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10

Number

Ratios and rates

Ratios and rates are used in many everyday situations to compare things. Ratios are used when mixing fertiliser, mixing cordial, using maps or reading house plans. Rates are used when calculating the price of petrol or mobile phone bills, calculating the cost of placing a classified advertisement and when calculating speed.

In this chapter you will: ■ ■ ■ ■ ■ ■ ■ ■ ■

write and simplify ratios in various forms solve problems involving ratios divide a quantity in a given ratio apply the unitary method to ratio problems interpret scales on maps and plans calculate rates from given information solve problems involving rates calculate speed given distance and time draw and interpret travel graphs, recognising concepts such as change of speed and change of direction convert rates from one set of units to another.

Wordbank ■ ■ ■ ■ ■ ■ ■

ratio An arrangement of numbers that compares two or more quantities of the same type. unitary method A method of finding a part of a quantity when given another part. scale The relationship between a scaled drawing, map or plan and the actual object it represents. actual Real. scaled As shown on a drawing, map or plan, usually much smaller but in proportion to what it represents. rate A measurement comparing quantities of different types, expressed using two units. speed A rate that compares distance travelled with time taken.

Think! Three friends combined to finance their new company. Jo invested $8000, Cathy invested $5000 and Rania invested $7000. After two years, their company made a profit of $30 000. How should the profit be shared among the three friends?

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Start up Worksheet 10-01 Brainstarters 10 Skillsheet 10-01 Simplifying fractions Skillsheet 1-01 Factors and divisibility

Skillsheet 4-01 Multiplying by 10, 100, 1000

1 Copy and complete: a 3m= c 2000 kg = e 360 cm = g 300 s = i 480 min = k 8.15 km = m 16 kg =

cm

b d f h j l n

t m min h m g

2h= 1.5 L = 18 g = 6.5 cm = 7500 mL = 850 mm = 45 min =

min mL mg mm L cm h

2 Find the highest common factor of each of the following pairs of numbers: a 12 and 18 b 48 and 36 c 20 and 64 d 50 and 25 e 35 and 21 f 16 and 40 3 Find the lowest common multiple of each of the following pairs of numbers: a 2 and 3 b 4 and 8 c 5 and 3 d 6 and 4 e 8 and 10 f 12 and 9 4 Calculate: a

3 --5

× 10

e 21 ×

2 --7

× 15

b

2 --3

f

8×

5 Calculate: a 0.2 × 10 d 4.05 × 100 g 2.125 × 1000

1 --2

b 1.7 × 10 e 0.4 × 10 h 6.21 × 1000

c

3 --4

× 12

d 6×

g

5 --6

× 12

h

1 --4

1 --3

× 20

c 0.95 × 100 f 1.9 × 100 i 4.7 × 100

Introducing ratios A ratio compares quantities that are of the same type, measured in the same units.

Mixing ingredients Ratio 1 : 1 : 1.25

Reading maps Scale ratio 1 : 20 000

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Reading house plans Scale ratio 1 : 100

We use a ratio when making a drink by mixing cordial and water. Nina likes her drink to have a strong flavour, Jerry prefers a weaker flavour, so he so she uses 1 part of cordial to 3 parts of water. uses 1 part of cordial to 5 parts of water.

1:5 1 part cordial to 5 parts water

1:3 1 part cordial to 3 parts water

Ratio of cordial to water in Nina’s drink is 1 : 3.

Ratio of cordial to water is 1 : 5.

A ratio can be used to compare the parts of a thing, regardless of the total amount. Nina makes enough drink for a party For a picnic, Nina makes the same by using 1 bucket of cordial to 3 buckets concentration (strength of flavour) of cordial of water. by using 1 cup of cordial to 3 cups of water. Water Cordial Cordial

Water

Ratio 1 : 3

The ratio of cordial to water in the drink for the party and the drink for the picnic is the same, even though the amounts are different.

Ratio 1 : 3

Ratios can be written in different ways: a cordial to water as 1 to 3 (in words) b cordial : water as 1 : 3 (using a colon ‘:’) cordial 1 c ---------------- as --- (as a fraction) water 3 The order in which we write a ratio is very important. Writing the ratio of cordial to water as 3 : 1 would mean 3 parts of cordial to 1 part of water. This would taste very strong! Of course, we should say that the ratio of cordial to water is 1 : 3 (that is 1 part of cordial to 3 parts of water). A ratio compares quantities of the same type given in the same units.

Note:

• a ratio has no units • a ratio does not tell us ‘how much’ there is.

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Example 1 For the box of chocolates shown below, find these ratios: a light chocolates to dark chocolates b round chocolates to square chocolates c chocolates in white paper to chocolates in brown paper

Solution a There are 5 light chocolates and 7 dark chocolates. So the ratio of light chocolates to dark chocolates is 5 : 7. b There are 7 round chocolates and 4 square chocolates. So the ratio of round chocolates to square chocolates is 7 : 4. c There are 5 chocolates in white paper and 7 chocolates in brown paper. So the ratio of chocolates in white paper to chocolates in brown paper is 5 : 7.

Example 2 Express each of the following quantities as a ratio. a a mass of 30 kg to a mass of 80 kg b a distance of 37 cm to a distance of 1 metre

Solution a A mass of 30 kg to a mass of 80 kg. Since the quantities are measured in the same units (kg) the ratio is simply 30 : 80, or 3 : 8. b A distance of 37 cm to a distance of 1 metre. The quantities are measured in different units so we must change all measurements to the same units before calculating the ratio. Because we usually use the smaller unit, we must change 1 metre to 100 centimetres. The ratio becomes 37 to 100, or 37 : 100.

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Exercise 10-01 Example 1

1 Find the ratio of shaded to unshaded parts for each of the following diagrams: a b c

d

e

f

2 For each diagram in Question 1, find the ratio of unshaded parts to the total number of parts. 3 a George likes to make a drink with a ratio of 1 part cordial to 4 parts water (or 1 : 4). Draw a diagram to show this ratio in a glass. b His father prefers to drink a weaker flavour with a ratio of 1 : 5. Show this on a diagram. c One day George’s father gave him a drink mixed with the ratio shown in the diagram. Estimate the ratio of cordial to water used. 4 Over the last 50 years the ratio of teachers to students in classrooms has improved. In the 1950s the ratio was sometimes as much as 1 : 45, in the 1970s the ratio was often 1 : 35 and, in the 1980s, the ratio was set at 1 : 30 for most classes. Now, in some classes, it is 1 : 24 or even 1 : 20. What does it mean to say that this ratio has improved? For whom has it improved? 5 In your classroom, find the ratio of: a teachers to students c chairs to tables

b females to males d students to chairs

6 In a classroom of 25 students, Mary found that 12 students had green eyes, 8 had blue eyes, 4 had brown eyes, and 1 had hazel eyes. Find the following ratios of student eye colours: a hazel to green b brown to hazel c blue to brown d brown to green R AT I OS AND R AT E S

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7 Out of every 100 people at the football, 60 were men, 25 were women, and the rest were children. Find the following ratios: a men to women b women to the total c children to men d women to children 8 Three classes were surveyed to find the ratio of students who wrote left-handed to those who wrote right-handed. In class A the ratio was 1 : 5, in class B the ratio was 2 : 3, and in class C the ratio was 1 : 6. If each class had between 21 and 26 students, which class had the most left-handed students? Example 2

9 Express each of these pairs of quantities as a ratio: a 13 g to 25 g b $17 to $100 c 5 litres to 12 litres d 150 km to 1 km e 16 goals to 3 goals f 100 m to 1000 m g 20 players to 15 players h 48 cars to 36 cars i 51 kg to 12 kg j 120 points to 180 points 10 Change the quantities to the same units first, then express each pair of quantities as a ratio: a 3 cents to $2.50 b 25 mm to 3 cm c 60 minutes to 3 hours d 10 years to 36 months e 14 days to 5 weeks f $3 to 500 cents g 50 cm to 2 m h 600 mL to 2 L i 0.5 km to 150 m j 2.5 kg to 750 g 11 Change the quantities in each of the following statements into a ratio. a For each teacher there were 25 students. b To cook steamed rice you need 1 cup of rice to 2 cups of boiling water. c Adventure cycling activities with schools require one teacher for every 4 students. d There are 11 girls for every 9 boys in this school. e No matter how much they won on Lotto, they agreed to share the prize equally. f She was twice as tall as he was.

