Chap 07

Tree

19.26 m1.52 m

1 m

ruler

To check whether her house

is in danger of being hit by a

gum tree falling during a

storm, Svetlana needs to

calculate how high the tree

is. It is too high to climb, so

she has taken the following

measurements on a sunny

day and drawn a rough

diagram to illustrate the

situation. She has also used a

metre ruler held vertically

from the ground and

measured the length of the

shadow cast on the ground.

How is Svetlana able to use

this information to calculate

the height of the tree?

This chapter will show you

a way in which this problem

can be solved.

7

66eTHINKING

Ratios andrates

278 M a t h s Q u e s t 8 f o r V i c t o r i a

READY?areyou

Are you ready?Try the questions below. If you have difficulty with any of them, extra help can be

obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon

next to the question on the Maths Quest 8 CD-ROM or ask your teacher for a copy.

Converting units of length, capacity and time

1 Convert each of the following to the units shown in brackets.

a 3 m (cm) b 5.2 m (mm) c 4.25 km (m) d 2 kg (g)

e 0.5 t (kg) f 6.4 L (mL) g 8.2 kL (L) h 4 min (s)

i 2 hours (minutes) j 4 weeks (days) k 2 years (months) l 2 years (weeks)

Highest common factor

2 Find the highest common factor of:

a 6 and 9 b 12 and 20 c 15 and 24 d 45 and 80.

Simplifying fractions

3 Simplify each of the following fractions.

a b c d

Finding and converting to the lowest common denominator

4 Write each of the following fraction pairs over the lowest common denominator.

a and b and c and d and

Converting a mixed number to an improper fraction

5 Write the following mixed numbers as improper fractions.

a 1 b 2 c 3 d 10

Multiplying decimals by 10, 100 and 1000

6 Perform each of the following multiplications.

a 1.237 × 10 b 0.084 × 10 c 0.284 × 100 d 0.000 784 × 1000

Multiplying a whole number by a fraction

7 Perform the following multiplications.

a 30 × b 60 × c 1 × 120 d × 96

Converting minutes to a fraction of an hour

8 Write each of the following as a fraction of 1 hour.

a 15 min b 40 min c 36 min d 54 min

7.1

1

2---

7.2

7.3

9

12------

20

25------

36

60------

72

100---------

7.4

1

3---

3

4---

3

8---

7

12------

2

3---

5

12------

7

10------

3

4---

7.5

3

4---

3

5---

7

10------

3

8---

7.6

7.7

2

3---

4

5---

1

4---

5

8---

7.9

C h a p t e r 7 R a t i o s a n d r a t e s 279

Introduction to ratiosRatios are used in many aspects of everyday life. They are used to compare quantities

of the same kind.

Where might you hear the following comparisons? What are the quantities being

compared?

1. Villeneuve’s car is twice as fast as Dick’s van.

2. By law there must be one teacher to every 15 students when on a school excursion.

3. You will need 2 buckets of water for every bucket of sand.

4. The fertiliser contains 3 parts of phosphorus to 2 parts of potassium.

5. The Tigers finished the season with a win : loss ratio of 5 to 2.

6. Mix 1 cup of water with 4 cups of flour.

In the last example we are considering the ratio of

water (1 cup) to flour (4 cups). We write 1 : 4 but say

‘one is to four’; we are actually mixing a total of 5

cups. Ratios can also be written in fraction form:

1 : 4 ⇔

Note: Since the ratios compare quantities of the same

kind, they do not have a name or unit of measurement.

That is, we write the ratio of water to flour as 1 : 4, not 1

cup : 4 cups. The order of the numbers in a ratio is

important. In the example of the ratio of water and flour,

1 : 4 means 1 unit (for example, a cup) of water to 4 units

of flour. The amount of flour is 4 times as large as the

amount of water. On the contrary, the ratio 4 : 1 means 4 units of water to 1 unit of flour,

which means the amount of water is 4 times as large as the amount of flour.

1

2---

1

4---

Look at the completed game of

noughts and crosses at right

and write the ratios of:

a crosses to noughts

b noughts to unmarked spaces.

THINK WRITE

a Count the number of crosses and the

number of noughts. Write the 2 numbers

as a ratio (the number of crosses must be

written first).

a 4 : 3

b Count the number of noughts and the

number of unmarked spaces. Write the

2 numbers as a ratio, putting them in the

order required (the number of noughts

must be written first).

b 3 : 2

1WORKEDExample


Before ratios are written, the numbers must be expressed in the same units of measure-

ment. Once the units are the same, they can be omitted. When choosing which of the

quantities to convert, keep in mind that ratios contain only whole numbers.

Introduction to ratios

1 Look at the completed game of noughts and crosses and write the ratios of:

a noughts to crosses

b crosses to noughts

c crosses to total number of spaces

d total number of spaces to noughts

e noughts in the top row to crosses in the bottom row.

2 Look at the coloured circles on the right and then write the following ratios.

a Black : red b Red : black

c Aqua : black d Black : aqua

e Aqua : red f Black : (red and aqua)

g Aqua : (black and red) h Black : total circles

i Aqua : total circles j Red : total circles

3 For the diagram shown, write the following

ratios.

a Shaded parts : unshaded parts

b Unshaded parts : shaded parts

c Shaded parts : total parts

SkillSH

EET 7.1

Converting units oflength, capacityand time

Rewrite the following statement as a ratio: 7 mm to 1 cm.

THINK WRITE

Express both quantities in the same

units. To obtain whole numbers,

convert 1 cm to mm (rather than 7 mm

to cm).

7 mm to 1 cm

7 mm to 10 mm

Omit the units and write the 2 numbers

as a ratio.

7 : 10

1

2

2WORKEDExample

1. Ratios compare quantities of the same kind.

2. The ratios themselves do not have a name or unit of measurement.

3. The order of the numbers in a ratio is important.

4. Before ratios are written, the numbers must be expressed in the same units of

measurement.

5. Ratios contain only whole numbers.

remember

7AWORKED

Example

1


4 In the bag of numbers shown to the right write the ratios of:

a even numbers to odd numbers

b prime numbers to composite numbers

c numbers greater than 3 to numbers less than 3

d multiples of 2 to multiples of 5

e numbers divisible by 3 to numbers not divisible by 3.

5 Rewrite each of the following statements as a ratio.

a 3 mm to 5 mm b 6 s to 19 s

c $4 to $11 d 7 teams to 9 teams

e 1 goal to 5 goals f 9 boys to 4 boys

g 3 weeks to 1 month h 3 mm to 1 cm

i 17 seconds to 1 minute j 53 cents to $1

k 11 cm to 1 m l 1 g to 1 kg

m 1 L to 2 kL n 7 hours to 1 day

o 5 months to 1 year p 1 km to 27 m

q 7 apples to 1 dozen apples r 13 pears to 2 dozen pears

s 3 females to 5 males t 1 teacher to 22 students

6 Out of 100 people selected for a school survey,

59 were junior students, 3 were teachers and

the rest were senior students. Write these ratios:

a teachers : juniors

b juniors : seniors

c seniors : teachers

d teachers : students

e juniors : other members of the survey.

7 Write each of the following, using a

mathematical ratio.

a In their chess battles, Lynda has won

24 games and Karen has won 17.

b There are 21 first-division teams and 17 second-division

teams.

c Nathan could long jump twice as far as Rachel.

d On the camp there were 4 teachers and 39 students.

e In the mixture there were 4 cups of flour and 1 cup of

milk.

f Elena and Alex ran the 400 m in the same time.

g The radius and diameter of a circle were measured.

h The length of a rectangle is three times its width.

i On Friday night 9 out of every 10 people enjoyed the

movie.

j The length of one side of an equilateral triangle com-

pared to its perimeter.

k The length of a regular hexagon compared to its perim-

eter.

l The number of correct options compared to incorrect

options in a multiple-choice question containing five

options.

2 9

74 5

6 8

WORKED

Example

2

SkillSHEET

7.1

Convertingunits oflength,

capacity andtime


8 A pair of jeans originally priced at $215 was

purchased for $179. What is the ratio of:

a the original price compared to selling price?

b the original price compared to the discount?

9 Matthew received a score of for his Maths

test. What is the ratio of:

a the marks received compared to marks lost?

b the marks lost compared to total marks poss-

ible?

10 For each comparison that follows, state whether

a ratio could be written and give a reason for

your answer. (Remember: Before ratios are

written, numbers must be expressed in the same

unit of measurement.)

a Anna’s mass is 55 kg. Her cat has a mass of 7 kg.

b Brian can throw a cricket ball 40 metres, and John can throw the same ball

35 metres.

c The cost of painting the wall was $55; its area is 10 m2.

d For a trip, the car’s average speed was 85 km/h; the trip took 4 h.

e Brett’s height is 2.1 m. Matt’s height is 150 cm.

f Jonathon apples cost $2.40 per dozen; Delicious apples cost $3.20 per dozen.

g Mary is paid $108; she works 3 days a week.

h David kicked 5 goals and 3 behinds; his team scored 189 points.

Simplifying ratios When the numbers in a ratio are multiplied or divided by the same number to obtain

another ratio, these two ratios are said to be equivalent. (This is similar to the process

of obtaining equivalent fractions.) For instance, the ratios 2 : 3 and 4 : 6 are equivalent,

as the second ratio can be obtained by multiplying both numbers of the first ratio by 2.

Ratios 10 : 5 and 2 : 1 are also equivalent, as the second ratio is obtained by dividing

both numbers of the first ratio by 5.

Like fractions, ratios are usually written in simplest form; that is, reduced to lowest

terms. This is achieved by dividing each number in the ratio by the highest common

factor (HCF).

97

100---------

Express the ratio 16 : 24 in simplest form.

THINK WRITE

Write the ratio. 16 : 24

÷ 8 ÷ 8

2 : 3Determine the largest number by which both 16 and

24 can be divided (that is, what is the highest

common factor)? It is 8.

Divide both 16 and 24 by 8 to obtain an equivalent

ratio in simplest form.

