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# eTHINKING 6

Ratios and
rates

7
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
How is Svetlana able to use
1m Tree this information to calculate
ruler the height of the tree?
1.52 m This chapter will show you
19.26 m
a way in which this problem
can be solved.
areyou278 Maths Quest 8 for Victoria

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.

## 7.1 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 1--- hours (minutes) j 4 weeks (days) k 2 years (months) l 2 years (weeks)
2

## 7.2 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.

## 7.3 Simplifying fractions

3 Simplify each of the following fractions.
9 20 36 72
a ------ b ------ c ------ d ---------
12 25 60 100

## 7.4 Finding and converting to the lowest common denominator

4 Write each of the following fraction pairs over the lowest common denominator.
1 3 3 7 2 5 7 3
a --- and --- b --- and ------ c --- and ------ d ------ and ---
3 4 8 12 3 12 10 4

## 7.5 Converting a mixed number to an improper fraction

5 Write the following mixed numbers as improper fractions.
a 1 3--- b 2 3--- 7
c 3 -----
- d 10 3---
4 5 10 8

## 7.6 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

## 7.7 Multiplying a whole number by a fraction

7 Perform the following multiplications.
a 30 × 2
--- b 60 × 4
--- c 1 --1- × 120 d 5
--- × 96
3 5 4 8

7.9
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
Chapter 7 Ratios and rates 279
Introduction to ratios
Ratios 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 1--- bucket of sand.
2
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 ⇔ 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.

WORKED Example 1
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 a 4:3
number of noughts. Write the 2 numbers
as a ratio (the number of crosses must be
written first).

## b Count the number of noughts and the b 3:2

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).
280 Maths Quest 8 for Victoria

7.1 Before ratios are written, the numbers must be expressed in the same units of measure-
HEET ment. Once the units are the same, they can be omitted. When choosing which of the
SkillS

Converting units of quantities to convert, keep in mind that ratios contain only whole numbers.
length, capacity
and time
WORKED Example 2
Rewrite the following statement as a ratio: 7 mm to 1 cm.
THINK WRITE
1 Express both quantities in the same 7 mm to 1 cm
units. To obtain whole numbers, 7 mm to 10 mm
convert 1 cm to mm (rather than 7 mm
to cm).
2 Omit the units and write the 2 numbers 7 : 10
as a ratio.

remember
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.

7A Introduction to ratios
WORKED 1 Look at the completed game of noughts and crosses and write the ratios of:
Example
1
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.
c Shaded parts : total parts
Chapter 7 Ratios and rates 281
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 2 9
4 5
d multiples of 2 to multiples of 5 7
6 8
e numbers divisible by 3 to numbers not divisible by 3.
WORKED 5 Rewrite each of the following statements as a ratio. 7.1 SkillS
Example
2
a 3 mm to 5 mm b 6 s to 19 s

HEET
c \$4 to \$11 d 7 teams to 9 teams Converting
e 1 goal to 5 goals f 9 boys to 4 boys units of
g 3 weeks to 1 month h 3 mm to 1 cm length,
capacity and
i 17 seconds to 1 minute j 53 cents to \$1 time
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.
282 Maths Quest 8 for Victoria

## 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?
97
9 Matthew received a score of --------
- for his Maths
100
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
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).

WORKED Example 3
Express the ratio 16 : 24 in simplest form.
THINK WRITE

## 1 Write the ratio. 16 : 24

2 Determine the largest number by which both 16 and ÷8 ÷8
24 can be divided (that is, what is the highest 2:3
common factor)? It is 8.
3 Divide both 16 and 24 by 8 to obtain an equivalent
ratio in simplest form.
Chapter 7 Ratios and rates 283
WORKED Example 4
Write the ratio of 45 cm to 1.5 m in simplest form.
THINK WRITE
1 Write the question. 45 cm to 1.5 m
2 Express both quantities in the same units by 45 cm to 150 cm
changing 1.5 m into cm. (1 m = 100 cm)
3 Omit the units and write the 2 numbers as a ratio. 45 : 150
4 Simplify the ratio by dividing both 45 and 150 by ÷ 15 ÷ 15
15 — the HCF. 3 : 10

## Graphics Calculator tip! Simplifying

ratios
Since simplifying ratios is similar to the process of
simplifying fractions, we can use the keys used
O
normally to operate with fractions. A ratio such as

CASI
16 : 24 can be entered as 16 ÷ 24. To simplify and Simplifying
obtain an answer in fractional form, we press MATH , ratios
select 1: Frac and then press ENTER . 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.

WORKED Example 5
Simplify the following ratios.
2 7 5
a --- : -----
- b --- : 5---
5 10 6 8

THINK WRITE
2 7
a 1 Write the fractions in ratio form. a --- : -----
-
5 10
2 Write equivalent fractions using the
lowest common denominator (in this
4 7
case, 10). ------ : -----
-
10 10
3 Multiply both fractions by 10. × 10 × 10
4 Check if the remaining whole 4:7
numbers that form the ratio can be
simplified. In this case they cannot.
Continued over page
284 Maths Quest 8 for Victoria

THINK WRITE
5
b 1 Write the fractions in ratio form. b --- : 5---
6 8
2 Write equivalent fractions using the
lowest common denominator (in this 20
------ : 15
------
case, 24). 24 24
× 24 × 24
3 Multiply both fractions by 24.
20 : 15
4 Check if the remaining whole
÷5 ÷5
numbers that form the ratio can be
4:3
simplified. In this case divide each
by the HCF of 5.

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.