Equivalent ratios Equivalent ratios are just equal ratios. For instance, we know that 2 : 4 = 1 : 2. But there are many others, such as 3 : 6 = 1 : 2, and 4 : 8 = 1 : 2. Notice that finding equivalent ratios is similar to finding equivalent fractions. A good example for illustrating equivalent ratios is the number of students in a class. Suppose there must be a ratio of 1 teacher to 25 students in each class. From this simple ratio we can make many equivalent ratios: • 1 teacher : 25 students, or 1 : 25 • 2 teachers : 50 students, or 2 : 50 • 3 teachers : 75 students, or 3 : 75 • 4 teachers : 100 students, or 4 : 100, etc. Using this ratio, how many teachers should there be if there are 200 students in Year 8?

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Example 3 1 Find the missing values to make each ratio equivalent to 2 : 3. 2:3 __ : 6 6 : __ __ : 12 10 : __ __ : 9

Solution Work out the number that the original ratio is being multiplied by each time and multiply both parts of the ratio by the same number.

×4

6 : ×5

: 12

10 :

4:6

⇒

× 30

×3

: 6

2:3 ×2

2 : 3

6:9 8 : 12 10 : 15

: 90

60 : 90

2 Find the missing values to make each of the following ratios equivalent to 40 : 30. 40 : 30 __ : 15 8 : __ __ : 3

Solution Work out the number that the original ratio is being divided by each time and divide both parts of the ratio by the same number. 40 : 30

÷2

40 : 30

8 :

÷ 10

÷5

: 15

: 3

⇒

20 : 15 8:6 4:3

Exercise 10-02 1 Copy and complete the following sets of equivalent ratios: a 1:3 b 5:2 c 4:7 2:6 10 : 4 8 : 14 3: 15 : 12 : : 15 : 10 : 49 10 : 40 : 40 : : 45 : 24 : 700

Example 3

d

3:5 : 10 12 : 21 : : 60 : 100

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Spreadsheet 10-01 Equivalent ratios

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e 24 : 12 12 : :4 6: :2 2:

f 48 : 64 : 32 12 : :8 3:

g 90 : 60 45 : : 20 18 : : 10 9:

h 28 : 42 : 21 4: :3

2 Write three equivalent ratios for each of the following. a 2:5 b 4:9 c 6:5 e 5:1 f 1:4 g 5:2 i 32 : 48 j 54 : 18 k 12 : 42

d 12 : 7 h 7:2 l 24 : 8

3 Copy and complete the following pairs of equivalent ratios. a 2:3 = 8: b 1:5 = 2: c d 4:7 = : 35 e 5 : 8 = 20 : f g 5 : 11 = : 66 h 3:4 = : 100 i j : 9 = 20 : 36 k 12 : = 3:1 l m : 45 = 6 : 9 n 24 : 12 = 4 : o p : 20 = 15 : 60 q 24 : 20 = 6 : r s 2:3:4 = 6: v

=

7 --8

:

21 ------

t

3:4:5 =

w

7 -----20

=

--------100

: 20 :

3:5 = : 15 7 : 12 = 49 : 2 : 1 = 10 : 17 : 34 = :2 16 : = 2:5 50 : 40 = : 20

u

2 --3

=

-----9

x

5 --3

=

25 ------

Simplifying ratios Skillsheet 10-01 Simplifying fractions

Ratios are usually simplified by dividing by a common factor. To find the simplest ratio, divide by the highest common factor (HCF). We can simplify 12 : 8 by dividing both parts of the ratio by 2 (because 2 divides evenly into 12 and 8). 12 : 8 =

12 -----2

: 8--2- = 6 : 4

However, this is not the simplest ratio. We can simplify again by dividing both parts by 2 again. 6:4 =

6 --2

: 4--2- = 3 : 2.

So 12 : 8 simplifies to 3 : 2. We can simplify 12 : 8 in one step if we divide both parts by 4, which is the highest common factor of 12 and 8. 12 : 8 =

12 -----4

: 8--4- = 3 : 2

When asked to simplify a ratio, we find the simplest possible form.

Example 4 Simplify each of the following: a 5 : 10 b 24 : 16

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c 12 : 20 : 16

Solution a 5 : 10 =

5 --5

-----: 10 5

Highest common factor is 5

= 1:2 On the calculator: b 24 : 16 =

24 -----8

5

-----: 16 8

12 -----4

=

Highest common factor is 8

= 3:2 On the calculator: c 12 : 20 : 16 =

10

a b/c

24

16 ------ : -----: 20 4 4

a b/c

16

=

d/c

(

SHIFT

a b/c 2nd F

)

Highest common factor is 4

= 3:5: 4 The calculator cannot be used here.

Example 5 Simplify the ratio of 2 hours to 1 day.

Solution First, change the values to the same units: 2 hours to 1 day = 2 hours to 24 hours = 2 : 24 -----= 2--2- : 24 2 = 1 : 12

Exercise 10-03 Example 4

1 Simplify each of the following ratios. a 10 : 100 b 12 : 24 e 18 : 12 f 56 : 24 i 87 : 87 j 123 : 321 m 38 : 14 n 120 : 65

c g k o

2 Simplify each of the following ratios. a 8 : 12 : 20 b 15 : 20 : 30 e 12 : 18 : 24 f 120 : 72 : 48

c 27 : 9 : 36 g 32 : 48 : 36

12 : 30 1000 : 100 51 : 17 16 : 56

d h l p

35 : 49 45 : 99 3 : 48 42 : 105

SkillBuilder 5-01 Ratio and variation

d 4 : 8 : 24 : 12 h 14 : 35 : 21 : 49

3 Change the quantities to the same units first, then express each pair of quantities as a ratio in simplest form: a 50 cm to 2 metres b 300 g to 1.2 kg c 5 days to 7 weeks d 30 min to 2 hours e 70 cents to $2.10 f 2 years to 6 months g 15 hours to 2 days h 20 mm to 1 metre i 4 tonnes to 350 kg j 25 min to 3 hours k 18 m to 1 km l 8 months to 4 years m 2 days to 8 hours n 75 cents to $5 R AT I OS AND R AT E S

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Spreadsheet 10-02 Simplifying ratios Example 5

CHAPTER 10

4 Mary was asked to find the ratio of white lollies to red lollies in a packet. If there were 8 white lollies and 10 red lollies, express this as a ratio in simplest form. 5 A common way of comparing countries is to use statistics with the same base number. For instance, when comparing the rate of infant mortality (the number of deaths of infants) the base number is 1000. • In the year 2000 the rate of infant mortality in Australia was 6 per 1000 live births (a ratio of 6 : 1000). • In Pakistan the rate was 74 per 1000 live births, or 74 : 1000. Express each of these ratios in simplest form. 6 At a book sale, the following books were sold in a morning: • 48 crime novels • 20 science fiction stories • 35 non-fiction books • 28 romance novels Express each of the following as a ratio in simplest form: a crime to romance b sci-fi to non-fiction c non-fiction to romance d sci-fi to crime e sci-fi to romance to non-fiction f crime to all the rest

Ratios with fractions and decimals Sometimes the quantities in ratios involve fractions or decimals. These ratios can also be simplified by making them into ratios involving whole numbers.

Example 6 Simplify

3 --5

: 1--3- .

Solution Multiply both parts of the ratio by a common multiple, preferably the lowest common multiple (LCM) of the denominators. LCM of 5 and 3 is 15. 3 --5

: 1--3- = =

3 1 --- × 15 : --5 3 45 15 ------ : -----5 3

× 15

= 9:5

Example 7 Simplify 0.7 : 0.05.