1

2

3

3WORKEDExample


Since simplifying ratios is similar to the process of

simplifying fractions, we can use the keys used

normally to operate with fractions. A ratio such as

16 : 24 can be entered as 16 ÷ 24. To simplify and

obtain an answer in fractional form, we press ,

select and then press . The calculations

for the ratios in worked examples 3 and 4 can be seen in

the screen shown opposite.

Recall that ratios must contain only whole numbers. Sometimes we are asked to

simplify ratios that have fractions or decimals in them.

If the ratio uses fractions, we write each fraction with a common denominator. Mul-

tiplying each fraction by this common denominator will reduce each part of the ratio to

a whole number as shown in the next worked example.

Write the ratio of 45 cm to 1.5 m in simplest form.

THINK WRITE

Write the question. 45 cm to 1.5 m

Express both quantities in the same units by

changing 1.5 m into cm. (1 m = 100 cm)

45 cm to 150 cm

Omit the units and write the 2 numbers as a ratio. 45 : 150

÷ 15 ÷ 15

3 : 10

Simplify the ratio by dividing both 45 and 150 by

15 — the HCF.

1

2

3

4

4WORKEDExample

Graphics CalculatorGraphics Calculator tip!tip! Simplifyingratios

CASI

O

SimplifyingratiosMATH

1: Frac� ENTER

Simplify the following ratios.

a : b :

Continued over page

THINK WRITE

a Write the fractions in ratio form. a :

Write equivalent fractions using the

lowest common denominator (in this

case, 10). :

× 10 × 10

4 : 7

Multiply both fractions by 10.

Check if the remaining whole

numbers that form the ratio can be

simplified. In this case they cannot.

2

5---

7

10------

5

6---

5

8---

12

5---

7

10------

2

4

10------

7

10------

3

4

5WORKEDExample


If the ratio uses decimals, we multiply by the smallest power of 10 that will produce a

whole number for both parts of the ratio.

THINK WRITE

b Write the fractions in ratio form. b :

Write equivalent fractions using the

lowest common denominator (in this

case, 24).:

× 24 × 24

20 : 15

÷ 5 ÷ 5

4 : 3

Multiply both fractions by 24.

Check if the remaining whole

numbers that form the ratio can be

simplified. In this case divide each

by the HCF of 5.

15

6---

5

8---

2

20

24------

15

24------

3

4

Write the following ratios in simplest form.

a 2.1 to 3.5 b 1.4 : 0.75

THINK WRITE

a Write the decimals in ratio form. a 2.1 : 3.5

× 10 × 10

21 : 35

÷ 7 ÷ 7

3 : 5

Both decimals have one decimal

place, so multiplying each by 10 will

produce whole numbers.

Simplify by dividing both numbers

by the HCF of 7.

b Write the decimals in ratio form. b 1.4 : 0.75

× 100 × 100

140 : 75

÷ 5 ÷ 5

28 : 15

Because 0.75 has two decimal

places, we multiply each decimal by

100 to produce whole numbers.

Simplify by dividing both numbers

by the HCF of 5.

1

2

3

1

2

3

6WORKEDExample

1. If each number in a ratio is multiplied or divided by the same number, the

equivalent (or equal) ratio is formed.

2. It is customary to write ratios in the simplest form. This is achieved by dividing

each number in the ratio by the highest common factor (HCF).

3. To form a ratio using fractions, convert the fractions so that they have a

common denominator and then write the ratio of the numerators.

4. Decimals can be easily changed into whole numbers if they are multiplied by

powers of 10 (that is, 10, 100, 1000 and so on).

remember


Simplifying ratios

1 Express each ratio in simplest form.

a 2 : 4 b 3 : 9 c 5 : 10 d 6 : 18

e 12 : 16 f 15 : 18 g 24 : 16 h 21 : 14

i 25 : 15 j 13 : 26 k 15 : 35 l 27 : 36

m 36 : 45 n 42 : 28 o 45 : 54 p 50 : 15

q 56 : 64 r 75 : 100 s 84 : 144 t 88 : 132

2 Complete the patterns of equivalent ratios.

a 1 : 3 b 2 : 1 c 2 : 3 d 64 : 32 e 48 : 64

2 : 6 4 : 2 4 : 6 __ : 16 24 : __

_ : 9 _ : 4 6 : _ __ : 8 12 : __

_ : 12 _ : 8 _ : 12 8 : __ __ : 8

5 : __ 20 : _ _ : 24 __ : 1 __ : __

3 Write the following ratios in simplest form.

a 8 cm to 12 cm b $6 to $18 c 50 s to 30 s

d 80 cm to 2 m e 75 cents to $3 f 2 h to 45 min

g 300 mL to 4 L h 500 g to 2.5 kg i 45 mm to 2 cm

j $4 to $6.50 k 2500 L to 2 kL l 2500 m to 2 km

m 30 cents to $1.50 n 2 h 45 min to 30 min o 200 m to 0.5 km

p 0.8 km to 450 m q 1 min to 300 s r 1.8 cm to 12 mm

s 3500 mg to 1.5 g t $1.75 to $10.50

4 Compare the following, using a mathematical ratio (in simplest form):

a The Hawks won 8 games and the Lions won 10 games.

b This jar of coffee costs $4 but that one costs $6.

c While Joanne made 12 hits, Holly made 8 hits.

d In the first innings, Ian scored 48 runs and Adam scored 12 runs.

e During the race, Rebecca’s average speed was 200 km/h while Donna’s average

speed was 150 km/h.

f In the basketball match, the Tigers beat the Magic by 105 points to 84 points.

g The capacity of the plastic bottle is 250 mL and the capacity of the glass container

is 2 L.

h Joseph ran the 600 m in 2 minutes but Maya ran the same distance in 36 seconds.

i In the movie audience, there were 280 children and 35 adults.

j On a page in the novel Moby Dick there are 360 words. Of these, 80 begin with a

vowel.

5 A serving of Weet Biscuit Cereal contains:

3.6 g of protein 0.4 g of fat

20 g of carbohydrate 1 g of sugar

3.3 g of dietary fibre 84 mg of sodium.

Find the following ratios in simplest form:

a sugar to carbohydrate b fat to protein

c protein to fibre d sodium to protein.

7BSkillSHEET

7.2

Highestcommon

factor

WORKED

Example

3

EXCEL Spreadsheet

Simplifyingratios

WORKED

Example

4

1

2---


6 In a primary school that has 910 students, 350 students are in the senior school and

the remainder are in the junior school. Of the senior school students, 140 are females.

There are the same number of junior males and junior females. Write the following

ratios in simplest form:

a senior students to junior students

b senior females to senior males

c senior males to total senior students

d junior males to senior males

e junior females to whole school population.


a to b to c :

d : e to f :

g : 1 h 1 to i 1 : 1

j 3 to 2 k 1 : 1 l 3 to 2


a 0.7 to 0.9 b 0.3 : 2.1 c 0.05 to 0.15 d 0.8 : 1

e 0.25 : 1.5 f 0.375 to 0.8 g 0.95 : 0.095 h 1 to 1.25

i 0.01 : 0.1 j 1.2 : 0.875 k 0.004 to 0.08 l 0.5 : 0.92

SkillSH

EET 7.3

Simplifying fractions

SkillSH

EET 7.4

Finding and converting to the lowest common denominator

WORKED

Example

5 1

3---

2

3---

5

7---

6

7---

1

4---

1

2---

SkillSH

EET 7.5

Converting a mixed number to an improper fraction

2

3---

5

6---

4

5---

9

20------

2

3---

3

5---

3

10------

2

3---

1

3---

1

4---

1

2---

1

3---

1

2---

3

5---

1

4---

4

5---

SkillSH

EET 7.6

Multiplying decimals by 10, 100 and 1000

WORKED

Example

6


9 Compare the following, using a mathematical ratio (in simplest form):

a Of the 90 000 people who attended the test match, 23 112 were females. Com-

pare the number of males to females.

b A Concorde jet cruises at 2170 km/h but a Cessna cruises at 220 km/h. Compare

their speeds.

c A house and land package is sold for $250 000. If the land was valued at

$90 000, compare the land and house values.

d In a kilogram of fertiliser, there are 550 g of phosphorus. Compare the amount

of phosphorus to other components of the fertiliser.

e Sasha saves $120 out of his take-home pay of $700 each fortnight. Compare his

savings with his expenses.

10 The table at right represents the selling

price of a house over a period of time.

a Compare the price of the house pur-

chased in December 1999 with the

purchase price in April 2003 as a

ratio in simplest form.

b Compare the price of the house pur-

chased in December 1999 with the

purchase price in December 2003 as

a ratio in simplest form.

c Compare the price of the house purchased in December 1999 with the purchase

price in March 2006 as a ratio in simplest form.

d ii How much has the value of the house increased from December 1999 to March

ii 2006?

ii Compare the increase obtained in part d i to the price of the house purchased in

ii December 1999 as a ratio in simplest form.

e Comment on the results obtained in part d.

11

a Toowoomba’s population is 80 000 and Brisbane’s population is 1.8 million. The

ratio of Toowoomba’s population to that of Brisbane is:

b When he was born, Samuel was 30 cm long. Now, on his 20th birthday, he is 2.1 m

tall. The ratio of his birth height to his present height is:

c The cost of tickets to two different concerts is in the ratio 3 : 5. If the more expensive

ticket is $110, the cheaper ticket is:

d A coin was tossed 100 times and Tails appeared 60 times. The ratio of Heads to Tails

was:

e Out of a 1.25 L bottle of soft drink, I have drunk 500 mL. The ratio of soft drink

remaining to the original amount is:

A 2 : 45 B 4 : 9 C 1 : 1.8 D 9 : 4 E none of these

A 3 : 7 B 1 : 21 C 7 : 10 D 1 : 7 E none of these

A $180 B $80 C $50 D $45 E $66

A 2 : 3 B 3 : 5 C 3 : 2 D 5 : 3 E 2 : 5

A 2 : 3 B 3 : 5 C 3 : 2 D 5 : 3 E 2 : 5

Date of sale Selling price

March 2006 $1.275 million

December 2003 $1.207 million

April 2003 $1.03 million

December 1999 $850 000

multiple choice

GAME time

Ratios

and

rates

— 001


ProportionA proportion is a statement of equality of two ratios.