WORKED Example 6
Write the following ratios in simplest form.
a 2.1 to 3.5 b 1.4 : 0.75
THINK WRITE
a 1 Write the decimals in ratio form. a 2.1 : 3.5
2 Both decimals have one decimal × 10 × 10
place, so multiplying each by 10 will 21 : 35
produce whole numbers. ÷7 ÷7
3 Simplify by dividing both numbers 3:5
by the HCF of 7.
b 1 Write the decimals in ratio form. b 1.4 : 0.75
2 Because 0.75 has two decimal × 100 × 100
places, we multiply each decimal by 140 : 75
100 to produce whole numbers. ÷5 ÷5
3 Simplify by dividing both numbers 28 : 15
by the HCF of 5.

remember
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).
Chapter 7 Ratios and rates 285

7B Simplifying ratios
WORKED 1 Express each ratio in simplest form. 7.2 SkillS
Example
3 a 2:4 b 3:9 c 5 : 10 d 6 : 18

HEET
e 12 : 16 f 15 : 18 g 24 : 16 h 21 : 14 Highest
i 25 : 15 j 13 : 26 k 15 : 35 l 27 : 36 common
factor
m 36 : 45 n 42 : 28 o 45 : 54 p 50 : 15
q 56 : 64 r 75 : 100 s 84 : 144 t 88 : 132
L Spre
2 Complete the patterns of equivalent ratios.

sheet
E
a 1:3 b 2:1 c 2:3 d 64 : 32 e 48 : 64 Simplifying
ratios
2:6 4:2 4:6 __ : 16 24 : __
_:9 _:4 6:_ __ : 8 12 : __
_ : 12 _:8 _ : 12 8 : __ __ : 8
5 : __ 20 : _ _ : 24 __ : 1 __ : __
WORKED 3 Write the following ratios in simplest form.
Example
4 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 1--- min to 300 s r 1.8 cm to 12 mm
2
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.
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.
286 Maths Quest 8 for Victoria

HEET
7.3
SkillS

6 In a primary school that has 910 students, 350 students are in the senior school and
Simplifying
fractions 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:
HEET
7.4 a senior students to junior students
SkillS

## Finding b senior females to senior males

and c senior males to total senior students
converting
to the d junior males to senior males
lowest
common e junior females to whole school population.
denominator
WORKED 7 Write the following ratios in simplest form.
Example

7.5 5 a 1
--- to 2
--- b 5
--- to 6
--- c 1
--- : 1---
HEET 3 3 7 7 4 2
SkillS

2
Converting d --- : 5--- e 4
--- to 9
------ f 2
--- : 3---
3 6 5 20 3 5
a mixed
number g 3
------ :1 h 1 2--- to 1
--- i 1 1--- : 1 1---
to an 10 3 3 4 2
improper
fraction j 3 1--- to 2 1--- k 1 : 1 3--- l 3 1--- to 2 4---
3 2 5 4 5

HEET
7.6 WORKED 8 Write the following ratios in simplest form.
SkillS

Example
Multiplying 6 a 0.7 to 0.9 b 0.3 : 2.1 c 0.05 to 0.15 d 0.8 : 1
decimals by
10, 100 e 0.25 : 1.5 f 0.375 to 0.8 g 0.95 : 0.095 h 1 to 1.25
and 1000 i 0.01 : 0.1 j 1.2 : 0.875 k 0.004 to 0.08 l 0.5 : 0.92
Chapter 7 Ratios and rates 287
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. Date of sale Selling price
a Compare the price of the house pur- March 2006 \$1.275 million
chased in December 1999 with the
purchase price in April 2003 as a December 2003 \$1.207 million
ratio in simplest form.
April 2003 \$1.03 million
b Compare the price of the house pur-
chased in December 1999 with the December 1999 \$850 000
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 multiple choice
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:
A 2 : 45 B 4:9 C 1 : 1.8 D 9:4 E none of these
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:
A 3:7 B 1 : 21 C 7 : 10 D 1:7 E none of these
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:
A \$180 B \$80 C \$50 D \$45 E \$66
d A coin was tossed 100 times and Tails appeared 60 times. The ratio of Heads to Tails
was:
GAME
A 2:3 B 3:5 C 3:2 D 5:3 E 2:5
time

e Out of a 1.25 L bottle of soft drink, I have drunk 500 mL. The ratio of soft drink Ratios
and
remaining to the original amount is: rates
A 2:3 B 3:5 C 3:2 D 5:3 E 2:5 — 001
288 Maths Quest 8 for Victoria

Proportion
A 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.
1 2
This can also be written as: --- = --- .
3 6
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.
3 9
Consider, for example, this proportion: --- = ------ . Again, 3 × 12 = 9 × 4.
4 12
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.

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

WORKED Example 7
Use the cross-multiplication method to determine whether the following pair of ratios is in
proportion: 6 : 9; 24 : 36.
THINK WRITE
6 24
1 Write the ratios in fraction form. --- ------
9 36
2 Perform a cross-multiplication. 6 × 36 = 216 24 × 9 = 216
3 Check whether the products are equal. 216 = 216
Therefore, the ratios are in proportion.