Solution Multiply by the appropriate power of ten. In this case, we multiply by 100 (we need to move the decimal point two places). 0.7 : 0.05 = 0.7 × 100 : 0.05 × 100 = 70 : 5 = 14 : 1

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NEW CENTURY MATHS 8

Exercise 10-04 1 Simplify each of these ratios:

Example 6

a

1 --3

: 2--5-

b

1 --4

: 1--3-

c

3 --4

: 2--3-

d

1 --2

: 3--8-

e

2 --5

3 : ----10

f

4 --5

: 1--2-

g

5 --8

: 1--4-

h

2 --3

: 1--2-

i

3 --4

7 : ----16

j

4 --5

: 1--2-

k

5 --6

: 2--5-

l

6 --5

: 2--3-

m

1 --2

: 1--3- : 1--4-

n

2 --5

7 : 3--4- : ----10

o

1 --6

: 3--5- : 2--3-

2 Simplify each of these ratios: a 2 1--2- : 4

b 20 : 1--2-

d 1 1--2- : 3--4-

e

g 1 1--2- : 2 1--3-

h 2 2--3- : 1 1--4-

3 Simplify each of these ratios: a 0.4 : 0.7 b d 0.9 : 1.8 e g 0.05 : 0.2 h j 0.375 : 0.25 k m 12 : 8.4 n p 1 : 0.2 : 0.03 q

5 --8

: 1 3--4-

c 6 : 1 1--2f

1 2--3- : 2 1--6-

i

1 1--3- : 4--5Example 7

1.3 : 0.8 0.6 : 0.8 0.25 : 0.5 0.005 : 0.5 6.3 : 7 2.4 : 1.2 : 3.6

c f i l o r

0.5 : 0.3 3.6 : 2.4 3.2 : 0.16 1.08 : 8.1 15 : 1.05 4.5 : 1 : 0.9

Applying ratios There are many applications of ratio in the modern world. Worksheet 10-02 Ratio recipes

Exercise 10-05 1 Designing rooms. Here is a bedroom design. The ratio of the design to the real bedroom is 1 : 50. This means that 1 cm on the design equals 50 cm in the real room. a By measuring the diagram, find: i the width of the real room (in centimetres). ii the length of the real room (in centimetres).

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b Calculate: i the perimeter of the real room, in metres. ii the area of the real room, in square metres. c By measurement find: i the length of the real bed iii the width of the door to the real bedroom

ii the width of the real desk iv the length of the real window

2 Making cordial. To mix a cordial drink 375 mL of cordial concentrate is topped up with water to make 2 L of ready-to-drink cordial. Find: a the amount of water used to make the cordial b the ratio of cordial to water, in its simplest form. 3 Enlarging letters. On dot paper or grid paper, draw these letters of the alphabet and then enlarge each letter using the given ratio. Ratio 1 : 3 Ratio 1 : 4

4 Comparing lengths. Write the lengths of these items. Find the ratio of the following pairs of lengths and simplify them if possible: a the ruler to the pencil b the ruler to the paperclip c the pencil to the paperclip d the eraser to the pencil e the shoe to the pencil

5 Gradient on roads. Road signs are used to tell car drivers, truck drivers and bicyclists how steep the roads are. They show the steepness of a hill by writing it as a ratio. For example, Bulli Pass is 1 : 6 and Mt Victoria pass is 1 : 8.

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NEW CENTURY MATHS 8

Rise Run

The gradient for a road is the ratio of the rise to the run of the hill. For instance, the ratio 1 : 5 means a rise of 1 metre for every 5 metres running along the horizontal. Find the gradient for each of the following hills. a b

1 Rise 5 Run Gradient of 1 : 5

1

1 6

5

c

d 1

1 3

4

e 1 10

f 1 7

Just for the record Screen ratios A normal (analogue) television screen has its length and width in the ratio 4 : 3 (approximately 1.33 : 1). This is called its aspect ratio. The ratio means that the screen is 4 units across and 3 units up (a ‘squarish’-looking screen).

1 2 3 4 5 6 7 8 9

1 2 4:3

3

(1.33 : 1)

1

16 : 9 (1.77 : 1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

2

3

4

Cinema screens and widescreen digital television sets, however, have an aspect ratio of 16 : 9 (approximately 1.77 : 1), giving a wider picture. Some major films are presented in a wide-screen format of 1.85 : 1 or 2.35 : 1.

When films and digital TV programs are shown on a 4 : television set, the picture needs to be ‘squeezed’ somehow, using a ‘pan-and-scan’ (full frame) format or a ‘letterbox’ format. Some DVD players allow you to choose the way you view the picture.

Find out how the ‘pan-and-scan’ and ‘letterbox’ formats work. R AT I OS AND R AT E S

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Dividing quantities in a given ratio Example 8 Worksheet 10-02 Ratio recipes Worksheet 10-03 Ratio problems

1 Elyse and Damian buy a $1.00 raffle ticket, each paying 50 cents, and win first prize of $50. How much should each person get if they share the winnings equally?

Solution They each put in 50 cents, so will share the prize equally, or in the ratio 1 : 1. The number of equal parts is 2 (or 1 + 1, described as 1 : 1). The first share =

1 --2

of $50

= $25 The second share will be the same, $25. (Check your answer: $25 + $25 = $50.) 2 Suppose that Elyse contributes 20 cents to the $1.00 raffle ticket and Damian contributes 80 cents. How much should each person get this time?

Solution The amount each person takes from the prize depends on the ratio of their contribution to the cost of the ticket. The ratio was 20 : 80 = 1 : 4 since one person put in four times the amount of the other. The number of equal parts is 5 (or 1 + 4, described as 1 : 4). The first share =

1 --5

of $50

= $10 The second share =

4 --5

of $50

= $40 (Check your answer: $10 + $40 = $50.) 3 Suppose Jane shares in buying the raffle ticket with Elyse and Damian. Elyse, Damian and Jane contribute 20 cents, 20 cents, and 60 cents respectively. How much does each get from a prize of $200?

Solution Step 1: State the three contributions as a ratio. They put in 20 : 20 : 60 = 2 : 2 : 6 = 1:1:3 Step 2: Find the number of equal parts. This is 5, since 1 + 1 + 3 = 5. Step 3: Work out how much each friend should get of the prize. Elyse’s share

=

1 --5

of $200

1 --5

of $200

3 --5

of $200

= $40 Damian’s share =

= $40 Jane’s share

=

= $120 (Check your answer: $40 + $40 + $120 = $200.)

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NEW CENTURY MATHS 8

To divide a quantity in a given ratio: • find the total number of parts • find the fraction of the amount for each share of the ratio • check that your answers add to the original amount.

Example 9 Divide $400 in the ratio 3 : 5.

Solution Total parts = 3 + 5 = 8 3 --8 5 --8

× 400 = $150 × 400 = $250

Check: $150 + $250 = $400

Exercise 10-06 1 Find the total number of parts if: a the ratio is 2 : 7 b the ratio is 4 : 1 c the ratio is 3 : 4 d the ratio is 7 : 4 e the ratio is 2 : 5 : 6 f the ratio is 1 : 2 : 3 2 Divide $500 in the ratio: a 4:1 b 2:3 c 7:3 d 1:9 e 5:5 f 3:1 3 Divide 450 kg in the ratio: a 4:5 b 3:2 c 2:7 d 1:4 e 9:1 f 2:1 4 Divide 720 cm in the ratio: a 1:3:5 b 2:4:3 c 5:3:4 d 3:4:1 e 1:2:3 f 11 : 7 : 6 5 Two friends share a prize of $90 in the ratio of 1 : 2. How much prize money does each receive? 6 Three friends find a large bag of lollies. They agree to share their discovery equally. There are 51 lollies in the bag. How many lollies does each friend get? 7 Three friends, Adam, Janelle and Derek, buy a Lotto ticket for $12. They contribute $3, $3 and $6 respectively and agree to share any prizes in the same ratio. How much prize money does each person get if they win $4000? 8 Share a $40 prize in the ratio 2 : 3. 9 Share a bag of 70 lollies in the ratio 1 : 1. 10 Share a $30 000 prize in the ratio 1 : 2. 11 Share a $1200 prize in the ratio 1 : 2 : 3. 12 Divide a piece of cheese of mass 1000 g into two pieces according to the ratio 1 : 4. 13 Share 34 pens in the ratio 9 : 8.