For example, we know from our study of equivalent ratios that 1 : 3 = 2 : 6. This state-

ment is a proportion.

This can also be written as: .

Observe that, in the proportion statement above, the products of the numbers,

diagonally across from each other, are equal. That is, 1 × 6 = 2 × 3. This observation

will be true for any other proportion.

Consider, for example, this proportion: . Again, 3 × 12 = 9 × 4.

The process of finding the products of the numbers diagonally across from each

other is called cross-multiplication. It is very useful in solving proportion problems

and other problems you will encounter later.

In general, if , then, using cross-multiplication, a × d = c × b.

1

3---

2

6---=

3

4---

9

12------=

a

b

c

d=

Use the cross-multiplication method to determine whether the following pair of ratios is in

proportion: 6 : 9; 24 : 36.

THINK WRITE

Write the ratios in fraction form.

Perform a cross-multiplication. 6 × 36 = 216 24 × 9 = 216

Check whether the products are equal. 216 = 216

Therefore, the ratios are in proportion.

16

9---

24

36------

2

3

7WORKEDExample

Find the value of a in the following proportion: .

THINK WRITE

Write the proportion statement. =

Cross-multiply and equate the products. a × 9 = 6 × 3

Solve for a by dividing both sides of

the equation by 9.9a = 18

=

a = 2

a

3---

6

9---=

1a

3---

6

9---

2

3

9a

9------

18

9------

8WORKEDExample


Proportion

1 Use the cross-multiplication method to determine whether the following pairs of ratios

are in proportion.

a 2 : 3; 8 : 12 b 4 : 7; 8 : 14 c 5 : 7; 10 : 14

d 5 : 8; 10 : 16 e ; f ;

g ; h ; i ;

j ; k ; l ;

2 Find the value of a in each of the following proportions.

a b c

d e f

The ratio of girls to boys on the school bus was 4 : 3.

If there were 28 girls, how many boys were there?

THINK WRITE

Let the number of boys be b and write a

proportion statement. (Since the first

number in the ratio represents girls,

place 28 (the number of girls) as the

numerator.)

=

Cross-multiply and equate the products. 4 × b = 28 × 3

Solve for b by dividing both sides by 4. 4b = 84

=

b = 21

Write the answer. There are 21 boys.

14

3---

28

b------

2

3

4b

4------

84

4------

4

9WORKEDExample

1. Proportion is a statement of equality of two ratios.

2. In any proportion, the products of the numbers, diagonally across from each

other, are equal.

3. In general, if , then, using cross-multiplication, a × d = c × b.a

b

c

d=

remember

7CMathcad

Proportion

WORKED

Example

7

EXCEL Spreadsheet

Proportion

EXCEL Spreadsheet

Proportion(DIY)

7

9---

21

25------

3

8---

12

32------

14

16------

5

9---

11

12------

7

8---

13

15------

6

7---

8

9---

24

27------

3

5---

6

8---

21

18------

49

42------

WORKED

Example

8 a

2---

4

8---=

a

6---

8

12------=

a

9---

2

3---=

3

a---

9

12------=

7

a---

14

48------=

10

a------

3

15------=


g h i

j k l

3 Write a proportion statement for each of the following situations.

a Ratio of boys to girls is 6 : 5. There

are n boys and 30 girls.

b Ratio of books to magazines is 7 : 4.

There are n books and 16 mag-

azines.

c Ratio of pens to pencils is 2 : 3.

There are n pens and 36 pencils.

d Ratio of pies to sausage rolls is 1: 4.

There are n pies and 100 sausage

rolls.

e Ratio of length to width is 3 : 1. The

length is n and the width is 4 metres.

f Ratio of lions to tigers is 4 : 3. There

are 12 lions and n tigers.

g Ratio of adults to children is 5 : 8.

There are 30 adults and n children.

h Ratio of red balls to white balls is

15 : 2. There are 45 red balls and n

white balls.

i Ratio of spectators to competitors is

50 : 1. There are 5000 spectators and

n competitors.

j Ratio of cats to dogs is 8 : 5. There

are 48 cats and n dogs.

4 Solve each of the following, using a proportion statement and the cross-multiplication

method.

a The ratio of boys to girls in a class is 3 : 4. If there are 12 girls, how many boys are

in the class?

b In a room the ratio of length to width is 5 : 4. If the width is 8 m, what is the length?

c The team’s win–loss ratio is 7 : 5. How many wins has it had if it has had 15 losses?

d A canteen made ham and chicken sandwiches in the ratio 5 : 6. If 20 ham

sandwiches were made, how many chicken sandwiches were made?

e The ratio of concentrated cordial to water in a mixture is 1 : 5. How much concen-

trated cordial is needed for 25 litres of water?

f The ratio of chairs to tables is 6 : 1. If there are 42 chairs, how many tables are there?

g The ratio of flour to milk in a mixture is 7 : 2. If 14 cups of flour are used, how much

milk is required?

h The ratio of protein to fibre in a cereal is 12 : 11. If there are 36 grams of protein,

what is the mass of fibre?

i In a supermarket, the ratio of 600 mL cartons of milk to litre cartons is 4 : 5. If there

are sixty 600 mL cartons, how many litre cartons are there?

j In a crowd of mobile-phone users, the ratio of men to women is 7 : 8. How many

women are there if there are 2870 men?

3

7---

a

28------=

12

10------

a

5---=

8

12------

a

9---=

35

7------

5

a

---=

24

16------

6

a

---=

30

45------

2

a

---=

WORKED

Example

9


5 While we know that only whole numbers are used in ratios, sometimes in a proportion

statement the answer can be a fraction or a mixed number. Consider the following

proportion:

=

then a × 4 = 7 × 6

4a = 42

a = 10.5 (or 10 )

Calculate the value of a in each of the following proportion statements. Write your

answer correct to 1 decimal place.

a b c d e

f g h i j

6 Write a proportion statement for each situation and then solve the problem. If neces-

sary, write your answer correct to 1 decimal place.

a A rice recipe uses the ratio of 1 cup of rice to 3 cups of water. How many cups of

rice can be cooked in 5 cups of water?

b Another recipe states that 2 cups of rice are required to serve 6 people. If you have

invited 11 people, how many cups of rice will you need?

c In a chemical compound there should be 15 g of chemical A to every 4 g of chemical

B. If my compound contains 50 g of chemical A, how many grams of chemical B

should it contain?

d A saline solution contains 2 parts of salt to 17 parts of water. How much water

should be added to 5 parts of salt?

e To mix concrete, 2 buckets of sand are needed for every 3 buckets of blue metal. For

a big job, how much blue metal will be needed for 15 buckets of sand?

7 Decide whether a proportion statement could be made, using each of the following

ratios:

a height : age b mass : age

c intelligence : age d distance : time

e cost : number f age : shoe size

g sausages cooked : number of people h eggs : milk (in a recipe)

i number of words : pages typed j length : area (of a square).

8

a If , then:

b If , then:

A p × q = l × m B p × l = q × m C p × m = l × q

D E none of these is true.

A x = 2 and y = 4 B x = 1 and y = 2 C x = 3 and y = 6

D x = 6 and y = 12 E all of these are true.

a

6---

7

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1

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a

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8

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a

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4

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a

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a

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9

10------=

5

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7

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8

a---

6

7---=

9

7---

a

6---=

13

6------

a

5---=

9

15------

7

a---=

7

8---

9

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multiple choice

p

q---

l

m----=

p

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l

q---=

x

3---

y

6---=


c If then, correct to the nearest whole number, x equals:

d The directions on a cordial bottle suggest mixing 25 mL of cordial with 250 mL of

water. How much cordial should be mixed with 5.5 L of water?

1 Find the ratio of unshaded parts : shaded parts in the diagram.

2 True or false? 5 weeks is to 2 months ⇔ 5 : 2

3

Of the 99 744 people who attended a tennis tournament, 32 854 were males. The best

ratio approximation when comparing the number of females to males is:

A 1 : 2 B 2 : 3 C 3 : 2 D 2 : 1 E 3 : 1.

4 Express the ratio 24 : 36 in simplest form.

5 Find 3 ratios equivalent to 2 : 3.

6 Rewrite the ratio $2.10 to 30 cents in simplest terms.

7 Use the cross-multiplication method to determine whether the pair of ratios , is

in proportion.

8 Find the value of b in the proportion = .

9 A grocer has apples and oranges in the ratio 3 : 5. If there are 420 apples, how many

oranges are there?

10 To make toffee, 1 cup of water is used for each 3.5 cups of sugar. If you use 10.5 cups

of sugar, how many cups of water will you need?

A 13 B 12 C 34 D 28 E 17

A 0.55 mL B 5.5 mL C 55 mL D 550 mL E 5500 mL

WorkS

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23

34------

x

19------=

1

multiple choice

12

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21

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1 The sides of a triangle are in the ratio 3 : 4 : 5. If the longest side of the

triangle measures 40 cm, what is the perimeter of the triangle?

2 A 747 airplane has a length of 70.7 m and a wing-

span of 64.4 m. A model of this plane has a wing-

span of 30 cm. How long is the model?

3 The triangle ABC at right is an isosceles right-

angled triangle. The length of AC is 20 cm. If the

lengths of AD and BD are the same and the

lengths of CE and BE are the same, what is the

length of DE?C E B

A

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Comparing ratios In some cases it is necessary to know which of two given ratios is larger (or smaller).

And sometimes it is necessary to know whether the given ratios are equal. In other

words, we may need to compare the ratios.

When we are measuring the steepness of various slopes or hills, we need to compare ratios.

Gradient is a word used in mathematics to describe the slope, or steepness, of a hill.

The gradient of a hill is calculated by finding the ratio between

any 2 points on a hill. This is also known as calculating .

Which is the larger ratio in the following pair?

3 : 5 2 : 3

THINK WRITE

Write each ratio in fraction form.

Change each fraction to the lowest common

denominator (which is 15).

Compare the fractions: since both fractions

have a denominator of 15, the larger the

numerator, the larger the fraction.

