Download Name of Lecturer: Mr. J.Agius Course: FCES

Name of Lecturer: Mr. J.Agius
Course: FCES
LESSON 28
Ratio and Proportion
28.1 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 28A
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
so
2cm : 1m = 2cm : 100cm = 2 : 100 = 1 : 50
2cm : 1m = 1 : 50
Learning Outcome 2 – Carry out Harder Numerical Calculations
Page 137
Name of Lecturer: Mr. J.Agius
Course: FCES
Exercise 28A
Express the following ratios in their simplest form:
1)
8 : 10
2)
32p : 96p
3)
48p : £2.88
4)
36 : 6
5)
60m, 40m
6)
£2, 20p
7)
0.6cm, 0.05cm
8)
0.4m, 1.6m
9)
15cm, 10cm
10) 750g, 2kg
11) 39 litres, 26 litres
12) 32 mph, 48 mph
Example 28B
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
Exercise 28B
Express the following ratios in their simplest form:
1) 18 : 24 : 36
2)
7 : 56 : 49
3) 144 : 12 : 24
4)
288 : 128 : 144
5) 42p : £1.05 : £2.10
Learning Outcome 2 – Carry out Harder Numerical Calculations
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Name of Lecturer: Mr. J.Agius
Course: FCES
We know that we can produce equivalent fractions by multiplying or dividing both numerator
and denominator by the same number,
2
4
=
3
6
So that
or
12
18
or
20
.
30
We can do the same with a ratio in the form 3 : 6.
3 : 6 = 6 : 12
and
2:
(i.e. multiplying both numbers by 2)
1
= 6 : 11
3
(i.e. multiplying both numbers by 3)
We can use this to simplify ratios containing fractions.
Example 28C
Express in their simplest form the ratios
a) 3 :
Answer
a) 3 :
b)
1
= 12 : 1
4
1
4
b)
2 4
:
3 5
(i.e. multiplying both numbers by 4)
2 4
2
4
:
= 15 
: 15  = 10 : 12 = 5 : 6
3 5
3
5
(i.e. multiplying both numbers by 15, work out and then divide by 2)
Exercise 28C
Express the following ratios in their simplest forms:
1)
2)
3)
4)
Learning Outcome 2 – Carry out Harder Numerical Calculations
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Name of Lecturer: Mr. J.Agius
Course: FCES
28.2 Relative Sizes
Example 28D
Which ratio is the larger, 6 : 5 or 7 : 6 ?
Answer
(We need to compare the sizes of
6
7
and
. So we express both with the same
5
6
denominator.)
6
36
=
5
30
so
6:5
and
7
35
=
6
30
is larger than 7 : 6
Exercise 28D
1)
Which ratio is the larger, 5 : 7 or 2 : 3?
2)
Which ratio is the smaller, 7 : 4 or 13 : 8?
3)
Which ratio is the larger,
5
7
or
?
8
12
Example 28E
Which of the ratios
Answer
4 : 6,
3
: 1, 12 : 16 are equal to one another?
4
4:6=2:3
3
:1=3:4
4
so
3
: 1 = 12 : 16
4
12 : 16 = 3 : 4
Exercise 28E
In the following set of ratios some are equal to one another. Identify the equal ratios.
1)
2)
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Name of Lecturer: Mr. J.Agius
Course: FCES
28.3 Problems
Example 28F
A family has 12 pets of which 6 are cats or kittens, 2 are dogs and the rest are birds. Find
the ratio of the numbers of a) birds to dogs
b) birds to pets.
Answer
There are 4 birds
a) Number of birds : number of dogs
=4:2
=2:1
b) Number of birds : number of pets
= 4 : 12
=1:3
Exercise 28F
In each question give your answer in its simplest form.
1)
Two brothers have £20 and £24 in their respective savings accounts.
Express these amounts as a ratio
2)
Miss Morgan has £320 in her current account, £400 in her deposit account, and £800
in her savings account.
Express these amounts as a ratio.
3)
A pound of grapes costs £1.60 and a pound of pears 72p. Write these prices as a ratio.
4)
The sides of a triangle are ¾, ½ and ¼. Writes these sides as a ratio in its simplest
form.
5)
A couple have 6 grandsons and 4 granddaughters. Find
a)
the ratio of the number of grandsons to that of granddaughters
b)
the ratio of the number of granddaughters to that of grandchildren.
6)
Rectangle A has length 12cm and width 6cm while rectangle B has length 8cm and
width 5cm. Find the ratio of
a)
the length of A to the length of B
b)
the area of A to the area of B
c)
the perimeter of A to the perimeter of B
d)
the size of an angle of A to the size of an angle of B
Learning Outcome 2 – Carry out Harder Numerical Calculations
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Name of Lecturer: Mr. J.Agius
Course: FCES
7)
Two angles of a triangle are 54 and 72. Find the ratio of the size of the third angle to
the sum of the first two.
8)
The width of a marigold flowerhead in a photograph is 9mm. In an enlargement the
width is 6cm. Write these widths as a ratio.
9)
The cost of a drug to a hospital is 75p and a rest home pays £4 for the same drug.
Write these prices as a ratio.
Mixed Exercise
1) Write each ratio in its simplest form.
a)
8:4
b)
15 : 20
c)
60 : 40
d)
12 : 36
e)
14 : 35
f)
8 : 12
g)
50 : 20
h)
96 : 36
i)
6 : 8 : 10
j)
40 : 60 : 80
k)
6 : 12 : 15
l)
48 : 60 : 108
2) Write each ratio in a common unit and then put it in its simplest form.
a) 15 mins : 2 hours
b) 30 mm : 2 cm
c) 750 g : 2 kg
d) €5 : 125 c
e)
f)
300 ml : 1.5 l
Learning Outcome 2 – Carry out Harder Numerical Calculations
1500 m : 10 km
Page 142