WORKED Example 8
a 6
Find the value of a in the following proportion: --- = --- .
3 9

THINK WRITE
a 6
1 Write the proportion statement. --- = ---
3 9
2 Cross-multiply and equate the products. a×9=6×3
3 Solve for a by dividing both sides of
9a = 18
the equation by 9.
9a 18
------ = ------
9 9
a=2
Chapter 7 Ratios and rates 289
WORKED Example 9
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
4 28
1 Let the number of boys be b and write a --- = ------
proportion statement. (Since the first 3 b
number in the ratio represents girls,
place 28 (the number of girls) as the
numerator.)
2 Cross-multiply and equate the products. 4 × b = 28 × 3
3 Solve for b by dividing both sides by 4. 4b = 84
4b 84
------ = ------
4 4
b = 21
4 Write the answer. There are 21 boys.

remember
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 a =
c
, then, using cross-multiplication, a × d = c × b.
b d

7C Proportion
WORKED 1 Use the cross-multiplication method to determine whether the following pairs of ratios Math
Example
are in proportion.
7
a 2 : 3; 8 : 12 b 4 : 7; 8 : 14 c 5 : 7; 10 : 14 Proportion
7 21 3 12
d 5 : 8; 10 : 16 e --- ; ------ f --- ; ------
9 25 8 32
L Spre
14 5 11 7 13 6
sheet
E

## g ------ ; --- h ------ ; --- i ------ ; ---

16 9 12 8 15 7 Proportion
8 24 3 6 21 49
j --- ; ------ k --- ; --- l ------ ; ------
9 27 5 8 18 42
L Spre
sheet
E

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

Example Proportion
8 a 4 a 8 a 2
a --- = --- b --- = ------ c --- = --- (DIY)
2 8 6 12 9 3
3 9 7 14 10 3
d --- = ------ e --- = ------ f ------ = ------
a 12 a 48 a 15
290 Maths Quest 8 for Victoria

3 a 12 a 8 a
g --- = ------ h ------ = --- i ------ = ---
7 28 10 5 12 9
35 5 24 6 30 2
j ------ = --- k ------ = --- l ------ = ---
7 a 16 a 45 a

## 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.
WORKED 4 Solve each of the following, using a proportion statement and the cross-multiplication
Example
9
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
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?
Chapter 7 Ratios and rates 291
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:
a 7
--- = ---
6 4
then a × 4 = 7 × 6
4a = 42
a = 10.5 (or 10 1--- )
2
Calculate the value of a in each of the following proportion statements. Write your
answer correct to 1 decimal place.
a 8 a 4 a 7 a 9 5 7
a --- = --- b --- = --- c --- = ------ d --- = ------ e --- = ------
7 5 6 5 3 10 9 10 a 10
8 6 9 a 13 a 9 7 7 9
f --- = --- g --- = --- h ------ = --- i ------ = --- j --- = ---
a 7 7 6 6 5 15 a 8 a
6 Write a proportion statement for each situation and then solve the problem. If neces-
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 multiple choice

p l
a If --- = ---- , then:
q m
A p×q=l×m B p×l=q×m C p×m=l×q
p l
D ---- = --- E none of these is true.
m q
x y
b If --- = --- , then:
3 6
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.
292 Maths Quest 8 for Victoria

23 x
c If ------ = ------ then, correct to the nearest whole number, x equals:
34 19
T 7.1 A 13 B 12 C 34 D 28 E 17
SHEE
d The directions on a cordial bottle suggest mixing 25 mL of cordial with 250 mL of
Work

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

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

1
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 multiple choice
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.
12 4
7 Use the cross-multiplication method to determine whether the pair of ratios ------ , --- is
in proportion. 15 5

21 3
8 Find the value of b in the proportion ------ = --- .
9 b
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?

QUEST
S
M AT H

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

E

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

A
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- D
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 C E B
length of DE?
Chapter 7 Ratios and rates 293
Have a laugh.
Solve the
Why are pianos ratio equations to find
the value of each letter
hard to open? and the puzzle code.

## 4 —45 6 6 4 —45 3 —43 7 —21 4 —51 4 —51 7 —21 1

2 4 —51 3

2 1 —21 6 3 —21 6 8 1

2 7 5 —21 6 4 —85 10 1

2 4 —85 4 6

3 : A = 15 : 24 G : 7 = 6 : 28 85 : 17 = M : 2 S : 5 = 16 : 40

B : 2 = 25 : 6 4 : H = 32 : 37 4 : 4 —21 = 8 : N 3:T=5:7

5 : 2 = 10 : C 26 : 60 = I : 12 O : 5 = 18 : 30 2 : 3 = U : 11

1 : 3 = D : 18 70 : 35 = 7 : J 7 : P = 35 : 57 21 : 24 = 7 : X

2 : 1 = 15 : E K : 11 = 30 : 132 1 : 2 = Q : 11 Z : 9 = 42 : 54

9:F=4:3 2—21 : L = 10 : 15 16 : 1 = 8 : R

## Why did the bees go on strike?

1
1 —2 3 8—31 4 —45 4 2 —21 4 —45 3 —43 7—21 4—51 4—51 7—21 1

2 6 —43 3 7 —212

1
1—2 11—52 2 9 11—52 2 6—43 5 —51 11—52 3 6—43 7 8 —31 3 7—21

## 4—51 5 —51 11 —52 2 7—31 6—43 2 1 —21 10 11 —52 8 6—43 2 4—51

294 Maths Quest 8 for Victoria

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.

WORKED Example 10
Which is the larger ratio in the following pair?
3:5 2:3
THINK WRITE
3 2
1 Write each ratio in fraction form. --- ---
5 3
9 10
2 Change each fraction to the lowest common ------ ------
denominator (which is 15). 15 15
9 10
3 Compare the fractions: since both fractions ------ < ------
have a denominator of 15, the larger the 15 15
numerator, the larger the fraction.
4 The second fraction is larger and it Therefore, 2 : 3 is the larger ratio.
corresponds to the second ratio in the pair.