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Example 9

Spreadsheet 10-03 Dividing quantities in a given ratio

Example 8

SkillBuilder 5-02 Dividing a given ratio

CHAPTER 10

14 Divide 48 kg in the ratio 3 : 5. 15 Two friends share a prize of $200 in the ratio of 3 : 2. How much more prize money does one friend receive than the other? 16 In Year 8, the ratio of boys to girls is 4 : 5. If there are 225 students in Year 8, find how many girls there are. 17 The ratio of girls to boys in Year 12 is 3 : 2. If there are 125 students in Year 12, find how many more girls there are than boys. 18 Adam needs to make 800 g of short-crust pastry. Flour and butter are needed in the ratio 3 : 1. How much flour is needed? 19 A company posts 1386 letters in a week. The ratio of local to overseas letters is 2 : 7. How many overseas letters are sent in a week? 20 A truck carries fruit and vegetable boxes in the ratio 5 : 7. If it carries a total of 7.5 tonnes, what mass of vegetables does it carry? 21 A quantity of 176 kg of an alloy is made from copper and zinc in the ratio 5 : 6. Find the mass of copper in the quantity of the alloy. 22 A 20 m cable is cut into three sections in the ratio 2 : 3 : 5. Find the length of each section. 23 When making mortar you mix sand and concrete in the ratio 6 : 1. You need 280 kg of mortar. How much sand will you need? 1 -. 24 A farmer planted 700 hectares of land with wheat, oats and corn in the ratio 1--6- : 3--4- : ----12 Find the area of land planted with each crop.

Ratios and the unitary method Worksheet 10-02 Ratio recipes

The unitary method will help us solve different types of ratio problems. The unitary method was applied to percentages at the end of Chapter 4. In the following problems you know one part of the ratio but not the total amount.

Example 10 Worksheet 4-11 The unitary method

In an English class the ratio of boys to girls is 5 : 6. If there are 15 boys in the class, how many girls are there?

Solution boys : girls = 5 : 6 = 15 : ? boys = 5 parts = 15 So 1 part = 15 ÷ 5 =3 girls = 6 parts =6×3 = 18 girls

So there are 18 girls in the class.

Exercise 10-07 1 Why is the method in Example 10 called the ‘unitary’ method?

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NEW CENTURY MATHS 8

2 a Two lengths of timber are in the ratio 4 : 7. The longer length is 56 cm. What is the shorter length? b In a rectangle, the ratio of the width to the length is 5 : 12. The length is 48 cm. Find the width. c In a school, the ratio of teachers to students is 1 : 18. If the college has 80 staff, how many students are there? d A tiler uses 4 green tiles to every 3 white ones. How many tiles are used altogether if 140 green tiles are used? e When making concrete, sand and cement are mixed in the ratio 4 : 1. If 120 kg of cement has been delivered, what mass of sand is needed? f The speed of two boats is in the ratio 7 : 4. The speed of the second boat is 10 km/h. Find the speed of the first boat. g An alloy contains copper and iron in the ratio 2 : 5. A quantity of alloy contains 20 kg of iron. What mass of copper does it contain? h Ali, Betty and Fiona share an amount of money in the ratio 10 : 8 : 7. If Betty received $72, how much did the others receive? i In a triangle, the lengths of the sides are in the ratio 3 : 4 : 5. If the longest side is 30 cm long, find the lengths of the other sides.

Example 10

Spreadsheet 10-04 Ratios and the unitary method

3 a The masses of two packets of detergent are in the ratio 3 : 10. i If the smaller packet has a mass of 1.5 kg, what is the mass of the larger one? ii If the larger packet costs $12.50 and the smaller packet costs $3.90, which packet is the cheaper per kilogram and by how much? b Enzo’s Produce Store buys fruit and vegetables in the ratio 5 : 7. The mass of the fruit is 8.5 tonnes. What is the total mass of produce ordered? c Simon and Joshua’s heights are in the ratio 8 : 9. If Joshua is 1.71 m tall, how tall is Simon? d In order to make up glue, the contents of two tubes, A and B, are mixed in the ratio 3 : 1. i If 15 mL of A is used, how much of B is needed? ii How much glue is made altogether if 15 mL of A is used. e A bushwalking rope is cut in the ratio 3 : 4. The longer piece is 116 m. What was the original length of the rope? f The heights of two buildings are in the ratio 7 : 6. If the small building is 160 m tall, how tall is the other building? g The directors of Centuryworld share its yearly profits in the ratio 1 : 2 : 3. The Managing Director, who received the greatest share, received $9 144 000. What was the profit for the year? h A triangle has sides in the ratio 2 : 5 : 4. If the shortest side is 18 cm, find: i the lengths of the other two sides ii the perimeter of the triangle. i Chemicals X, Y and Z are mixed in the ratio 3 : 7 : 10. If 5 mL of X is used, how much of each of the other chemicals will be used?

Scale drawings One very important application of ratio is the use of scale drawings for maps and house plans. When compared, the lengths on each map or drawing are in the same ratio as the lengths in the real world. This ratio allows us to measure real distances by using the map or diagram. R AT I OS AND R AT E S

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Worksheet 10-04 Scale drawings CHAPTER 10

Example 11 Here is a scale drawing of a clock. The scale shows the ratio of the drawing to its real size. Measure the length of the clock, and then work out its actual length.

XI

XII

I II

X

Solution The scaled length of the clock is 5 cm. 1 part = 5 cm Actual length = 6 parts =5×6 = 30 cm The actual length of the clock is 30 cm.

III

IX

Scale 1 : 6 IV

VIII VII

VI

V

Example 12 This is a scale drawing of a small screw. Find its actual length.

Solution

Scale 5 : 1

The length of the drawn screw is measured at 40 mm. The scale of 5 : 1 means that the real screw is smaller than the one drawn. 5 parts = 40 mm Actual length = 1 part = 40 ÷ 5 = 8 mm The actual length of the screw is 8 mm.

Exercise 10-08 1 Measure the length of each scaled-down image below, and then use the scale ratio to work out the actual length of the object shown. a Fish 1 : 3 b House 1 :100

1:4

th

ng

Le

gt

h

c Pen

d Tennis racquet 1 : 20

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Length

Length

Le n

Example 11

2 The plans of a house are drawn to a scale of 1 : 100. Find the actual length for each of the following measurements on the plan in: i centimetres ii metres. a length of the lounge room, 4 cm b length of the kitchen bench, 2 cm c width of the door, 0.9 cm d width of the garage, 2.75 cm e length of the side wall, 17 cm f width of the window, 1.2 cm 3 This house plan is drawn to a scale of 1 : 100.

WC Laundry Second Bedroom

Bath

Main Bedroom Kitchen

Hall

Lounge Dining Scale 1 : 100

a Find the dimensions (length × width) of the following rooms, in metres. (Measure the distances from the inside walls.) i the main bedroom ii the lounge room iii the bathroom iv the second bedroom v the kitchen b Find the area (in square metres) of: i the main bedroom ii the lounge room iii the kitchen

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Example 12

4 Measure the length of each magnified image below, and then use the scale ratio to work out the actual length of the object shown. a Flea 100 : 1 b Microchip 3 : 1

Le t ng h

Le

4:3

th

d Bacterium

100 : 1

Length

c Nut

ng

th

ng

Le

Worksheet 10-05 Map of Adelaide

Map scales Example 13 Write this scale as a ratio:

0

2

4

6

8

km 10

Solution The length of the segment from 0 to 10 on the diagram is 5 cm. So, 5 cm on the map represents 10 km of actual distance. To write the scale in the useful form 1 : actual measure we must convert to the same units and simplify: 5 cm : 10 km (measured and read from the diagram) 5 cm : 10 × (1000 × 100) cm (convert each measure to the same units) 5 cm : 1 000 000 cm 1 : 200 000 (simplifying the ratio) The scale is 1 : 200 000.

Example 14 A map has a scale of 1 : 25 000. a What is the actual distance if the scaled distance is 4 cm? b What is the scaled distance if the actual distance is 3.5 km?

