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Ratios
Ratio
3
What is a ratio?
3
Simplifying ratios
4
Ratios
7
Equivalent ratios
9
Problems involving ratio
11
Dividing a quantity in a given ratio
13
Similar figures
17
Suggested responses to activities
4930CF: 7 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
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4930CF: 7 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Ratio
What is a ratio?
A ratio compares two or more quantities of the same kind in a special order.
For example, if you have four red smarties and seven yellow smarties,
then the ratio is written as:
Red smarties:Yellow smarties = 4:7
Figure 1: Example of ratios
We read this as ‘the ratio of red smarties to yellow smarties is 4 to 7’. The
numbers 4 and 7 are called the terms of the ratio.
Look at the following advertisement:
Great bargains!!
Melbourne—2 nights from just $259
Gold Coast—5 nights from just $319
Adelaide—2 nights from just $369
Cairns—5 nights from just $589
What is the cost ratio of the Melbourne trip compared to the Adelaide trip?
The answer is 259:369.
If we wrote 369:259, it would be wrong, as this would be the Adelaide cost
compared to the Melbourne price.
4930CF: 7 Applying Everyday Mathematics
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The order of the numbers in a ratio is very important.
Simplifying ratios
Now let’s look at simplifying ratios. We usually simplify ratios in the same
way that we simplify fractions.
In comparing two quantities we often do not know the actual size of either.
For example, Kate says she has ‘four times as much money as Bjorn.’
There are an infinite number of actual ratios that could describe this
situation, such as:
(a) $8: $2
(b) $20: $5
(c) $40: $10
(d) $120: $30
(e) 16c: 4c
In all of these, the first term is four times the second term and the simplest
ratio to express this is 4:1. In this way ratios can be simplified like
equivalent fractions.
That is:
1
2
Simplify:
(a)
8:6 ________________
(b)
36:12 ______________
(c)
9:12:3 ______________
A fruit drink contains 50% pineapple juice, 20% orange juice and the
rest is water. What is the ratio of:
(a) Pineapple to orange juice? _________________________________
(b) Orange juice to water? ____________________________________
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(a)
8:6 = 4:3 (Divide both numbers by 2.)
(b)
36:12 = 3:1 (Divide both numbers by 12, the highest factor
of both.)
9:12:3 = 3:4:1 (Divide each number by 3.)
P:O = 50:20 = 5:2 (Divide both number by 10.)
(c)
(a)
(b)
O:W = 20:30 = 2:3 (Divide both numbers by 10 noting that water
must make up 30% of the juice giving a total of 100%.)
Note: For large terms you may need to divide the numbers several times.
For example, to simplify:
A ratio is in its simplest form when its terms are whole numbers that have
no common factors.
Activity 7A
Write the answers to the following problems in your workbook.
1
There are four blue-eyed people and one brown-eyed person in my family. Write down
the ratio for blue-eyed people to brown-eyed people.
2
A university student has three mathematics texts, four English texts and two
psychology texts. What is the ratio of:
(a) Mathematics texts to English texts?
(b) Psychology texts to mathematics texts?
3
A survey of traffic in the main street stated: ‘There are five trucks for every fourteen
cars’.
(a) What is the ratio of trucks to cars?
(b) Does the statement mean that only five trucks and fourteen cars were counted?
4930CF: 7 Applying Everyday Mathematics
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4
A proportion is a statement that two or more ratios are equal. State whether these
proportion statements are true or false:
(a) 6:12 = 1:2
(b) 49:7 = 9:1
(c) 8:32 = 4:1
(d) 27:6 = 9:2
(e) 24:36 = 8:13
(f) 2:4:6 = 1:2:3
5
Write the following ratios in their simplest form:
(a) 4:8
(b) 20:15
(c) 18:16
(d) 22:12
(e) 2.4:3.6 [Hint: multiply both terms by 10 then simplify.]
(f) 15:10:100
(g) 200:40:80
6
Alec and Bertha invest $4200 and $2800 each in a business. What is the ratio of Alec’s
share to Bertha’s share?