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.
vertical distance
The gradient of a hill is calculated by finding the ratio --------------------------------------------- between
horizontal distance
rise
any 2 points on a hill. This is also known as calculating -------- .
run

WORKED Example 11
A
Find the gradient of the hill (AB),
Vertical
if AC = 2 m and BC = 10 m. distance
B
Horizontal distance C

THINK WRITE
vertical distance
horizontal distance
2
2 Vertical distance is 2 m and horizontal Gradient = ------
distance is 10 m. Substitute these 10
values into the formula.
1
3 Simplify by dividing both numerator Gradient = ---
and denominator by 2. 5
Chapter 7 Ratios and rates 295
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 1--- , and the gradient of hill B is 1--- ,
2 5

## we can conclude that hill A is steeper, as 1

--- > 1
--- .
2 5

remember
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
vertical distance
the ratio --------------------------------------------- . (The distances are measured between any 2
horizontal distance
points on the slope.)
296 Maths Quest 8 for Victoria

7D Comparing ratios
WORKED 1 Which is the greater ratio in each of the following pairs?
d Example
hca
10
a 1 : 4, 3 : 4 b 5 : 9, 7 : 9 c 6 : 5, 2 : 5
Mat

d 3 : 5, 7 : 10 e 7 : 9, 2 : 3 f 2 : 5, 1 : 3
Comparing
ratios g 2 : 3, 3 : 4 h 5 : 6, 7 : 8
i 5 : 9, 7 : 12 j 9 : 8, 6 : 5
L Sp he 2 In each of the following cases, decide which netball
et
EXCE

## team has the better record.

Comparing
ratios a St Marys won 2 matches out of 5. Colac won 5 out
of 10.
reads b Bright 50 won 13 out of 18. Corio won 7 out of 12.
L Sp he
c Seymour won 12 out of 20. Geelong won 14 out of
et
EXCE

Comparing 25.
ratios d Bell Post Hill won 8 out of 13. Bairnsdale won 13 out
(DIY)
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?
WORKED 4 a Find the gradient of each of the hills represented by the
Example
11
following triangles.
i A ii C iii E

2m
3m 2m
F
5m
B D
3m 3m
b Which slope has the largest gradient? iv G
c Which slope has the smallest gradient? 1.5 m
d List the hills in order of increasing steepness. H
7m
5 Draw triangles that demonstrate a gradient of:
2 3 4 3 2
a --- b --- c --- d --- e ---
1 1 3 2 5
6 multiple choice
L
a If the gradient of LN in the triangle at right is 1, then:
Aa>b B a<b Ca=b a
Da=1 E b=1
5 a
b If --- > --- , then a could be: N
b
6 5
A 4 B 5 C 6 D 7 E all of these numbers.
a 3
c If --- < --- , then:
b 5
Aa<3 B b<5 Ca<b Db<a E a=2
Chapter 7 Ratios and rates 297
Increasing and decreasing in a given
ratio
Consider the following products and their results.
5
20 × --- = 25 ⇒ The result (25) is larger than the original (20).
4
5
20 × --- = 20 ⇒ The result is the same as the original (20).
5
4
20 × --- = 16 ⇒ The result (16) is smaller than the original (20).
5
5
Observe that multiplying by a ratio greater than 1 (that is, --- ) caused an increase of a
4
5
quantity, multiplying by a ratio equal to 1 (that is, --- ) resulted in the quantity being
5
4
unchanged and multiplying by a ratio smaller than 1 (that is, --- ) caused a decrease of a
5
quantity.
Generally, multiplying by a ratio greater than one increases the quantity, while
multiplying by a ratio less than one decreases the quantity.

WORKED Example 12
Increase \$25 in the ratio 6 : 5.
THINK WRITE
1 The given ratio when expressed as a New amount = \$25 × 6
---
5
fraction (that is, 6--- ) is greater than 1.
5
So, to increase the amount, multiply it
by this ratio.
2 Evaluate. New amount = \$30

WORKED Example 13
Decrease 55 kg in the ratio 3 : 5.
THINK WRITE
1 The given ratio when expressed as a New mass = 55 kg × 3
---
5
fraction (that is, --3- ) is less than 1 and
5
therefore will cause a decrease in the
amount. So multiply the mass by this
ratio.
2 Evaluate. New mass = 33 kg
298 Maths Quest 8 for Victoria

WORKED Example 14
In recent years, bus fares have increased by one-fifth.
If the old fare was \$3, what is the new fare?
THINK WRITE
New fare = old fare + 1
--- of old fare
The old fare can be considered as 5
1
New fare = 1 + 1
---
a whole. Find the new fare as a fraction 5

## of the old fare (it is 1

--- bigger). New fare = 5
--- + 1
---
5 5 5

New fare = 6
--- of the old fare
5

## 2 Multiply the old fare by the ratio 6--- . New fare = 3 × 6

---
5 5
18
New fare = ------
5
3 Evaluate. New fare = \$3.60

Note: There is an alternative solution to worked example 14. We could first calculate
the increase in dollars and cents by finding 1--- of \$3 (since the increase was 1--- of the old
5 5
fare). We would then add the increase to the old fare to obtain the new fare.
Check this calculation:
Increase = 1
--- × \$3
5
= \$0.60
New fare = old fare + increase
= \$3 + \$0.60
= \$3.60 (same answer as before)

remember
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.

## Increasing and decreasing in

7E a given ratio
WORKED 1 Increase the following quantities in the ratio given.
HEET
7.7 Example
a Increase \$50 in the ratio 3 : 2. b Increase \$75 in the ratio 6 : 5.
12
SkillS

Multiplying
c Increase \$120 in the ratio 5 : 3. d Increase 24 mm in the ratio 7 : 4.
a whole e Increase 72 cm in the ratio 7 : 6. f Increase 155 m in the ratio 8 : 5.
number by g Increase 36 mg in the ratio 5 : 2. h Increase 84 g in the ratio 8 : 7.
a fraction
i Increase 132 kg in the ratio 13 : 12. j Increase 450 cm3 in the ratio 9 : 5.
Chapter 7 Ratios and rates 299
WORKED 2 Decrease the following quantities in the ratio given. Math
Example
a Decrease \$20 in the ratio 1 : 2. b Decrease \$60 in the ratio 2 : 3.