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Solution a Scale 1 : 25 000 Actual distance = 4 × 25 000 cm = 100 000 cm = 1000 m = 1 km b 3.5 km = 3500 m = 350 000 cm Scaled distance = 350 000 cm ÷ 25 000 = 140 cm

(dividing 100 000 by 100) (dividing 1000 by 1000) (multiplying 3.5 by 1000) (multiplying 3500 by 100)

Multiply by scale Scaled distance

Actual distance

Divide by scale

Exercise 10-09 1 For each map scale given as a diagram in parts a to f below, write the scale in the form 1 : actual measure. a b metres kilometres 0

c

100

200

0

300

500

d 0

0 1000

1

1

2

3

4

Example 13

5

1500 metres 2

3 km

e

0

1

2

3

4 km

f

0

150

300

450

600

750

metres

2 A bushwalking map has a scale of 1 : 25 000. Find: a the actual distance for each of the following scaled distances. i 2 cm ii 3 cm iii 6 cm iv 5.5 cm v 9.5 cm b the scaled distance for the following actual distances. i 10 km ii 2.5 km iii 5 km iv 4 km v 7 km

Example 14

vi 2.75 cm vi 0.5 km

3 On a map using a scale of 1 : 10 000 000, the Nile (the world’s longest river) would be the length of an average shoe lace (about 66.7 cm). How many kilometres long is the Nile River? 4 The town of Gilgandra is 66 km north of Dubbo. A map uses a scale of 1 : 100 000. How long would the scaled distance be between the two locations? 5 Lord Howe Island is 2.8 km wide. How long would its scaled width on a map be if a scale of 1 : 50 000 was used? R AT I OS AND R AT E S

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6 Below is a map of Nambucca Heads using a scale of 1 : 40 000. 1

N NAMBUCCA

Water Towers

Coronation Park

ST

PO

WE

PIG

Anglican Church

GO TT

ST

Foreshore Caravan Park

4

5

6

Rotary Lookout

Shelley Beach Lookout Lagoon

Causeway

OCEAN

Police Stn

3

PACIFIC

ET

FOREST Catholic Church

SOUTH

HEADS

2

STATE

ST RE

NAMBUCCA

Cemetery

7

Island Golf Course

8 A

B

C

D

E

F

G

H

I

J

a Find the direct distance between: i the Water Towers (C2) and the Catholic Church (F3) ii the Anglican Church (G4) and Coronation Park (I4) iii Rotary Lookout (H6) and Shelley Beach Lookout (J6) b Find the length of: i West Street (E4) ii Piggott Street (D5) c How long is the lagoon (I7)? d What are the dimensions of the cemetery (J2)? e How long is the causeway leading to the Island Golf Course (B7)? f To train for the ‘City to Surf’ run you decide to run 8 km three times a week. What distance will this be on the map? Outline a possible course for your training run, starting and finishing at the Foreshore Caravan Park (D6).

Working mathematically Applying strategies: Scale drawings 1 Measure the length and width of any large rectangle (for example a paved area, a brick wall, a blackboard, the classroom floor). On graph paper make a scale drawing of your rectangle. Include the scale used on your drawing and list the real measurements made. 2 Choose one of the following activities. a Measure the length and width of the school reception area or foyer. Suppose there are plans to enlarge the length and width of this area by 50 per cent. Draw a scaled plan of the enlarged area, showing the location and size of key features. Include a list of your measurements, and the scale used.

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NEW CENTURY MATHS 8

b Draw a scale plan of your maths classroom. Remember to measure the location of immovable objects such as windows, doors, and blackboards. Other useful measurements are the length and width of the room and, possibly, the diagonal distance across the room. Include a list of your measurements, and the scale used. c Draw a scale plan of your bedroom. Remember to measure the location of immovable objects such as windows and doors. Other useful measurements are the length and width of the room and, possibly, the diagonal distance across the room. Include a list of your measurements, and the scale used.

Skillbank 10 Comparing fractions, decimals and percentages To compare or order fractions, we express them with a common denominator first. To compare or order decimals, we express them with the same number of decimal places first. 1 Examine these examples. a Which fraction is smaller:

4 -----10

or 3--8- ?

Using a common denominator of 80 (8 × 10). 4 4×8 32 ------ = --------------- = -----10 10 × 8 80 3 3 × 10 30 --- = --------------- = -----8 8 × 10 80 32 ------ < -----By comparing numerators: 30 80 80 ∴

3 --8

or

Using a common denominator of 40 (the LCM of 10 and 8). 4 4×4 16 ------ = --------------- = -----10 10 × 4 40 3 3×5 15 --- = ------------ = -----8 8×5 40 16 ------ < -----By comparing numerators: 15 40 40 ∴

is smaller.

3 --8

SkillTest 10-01 Comparing fractions, decimals and percentages

is smaller.

b Write these decimals in ascending order: 0.407, 0.47, 0.047, 0.4. Express all decimals with three decimal places by inserting zeros at the end where necessary. 0.407 0.47 0.047 0.4 become: 0.407 0.470 0.047 0.400 In ascending order (smallest to largest), this is 0.047, 0.400, 0.407, 0.470, or: 0.047, 0.4, 0.407, 0.47 2 Now find the smaller value in each pair: a

2 --3

and

3 --5

b

1 --4 3 --7

and and

1 --3 4 -----10

c 0.15 and 0.105

d 3.826 and 3.68

e

f

g 2.87 and 2.817

h 0.5301 and 0.503

i

5 --6 1 --5

and and

7 --8 2 --9

3 Write the numbers in each of these sets in ascending order. a 0.81, 0.082, 0.821 b 3.5, 3.51, 3.55, 3.513 c

2 1 2 --- , --- , --3 6 5

d 0.007, 0.07, 0.7, 0.707

e 10.49, 10.409, 10.4, 10.04

f

g 0.345, 0.045, 0.5, 0.43

h

3 --- , 8 1 --- , 4

2 ------ , 10 4 ------ , 10

1 --2 3 --5

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To compare fractions with decimals or percentages, or to order fractions, decimals and percentages, express them all as decimals first. 4 Examine this example. ˙ , 3--- . ------ , 0.6 Place these amounts in descending order: 68%, 13 20 5 ------ = 0.65, 0.6˙ = 0.666…, As decimals, 68% = 0.68, 13

3 --5

20

= 0.6.

In descending order (largest to smallest), this is 0.68, 0.666 …, 0.65, 0.6, or: 3 ------ , --- . 68%, 0.6˙ , 13 20

5

5 Write the numbers in each of these sets in ascending order. a 0.25, 1--6- , 16%, d

3 --- , 4

1 --5

0.639, 55%,

2 --5

b 27%, 1--3- , 0.4, 0.28

c 0.05, 50%, 6%,

e 69%, 0.609, 2--3- , 0.6

f

2 --- , 9

1 --8

0.105, 17%, 22.5%

Rates Unlike a ratio, a rate compares quantities of different types that are measured in different units. For instance, speed compares the distance travelled and the time taken. A speed of 60 kilometres per hour means that 60 kilometres are covered each hour. A mobile phone call may be charged at $1.30 per minute, petrol may cost 95.5c per litre, and $1 Australian may be equal to 56c US. All of these are rates. There is a special way of writing rates: • A speed of 60 kilometres per hour is written as 60 km/h. • A call rate of $1.30 per minute is written as $1.30/min We normally express a rate ‘per single unit’ (per means ‘for each’).

Example 15 Write each of these as a rate: a a factory produces 80 cars in 4 hours

Solution

a 80 cars in 4 hours = 80 ÷ 4 = 20 cars/h

b a ham costing $40 has a mass of 8 kg. b $40 for 8 kg = 40 ÷ 8 = $5/kg

Exercise 10-10 1 Here are some rates. Write the units used in each: a typing speed b heartbeat rate c phone charges d cost of meat e wages f cost of petrol g cricket run rate h population growth i population density j fuel consumption Example 15

2 Write each of the following as a rate: a 20 sheep in 1 hour c $4.50 for 1 kg e 100 metres in 5 seconds g 160 marks in 4 tests

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b d f h

9.8 metres in 1 second 25 students for each teacher 136 points in 4 games 32 articles in 4 hours

i k m o q s

180 cars in 2 days 259 metres in 7 seconds 9000 revolutions in 6 minutes 400 km in 8 hours 36 runs in 8 overs $126 for 12 hours

j l n p r t

4800 boxes in 8 hours 46 000 bottles in 50 hours $175 for 5 hours $16.50 for 6 kg 240 km using 30 litres 5000 kg for 100 hectares

3 In your own words, explain what is meant by each of these rates: a a speed of 100 km/h b a traffic survey of 150 cars/h c petrol consumption of 10.3 L/100 km d a factory producing 600 bottles/day e a freight rate of 45c/kg f a farmer keeping 60 sheep/hectare g 240 passengers/flight h a factory manufacturing 920 toys/day i a pump supplying 230 L/hour

Using rates

Worksheet 10-06 Rates problems

Example 16 1 A river is flowing at a speed of 6 kilometres per hour. How far would a log on the river travel in 3 hours?

