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Name of Lecturer: Mr. J.Agius
Course: HVAC1
Lesson 10
Chapter 3: Ratios & Proportion

Simplifying Ratios
Suppose that Peter makes a model of his father’s boat. If the model is 1m long while the actual boat is
20m long, we say that the ratio of the length of the model to the length of the actual boat is 1m : 20m or,
more simply, 1 : 20. We can also write the ratio as the fraction
1
20.
If Peter built a larger model, which was 2m long, then the ratio would be
length of mod el
2m 1
=
=
length of actual boat 20m 10
Or
length of model : length of boat = 1: 10
Ratios are therefore comparisons between related quantities.
Example 1
Express the ratios
a) 24 to 72
b) 2cm to 1m in their simplest form.
Answer
a)
24 3 1
= =
72 9 3
or
24 : 72 = 3 : 9 = 1: 3
so
24: 72 = 1 : 3
(dividing both numbers by 8 and then by 3)
b) (Before we can compare 2cm and 1m they must be expressed in the same unit.)
2cm
2cm
1
=
=
1m
100cm 50

or
2cm : 1m = 2cm : 100cm = 2 : 100 = 1 : 50
so
3 Ratios & Proportion
2cm : 1m = 1 : 50
Page 1
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Example 2
Simplify the ratio 24 : 18 : 12
Answer
(As there are three numbers involved, this ratio cannot be expressed as a single fraction)
To simplify such ratio we have to find that number which divides both these 3 numbers. In this
case 6 divides all the 3 numbers. So
24 : 18 : 12 = 4 : 3 : 2
We know that we can produce equivalent fractions by multiplying or dividing both numerator
and denominator by the same number,
2
4
12
20
So that
=
or
or
.
3
6
18
30
We can do the same with a ratio in the form 3 : 6.
3 : 6 = 6 : 12
(i.e. multiplying both numbers by 2)
1
And
2 : = 6 : 11
(i.e. multiplying both numbers by 3)
3
We can use this to simplify ratios containing fractions.
Example 3
Express in their simplest form the ratios
a) 3 :
1
4
b)
2 4
:
3 5
Answer
1
= 12 : 1
4
a)
3:
b)
2 4
2
4
:
= 15 
: 15 
3 5
3
5
(i.e. multiplying both numbers by 4)
= 10 : 12 = 5 : 6
(i.e. multiplying both numbers by 15, work out and then divide by 2)
3 Ratios & Proportion
Page 2
Name of Lecturer: Mr. J.Agius

Course: HVAC1
Relative Sizes
Example 4
Which ratio is the larger, 6 : 5 or 7 : 6 ?
Answer
(We need to compare the sizes of
6
36
=
5
30
so
6
7
and . So we express both with the same denominator.)
5
6
and
7
35
=
6
30
6:5
is larger than 7 : 6
4 : 6,
3
: 1, 12 : 16 are equal to one another?
4
Example 5
Which of the ratios
Answer
4:6=2:3
3
:1=3:4
4
so
3
: 1 = 12 : 16
4
3 Ratios & Proportion
12 : 16 = 3 : 4
Page 3
Name of Lecturer: Mr. J.Agius
Course: HVAC1
Simplifying Ratios
Exercise 1
Express the following ratios in their simplest form:
a)
d)
g)
j)
m)
p)
8 : 10
36 : 6
0.6cm : 0.05cm
750g : 2kg
8:4
14 l : 35 000 ml
b)
e)
h)
k)
n)
q)
32c : 96c
60m : 40m
0.4m : 1.6m
39 litres : 26 litres
15 minutes : 20 minutes
96 : 36
c)
f)
i)
l)
o)
r)
48c : €2.88
£2 : 20p
15cm : 10cm
32 mph : 48 mph
12 cm : 360 mm
18 : 72
Exercise 2
Express the following ratios in their simplest form:
a)
18 : 24 : 36
b)
7 : 56 : 49
c)
144 : 12 : 24
d)
40 : 60 : 80
e)
6 : 12 : 15
f)
48 : 60 : 108
g)
288 : 128 : 144
h)
42c : €1.05 : €2.10
i)
500g : 2.5 kg : 1200g
c)
1 1 1
: :
6 8 12
Exercise 3
Express the following ratios in their simplest forms:
a)
d)
6
7
1 5 2
: :
2 6 3
5:
b)
e)
1 1
:
2 8
4 3
:
9 7
f)
:2:
Exercise 4
1)
Which ratio is the larger, 5 : 7 or 2 : 3?
3)
Which ratio is the larger,
5
7
or
?
8
12
2)
Which ratio is the smaller, 7 : 4 or 13 : 8?
4)
Which ratio is the smaller, 3 : 7 or 4 : 9?
Exercise 5
In the following set of ratios some are equal to one another. Identify the equal ratios.
1)
6 : 8, 24 : 32,
3 Ratios & Proportion
3
:1
4
2)
2
: 3 , 4 : 18, 2 : 6
3
Page 4