13
c Decrease \$140 in the ratio 3 : 5. d Decrease 36 mm in the ratio 5 : 6. Increasing
e Decrease 75 cm in the ratio 1 : 3. f Decrease 220 m in the ratio 3 : 4. and
decreasing
g Decrease 45 mg in the ratio 4 : 5. h Decrease 64 g in the ratio 7 : 8. in a given
i Decrease 99 kg in the ratio 7 : 9. j Decrease 840 cm3 in the ratio 11:12. ratio
L Spre
3 In each case decide whether the change will be an increase or a decrease and write the

sheet
E
new quantity. Increasing
and
a Change \$20 in the ratio 5 : 2. b Change \$55 in the ratio 2 : 5. decreasing
c Change \$50 in the ratio 9 : 10. d Change 54 m in the ratio 5 : 9. in a given
e Change 200 L in the ratio 6 : 5. f Change 450 km in the ratio 10 : 9. ratio
2
g Change 56 cm in the ratio 7 : 8. h Change 120 mg in the ratio 8 : 5.
L Spre
i Change 12.5 kg in the ratio 4 : 5. j Change \$6.80 in the ratio 3 : 2. XCE ad

sheet
E
Increasing
4 Council rates this year have increased in the ratio 5 : 4. If the old rates were \$1200, and
what are the new rates? decreasing
in a given
ratio (DIY)
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?

## WORKED 9 In each case, calculate the new quantity.

Example
14
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 multiple choice
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
300 Maths Quest 8 for Victoria

b The volume of air in a balloon was changed in the ratio a : b. The volume would
decrease if:
A a>b B a=b C a<b D b<a E a = 2 and
b=1
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 7 : 20 B 20 : 7 C 35 : 1 D 1 : 35 E 27 : 20

QUEST
S
M AT H

2 3
NG

1 To make two --- -cup servings of cooked rice, you add --- of a cup of rice,
3 4
E

CH LL
A 1 1
--- teaspoon of salt and 1 teaspoon of butter to 1 --- cups of water. How
4 2
2
many --- -cup servings of cooked rice can you make from a bag con-
3
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?
Chapter 7 Ratios and rates 301
Dividing in a given ratio
Consider 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.

WORKED Example 15
Share the amount of \$500 000 in the ratio 3 : 7.
THINK WRITE
1 Determine how many shares (parts) are in Total number of parts = 3 + 7
the ratio. Total number of parts = 10
3
2 The first share represents 3 parts out of a First share = ------ × \$500 000
10
3 First share = \$150 000
total of 10, so find ------ of the total amount.
10
3 The second share is the remainder, so Second share = \$500 000 − \$150 000
subtract the first share amount from the total Second share = \$350 000
amount.

Note: In worked example 15, the second share represents 7 parts out of the total of 10.
7
So the alternative way to calculate the size of the second share would be to find -----
- of
10
the total amount.
Check this calculation: Second share = 7
------ × \$500 000.
10
= \$350 000 (same answer as before)

WORKED Example 16
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
1 Find the total number of parts. Total number of parts = 1 + 2 + 4
Total number of parts = 7
2
2 There are 2 parts of sand to be used in Amount of sand = --- × 4.2 m2
7
2 Amount of sand = 1.2 m2
the mixture, so find --- of the total
7

remember
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.
302 Maths Quest 8 for Victoria

HEET
7.7 7F Dividing in a given ratio
SkillS

Multiplying 1 Write the total number of parts for each of the following ratios.
a whole
number by a 1:2 b 2:3 c 3:1 d 3:5 e 4:9
a fraction f 5:8 g 6:7 h 9 : 10 i 1:2:3 j 3:4:5
WORKED 2 Share the amount of \$1000 in the following ratios.
L Sp he Example
a 2:3 b 3:1 c 1:4 d 1:1 e 3:5
et
EXCE

15
Dividing f 5:3 g 3:7 h 9:1 i 7 : 13 j 9 : 11
in a given
ratio 3 If Nat and Sam decided to share their lottery winnings of \$10 000 in the following
ratios, how much would each receive?
reads a 1:1 b 2:3 c 3:2 d 3:7 e 7:3
L Sp he
et
EXCE

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

Dividing
in a given 4 Rosa and Mila bought a lottery ticket costing \$10. How should they share the first
ratio (DIY) 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?
WORKED 5 Concrete mixture is made up of 1 part cement, 2 parts sand and 4 parts blue metal.
Example
16
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
Chapter 7 Ratios and rates 303
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 multiple choice
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:
A 3 cm2 B 5 cm2 C 8 cm2 D 10 cm2 E 16 cm2

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

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?
7 5 7 3 5
A ------ B --- C --- D --- E ------
12 7 5 5 12

d A television channel that telecasts only news, movies and sport does so in the ratio
SHEE
T 7.2
2 : 3 : 4 respectively. How many movies, averaging a length of 1 --1- hours, would be

Work
2
shown during a 24 hour period?
A2 B 3 C4 D5 E 6

2
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.

50 6
3 Find the value of n in the proportion ------ = --- .
n 9
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?

## 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?
304 Maths Quest 8 for Victoria

History of mathematics
T H E R H I N D PA P Y R U S ( c . 1 8 5 0 BC)

## During this time . . .

The Sumerians built the
first cities, invented
wheels from date palm
trunks.
Papyrus reeds were
used to make boats,
The Bronze Age began.