Solution The rate is 6 km/h. Each hour the log would travel 6 kilometres. In 3 hours it travel 6 × 3 = 18 kilometres. 2 A soft drink manufacturer makes aluminium cans at the rate of 150 cans/min. a How long would it take to produce 1000 cans? b How many cans are produced in an 8-hour shift at the factory?

Solution a The factory produces 150 cans in 1 minute. 1000 cans take

1000 -----------150

minutes = 6 2--3- minutes

= 6 minutes and 40 seconds b The factory produces 150 cans per minute. Number of cans in 1 hour = 150 × 60 = 9000 Number of cans in 8 hours = 9000 × 8 = 72 000 cans Note that rate problems are solved by either multiplying or dividing. One simple rule for deciding whether to multiply or to divide is to examine the position of the required quantity in the rate, written in the form x/y. • To find the first quantity x, you multiply. • To find the second quantity y, you divide. Example 16, Question 2, used the rate cans/min. • To find the number of cans (in part b), you multiply. • To find the time in minutes (in part a), you divide. R AT I OS AND R AT E S

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Exercise 10-11 Example 16

1 Ben types 55 words per minute. How many words can he type in 20 minutes? 2 Maria is paid $12.50 per hour. How much will she earn if she works 38 hours in a week? 3 A footballer scores at a rate of 8 points per game. How many points will he score in 12 games? 4 A Concorde aircraft travels at 900 km/h. How far will it travel in 20 minutes? 5 Sue runs 100 metres in 12 seconds. How far will she run in one minute at the same rate? 6 A tap drips water at a rate of 15 mL/hr. How much water would be wasted in one week? 7 A farmer can graze 15 sheep per hectare. If he has 65 hectares set aside for sheep, how many sheep can he graze? 8 Mia cycles at 12 km per hour for 4 hours. How far does she travel? 9 An aeroplane can carry a total of 450 passengers per flight. How many flights will it take to carry 2700 passengers? 10 A flywheel rotates at a rate of 1200 revolutions per minute. How many revolutions does it make in 20 seconds? 11 A wheel is timed to make two revolutions per second. a How many revolutions does it make in 15 seconds? b How many revolutions does it make in 5.5 seconds? c How long does it take to make 80 revolutions? 12 The temperature in Mittagong at 7:00am is 8°C. If the temperature rises at a rate of 3°C/h, find: a the temperature at 10:00am b the temperature at 2:00pm c the time at which the temperature will be 23°C. 13 Petrol costs 95 c/L. If Robert had $20 in his pocket, how much petrol could he buy (correct to the nearest litre)? 14 The Wong family needs to buy new carpet for the family home. The carpet costs $125/m and the carpet layers charge $480 to lay it. How much will it cost to carpet the house if the Wongs need 28 metres of carpet? 15 A farmer uses fertiliser at a rate of 30 kg/ha. How many hectares can she cover if she has 145 kg of fertiliser? 16 In an experiment, the temperature rose at the rate of 5°/s. How long would a scientist need to wait for the temperature to rise by 35° if it continued rising at the same rate? 17 Zac has 504 cows in a 42 hectare paddock. a How many cows per hectare does he have? b How many cows could he keep in a 40 hectare paddock? c How many hectares would he need to keep 840 cows?

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18 Jane is filling the tank shown on the right from a pipe which supplies water at the rate of 1.25 litres per second. How long will it take to fill the tank: a in seconds? b in minutes? c in hours?

6750 L

19 Two brothers, David and Bobby, take turns mowing a rectangular lawn, 60 m long and 40 m wide. a Find the area of the lawn. b When David cuts the grass, he does it in one hour using an old lawn mower. How many square metres of lawn does he cut in one minute? c Bobby uses a newer lawn mower and it takes him 40 minutes when he mows the lawn. How many square metres of lawn does he cut in one minute? d If both brothers start mowing at the same time, how long will it take them to mow the lawn together. 20 Kim and Sonya went on holiday. They filled their car with petrol when the reading on the odometer was 0 3 4 5 6 8 . Next time they filled the car, the odometer reading was 0 3 5 2 1 8 . On this second filling, petrol cost 95 cents per litre and they needed to pay $47.50. How many km/L had the car done between the first filling and the second filling.

Speed (a special rate) Imagine you are travelling in a car on a highway with the speedometer measuring your speed at 100 km per hour. What does this mean? It means that, at this speed, you will travel a distance of 100 km in 1 hour. If you maintain this speed, after 2 hours you will have travelled 200 km, after 3 hours you will have travelled 300 km … and so on. Average speed is calculated by dividing the distance travelled by the time taken.

Worksheet 10-07 What’s my speed?

distance travelled Average speed = -----------------------------------------time taken

Example 17 Find the average speed for each of the following: a A car which travels 270 km in 3 hours. b A train which travels 80 km in 20 minutes.

Solution a Using the formula: distance travelled Speed = -----------------------------------------time taken 270 = --------3 = 90 km per hour (or 90 km/h) R AT I OS AND R AT E S

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b Speed is usually in kilometres per hour, so we need to write 20 minutes as a fraction of an hour: 20 -----60

=

1 --3

h

distance travelled Speed = -----------------------------------------time taken 80 = ----1 --3

= 80 × 3 = 240 km/h

Exercise 10-12 Example 17

1 Find the average speed for each of the following: a A car travels a distance of 600 km in 6 hours. b A horse rider travels a distance of 15 km in 3 hours. c A jet travels 2400 km in 5 hours. d A bushwalker walks 25 km in 5 hours. e A cyclist rides 85 km in 5 hours. f A jet travels 2100 km in 3 hours. g An athlete runs 10 000 m in 0.5 hours. h A walker travels 35 km in 8.75 hours. i A motorbike travels 250 km in 2.5 hours. j A car travels 280 km in 3.5 hours. 2 Find the speed for each of the following (either in km/h or m/s) to the nearest whole number. a A racing cyclist pedals 100 km in 2 hours and 30 minutes. b A car travels 450 km in 4 hours and 45 minutes. c A truck travels 390 km in 3 hours and 15 minutes. d A bushwalker travels 15 km in 4 hours and 30 minutes. e A car travels 90 km in 45 minutes. f A motorbike on the race track travels 250 km in 1 hour and 15 minutes. g A kangaroo travels 1.5 km in 15 minutes. h A boat travels 15 km in 1 hour and 40 minutes. i An athlete runs 200 m in 20 seconds. j Another athlete runs 800 m in 2 minutes. 3 Jack and Jill walk to school. The distance is 3000 metres and it usually takes about 30 minutes. Find their (average) speed in: a metres per minute b kilometres per hour. 4 Peter walked to school in 15 minutes. He estimated that the distance was about 1500 metres. Find his speed in: a metres per minute b kilometres per hour. 5 A bird flies 348 m in 6 seconds. Find its speed in metres per second. 6 Mrs Hassam lives 20 kilometres from the nearest railway station and it takes her 30 minutes to drive there in her car. Find her average speed in kilometres per hour.

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7 Find the distance travelled for each of the following: a A car travels for 8 hours at an average speed of 90 km/h. b A car travels for 3 hours at an average speed of 100 km/h. c A cyclist travels for 2 hours at an average speed of 39 km/h. d A bushwalker travels for 7 hours and 30 minutes at an average speed of 4 km/h. e A runner travels for 2 hours and 30 minutes at an average speed of 12 km/h. f A truck travels for 14 hours at an average speed of 95 km/h. 8 Majid can walk at a speed of 6 km/h. How far can he walk in 30 minutes? 9 A bus travels at 75 km/h for 5 hours, and then travels a further distance of 200 km over 4 hours. Find the average speed for the whole journey. 10 A car travels for 3 hours at 60 km/h and then travels a further distance of 100 km at 50 km/h. What is the average speed for the trip? 11 How long does it take a go-cart to travel 60 kilometres around a circuit at a speed of 40 km/h. 12 We normally measure the speed of an object in km/h or m/s. However, there are other ways of measuring speed such as the knot and the mach. Find what these special terms mean and where they are used.

