# Equivalent ratios

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Definition

There are 15 red discs and 5 green discs in  a bag. The ratio of the discs is red : green = 15 : 5.

The diagram shows how we can put the discs into five rows, each row containing 3 red discs and one green disc. For every 3 red discs, there is 1 green disc, i.e. red : green = 3 : 1.If we divide each term in 15 : 5 we also get 3 : 1. We say that 3 : 1 and 15 : 5 are equivalent ratios.

We find equivalent ratios similar to the way we find equivalent fractions. We can:

·       multiply each term of the ratio by the same number;

·       divide each term of the ratio by a common factor.

Thus:    2 : 5 = 6 : 15        [multiply by 3]

20 : 16 = 10 : 8      [divide by common factor 2]

Simplifying ratios

Recall that when we simplify a fraction, we reduce until the numerator and denominator have no common factor other than 1. We do the same thing when we simplify a ratio. We reduce it to its simplest equivalent ratio, to a ratio where the only common factor shared by the terms is 1.

To simplify 36 : 48 we have: 36 : 48 = 6 : 8     [divide by common factor 6]

= 3 : 4     [divide by common factor 2]

or:   36 : 48 = 3 : 4    [divide by HCF 12]

Example 1   Determine if each of the following ratios are equivalent.

a.   12 : 24 = 1 : 2

b.    20 : 25 = 5 : 10

c.    4 : 8 : 16 = 1 : 2 : 4

a.    Divide each term by the HCF 12.

12 : 24 = 12 ÷ 12 : 24 ÷ 12

= 1 : 2

TRUE

b.    Divide each term by the HCF 5.

20 : 25 = 20 ÷ 5 : 25 ÷ 5

= 4 : 5

5 : 10

FALSE

c.    Divide each term by the HCF 4.

4 : 8 : 16 = 4 ÷ 4 : 8 ÷ 4 : 16 ÷ 4

= 1 : 2 : 4

TRUE

Example 2        Find the missing number to make each pair of ratios equivalent.

a.     5 : 6  = ? : 24

b.    20 : 30  =  2 : ?

a.     24 = 4 ×  6

missing number = × 5

= 20

b.     2 = 20 × 10

missing number = 30 × 10

= 3

Example 3       Simplify each of the following ratios.

a.      12 : 36

b.      24 : 9

c.       8 : 12 : 28

a.    12 : 36 = 1 : 3                [divide each term by HCF 12]

b      24 : 9 = 8 : 3                 [divide each term by HCF 3]

8 : 12 : 28 = 2 : 3: 7       [divide each term by HCF 4]

Example 4        Simplify each of the following ratios.

a.           20cm : 200m

b.          1.2 : 0.02

c.

a.         20cm : 200m = 20cm : 200 ´ 100cm

= 20 : 20 000

= 1 : 1000

b.         1.2 : 0.02 = 1.2 ´ 100 : 0.02 ´ 100

= 120 : 2

= 60 : 1

c.                [multiply by LCM]

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### Question 1/5

1. 4 : 25 is equivalent to

 12 : 73 16 : 100 25 : 4 73 : 12

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