The ancient Egyptians differed from the The Rhind papyrus was named after the
ancient Greeks in that Egyptians thought Scottish Egyptologist, A. Henry Rhind, who
about mathematics in a practical, rather than bought the 6-m scroll in 1858. A scribe
an abstract, way. They didn’t like fractions named Ahmes is believed to have copied it in
that had numerators other than one (except around 1650 BC from a document originally
the fraction two-thirds for reasons still written about 200 years before that. This
unknown). They found that fractions with papyrus shows a method for multiplying
numerators of one, unit fractions, were easy numbers using only addition and subtraction.
to multiply, since the numerator would Also known as the aha papyrus (aha meaning
unknown quantity to be determined, an early
always be one; for example, 1
--- × 1
--- = 1--- .
2 3 6 pronumeral), it is now in the British Museum
The Egyptians developed ingenious in London.
methods to avoid using any fraction other Questions
than those with a numerator of one. Solutions 1. Which numerator did the Egyptians use in
to many Egyptian problems concerned with their calculations with fractions?
beer and bread were recorded on papyri. The 2. Which fraction was an exception to this
most famous of these is the Rhind papyrus, rule?
which contains 84 problems and their 3. What practical problems did most of the
solutions including the calculation of the solutions deal with?
ancient Egyptian value for pi (π) of 3.1605. Research
A part of the papyrus is shown in the How was Egyptian multiplication done with
photograph above. only addition and subtraction?
Chapter 7 Ratios and rates 305
Scale drawing
You 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.
er
CASPIAN
hine

Riv
Riv
SU R DEPRESSION

er

er
D E i ve

r
Rive
TI r Don
ets Riv
C er
River

CA River

MT
RP Dni Astrakhan

Ri
be AT ester

S
Danu n

ve
Do

r
River HI Riv
River AN er

za
Tis
r Grossglockner Sea of

MT
3801 m Azov
ALPS

S
CASPIAN SEA
VE MT

Crimea CAUCASUS
River MTS
LE S

IAN
AP TRANSYLVAN Mt Elbrus 5642 m
BI

EN
ALPS
T

NE RI River
D

## S AT Danube BLACK SEA

IN

IC
AR

SE BALKAN MTS
IC

Elba A AL
Crkvice PS
Balkan
Peninsula
TYRRHENIAN Vesuvius
SEA 1281 m
Mt Olympus ASIA
2917 m AEGEAN
IONIAN
AN SEA
Mt Etna
3323m 1 : 28 000 000
0 250 500 750 1000 km
Malta Cape Tainaron Rhodes
SE 1 centimetre on the map represents
A Cyprus
280 kilometres on the ground.
Crete Conformal Conic Projection

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.

WORKED Example 17
Rewrite the ratio scale 1 : 10 000, using the most appropriate units.
THINK WRITE
1 Choose a unit for the ratio (usually cm); 1 : 10 000
it must be the same for both numbers. 1 cm ⇔ 10 000 cm
2 Since 10 000 is quite a large number,
1 cm ⇔ 10 000 ÷ 100
convert to a suitable unit. In this case it
1 cm ⇔ 100 m
is most appropriate to convert to
metres. Therefore divide by 100.

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.
306 Maths Quest 8 for Victoria

WORKED Example 18
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
1 Let the height of the tree be h. h = height of the tree
1 5
2 The ratio of the length of the drawing to --------- = ---
100 h
5
the actual length of the tree is --- . At the
h
same time we know that the scale used
1
for the drawing is --------- . Equate these
100
two ratios to form a proportion.
3 Cross-multiply. 1 × h = 5 × 100
4 Find the value of h. (The units of length 1 × h = 500 cm
for the object must be the same as the
units of length used for the drawing.)
5 Convert the height to the more The actual height of the tree is 5 m.
appropriate units and write the answer.

## 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 2--- . To reduce an original shape by using a
1
scale factor of 1 : 2, we would multiply each length of the shape by 1--- .
2
(Compare this with increasing or decreasing an amount in a given ratio. Can you see
any resemblance?)

WORKED Example 19
Find the new dimensions of a rectangle 3 cm × 2 cm if it is enlarged using a scale factor of
3 : 1.
THINK WRITE
3
1 Each dimension has to be enlarged by New length = 3 × ---
1
3
the scale factor --- . So multiply each, = 9 cm
1 3
New width = 2 × ---
3 1
length and width by --- to find the
1 = 6 cm
dimensions of the rectangle after the
enlargement.
2 Write the answer. The enlarged rectangle is 9 cm × 6 cm.
Chapter 7 Ratios and rates 307
remember
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.

7G Scale drawing

WORKED 1 Rewrite the following ratio scales using the most appropriate units. 7.8 SkillS
Example
17
a 1 : 10 b 1 : 100 c 1 : 1000 d 1 : 10 000 e 1 : 100 000

HEET
f 1 : 5000 g 1 : 60 000 h 1 : 400 i 1 : 750 000 j 1 : 2 200 000 Converting
units of
2 Write the following as ratio scales. length,
capacity
a 1 cm ⇔ 20 cm b 1 cm ⇔ 50 000 cm and time
c 1 cm ⇔ 10 m d 1 cm ⇔ 200 m Math
e 1 cm ⇔ 5 km f 1 cm ⇔ 50 km

g 1 cm ⇔ 5.5 km h 2 cm ⇔ 2 km Ratio
i 2 cm ⇔ 2.5 km j 3 cm ⇔ 6 km scales

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

a Find the actual distance between:

sheet
E
i two buildings that are 3 cm apart on the plan Map
scales
ii two streets that are 5 cm apart on the plan
iii two houses that are 2 mm apart on the plan
L Spre
iv two parks that are 6.5 cm apart on the plan XCE ad

sheet
E
v the town hall and the high school that are 7.5 mm apart on the plan Map
vi two cinemas that are 2.8 mm apart on the plan. scales
b Find the distances on the plan between: (DIY)
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.

