spring and have two bottles: a 3-litre bottle and a 5-litre bottle. How can you measure exactly 4 litres?
SELF-ASSESSMENT
anayaeducacion.es Answer key.
1 Explain why the invention of the metric decimal
system was necessary.
7 Convert to simple amounts:
a) 2 km2 15 hm2 23 dam2 = … m2 b) 35 m2 12 dm2 9 cm2 = … dm2
2 Give the appropriate unit for measuring these
magnitudes: a) The width of a football field. b) The thickness of a sheet of paper. c) The capacity of a perfume bottle. d) The weight of a lorry’s load.
8 Calculate:
a) (3 hm 5 dam 6 m) + (2 dam 5 m 8 dm) b) (3 L 4 dL 5 cL ) – (8 dL 5 cL 3 mL ) 9 Calculate:
a) (3 km 8 hm 5 m) · 4 b) (5 m2 14 dm2 25 cm2) · 8
3 Copy and complete in your notebook:
a) 5.2 km = … hm c) 0.07 m = … cm
b) 18 hm = … m d) 345 mm = … cm
10 A lorry transports 8 pallets of coffee. Each pallet
contains 60 boxes and each box contains 75 coffee bags. Each bag contains 250 grams of coffee. How many tons of coffee does the lorry transport?
4 Express as complex amounts:
a) 2 537 m c) 0.856 kg
b) 35.42 daL d) 2 348 mm
5 Express as simple amounts:
a) 3 hm 8 dam 4 m 5 dm b) 5 L 6 dL 7 cL c) 5 kg 7 dag 8 g 6 Copy and complete in your notebook:
a) 5 hm2 = … ha b) 3.5 hm2 = … m2 c) 3 450 mm2 = … cm2
Commitment
5L
11
A broken tap loses one drop of water per second. We estimate that the volume of each drop equals 0.05 mL. How much water does the tap lose in one day?
12 We want to cover a
cube-shaped package with cloth. The edge of the package measures half a metre.
0.5 m
We need 50 % extra cloth for stitches and overlapping fabric. How much cloth do we need?
Watch the video for target 12.4. Think of something you can do to contribute to achieve that goal. Make a commitment to put your idea into practice.
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