Check your answers with the suggested responses at the end of the topic.
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4930CF: 7 Applying Everyday Mathematics
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Ratios in real life
It is important that you compare things of the same kind. For example,
length must be compared with length, not with weight.
In comparing things we usually make sure that they are measured in the
same units.
Figure 2: Length and weight
Example 1
Simplify 18 kg:9 kg.
The units are the same, so 18:9 = 2:1
Example 2
This example has different units. Change the hours to minutes.
2 ½ hours: 50 minutes
= 150 minutes: 50 minutes
= 3: 1
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(Remember, there are 60 minutes in
1 hour and 30 minutes in ½ hour.)
Remember, simplify by dividing.
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Activity 7B
Write the answers to the following problems in your workbook.
Write the following ratios in their simplest form, after first changing to the same units.
(a)
1 m:50 cm
(There are 100 centimetres in 1 metre.)
(b)
1 ½ hours:30 minutes
(c)
2 days:6 hours
(d)
$1.50:70 cents
(e)
$14:$7.50
(f)
1 m:12 cm
(g)
2 g:2 kg
(There are 1000 grams in 1 kilogram.)
(h)
5 cm:20 mm
(There are 10 millimetres in 1 centimetre.)
(i)
3 L:500 mL
(There are 1000 millilitres in 1 litre.)
(Remember, 24 hours in one day.)
Check your answers with the suggested responses at the end of the topic.
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4930CF: 7 Applying Everyday Mathematics
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Equivalent ratios
As well as simplifying ratios, you can change them by multiplying the
terms by the same number.
Here’s an example.
Find the missing term in this proportion statement:
8:5 = 24:
Answer
(Multiply both terms by 3, since 24 = 8 3)
Therefore  = 15
Another example
From a box of smarties take out three orange smarties and two brown
smarties. If you double the number of orange smarties to six, what would
you have to do to keep the ratio of orange smarties to brown smarties the
same?
Answer
Orange:brown = 3:2
3:2 = 6:
(Multiply both terms by 2, since 6 = 3 × 2)
So you would need four brown smarties to keep the ratio of orange smarties
to brown smarties the same.
4930CF: 7 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
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Activity 7C
Write the answers to the following problems in your workbook.
Complete the following ratios by filling in the blank squares:
(a) 2:5 = 4:
(b) 8:1 = 24:
(c) 3:2 = :6
(d) 3:4 = :40
(e) 2:7 = 12:
(f) 6:5 = 30:
(g) 9:36 = 36:
(h) 1:4 = :12
(i) 2:3:4 = :15:
Check your answers with the suggested responses at the end of the topic.
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Problems involving ratio
In the first type of ratio problem we will look at, we are told the ratio of
two quantities. We are then given one quantity and asked to find the other.
Examples
1
In a group of people, the ratio of people with brown eyes to blue eyes is
5:3. If there are 21 blue-eyed people, how many brown-eyed people are
there?
2
In a tennis club the ratio of men to women is 4:7. If there are 32 men,
how many women are there?
Answers
1
Brown:blue = 5:3
That is, 5:3 = :21
(Multiply both terms by 7, since 21 = 3 × 7)
So there are 35 brown-eyed people.
2
(Multiply both terms by 8, since 32 = 4 × 8)
So there are 56 women in the club.
4930CF: 7 Applying Everyday Mathematics
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Activity 7D
Write the answers to the following problems in your workbook.
1
The ratio of adults to children watching a soccer game is 10:3. If there are 60 adults
how many children are there?
2
Daniel and Tracey divide some money in the ratio 2:3. If Tracey has $2.70 how much
does Daniel have?
3
A mortar mix is made up of sand and cement in the ratio 4: 1. If 12 buckets of sand is
used how much cement is needed?
4
The ratio of a boy’s weight to that of his father is 3:7. If the boy weighs 33 kg, how
much does his father weigh?
5
The ratio of teachers to students in a TAFE college is 1:15. In a college of 960
students, how many teachers are there?