Travel graphs Travel graphs also compare distance and time. The slope or steepness of the graph indicates speed.

Worksheet 10-08 Jane’s diary

Example 18 Janet’s cycling trip

70

Distance from home (km)

This graph shows Janet’s cycling trip. a At what time did Janet leave home? b When was Janet’s first stop? How far from home was she? c Find her average speed over the first two hours. d How far from home is Janet when she decides to return home? e How far does she travel altogether? f Find her average speed during her trip home. g For how long did Janet stop altogether in the day?

60 50 40 30 20 10 0 9:00 10:00 11:00 12:00 1:00 2:00 3:00 4:00 am am am noon pm pm pm pm Time R AT I OS AND R AT E S

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Solution a 9:00am b Each stop is indicated by a horizontal section of the graph (time passes but no distance is travelled). Janet’s first stop is at 11:00am, when she is 40 km from home. distance c Average speed = ------------------time 40 km = ----------------2 hours = 20 km/h Janet’s average speed over the first two hours is 20 km/h. d The point where Janet heads for home is when the graph changes direction. She is 65 km from home when she decides to return (at 1:30pm). e 65 + 65 = 130 km Janet travels 130 km altogether. 65 km f Average speed = -------------------2 1--2- hours = 26 km/h Janet’s average speed during her trip home was 26 km/h.

g

1 --2

h + 1 h = 1 1--2- h

Janet stopped for a total of 1 1--2- hours. On a travel graph: • a horizontal (flat) line indicates a stop • the steeper the slope of the graph, the greater the speed • the part of the graph going downwards indicates a change of direction or a return trip.

Example 19 Draw a travel graph to represent the following walking trip: Adam leaves camp at 8:00am and walks at an average speed of 4 km/h for 2 hours. He stops for 1--2- an hour for a snack and to enjoy the view. He walks a further 11 km in the next 2 1--2- hours to meet his friends. Adam’s travel graph Summarise the data. Depart: 8:00am By 10:00am: He has walked 2 × 4 = 8 km from home 10:00 to 10:30am: He stops 10:30am to 1:00pm: He walks a further 11 km ∴ At 1:00pm he is 8 + 11 = 19 km from home.

Distance from home (km)

Solution 20 16 12 8 4 0 8:00 9:00 10:00 11:00 12:00 1:00 am am am am noon pm Time

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Exercise 10-13

8 6 4 2 0 10:00 11:00 12:00 1:00 2:00 3:00 4:00 am am noon pm noon pm pm Time Merrill’s trip Distance from home (km)

2 Merrill drives from Sydney to Canberra, stopping to visit friends in Goulburn. a How far is Canberra from Sydney? b How long does the trip take? c How long does Merrill stop at Goulburn? d Find her average speed for the journey (excluding stops). e How far is it from Goulburn to Canberra?

Example 18

Thorald’s walk Distance from home (km)

1 Thorald walks to visit his friend Emil. This graph shows his journey. a How long did it take Thorald to walk to Emil’s house? How far was it from his own house? b Find his average speed for the entire walk. c How far does Thorald walk altogether? d Between what times does Thorald stop on his walk?

300 200 100 0 8:00 9:00 10:00 11:00 12:00 am am am am noon Time

Distance from home (km)

3 This graph shows a A cycling trip cyclist’s day trip. a At what times did the 28 speed of the cyclist change? 24 b At what time did the 20 cyclist start to return home? 16 c How far did the cyclist travel altogether on 12 this day? d How long did the 8 cyclist spend 4 ‘on the road’? e Find the cyclist’s 0 average speed for 10:00 am 11:00 am 12:00 noon 1:00 pm the day. Time f Find the cyclist’s average speed for each of these four stages: i 10:00am to 11:00am ii 11:00am to 12:15pm iii 12:45pm to 2:00pm iv 2:00pm to 3:00pm

2:00 pm

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3:00 pm

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Sa m

Distance from Macquarie (km)

4 Brian and Sam travel Brian and Sam’s journeys between Bligh and Macquarie. Bligh 240 a How far is it Bri an between the 200 two towns? b At what time do 160 Brian and Sam 120 pass each other? How far are they 80 from Macquarie? c Who has the 40 faster means of transport, Brian Macquarie 0 8:00am 9:00am 10:00am 11:00am 12:00 noon 1:00pm or Sam? Time d At what times is Sam stationary (not moving)? e Find the average speed of each person (excluding stops). 5 On the same graph, show the distance travelled by each of the following in 2 hours: a a person walking at 4 km/h b a cyclist travelling at 18 km/h c a car travelling at 60 km/h Example 19

6 A triathlete has the following training program for cycling and running: • Start 5:00am • Run 12 km in 2 hours • Rest 1--2- hour • Run a further 8 km in the next hour • Rest 1--2- hour • Pick up bike and cycle home, arriving at 10:00am. Draw the graph of this training session. 7 Judy and Keith decide to go to Melbourne on holiday. The following is a description of their drive from Sydney to Melbourne. Draw a graph of this information. • Depart from Sydney: 6:00am • Stop at Goulburn (193 km) at 8:00am. Breakfast 45 minutes. • Stop at Gundagai (190 km) at 11:00am. Morning tea/petrol 1--2- hour. • Stop at Albury (183 km) at 1:30pm. Lunch 1 hour. • Stop at Seymour (204 km) at 4:15pm. Snack/Petrol 15 minutes. • Arrive in Melbourne 103 km at 6:00pm.

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Converting units It is often useful to be able to change rates from one set of units to a different set. For example, to change kilometres/hour to metres/second, or to change litres/day to litres/second.

Example 20 Change $300/week to $/year.

Solution $300/week = $300 × 52/year = $15 600/year

(52 weeks in 1 year)

Example 21 1 Change 10 m/s into km/h.

Solution 10 m/s = 10 × 60 m/min = 600 m/min = 600 × 60 m/h = 36 000 m/h 36 000 = ----------------- km/h 1000 = 36 km/h

(60 seconds in 1 minute) (60 minutes in 1 hour) (1000 m in 1 kilometre)

2 Change 2 mL/minute to litres/day.

Solution

2 mL/minute = 2 × 60 mL/h = 120 mL/h = 120 × 24 mL/day = 2880 mL/day 2880 = ------------ L/day 1000 = 2.88 L/day

(60 minutes in 1 hour) (24 hours in 1 day) (1000 mL in 1 L)

3 Change 30 km/h into m/s.

Solution 30 km/h = 30 × 1000 m/h (1000 metres in 1 kilometre) = 30 000 m/h 30 000 = ----------------- m/min (60 minutes in 1 hour) 60 = 500 m/min 500 = --------- m/s 60

(60 seconds in 1 minute)

= 8 1--3- m/s

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Exercise 10-14 Example 20

Example 21

CAS 10-01 Rate conversions

1 Convert: a 5 m/s to m/h d 20 mL/min to mL/h g 5 kg/m2 to kg/ha j 0.5 km/min to km/h

b e h k

30 m/s to m/h 40 mL/minute to mL/h 2.5 tonnes/day to kg/day 15 sheep/h to sheep/day

c f i l

$10/kg to $/g 5 mL/s to mL/h $750/week to $/year $60/day to $/week

2 Change: a 5 m/s to km/h d 40 mL/min to L/h g 25 g/m2 to kg/ha j 5 kg/m to g/cm

b e h k

35 cm/s to m/h 40 L/minute to mL/s 2.5 tonnes/h to kg/day 8 km/L to m/mL

c f i l

10 cents/g to $/kg 5 mL/s to L/day 110 km/h to m/s 70c/min to $/h

3 Each speed below is given in metres per second. Change each speed into kilometres per hour. (Round your answers to two decimal places.) a An African cheetah was measured running at 27 m/s. b A German peregrine falcon dived at 97 m/s. c A Tanzanian snake can travel at a speed of 3.3 m/s. d A sailfish off the coast of Florida was estimated to swim 30 m/s. e A racing cyclist rode at 23 m/s.

