Equivalent ratios

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 = 4 × 5

                                                        = 20

 

  b.     2 = 20 × 10

                                 missing number = 30 × 10

                                                         = 3

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]

 

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

 

                         a.           20cm : 200m

 b.          1.2 : 0.02

 c.             

 

                                                        = 20 : 20 000

                                                        = 1 : 1000

 

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

                                                   = 120 : 2

                                                    = 60 : 1

 

                       c.                [multiply by LCM]