6
A type of solder is made by mixing lead and tin in the ratio 2:3. How much lead would
be needed to be mixed with 150 g of tin?
7
At a college the ratio of female students to male students is 5:4. If there are 420 male
students, find:
(a) the number of female students
(b) the total enrolment of the College.
8
A sum of money is divided between three friends in the ratio 8:3:5. If the smaller share
is $15, what are the other shares and how much was there to be divided?
Check your answers with the suggested responses at the end of the topic.
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4930CF: 7 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Dividing a quantity in a given ratio
The second type of ratio problem involves dividing or sharing a quantity in a
given ratio. In this case the total quantity is known.
Examples
1
Ahmed and Belinda do some odd jobs for their neighbour, who pays
them $40. If Ahmed works for two hours and Belinda works for
three hours find:
(a) the ratio of Ahmed’s hours to Belinda’s hours
______________________________________________________
(b) what fraction of the total hours did:
(i) Ahmed work? ______________________________________
(ii) Belinda work? ______________________________________
(c) How much should each get?
______________________________________________________
______________________________________________________
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In a town the ratio of adults to children is 2: 5. If there are 9100
people in the town how many children are there?
A dough mix is made by mixing flour, water and salt in the ratio 10:9:1.
How much of each ingredient is needed to make 400 g of dough?
4930CF: 7 Applying Everyday Mathematics
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Answers
1
(a) 2:3
(b) (i)
(Note: total hours = 2 + 3 = 5)
(ii)
(c) Ahmed worked for
of the time so should get
Belinda worked for
of the time so should get
of 40.
of 40.
Therefore:
2
Ahmed should get
× 40 = $16
Belinda should get
× 40 = $24.
The total population is 9100.
Adults:children = 2:5
Note: There is no need to find the number of adults as the question only
asks for the number of children.
3
14
Flour is
of the mix.
Water is
of the mix.
4930CF: 7 Applying Everyday Mathematics
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Salt is
of the mix.
Activity 7E
Write the answers to the following problems in your workbook.
1
A class contains 14 boys and 16 girls.
(a) What is the ratio of boys to girls?
(b) What is the ratio of boys to the total number of students in the class?
(c) What fraction of the class are boys?
(d) What fraction of the class are girls?
2
Concrete is made by mixing cement, sand and gravel in the ratio 1:3:4.
(a) What fraction of concrete is cement?
(b) What fraction of concrete is gravel?
(c) How much gravel is needed to make 64 kg of concrete?
3
Share these amounts in the ratio given:
(a) 15 pens in the ratio 1:2
(b) $60 in the ratio 3:2
(c) 420 kg in the ratio 3:4
(d) $2.50 in the ratio 8:2.
4
A chemical solution is made by mixing an acid with water in the ratio 1:24. How much
acid is needed for a 200 mL solution?
4930CF: 7 Applying Everyday Mathematics
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5
Company A and Company B wish to invest in the following business opportunity
in the ratio 5:6. How much money does each company need to put into the deal?
NOWRA REAL ESTATE
$88 000 o.n.o.
2 fabulous commercial strata suites.
Suit professional or small business
Prime position.
BARGAIN.
6
Frank and Nigel pooled their savings to start up a lawn mowing business.
Frank invested $6000 while Nigel invested $4000.
(a) Find the ratio of their investments, in simplest form.
(b) In their first month of trading they made a profit of $270. How much should
each receive, if this is divided up in the ratio of their investments?
7
Sarah and Rebecca contributed $7 and $3 (respectively) to buy a $10 raffle ticket
which had a $25 000 first prize. They won! How much should each receive?
8
A bread mixture is made by mixing flour, water and honey in the ratio 6:3:1.
What quantity of each ingredient is needed to make a 450 g loaf?
9
A recipe for a punch requires fruit juice and soft drink to be mixed in the ratio 5:3.
To make 10 litres of punch,
(a) how much fruit juice is needed?
(b) how many 1.25 L bottles of soft drink will need to be purchased?