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

Example
18
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?
308 Maths Quest 8 for Victoria

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: Kitchen/Living Bedroom 1
a the dimensions of the lounge
Lounge
b the width of the hallway Linen
c the length of the garage
d the floor area of the kitchen/ Laundry Bath Bedroom 3 Bedroom 2
living room WC
e the total floor area of the largest
Scale 1:200
bedroom. Garage
(Hint: Use a ruler to find the
required lengths on the plan first.)

WORKED 6 Find the new dimensions of the following shapes if they are enlarged using the scale
Example
19
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 multiple choice
a On a map the scale is given as ‘5 cm represents 2.5 km’. In ratio form, this would
be:
A 1 : 0.5 B 1:5 C 1 : 50 D 1 : 5000 E 1 : 50 000
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 2:1 B 3:1 C 1:2 D 2:3 E 3:2
Chapter 7 Ratios and rates 309
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?
A 6 km
B 7 km C
C 8 km
D 9 km
E 2 km

A
B

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 1 : 120 B 1 : 40 Bampton
C 1 : 120 000 D 1 : 4 000 000 Cordwell
E 1 : 1 200 000

## THINKING How high is that tree?

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 1m
measurements on a sunny day? ruler Tree
To calculate the height of the tree,
produce a scale drawing of the tree 1.52 m
19.26 m
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!
310 Maths Quest 8 for Victoria

Rates
Whether 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.

WORKED Example 20
Express the following statement using a rate in simplest form:
The 30 litre container was filled in 3 minutes.
THINK WRITE
30 L
1 A suitable rate would be litres per minute Rate = -------------
(L/min). Put the capacity of the container in 3 min
10 L
the numerator and the time in which it was Rate = -------------
filled in the denominator of the fraction. 1 min
2 Simplify the fraction. Rate = 10 L/min

WORKED Example 21
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
1 The rate is given in \$ per hour. So it actually Payment per 1 hour = \$8.50
tells us the amount of money earned in each
hour; that is, the hourly payment.
2 State the number of hours worked. Hours worked = 6
3 To find the total payment, multiply the hourly Total payment = \$8.50 × 6
payment by the total number of hours worked. Total payment = \$51

remember
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).
Chapter 7 Ratios and rates 311

7H Rates

1 What quantities (such as distance, time, volume) are changing if the units of rate are:
a km/h? b cm3/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
WORKED 3 Express each of the following statements using a rate in simplest form.
Example
20

## 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.
312 Maths Quest 8 for Victoria

## 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.
WORKED 4 Sima is paid \$15.50 per hour. At this rate, how much does she earn in a day on which
Example
21
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?
Chapter 7 Ratios and rates 313
17 multiple choice
a A case containing 720 apples was bought for \$180. The cost could be written as:
A 30 cents each B 20 cents each C \$3.00 per dozen
D \$2.00 per dozen E \$2.80 for 10
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.)
A 65 B 68.2 C 71.8 D 73.2 E 74.1
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)?
A \$1400 B \$1450 C \$1500 D \$1550 E \$1600
d A tank of capacity 50 kL is to be filled by a hose whose flow rate is 150 L/min. If GAME
the tap is turned on at 8 am, when will the tank be filled?

time
A Between 1.00 pm and 1.30 pm B Between 1.30 pm and 2.00 pm Ratios
C Between 2.00 pm and 2.30 pm D Between 2.30 pm and 3.00 pm and rates
— 002
E Between 3.00 pm and 3.30 pm

Speed
If 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:
distance travelled
Average speed = ---------------------------------------------
time taken
distance
Corresponding to the ratio ------------------- , the units of measurement for speed contain units
time
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).

WORKED Example 22
Calculate the average speed of a train which travels 550 km in 10 h.
THINK WRITE
distance travelled
1 Write the formula for calculating average Average speed = ------------------------------------------
speed. time taken
550 km
2 Substitute the values for the distance and Average speed = ------------------
time into the formula and evaluate. 10 h
55 km
Average speed = ---------------
1h
Average speed
3 Write the answer with the appropriate unit. Average speed = 55 km/h
314 Maths Quest 8 for Victoria

WORKED Example 23
Express the speed of 60 km/h in m/s.
THINK WRITE
1 Write the speed in fraction form. distance
Speed = -------------------
time
60 km
= ---------------
1h
2 Convert the given units to the required 60 × 1000 m
= ------------------------------- Speed
units — change kilometres to metres 1 × 60 × 60 s
(that is, multiply by 1000) and hours to 60000 m
seconds (that is, multiply by 3600). = ---------------------
3600 s