<

The second fraction is larger and it

corresponds to the second ratio in the pair.

State your conclusion.

Therefore, 2 : 3 is the larger ratio.

13

5---

2

3---

29

15------

10

15------

39

15------

10

15------

4

10WORKEDExample

vertical distance

horizontal distance---------------------------------------------

rise

run--------

Find the gradient of the hill (AB),

if AC = 2 m and BC = 10 m.

THINK WRITE

Write the rule for finding the gradient. Gradient =

Vertical distance is 2 m and horizontal

distance is 10 m. Substitute these

values into the formula.

Gradient =

Simplify by dividing both numerator

and denominator by 2.

Gradient =

1vertical distance


22

10------

31

5---

11WORKEDExample

Horizontal distance

Vertical

distance

A

CB


Since the gradient is the measure of the steepness, when we need to compare the

steepness of two (or more) hills, all we need to do is compare the gradients of these

hills. The larger the gradient, the steeper the hill.

For example, if the gradient of hill A is found to be , and the gradient of hill B is ,

we can conclude that hill A is steeper, as > .

1

2---

1

5---

1

2---

1

5---

1. To compare ratios, write them in fraction form first and then compare the 2

fractions by writing them with a common denominator.

2. Gradient is a measure of the steepness of the slope and is calculated by finding

the ratio . (The distances are measured between any 2

points on the slope.)

vertical distance


remember


Comparing ratios

1 Which is the greater ratio in each of the following pairs?

a 1 : 4, 3 : 4 b 5 : 9, 7 : 9 c 6 : 5, 2 : 5

d 3 : 5, 7 : 10 e 7 : 9, 2 : 3 f 2 : 5, 1 : 3

g 2 : 3, 3 : 4 h 5 : 6, 7 : 8

i 5 : 9, 7 : 12 j 9 : 8, 6 : 5

2 In each of the following cases, decide which netball

team has the better record.

a St Marys won 2 matches out of 5. Colac won 5 out

of 10.

b Bright 50 won 13 out of 18. Corio won 7 out of 12.

c Seymour won 12 out of 20. Geelong won 14 out of

25.

d Bell Post Hill won 8 out of 13. Bairnsdale won 13 out

of 20.

3 In a cricket match, Jenny bowled 5 wides in her 7 overs and

Lisa bowled 4 wides in her 6 overs. Which bowler had the

higher wides per over ratio?

4 a Find the gradient of each of the hills represented by the

following triangles.

b Which slope has the largest gradient?

c Which slope has the smallest gradient?

d List the hills in order of increasing steepness.

5 Draw triangles that demonstrate a gradient of:

a b c d e

6

a If the gradient of LN in the triangle at right is 1, then:

A a > b B a < b C a = b

D a = 1 E b = 1

b If > , then a could be:

A 4 B 5 C 6 D 7 E all of these numbers.

c If < , then:

A a < 3 B b < 5 C a < b D b < a E a = 2

7DWORKED

Example

10

EXCE

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Comparingratios

EXCE

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Comparingratios (DIY)

Mat

hcad

Comparingratios

WORKED

Example

11

3 m

3 m

A

B3 m

2 m

C

D

2 m

5 m

E

F

i ii iii

1.5 m

7 mH

Giv

2

1---

3

1---

4

3---

3

2---

2

5---

multiple choiceL

Nb

a

5

6---

a

5---

a

b---

3

5---


Increasing and decreasing in a given ratio

Consider the following products and their results.

20 × = 25 ⇒ The result (25) is larger than the original (20).

20 × = 20 ⇒ The result is the same as the original (20).

20 × = 16 ⇒ The result (16) is smaller than the original (20).

Observe that multiplying by a ratio greater than 1 (that is, ) caused an increase of a

quantity, multiplying by a ratio equal to 1 (that is, ) resulted in the quantity being

unchanged and multiplying by a ratio smaller than 1 (that is, ) caused a decrease of a

quantity.

Generally, multiplying by a ratio greater than one increases the quantity, while multiplying by a ratio less than one decreases the quantity.

5

4---

5

5---

4

5---

5

4---

5

5---

4

5---

Increase $25 in the ratio 6 : 5.

THINK WRITE

The given ratio when expressed as a

fraction (that is, ) is greater than 1.

So, to increase the amount, multiply it

by this ratio.

New amount = $25 ×

Evaluate. New amount = $30

1

6

5---

6

5---

2

12WORKEDExample

Decrease 55 kg in the ratio 3 : 5.

THINK WRITE

The given ratio when expressed as a

fraction (that is, ) is less than 1 and

therefore will cause a decrease in the

amount. So multiply the mass by this

ratio.

New mass = 55 kg ×

Evaluate. New mass = 33 kg

1

3

5---

3

5---

2

13WORKEDExample


Note: There is an alternative solution to worked example 14. We could first calculate

the increase in dollars and cents by finding of $3 (since the increase was of the old

fare). We would then add the increase to the old fare to obtain the new fare.

Check this calculation:

Increase = × $3

= $0.60

New fare = old fare + increase

= $3 + $0.60

= $3.60 (same answer as before)

Increasing and decreasing in a given ratio

1 Increase the following quantities in the ratio given.

a Increase $50 in the ratio 3 : 2. b Increase $75 in the ratio 6 : 5.

c Increase $120 in the ratio 5 : 3. d Increase 24 mm in the ratio 7 : 4.

e Increase 72 cm in the ratio 7 : 6. f Increase 155 m in the ratio 8 : 5.

g Increase 36 mg in the ratio 5 : 2. h Increase 84 g in the ratio 8 : 7.

i Increase 132 kg in the ratio 13 : 12. j Increase 450 cm3 in the ratio 9 : 5.

In recent years, bus fares have increased by one-fifth. If the old fare was $3, what is the new fare?

THINK WRITE

The old fare can be considered as

a whole. Find the new fare as a fraction

of the old fare (it is bigger).

New fare = old fare + of old fare

New fare = 1 +

New fare = +

New fare = of the old fare

Multiply the old fare by the ratio . New fare = 3 ×

New fare =

Evaluate. New fare = $3.60

1

1

5---

1

5---

1

5---

5

5---

1

5---

6

5---

26

5---

6

5---

18

5------

3

14WORKEDExample

1

5---

1

5---

1

5---

1. Multiplying by a ratio greater than one results in an increase in the amount

being calculated.

2. Multiplying by a ratio smaller than one causes a decrease of the amount.

remember

7E

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WORKED

Example

12


2 Decrease the following quantities in the ratio given.

a Decrease $20 in the ratio 1 : 2. b Decrease $60 in the ratio 2 : 3.

c Decrease $140 in the ratio 3 : 5. d Decrease 36 mm in the ratio 5 : 6.

e Decrease 75 cm in the ratio 1 : 3. f Decrease 220 m in the ratio 3 : 4.

g Decrease 45 mg in the ratio 4 : 5. h Decrease 64 g in the ratio 7 : 8.

i Decrease 99 kg in the ratio 7 : 9. j Decrease 840 cm3 in the ratio 11:12.

3 In each case decide whether the change will be an increase or a decrease and write the

new quantity.

a Change $20 in the ratio 5 : 2. b Change $55 in the ratio 2 : 5.

c Change $50 in the ratio 9 : 10. d Change 54 m in the ratio 5 : 9.

e Change 200 L in the ratio 6 : 5. f Change 450 km in the ratio 10 : 9.

g Change 56 cm2 in the ratio 7 : 8. h Change 120 mg in the ratio 8 : 5.

i Change 12.5 kg in the ratio 4 : 5. j Change $6.80 in the ratio 3 : 2.

4 Council rates this year have increased in the ratio 5 : 4. If the old rates were $1200,

what are the new rates?

5 Land values have increased in the ratio 7 : 5. What is the new value of a block of

land formerly valued at $54 000?

6 Gate takings for AFL games decreased in the ratio 5 : 8. In the previous year the gate

takings were $1.2 million. What were they this year?

7 The capacity of water in a small dam was 36 000 kL last week. If it has since

decreased in the ratio 7 : 12, what is its capacity now?

8 A newsagent recently changed his order of newspapers in the ratio 7 : 6. What is his

new order if he previously ordered 390 papers?

9 In each case, calculate the new quantity.

a A former fare of 80 cents increased by one-fifth.

b A former rate of pay of $4.50 per hour was increased by two-fifths.

c A former mass of 80 kg increased by one-quarter.

d A former value of $1200 increased by one-third.

e A former volume of 36 cm3 increased by two-thirds.

f A former fare of $1.20 decreased by one-fifth.

g A former mass of 90 kg decreased by one-sixth.

h A former cost of $15 decreased by two-fifths.

i A former speed of 84 km/h decreased by one-twelfth.

j A former number of 480 in an audience decreased by three-quarters.

10 Due to a pay rise, Lena’s salary was increased in the ratio 6 : 5.

a If her old salary was $45 000, what is her new salary?

b If her new salary is $48 000, what was her old salary?

11

a The dimensions of a rectangle 8 cm by 5 cm were increased in the ratio 2 : 1. The

new rectangle has an area of:

A 40 cm2 B 80 cm2 C 20 cm2 D 160 cm2 E 120 cm2

EXCEL Spreadsheet

Increasingand

decreasingin a given

ratio

EXCEL Spreadsheet

Increasingand

decreasingin a givenratio (DIY)

WORKED

Example

13

Mathcad

Increasingand

decreasingin a given

ratio

WORKED

Example

14

multiple choice


b The volume of air in a balloon was changed in the ratio a : b. The volume would

decrease if:

c During a year the price of a trolley of

groceries increased in the ratio 8 : 5.

How much would I have paid for a

trolley of groceries at the beginning

of the year if I paid $120 at the end

of the year?