Power plus 1 Write each of the scales below as a ratio in its simplest form: Source map or plan

Scale

Great Britain road atlas 0

10

20

30

40 km

Devon and Cornwall map 0

10 km

Paris street map 0

Ontario (Canada) road atlas

400 m

0

10

20 miles

0

10

20

30 km

0

20

40

60

Australian road atlas 80 km

2 Australia’s annual birth rate in 1995 was approximately 15 per 1000. Give that the 1995 population was 18 054 000, how many babies were born in 1995? 3 Singapore has a population of 2 733 000. The population density of Singapore is 4224 inhabitants/km2. a What is the area of this very crowded country? b Australia’s population density is 2.35 inhabitants/km2. The area of Australia is 7 682 300 km2. If Australia were populated as densely as Singapore, what would the population be?

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NEW CENTURY MATHS 8

4 A spider moves at 1 cm/s. If the spider is in the back left-hand corner of your classroom, find how long (in minutes) it will take to reach: a the nearest person b you c the blackboard d the door of the classroom 5 By writing each price as a rate (cents/kg, cents/g, cents/mL, etc), find out which item is the best value for money each time: 2L 375 mL × 6 a b 750 g 375 g

$0.99

$1.99

440 g

c

$1.78

$2.99

$2.25

735 g d

235 g

175 g

$1.21

115 g

$1.19

$1.71

$2.19

6 Work out a formula for converting speed expressed as x m/s to y km/h.

Cyclist’s journey 100 Distance (km)

7 Here is a travel graph of a bike rider’s journey. Write a story about her ride, based on the information in the graph.

80 60 40 20 0

2

3 4 Time (hours)

5

6

Bushwalking trips

B Distance (km)

8 Match each of the stories below with the most likely of these graphs. a The bushwalker was fast at first, but then got slower after lunch. b The bushwalker kept a constant speed all day. c The bushwalker was slow at first, but got faster after lunch.

1

A

C Time (hours) R AT I OS AND R AT E S

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Language of maths actual equivalent ratio simplify term

Worksheet 10-09 Ratios and rates crossword

compare part scale slope travel graph

direction plan scale drawing speed unit

divide rate scaled stationary unitary method

1 Which word in the list above means ‘not moving’. 2 List two differences between a ratio and a rate. 3 What is meant by the scale of a map or plan? 4 What does an average speed of 85 km/h actually mean? 5 Explain what is meant when something or someone is given ‘a rating’. Explain whether this has the same meaning as as ‘rate’ in mathematics. 6 How is a change in speed represented on a travel graph?

Topic overview • Give examples of places in which ratios and rates are used. • What did you learn in this topic? • What did you find most difficult about this topic? Discuss any problems with your teacher or a friend. • Copy this overview and add anything else you think it needs.

RATIOS Problems E_________ U _ _ _ _ _ _ method

S__________ F________

S _ _ _ _ drawings and maps

D_______ RATES 60 40 20

Problems

0

80 100 120 140 km/h 160 180

T _ _ _ _ _ graphs S____

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NEW CENTURY MATHS 8

Chapter 10

Review

Topic test Chapter 10

1 Find the ratio of shaded parts to unshaded parts in each of these diagrams: a b

Ex 10-01

2 Copy and complete the following equivalent ratios: a 2:5 b 4:7 4 : 10 8 : 14 6: 12 : : 20 : 28 10 : 10 20 : : 30 : 42

Ex 10-02

3 Write three equivalent ratios for each of the following: a 5:6 b 21 : 4 4 Copy and complete the following equivalent ratios: a 2:3 = 8: b 1:5 = 2: d 3:4:5 =

: 20 :

e

2 --3

=

Ex 10-02

c

3 --7 Ex 10-02

c 2:3:4 = 6: f

--------9

5 Simplify each of the following ratios: a 12 : 21 b 25 : 75 d 25 : 45 e 18 : 6 : 24

7 --8

=

:

21 --------Ex 10-03

c 6 : 36 f 5 : 25 : 100

6 Change the quantities to the same units first, then express each pair of quantities as a ratio in simplest form: a 5 km to 2000 metres b 2 kg to 3000 grams c $25 to 100c d 18 months to 4 years e 7 days to 5 weeks f 400 metres to 2 km

Ex 10-03

7 Simplify each of the following ratios:

Ex 10-04

a

1 --3

1 --2

:

e

4 --5

: 2--3-

i

0.98 : 0.245

b

1 --9

f

3 1--3- : 2 2--5-

g 0.35 : 0.45

h 0.08 : 1.2

j

1.5 : 6

k 91 : 5.6

l

:

1 --3

c

3 --4

:

9 -----16

d

1 1--2-

: 2 1--2-

2.7 : 1.8

8 The recipe for making 1 dozen muffins requires the following ingredients: • 1 cup of plain flour • 3 eggs • 4 teaspoons of baking powder •

1 --4

cup of sugar

•

3 --4

Ex 10-05

cup milk

• 4 tablespoons of melted butter.

Rewrite the recipe showing the ingredients needed to make 48 muffins. 9 Two friends win a prize of $100. They decide to share the prize in the ratio 2 : 3, because the ticket cost $5 and one contributed $2 and the other contributed $3. How much prize money should each receive? R AT I OS AND R AT E S

Ex 10-06

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CHAPTER 10

Ex 10-06

10 Share $6000 in the ratio 1 : 2 : 3.

Ex 10-07

11 a The ratio of girls to boys in Year 8 is 4 : 3. If there are 75 boys in Year 8, find how many girls there are. b Donald and Evelyn invest in a business in the ratio 5 : 7. If Evelyn invested $63 000, how much did Donald invest?

Ex 10-08

12 Measure the length of each scale drawing below, and then use the ratio to work out the actual length of the object shown. a Fish

Scale 1: 10

b Frog

Scale 1: 4 0

500 m

Ex 10-09

13 On a tourist map of Sydney the scale is given by a Write this scale as a simplified ratio. b Find the actual distance between the following places, given the scaled distance: i Circular Quay station to the Opera House (2.5 cm) ii Pyrmont Bridge to NSW Parliament House (4.4 cm) c Find the scaled distance between the following places given the actual distance: i Art Gallery of NSW to Sydney Tower (875 m) ii Circular Quay to Central Railway station (2.5 km).

Ex 10-10

14 Write each of the following as a rate: a $10.50 for 3 kg c 220 km in 2 hours e 425 marks in 5 tests

Ex 10-11

b 512 points in 4 games d $56.40 for 4 hours f 260 runs in 50 overs

15 a Mince is $3.99/kg. How much is it for 5 kg? b Amy drives for 2 1--2- hours at 90 km/h. How far does she travel? c Daria earns $13.25/h. How much is she paid for 38 hours work? d How many litres of petrol can you buy with $25 if petrol is 95c/L? e Fertiliser is used at 20 kg/ha. How many hectares can be covered with 150 kg of fertiliser?

Ex 10-12

16 Find the average speed for each of the following: a A car travels a distance of 280 km in 4 hours. b A horse travels a distance of 30 km in 3 hours. c A jet travels a distance of 1200 km in 2 hours. d A person walks 25 km in 5 hours.

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NEW CENTURY MATHS 8

17 Find the speed for each of the following, after changing all units to kilometres and hours. a A cyclist travels a distance of 45 km in 2 hours and 15 minutes. b A car travels a distance of 135 km in 1 hour and 30 minutes.

Ex 10-12

18 Find the distance travelled by: a a car which travels for 2 hours at an average speed of 85 km/h b a truck which travels for 5 hours and 30 minutes at an average speed of 45 km/h c a jet which travels for 1 hour at an average speed of 800 km/h.

Ex 10-12

19 Keith and Kent decided to go bushwalking. This travel graph shows their walk. a How far did Keith and Kent walk? b How many stops did they make? c Find their average speed in the first 1 1--2- hours. d At what time did they start back? e Between what times are Keith and Kent walking fastest?

Ex 10-13

Distance from camp (km)

Keith and Kent’s bushwalk

10 8 6 4 2 0 10:00 11:00 12:00 1:00 2:00 3:00 4:00 5:00 6:00 am am noon pm pm pm pm pm pm Time

20 a Bill and Jill walked a distance of 2400 metres to school in 30 minutes. Find their (average) speed in: i metres per minute ii kilometres per hour. b Find the speed of each animal below, first in metres per second, and then in kilometres per hour. i A farm dog ran 300 metres in 40 seconds. ii A kangaroo travelled a distance of 150 metres in 12 seconds.

R AT I OS AND R AT E S

309

Ex 10-14

CHAPTER 10