10 A metal alloy is made by mixing copper and nickel in the ratio 7:5. What mass of each
metal is contained in a 6 kg block of the alloy?
Check your answers with the suggested responses at the end of the topic.
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4930CF: 7 Applying Everyday Mathematics
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Similar figures
When we enlarge a photograph, the enlargement is the same shape as the
original photo. We say that the enlargement and the original are similar
figures.
Two figures are similar if they have the same shape.
1
(a)
A photograph is 15 cm long and 9 cm wide. If the photograph is to
be enlarged so that the length is 30 cm, what will the width be?
___________________________
(b)
(c)
2
A photograph is enlarged. If the length is tripled, what happens to
its width?
___________________________
When a photograph is enlarged is it possible to increase the length
without changing the width?
___________________________
The two rectangles shown are similar.
What is the ratio of:
(a) the lengths
____________
(b) the widths
____________
1
(a)
18 cm.
2
(b)
(c)
(a)
It is tripled.
No.
8:16 = 1:2
(b)
3:6 = 1:2
4930CF: 7 Applying Everyday Mathematics
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In similar figures all matching angles are equal and all matching sides are in
the same ratio.
Example
The diagram shows two figures with equal angles marked by the same
symbol.
(a) What is the ratio of corresponding (matching) sides?
(b) Are the figures similar?
Answer
(a)
(b) Since matching angles are equal and matching sides are in the same
ratio the figures are similar.
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4930CF: 7 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Activity 7F
Write the answers to the following problems in your workbook.
1
By checking ratios of matching sides, join each pair of similar rectangles. The first has
been done for you.
4930CF: 7 Applying Everyday Mathematics
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After checking that matching angles are equal and that their matching sides are in the
same ratio, join each pair of similar figures.
Check your answers with the suggested responses at the end of the topic.
4930CF: 7 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
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Suggested responses to activities
Activity 7A
1
4:1
2
(a) 3:4
3
4
(b) 2:3
(a) 5:14
(b) No. The simplified ratio is 5:14. There may have been 10 trucks
and 28 cars, or 50 trucks and 140 cars etc.
(a) T
(b)
(c)
(d)
(e)
F
F
T
F
5
(f) T
(a) 1:2
6
(b) 4:3
(c) 9:8
(d) 11:6
(e) 2:3
(f) 3:2:20
(g) 5:1:2
4200:2800 = 3:2.
Activity 7B
(a) 2:1
(b) 3:1
(c) 8:1
(d) 15:7
(e) 28:15
(f) 25:3
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(g) 1:1000
(h) 5:2
(i) 6:1
Activity 7C
(a) 10
(b) 3
(c) 9
(d) 30
(e) 42
(f) 25
(g) 144
(h) 3
(i) 10, 20.
Activity 7D
1
18 children
2
$1.80
3
Three buckets of cement.
4
77 kg
5
64 teachers
6
100 g of lead
7
(a) 525 female students
(b) 945 students altogether.
Other shares: $40 and $25. Total = 40 + 15 + 25 = $80.
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Activity 7E
1
(a) 14:16 = 7:8
(b) 14:30 = 7:15
(c)
(d)
4930CF: 7 Applying Everyday Mathematics
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2
(a)
(b)
4
(c) 32 kg of gravel.
(a) 5 Pens:10 Pens
(b) $36:$24
(c) 180 kg:240 kg
(d) $2:$0.50.
8 mL of acid.
5
$40 000 (A) and $48 000 (B)
6
7
(a) 3:2
(b) $162 (Frank) and $108 (Nigel)
$17 500 (Sarah) and $7500 (Rebecca)
8
270 g (flour), 135 g (water), 45 g (honey)
9
(a)
3
(b) 3 Bottles of soft drinks (to make
24
of soft drinks)
4930CF: 7 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996
Activity 7F
1
4930CF: 7 Applying Everyday Mathematics
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2
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4930CF: 7 Applying Everyday Mathematics
 DET NSW, 2005/008/012/06/2006 P0025996