A $75 B $100

C $120 D $140

E $192

d Over a period of time Mera’s height

changed from 140 cm to 160 cm. Her

height increased in the ratio:

A 7 : 1 B 8 : 1

C 7 : 8 D 8 : 7

E 20 : 1

e A student increased his Mathematics

marks by 20%. This is equivalent to

an increase in the ratio:

A 5 : 1 B 6 : 5

C 6 : 1 D 20 : 1

E 1 : 5

f The price of petrol has increased by 35% over the past few months. This is equiv-

alent to an increase in the ratio:

A a > b B a = b C a < b D b < a E a = 2 and

b = 1

A 7 : 20 B 20 : 7 C 35 : 1 D 1 : 35 E 27 : 20

MA

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1 To make two -cup servings of cooked rice, you add of a cup of rice,

teaspoon of salt and 1 teaspoon of butter to 1 cups of water. How

many -cup servings of cooked rice can you make from a bag con-

taining 12 cups of rice?

2 During a sale, the price of a television was reduced in the ratio 4 : 5.

After the sale, the price of the television was increased to the original

price.

a By what ratio was the sale price increased to restore the original

price?

b What percentage increase is this?

2

3---

3

4---

1

4---

1

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2

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Dividing in a given ratioConsider the following situation: Isabel and Rachel decided to buy a $10 lottery ticket.

Isabel had only $3, so Rachel put in the other $7. The ticket won the first prize of

$500 000. How are the girls going to share the prize? Is it fair that they get equal

shares?

In all fairness, it should be shared in the ratio 3 : 7.

Note: In worked example 15, the second share represents 7 parts out of the total of 10.

So the alternative way to calculate the size of the second share would be to find of

the total amount.

Check this calculation: Second share = × $500 000.

= $350 000 (same answer as before)

Share the amount of $500 000 in the ratio 3 : 7.

THINK WRITE

Determine how many shares (parts) are in

the ratio.

Total number of parts = 3 + 7

Total number of parts = 10

The first share represents 3 parts out of a

total of 10, so find of the total amount.

First share = × $500 000

First share = $150 000

The second share is the remainder, so

subtract the first share amount from the total

amount.

Second share = $500 000 − $150 000

Second share = $350 000

1

2

3

10------

3

10------

3

15WORKEDExample

7

10------

7

10------

Concrete mixture for a footpath was made up of 1 part of cement, 2 parts of sand and 4 parts of blue metal. How much sand was used to make 4.2 m2 of concrete?

THINK WRITE

Find the total number of parts. Total number of parts = 1 + 2 + 4

Total number of parts = 7

There are 2 parts of sand to be used in

the mixture, so find of the total

amount of concrete made.

Amount of sand = × 4.2 m2

Amount of sand = 1.2 m2

1

2

2

7---

2

7---

16WORKEDExample

To share a certain amount in a given ratio, find the total number of shares (parts)

first. The size of each share is given by the fraction this share represents out of the

total number of shares.

remember


Dividing in a given ratio

1 Write the total number of parts for each of the following ratios.

a 1 : 2 b 2 : 3 c 3 : 1 d 3 : 5 e 4 : 9

f 5 : 8 g 6 : 7 h 9 : 10 i 1 : 2 : 3 j 3 : 4 : 5

2 Share the amount of $1000 in the following ratios.

a 2 : 3 b 3 : 1 c 1 : 4 d 1 : 1 e 3 : 5

f 5 : 3 g 3 : 7 h 9 : 1 i 7 : 13 j 9 : 11

3 If Nat and Sam decided to share their lottery winnings of $10 000 in the following

ratios, how much would each receive?

a 1 : 1 b 2 : 3 c 3 : 2 d 3 : 7 e 7 : 3

f 1 : 4 g 9 : 1 h 3 : 5 i 12 : 13 j 23 : 27

4 Rosa and Mila bought a lottery ticket costing $10. How should they share the first

prize of $50 000 if their respective contributions were:

a $2 and $8? b $3 and $7? c $4 and $6?

d $5 and $5? e $2.50 and $7.50?

5 Concrete mixture is made up of 1 part cement, 2 parts sand and 4 parts blue metal.

a How much sand is needed for 7 m3 of concrete?

b How much cement is needed for 3.5 m3 of concrete?

c How much blue metal is required for 2.8 m3 of concrete?

d How much sand is used for 5.6 m3 of concrete?

e How much cement is needed to make 8.4 m3 of concrete?

6 Three friends buy a Lotto ticket costing

$20. How should they share the first prize

of $600 000 if they each contribute:

a $3, $7 and $10?

b $6, $6 and $8?

c $1, $8 and $11?

d $5, $6 and $9?

e $5, $7.50 and $7.50?

7 Three angles of a triangle are in the ratio 1 : 2 : 3. What is the magnitude of each angle?

8 In a school, the ratio of girls in Years 8, 9 and 10 is 6 : 7 : 11. If there are 360 girls in

the school:

a how many Year 8 girls are there?

b how many more Year 10 girls are there than Year 8 girls?

9 In a moneybox, there are 5 cent, 10 cent and 20 cent coins in the ratio 8 : 5 : 2. If there

are 225 coins altogether:

a how many 5 cent coins are there?

b how many more 10 cent coins than 20 cent coins are there?

c what is the total value of the 5 cent coins?

d what is the total value of the coins in the moneybox?

10 In a family, 3 children receive their allowances in the ratio of their ages, which are

15 years, 12 years and 9 years. If the total of the allowances is $60, how much does

each child receive?

7F

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EXCE

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Dividingin a givenratio

EXCE

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Dividingin a givenratio (DIY)

WORKED

Example

15

WORKED

Example

16


11 The angles of a quadrilateral are in the ratio 2 : 3 : 4 : 6. What is the difference in mag-

nitude between the smallest and largest angles?

12

a A square of side length 4 cm has its area divided into two sections in the ratio 3 : 5.

The area of the larger section is:

b A block of cheese is cut in the ratio 2 : 3. If the smaller piece is 150 g, the mass of

the original block was:

c Contributions to the cost of a lottery ticket were $1.75 and $1.25. What fraction of

the prize should the larger share be?

d A television channel that telecasts only news, movies and sport does so in the ratio

2 : 3 : 4 respectively. How many movies, averaging a length of 1 hours, would be

shown during a 24 hour period?

1 A class of 30 students was surveyed and it was found that 16 students had dogs as

pets, 9 students had cats as pets and the remainder had no pets. Write down the ratio

of cats : no pets.

2 True or false? The ratio 85 : 105 in simplest form is 17 :21.

3 Find the value of n in the proportion .

4 The ratio of blue-eyed students to brown-eyed students in a class is 4 : 7. If there are

16 blue-eyed students, how many brown-eyed students are there?

5 Which is the greater ratio, 3 : 4 or 5 : 7?

6 Which is the smaller ratio, 8 : 5 or 12 : 9?

7 Increase 260 cm3 in the ratio 9 : 7. (Write your answer correct to 2 decimal places.)

8 Decrease 55 g in the ratio 5 : 6. (Write your answer correct to 2 decimal places.)

9 Jack and Jill share the cost of a $10 lottery ticket in the ratio $6 and $4. If they win

first prize of $950 000, how should they share first prize?

10 Cordial can be made up with 7.5 parts of water and 1.5 parts of cordial. What fraction

of water should there be?

A 3 cm2B 5 cm2

C 8 cm2D 10 cm2

E 16 cm2

A 75 g B 200 g C 300 g D 375 g E 450 g

A B C D E

A 2 B 3 C 4 D 5 E 6

multiple choice

7

12------

5

7---

7

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3

5---

5

12------

WorkS

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2

50

n

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9---=


History of mathematicsT H E R H I N D PA P Y R U S ( c . 1 8 5 0 B C )

The ancient Egyptians differed from the

ancient Greeks in that Egyptians thought

about mathematics in a practical, rather than

an abstract, way. They didn’t like fractions

that had numerators other than one (except

the fraction two-thirds for reasons still

unknown). They found that fractions with

numerators of one, unit fractions, were easy

to multiply, since the numerator would

always be one; for example, × = .

The Egyptians developed ingenious

methods to avoid using any fraction other

than those with a numerator of one. Solutions

to many Egyptian problems concerned with

beer and bread were recorded on papyri. The

most famous of these is the Rhind papyrus,

which contains 84 problems and their

solutions including the calculation of the

ancient Egyptian value for pi (π) of 3.1605.

A part of the papyrus is shown in the

photograph above.

The Rhind papyrus was named after the

Scottish Egyptologist, A. Henry Rhind, who

bought the 6-m scroll in 1858. A scribe

named Ahmes is believed to have copied it in

around 1650 BC from a document originally

written about 200 years before that. This

papyrus shows a method for multiplying

numbers using only addition and subtraction.

Also known as the aha papyrus (aha meaning

unknown quantity to be determined, an early

pronumeral), it is now in the British Museum

in London.

Questions

1. Which numerator did the Egyptians use in

their calculations with fractions?

2. Which fraction was an exception to this

rule?

3. What practical problems did most of the

solutions deal with?

Research

How was Egyptian multiplication done with

only addition and subtraction?

1

2---

1

3---

1

6---

During this time . . .

The Sumerians built the

first cities, invented

writing and made

wheels from date palm

trunks.

Papyrus reeds were

used to make boats,

baskets and paper.

The Bronze Age began.


Scale drawingYou have often seen a scale, stated on a map or a plan. Refer to an atlas and you will

see an example of a scale.

Scale drawings are very valuable in that they show a true picture (or model) of the

real article. That is, an object and its scale drawing are similar figures — exactly the

same in shape, but different in size.

Plans and maps often use a ratio scale (that is, a scale written as a ratio). The first

number in a ratio represents the drawing, while the second number represents the

real object.

For example, let us suppose the ratio scale on a plan is written as 1 : 100. This means

that any distance measured on the plan is 100 times bigger in real life; that is, 1 cm on

a plan corresponds to 100 cm (or 1 m) in real life. This can also be written as 1 cm ⇔ 1 m.

Scales can be used to find the size of a real object, given the size of its drawing, and

vice versa. When using the ratio scales, it is important to remember that the units of

length for both, the drawing and the real object, are always the same. For example, if

the length of the drawing is given in mm, the length of the object, found using the ratio

scale, will also be in mm. Once the answer is obtained, it can then be changed (if

necessary) to more appropriate units.