## 3 Simplify. Speed = 16 2--- m/s

3

WORKED Example 24
At right is a graph showing a bushwalker’s hike.
Distance from campsite (km)
Write a short story describing his hike. 20
18
16
14
12
10
8
6
4
2
0
0 1 2 3 4 5 4 5 6 7 8 9 10 11
THINK WRITE
Time (hours)
1 Label the relevant points onto the
Distance from campsite (km)

graph. 20 E
18 C
16
14 D
12
10
8 A
6 B
4
2 F
0
0 1 2 3 4 5 4 5 6 7 8 9 10 11
Time (hours)
2 Determine the coordinates of the first Coordinate A (2, 7)
point from the origin and interpret this The bushwalker has travelled for 2 hours and he
point in terms of the problem (that is, is 7 km away from the campsite.
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.
Chapter 7 Ratios and rates 315
THINK WRITE
3 Determine the coordinates of the Coordinate B (3, 7)
second point from the origin and The bushwalker is still 7 km from the campsite,
interpret this point in terms of the 1 hour later. He may have stopped for a break
problem. or a meal. A horizontal line on the graph
Note: A horizontal line on the graph indicates the hiker has not moved.
indicates the hiker has not moved.
4 Determine the coordinates of the next Coordinate C (5, 16)
leg of the journey and interpret this The bushwalker is 16 km from the campsite. It
point in terms of the problem. Compare has taken him 2 hours to travel 9 km. He has
this section of the graph with that in covered this leg of the journey at a faster pace
step 2. than the first leg.
5 Determine each of the remaining Coordinate D (6, 16)
coordinates and highlight any important The bushwalker has taken another 1-hour break.
points. 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)
the campsite.

remember
1. Average speed is an important rate that can be calculated if the distance
travelled and the time taken are known.
distance travelled
2. Average speed = ------------------------------------------
time taken
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.

## 7I Speed 7.9 SkillS

HEET

Converting
WORKED 1 Calculate the average speed (in km/h) in each of the following cases. minutes to
Example
22
a A car travels 400 km in 5 h. a fraction
of an hour
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.
316 Maths Quest 8 for Victoria

d 2 Calculate the average speed (in km/h) for each of the following.
hca
a A car travels 40 km in 30 min.
Mat

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

converter 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.
reads h A racing car completes a lap of 3 km in 1 min.
L Sp he
i A rally driver completes a course of 1500 km
et
EXCE

Speed in 18 h 45 min.
converter 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.
WORKED 5 Express each of the following speeds in m/s.
Example
23
a 36 km/h b 72 km/h c 54 km/h
d 90 km/h e 180 km/h

Distance (km)

## Geoff’s trip on his motorbike. 210

a Calculate Geoff’s average speed. 140
b If he continued at this average speed, how far
70
would he travel in 7 h?
c If he continued at this speed, how long (to the nearest 0
quarter-hour) would it take him to ride 400 km? 0 1 2 3 4
Time (h)
7 At 8 am, Rob sets off from a starting point to
canoe down a river. At 10 am, Kate sets off
40
from the same starting point in her motorboat.
Distance (km)

## The graph below shows their journeys. 30

a What is Rob’s average speed? b
20 Ro
b When Kate passes Rob:
te
Ka

## i what time is it? 10

ii how far from the starting point are they?
0
c Rob and Kate have agreed to meet at a point 8 am 10 am 12 noon 2 pm
80 km from the starting point. At what time 9 am 11 am 1 pm
Time (hours)
will Kate arrive at this meeting point?
Chapter 7 Ratios and rates 317
WORKED 8 Below is a graph, showing a bushwalker’s hike. Write a short story describing her hike.
Example
24
20

Distance (km)
16
12
8
4
0
0 1 2 3 4 5 6
Time (h)

9 multiple choice
a Over a short distance, an emu can reach a speed of 30 m/s. This is approximately:
A 60 km/h B 80 km/h C 90 km/h D 100 km/h E 110 km/h
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:
A 3 km/h B 4.5 km/h C 5 km/h D 6 km/h E 9 km/h
c The graph of a car’s journey is shown at right.
Which of the following is true?

Distance (km)
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. Time (h)
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.
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? T 7.3
SHEE
A 500 km from Sydney B 450 km from Sydney

Work
C 600 km from Sydney D 600 km from Melbourne
E 550 km from Melbourne

QU EST
S
M AT H

NG

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

E

CH LL average speed of 20 km/h. What was his average speed for the whole
A
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?
318 Maths Quest 8 for Victoria

summary
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.

WORD LIST
steepness diagonally total number of whole numbers
order across from parts equality
ratio each other drawing equivalent
decrease increase quantities common
highest common the same units of actual object denominator
factor measurement different
Chapter 7 Ratios and rates 319

CHAPTER
review
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 7A
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 7B
d \$2 : 60 cents e 20 s : 1 1--- min 1 1
f -----
- : ---
2 12 3
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.
n 20 n 5 2 8
7C
a --- = ------ b ------ = --- c --- = ---
3 15 28 7 3 n
4 12 6 5 3 n
d --- = ------ e --- = --- f ------ = ---
5 n n 8 10 4
4 The directions for making lime cordial require the mixing of 1 part cordial to 6 parts of
water. 7C
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?
4 2 7 5
7D
a --- , --- b ------ , ---
5 3 12 8
6 Place a number in the box to make a ratio greater than 3 : 2 but less than 2 : 1.
K:6 7D
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 7D
points is 10 m, and the vertical distance is 4 m.
a Calculate the gradients of slide A and slide B.
4
8 a Reduce 70 kg in the ratio --- .
5
b Increase 90 kg in the ratio 10 : 9.
7E
9 Seraphima’s salary of \$38 000 has recently been increased by one-fifth. What is her new
salary? 7E
10 multiple choice
7E
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
320 Maths Quest 8 for Victoria

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

7F b Share \$720 in the ratio 7 : 5.
12 Three people share a Lotto prize
7F 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.

7G
14 Change each of the following to a ratio scale:
7G 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.
7G 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
7H L/100 km.
17 David’s car has a fuel consumption rate of 12 km/L, and Susan’s car has a fuel consumption
7H 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
19 multiple choice
7I 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
7I 2 h 30 min. Calculate its speed in:
a km/h b m/s.
21 A cyclist riding at 12 km/h completes
7I a race in 3 h 45 min.
a What is the distance of the race?
test
yourself b At what speed would he have to ride
CHAPTER
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