RiverRiv

er

River

Donets

River

Don

RiverTisza

River

DniesterRiver

River

River

Riv

er

River

Danube

r

River

hin

e

er

Danube

Riv

er

A N

SE

A

Sea of

Azov

BLACK SEA

AEGEAN

SEAIONIAN

SEA

TYRRHENIAN

SEA

CASPIAN SEA

AD

RIATIC

SEA

ALPS

Mt Olympus2917 m

CARPATHIAN

DIN

ARIC

ALPS

VELE

BIT

MTSAPENNINES

ALPS

SUDETIC

MTS

Vesuvius1281 m

Mt Etna3323m

CAUCASUS MTSMt Elbrus 5642 m

MTS

TRANSYLVANIAN

Grossglockner3801 m

BALKAN MTS

Teulada

Cape Tainaron

Balkan

Peninsula

Crimea

CASPIANDEPRESSION

A S I A

Malta

Elba

CreteCyprus

Rhodes

500 km750 10000 250

1 centimetre on the map represents

280 kilometres on the ground.

1 : 28 000 000

Crkvice

Astrakhan

Conformal Conic Projection

Rewrite the ratio scale 1 : 10 000, using the most appropriate units.

THINK WRITE

Choose a unit for the ratio (usually cm);

it must be the same for both numbers.

1 : 10 000

1 cm ⇔ 10 000 cm

Since 10 000 is quite a large number,

convert to a suitable unit. In this case it

is most appropriate to convert to

metres. Therefore divide by 100.

1 cm ⇔ 10 000 ÷ 100

1 cm ⇔ 100 m

1

2

17WORKEDExample


A scale factor is used when we are enlarging or reducing figures.

To obtain the lengths of the enlarged (or reduced) shape, a scale factor is written as a

fraction and then each length of the original shape is multiplied by that fraction.

For example, to enlarge an original shape by using a scale factor of 2 : 1, we would

multiply each dimension of the shape by . To reduce an original shape by using a

scale factor of 1 : 2, we would multiply each length of the shape by .

(Compare this with increasing or decreasing an amount in a given ratio. Can you see

any resemblance?)

If a scale of 1 : 100 is used to draw a tree, what is the actual height of a tree whose height

in the drawing is 5 cm?

THINK WRITE

Let the height of the tree be h. h = height of the tree

The ratio of the length of the drawing to

the actual length of the tree is . At the

same time we know that the scale used

for the drawing is . Equate these

two ratios to form a proportion.

Cross-multiply. 1 × h = 5 × 100

Find the value of h. (The units of length

for the object must be the same as the

units of length used for the drawing.)

1 × h = 500 cm

Convert the height to the more

appropriate units and write the answer.

The actual height of the tree is 5 m.

1

2

5

h---

1

100---------

1

100---------

5

h---=

3

4

5

18WORKEDExample

2

1---

1

2---

Find the new dimensions of a rectangle 3 cm × 2 cm if it is enlarged using a scale factor of

3 : 1.

THINK WRITE

Each dimension has to be enlarged by

the scale factor . So multiply each,

length and width by to find the

dimensions of the rectangle after the

enlargement.

New length =

= 9 cm

New width =

= 6 cm

Write the answer. The enlarged rectangle is 9 cm × 6 cm.

1

3

1---

3

1---

33

1---×

23

1---×

2

19WORKEDExample


Scale drawing

1 Rewrite the following ratio scales using the most appropriate units.

a 1 : 10 b 1 : 100 c 1 : 1000 d 1 : 10 000 e 1 : 100 000

f 1 : 5000 g 1 : 60 000 h 1 : 400 i 1 : 750 000 j 1 : 2 200 000

2 Write the following as ratio scales.

a 1 cm ⇔ 20 cm b 1 cm ⇔ 50 000 cm

c 1 cm ⇔ 10 m d 1 cm ⇔ 200 m

e 1 cm ⇔ 5 km f 1 cm ⇔ 50 km

g 1 cm ⇔ 5.5 km h 2 cm ⇔ 2 km

i 2 cm ⇔ 2.5 km j 3 cm ⇔ 6 km

3 On a town plan, the scale is given as 1 : 100 000.

a Find the actual distance between:

i two buildings that are 3 cm apart on the plan

ii two streets that are 5 cm apart on the plan

iii two houses that are 2 mm apart on the plan

iv two parks that are 6.5 cm apart on the plan

v the town hall and the high school that are 7.5 mm apart on the plan

vi two cinemas that are 2.8 mm apart on the plan.

b Find the distances on the plan between:

i two intersections that are 1 km apart

ii two department stores that are 200 m apart

iii an airport and a motel that are 15 km apart

iv a service station and a bank that are 2.5 km apart

v the northern and southern extremities that are 22 km apart

vi two shops that are 500 m apart.

4 If a scale of 1 : 50 is used to make each of the following drawings:

a what would be the width of the drawing of a house that is 25 m wide?

b what would be the height of the drawing of a man who is 2 m tall?

c what would be the dimensions of the drawing of a rectangular pool 50 m by 20 m?

d what would be the dimensions of the drawing of a table 4 m by 1.5 m?

1. Ratio scales are often used on maps and plans. The first number in a ratio scale

refers to the drawing; the second number refers to the real object.

2. Ratio scales can be used for finding the size of the real object when the size of

the drawing is known, and vice versa. The given length and the length obtained

using the ratio scale are always in the same units.

3. Scale factors are used for enlarging or reducing figures. To find the new

dimensions of the object after enlargement or reduction, all the original

dimensions of the object are multiplied by the scale factor.

remember

7GWORKED

Example

17

SkillSHEET

7.8

Convertingunits oflength,

capacityand time

Mathcad

Ratioscales

EXCEL Spreadsheet

Mapscales

EXCEL Spreadsheet

Mapscales(DIY)

WORKED

Example

18


e what is the actual height of a tree whose height in the drawing is 10 cm?

f what is the actual length of a truck whose length in the drawing is 25 cm?

g what are the dimensions of a television set whose dimensions on the drawing are

10 mm by 8 mm?

h what are the dimensions of a bed whose dimensions on the drawing are 4.5 cm long

by 3 cm wide by 1.2 cm high?

5 Referring to the plan of the house

shown at right, find:

a the dimensions of the lounge

b the width of the hallway

c the length of the garage

d the floor area of the kitchen/

living room

e the total floor area of the largest

bedroom.

(Hint: Use a ruler to find the

required lengths on the plan first.)

6 Find the new dimensions of the following shapes if they are enlarged using the scale

factor shown.

a Rectangle 4 cm × 3 cm (scale factor 2 : 1)

b Square of side length 2 cm (scale factor 3 : 1)

c Square of side length 3 cm (scale factor 3 : 2)

d Circle with radius 1.5 cm (scale factor 4 : 1)

e Equilateral triangle with side length 6 cm (scale factor 4 : 3)

f Isosceles triangle with sides 4 cm, 4 cm, 7 cm (scale factor 3 : 2)

7 Calculate the new dimensions of the following shapes when they are reduced with the

scale factor shown.

a Rectangle 6 cm × 4 cm (scale factor 1 : 2)

b Square of side length 3 cm (scale factor 2 : 3)

c Circle with diameter 20 cm (scale factor 1 : 5)

d Equilateral triangle with side length 4 cm (scale factor 3 : 4)

e Isosceles triangle with sides 5 cm, 5 cm and 2 cm (scale factor 2 : 5)

8 Draw the original and the reduced shapes, described in question 7 parts a and b.

9

a On a map the scale is given as ‘5 cm represents 2.5 km’. In ratio form, this would

be:

b A rectangle 8 cm by 6 cm is enlarged so that the new rectangle has an area of

108 cm2. The scale factor used was:

A 1 : 0.5 B 1 : 5 C 1 : 50 D 1 : 5000 E 1 : 50 000

A 2 : 1 B 3 : 1 C 1 : 2 D 2 : 3 E 3 : 2

Lounge

Kitchen/Living

Linen

WC

Laundry

Garage

Bedroom 3 Bedroom 2

Bedroom 1

Scale 1:200

Bath

WORKED

Example

19

multiple choice


c A national park ranger is plan-

ning a survey. The scale on her

map is 1 : 20 000. She plans to

walk from A to B to C and then

back to A, as shown on her map.

How far will she walk in total?

d This line from a map shows the 2 towns of Bampton and Cordwell. The towns are

actually 120 km apart. The scale on the map is:

A 6 km

B 7 km

C 8 km

D 9 km

E 2 km

A 1 : 120 B 1 : 40

C 1 : 120 000 D 1 : 4 000 000

E 1 : 1 200 000

Consider Svetlana’s problem from

the start of the chapter. She has

taken some measurements and

drawn a rough sketch of the

situation as illustrated at right.

1 Why did she take the

measurements on a sunny day?

To calculate the height of the tree,

produce a scale drawing of the tree

triangle.

2 Consider the imaginary line running from the top of the tree or the metre ruler

to the end of the shadow along the ground. Explain why the angle that this line

makes with the ground is the same for each triangle.

3 Draw a scale drawing of the metre-ruler triangle and measure the angle.

4 Use the measured angle and produce a scale drawing of the tree triangle.

5 What measurement do you obtain for the height of the tree?

6 Try this investigation again by obtaining your own measurements to calculate

the height of a tree or other tall object. Remember that you need a sunny day!

C

B

A

Cordwell

Bampton

THINKING How high is that tree?

Tree

19.26 m1.52 m

1 m

ruler


RatesWhether we like it or not, most things in life change, either over a short or long period

of time. The concept of rates is very important and useful for measuring and com-

paring how different quantities change.

We usually write a rate using the word per or a slash, /. For example, the speed of a

car is measured in kilometres per hour (km/h). The cost of flour can be measured in

dollars per kilogram ($/kg). It is customary to write the rates in simplest form.

A rate is considered to be in its simplest form if it is per one unit.

For example, the rate 20 metres per 8 minutes can be expressed in simplest form as

2.5 metres per minute.

Express the following statement using a rate in simplest form:The 30 litre container was filled in 3 minutes.

THINK WRITE

A suitable rate would be litres per minute

(L/min). Put the capacity of the container in

the numerator and the time in which it was

filled in the denominator of the fraction.

Rate =

Rate =

Simplify the fraction. Rate = 10 L/min

130 L

3 min-------------

10 L

1 min-------------

2

20

WORKED

Example

Joseph is paid $8.50 per hour as a casual worker. At this rate, how much does he receive for 6 hours of work?

THINK WRITE

The rate is given in $ per hour. So it actually

tells us the amount of money earned in each

hour; that is, the hourly payment.

Payment per 1 hour = $8.50

State the number of hours worked. Hours worked = 6

To find the total payment, multiply the hourly

payment by the total number of hours worked.

Total payment = $8.50 × 6

Total payment = $51

1

2

3

21WORKEDExample

1. Rates are used to measure and compare the changes in different quantities.

2. Rates are usually written using ‘per’ or a slash (/).

3. Rates are considered to be in simplest form if they are expressed per one unit

(for example, per minute, per hour, per kg and so on).

remember

C h a p t e r 7 R a t i o s a n d r a t e s

311

Rates

1

What quantities (such as distance, time, volume) are changing if the units of rate are:

a

km/h?

b

cm

3

/sec?

c

L/km?

d

$ per h?

e

$ per cm?

f

kL/min?

g

cents/litre?

h

$ per dozen?

i

kg/year?

j

cattle/hectare?

2

What units would you use to measure the changes taking place in each of the

following situations.

a A rainwater tank being filled b A girl running a sprint race

c A boy getting taller d A snail moving across a path

e An ink blot getting larger f A car consuming fuel

g A batsman scoring runs h A typist typing a letter

3 Express each of the following statements using a rate in simplest form.

a A lawn of 600 m2 was mown in 60 min.

b A tank of capacity 350 kL is filled in 70 min.

c A balloon of volume 4500 cm3 was inflated in 15 s.

d The cost of 10 L of fuel was $13.80.

e A car used 16 litres of petrol in travelling 200 km.

f A 12 m length of material cost $30.

g There were 20 cows grazing in a paddock that was 5000 m2 in area.

h The gate receipts for a crowd of 20 000 people were $250 000.

i The cost of painting a 50 m2 area was $160.

j The cost of a 12 minute phone call was $3.00.

k The team scored 384 points in 24 games.

7H

WORKED

Example

20


l Last year 75 kg of fertilizer cost $405.

m The winner ran the 100 m in 12 s.

n To win, Australia needs to make 260 runs in 50 overs.

o For 6 hours work, Bill received $159.

p The 5.5 kg parcel cost $19.25 to post.

q Surprisingly, 780 words were typed in 15 minutes.

r From 6 am to 12 noon, the temperature changed from 10°C to 22°C.

s When Naoum was 10 years old he was 120 cm tall. When he was 18 years old he

was 172 cm tall.

t A cyclist left home at 8.30 am and at 11.00 am had travelled 40 km.

4 Sima is paid $15.50 per hour. At this rate, how much does she earn in a day on which

she works 7 hours?

5 A basketball player scores, on average, 22 points per match. How many points will he

score in a season in which he plays 18 matches?

6 Water flows from a hose at a rate of 3 L/min. How much water will flow in 2 h?

7 A car’s fuel consumption is 11 L/100 km. How much fuel would it use in travelling

550 km?

8 To make a solution of fertiliser, the directions recommend mixing 3 capfuls of

fertiliser with 5 L of water. How many capfuls of fertiliser should be used to make

35 L of solution?

9 Anne can type 60 words per minute. How long will she take to type 4200 words?

10 Marie is paid $42 per day. For how long will she have to work to earn $504?

11 The rate of 1 teacher per 16 students is used to staff a school. How many teachers will

be required for a school with 784 students?

12 Land is valued at $42 per m2. How much land could be bought for $63 000?

13 On average, a test bowler took 1 wicket every 4.5 overs. How many wickets did he

take in a season in which he bowled 189 overs?

14 Tea bags in a supermarket can be bought for $1.45 per pack (pack of 10) or for $3.85

per pack (pack of 25). Which is the cheaper way of buying the tea bags?

15

Car A uses 41 L of petrol in travelling 500 km. Car B uses 34 L of petrol in travelling

400 km. Which car is the more economical?

16 Coffee can be bought in 250 g jars for $9.50 or in 100 g jars for $4.10. Which is the

cheaper way of buying the coffee and how large is the saving?

WORKED

Example

21


17

a A case containing 720 apples was bought for $180. The cost could be written as:

b Mark, a test cricketer, has a batting strike rate of 68, which means he has made 68

runs for every 100 balls faced. What is Steve’s strike rate if he has faced 65 overs

and has made 280 runs? (Note: Each over contains 6 balls.)

c A carport measuring 8 m × 4 m is to be paved. The paving tiles cost $36 per m2 and

the tradesperson charges $12 per m2 to lay the tiles. How much will it cost to pave

the carport (to the nearest $50)?

d A tank of capacity 50 kL is to be filled by a hose whose flow rate is 150 L/min. If

the tap is turned on at 8 am, when will the tank be filled?

SpeedIf we travelled from Sydney to Melbourne and someone asked our speed for the trip,

we could answer that it was continually changing and therefore we didn’t know.

However, we could tell them our average speed for the trip.

The average speed is an important rate and can be calculated using the formula:

Corresponding to the ratio , the units of measurement for speed contain units

of length in the numerator and units of time in the denominator (for example, km/h,

m/min, cm/s).

The speed of a car is usually measured in km/h. If we were measuring the speed of

an athlete running a distance of 200 m, it would be more appropriate to use the unit m/s

(metres per second).

A 30 cents each B 20 cents each C $3.00 per dozen

D $2.00 per dozen E $2.80 for 10

A 65 B 68.2 C 71.8 D 73.2 E 74.1

A $1400 B $1450 C $1500 D $1550 E $1600

A Between 1.00 pm and 1.30 pm B Between 1.30 pm and 2.00 pm

C Between 2.00 pm and 2.30 pm D Between 2.30 pm and 3.00 pm

E Between 3.00 pm and 3.30 pm

multiple choice

GAME time

Ratios

and rates

— 002

Average speeddistance travelled

time taken---------------------------------------------=

distance

time-------------------

Calculate the average speed of a train which travels 550 km in 10 h.

THINK WRITE

Write the formula for calculating average

speed.

Average speed =

Substitute the values for the distance and

time into the formula and evaluate.

Average speed =

Average speed =

Average speed

Write the answer with the appropriate unit. Average speed = 55 km/h

1distance travelled

time taken------------------------------------------

2550 km

10 h------------------

55 km

1 h---------------

3

22

WORKED

Example

314

M a t h s Q u e s t 8 f o r V i c t o r i a

Express the speed of 60 km/h in m/s.

THINK WRITE

Write the speed in fraction form.Speed =

=

Convert the given units to the required

units — change kilometres to metres

(that is, multiply by 1000) and hours to

seconds (that is, multiply by 3600).

= Speed

=

Simplify. Speed = 16 m/s

1 distance

time -------------------

60 km

1 h---------------

2 60 1000× m

1 60 60×× s-------------------------------

60000 m

3600 s---------------------

32

3---

23WORKEDExample

At right is a graph showing a bushwalker’s hike.

Write a short story describing his hike.

THINK WRITE

Label the relevant points onto the

graph.

Determine the coordinates of the first

point from the origin and interpret this

point in terms of the problem (that is,

how far has the hiker travelled and how

long has it taken him).

Note: The first value of the coordinate

refers to time taken and the second

value of the coordinate refers to the

distance from the campsite.

Coordinate A (2, 7)

The bushwalker has travelled for 2 hours and he

is 7 km away from the campsite.

Time (hours)

Dis

tance

fro

m c

ampsi

te (

km

)20181614121086420

0 1 32 54 4 5 6 7 8 9 1011

1

Time (hours)

A

B

CD

E

F

Dis

tance

fro

m c

ampsi

te (

km

)

20181614121086420

0 1 32 54 4 5 6 7 8 9 1011

2

24WORKEDExample


Speed

1 Calculate the average speed (in km/h) in each of the following cases.

a A car travels 400 km in 5 h.

b A jet travels 1600 km in 2 h.

c The winner of a motorbike race takes 3 h to travel 450 km.

d A marathon runner takes 3 h to run 42 km.

e A train travels 495 km in 11 h.

f A cyclist rides 66 km in 4 h.

g An aeroplane travels 875 km in 3.5 h.

h A triathlete runs 5000 m in half an hour.

i A bus travels 532 km in 5.6 h.

j Sound travels 305 km in a quarter of an hour.

THINK WRITE

Determine the coordinates of the

second point from the origin and

interpret this point in terms of the

problem.

Note: A horizontal line on the graph

indicates the hiker has not moved.

Coordinate B (3, 7)

The bushwalker is still 7 km from the campsite,

1 hour later. He may have stopped for a break

or a meal. A horizontal line on the graph

indicates the hiker has not moved.

Determine the coordinates of the next

leg of the journey and interpret this

point in terms of the problem. Compare

this section of the graph with that in

step 2.

Coordinate C (5, 16)

The bushwalker is 16 km from the campsite. It

has taken him 2 hours to travel 9 km. He has

covered this leg of the journey at a faster pace

than the first leg.

Determine each of the remaining

coordinates and highlight any important

points.

Coordinate D (6, 16)

The bushwalker has taken another 1-hour break.

Coordinate E (7, 18)

The bushwalker is now 18 km from the

campsite. It has taken him 7 hours to reach this

point.

Coordinate F (11, 0)

It has taken the bushwalker 4 hours to return to

the campsite.

3

4

5

1. Average speed is an important rate that can be calculated if the distance

travelled and the time taken are known.

2.

3. Speed is measured in units such as km/h, m/min, cm/s and so on.

4. Horizontal lines on distance–time graphs imply 0 distance travelled.

Average speeddistance travelled

time taken------------------------------------------=

remember

7IWORKED

Example

22

SkillSHEET

7.9

Convertingminutes toa fraction

of an hour


2 Calculate the average speed (in km/h) for each of the following.

a A car travels 40 km in 30 min.

b A car travels 200 km in 2 h 30 min.

c A car travels 60 km in 45 min.

d A jet travels 2625 km in 3 h 30 min.

e A walker walks 10 km in 1 h 15 min.

f A motorbike travels 75 km in 40 min.

g A horse gallops 3 km in 3 min.

h A racing car completes a lap of 3 km in 1 min.

i A rally driver completes a course of 1500 km

in 18 h 45 min.

Note: Express the speed as a whole number.

j A cyclist rides 500 m in 1 min.

3 Find the distance travelled in each of the following.

a A cyclist rides for 2 h at a speed of 18 km/h.

b A car travels at 85 km/h for 3 h.

c Malka rides her motorbike for 2 h 30 min at 140 km/h.

d The Sunlander travels at 80 km/h for 5 h 45 min.

e An aeroplane travels at 360 km/h for 1 hour 15 min.

4 Find the time taken for each of the following trips.

a A car travels 400 km at 80 km/h.

b A cyclist rides for 50 km at 10 km/h.

c A train travels 560 km at 80 km/h.

d A jet travels 3000 km at 750 km/h.

e A woman jogs for 5 km at 10 km/h.

5 Express each of the following speeds in m/s.

a 36 km/h b 72 km/h c 54 km/h

d 90 km/h e 180 km/h

6 The graph at right is a distance–time graph, showing

Geoff’s trip on his motorbike.

a Calculate Geoff’s average speed.

b If he continued at this average speed, how far

would he travel in 7 h?

c If he continued at this speed, how long (to the nearest

quarter-hour) would it take him to ride 400 km?

7 At 8 am, Rob sets off from a starting point to

canoe down a river. At 10 am, Kate sets off

from the same starting point in her motorboat.

The graph below shows their journeys.

a What is Rob’s average speed?

b When Kate passes Rob:

i what time is it?

ii how far from the starting point are they?

c Rob and Kate have agreed to meet at a point

80 km from the starting point. At what time

will Kate arrive at this meeting point?

Mat

hcad

Speedconverter

EXCE

L Spreadsheet

Speedconverter

WORKED

Example

23

Time (h)

Dis

tance

(km

)

0

70

140

210

280

0 1 2 3 4

Time (hours)

Rob

Kat

e

Dis

tance

(km

)

0

10

20

30

40

8 am9 am

10 am11 am

12 noon1 pm

2 pm


8 Below is a graph, showing a bushwalker’s hike. Write a short story describing her hike.

9

a Over a short distance, an emu can reach a speed of 30 m/s. This is approximately:

b On his way from home to school (a distance of 2.5 km), a boy rides at 6 km/h for

20 min and walks the rest of the way at 3 km/h. His average speed for the whole trip is:

c The graph of a car’s journey is shown at right.

Which of the following is true?

d One route from Melbourne to Sydney is 950 km. Car A leaves Melbourne travelling

at 90 km/h and Car B leaves Sydney travelling on the same route at 100 km/h.

Where will the cars pass each other?

A 60 km/h B 80 km/h C 90 km/h D 100 km/h E 110 km/h

A 3 km/h B 4.5 km/h C 5 km/h D 6 km/h E 9 km/h

A The car travelled at a constant speed for the

journey.

B The car travelled some distance, then stopped

and then travelled the rest of the journey at a

slower pace.

C The car travelled some distance, slowed down and

then increased its speed.

D The car went very fast for some time and then slowed down.

E The car travelled a certain distance, then stopped and then returned home.

A 500 km from Sydney B 450 km from Sydney

C 600 km from Sydney D 600 km from Melbourne

E 550 km from Melbourne

WORKED

Example

24

Time (h)

Dis

tance (

km

)

0

4

8

16

12

20

0 1 2 3 4 5 6

multiple choice

Time (h)D

ista

nce (

km

)

WorkS

HEET 7.3

MA

TH

S Q

UEST

CHALL

EN

GE

CHALL

EN

GE

MA

TH

S Q

UEST

1 Lachlan was driven from Richmond to Kinglake National Park, a

distance of 60 km, at an average speed of 80 km/h. He cycled back at an

average speed of 20 km/h. What was his average speed for the whole

journey? (Hint: It is not 50 km/h.)

2 The speed of the Discovery space shuttle while in orbit was 17 400 miles

per hour. What is this in km/h? (1 kilometre = 0.62 miles)

3 The rate of ascent for the Discovery space shuttle is 71 miles in

8.5 minutes.

a What speed is this in km/min?

b What speed is this in km/h?


Copy the sentences below. Fill in the gaps by choosing the correct word or

expression from the word list that follows.

1 Ratios compare of the same kind.

2 Ratios contain only and do not contain a name or any units.

3 The of the numbers in a ratio is important.

4 Before ratios are written, the numbers must be expressed in .

5 Ratios can be simplified by dividing each number by the .

6 If each number in the ratio is multiplied/divided by the same number, the

resultant ratio is to the original ratio.

7 Proportion is a statement of of the two ratios.

8 In any proportion the products of the are equal.

The process of finding such products is called cross-multiplication.

9 To compare the ratios, express them in a fractional form, and then

compare fractions by writing them with a .

10 Gradient measures the of a hill and can be calculated by

finding the ratio of the vertical distance and the horizontal distance

between any two points on the hill.

11 The larger the is, the steeper the hill.

12 Multiplying a quantity by a ratio greater than 1 causes it to ;

multiplying by a ratio smaller than 1 causes a quantity to .

13 To share a quantity in a given ratio, first the must be found.

14 Ratio scales are used in maps and plans. The first number represents the

, and the second number represents the .

15 To obtain the dimensions of an enlarged or reduced object, the original

dimensions are multiplied by the , expressed as a fraction.

16 Rates are used to measure and compare the changes in and

are written using the word per, or a slash (/).

17 Average speed is an important rate and is calculated by finding the

of the total distance travelled and the time taken.

summary

W O R D L I S T

scale factor

steepness

order

ratio

decrease

highest common

factor

numbers

diagonally

across from

each other

increase

the same units of

measurement

gradient

total number of

parts

drawing

quantities

actual object

different

quantities

whole numbers

equality

equivalent

common

denominator


1 On a farm there are 5 dogs, 3 cats, 17 cows and 1 horse. Write the following ratios.

a cats : dogs b horses : cows c cows : cats

d dogs : horses e dogs : other animals

2 Express each of the following ratios in simplest form.

a 8 : 16 b 24 : 36 c 35 mm : 10 cm

d $2 : 60 cents e 20 s : 1 min f :

g 4 : 10 h 56 : 80 i 2 hours : 40 min

j 1.5 km : 400 m

3 Find the value of n in each of the following proportions.

a b c

d e f

4 The directions for making lime cordial require the mixing of 1 part cordial to 6 parts of

water.

a Express this as a ratio.

b How much cordial would you have to mix with 9 L of water?

5 Which is the larger ratio?

a , b ,

6 Place a number in the box to make a ratio greater than 3 : 2 but less than 2 : 1.

K : 6

7 The horizontal and the vertical distances between the top and bottom points of slide A are

3 m and 2 m respectively. For slide B the horizontal distance between the top and bottom

points is 10 m, and the vertical distance is 4 m.

a Calculate the gradients of slide A and slide B.

b Which slide is steeper? Justify your answer.

8 a Reduce 70 kg in the ratio .

b Increase 90 kg in the ratio 10 : 9.

9 Seraphima’s salary of $38 000 has recently been increased by one-fifth. What is her new

salary?

10

Note: There may be more than one correct answer.

An amount of money is changed in the ratio a : b. Which of the following will see the

amount increased?

A a = 2, b = 3 B a = 1, b = 2 C a = 3, b = 2 D a = 2, b = 1 E a = 2, b = 2

7A

CHAPTERreview

7B1

2---

1

12------

1

3---

7Cn

3---

20

15------=

n

28------

5

7---=

2

3---

8

n---=

4

5---

12

n------=

6

n---

5

8---=

3

10------

n

4---=

7C

7D4

5---

2

3---

7

12------

5

8---

7D

7D

7E4

5---

7E

7Emultiple choice


11 a Divide $25 in the ratio 2 : 3.

b Share $720 in the ratio 7 : 5.

12 Three people share a Lotto prize

of $6600 in the ratio 4 : 5 : 6. What is

the difference between the smallest and

largest shares?

13 A map has a scale of 1 cm ⇔ 250 km. Write this scale as a ratio.

14 Change each of the following to a ratio scale:

a 1 cm represents 1 m b 1 cm represents 1 km

c 2 cm represents 5 km d 3 cm represents 60 km.

15 On a town plan the scale is given as 1 : 50 000.

a How far apart are two buildings that are 2 cm apart on the plan?

b A statue in a park is 5 km from the town boundary. How far would this be on the plan?

16 A car travels 840 km on 72 litres of petrol. Find the fuel consumption of the car in

L/100 km.

17 David’s car has a fuel consumption rate of 12 km/L, and Susan’s car has a fuel consumption

rate of 11 km/L.

a Which car is more economical?

b How far can David’s car travel on 36 L of fuel?

c How much fuel (to the nearest litre) would Susan’s car use in travelling 460 km?

18 A 1 kg packet of flour costs $2.80 and a 750 g packet costs $2.20. Which is the cheaper way

of buying flour?

19

Note: There may be more than one correct answer.

A car is travelling from Sydney to Brisbane, a distance of 1050 km. If the car departs

Sydney at 6.00 am and needs to arrive in

Brisbane by 6.00 pm, which of the

following speeds will get the car there in

time?

A 80 km/h B 90 km/h

C 100 km/h D 110 km/h

E 105 km/h

20 A 737 jet travels a journey of 2250 km in

2 h 30 min. Calculate its speed in:

a km/h b m/s.

21 A cyclist riding at 12 km/h completes

a race in 3 h 45 min.

a What is the distance of the race?

b At what speed would he have to ride

to complete the race in 3 h?

7F

7F

7G7G

7G

7H

7H

7H

7I multiple choice

7I

7I

testtest

CHAPTER

yourselfyourself

7

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Chap 07