What is a Ratio?

A ratio is just another way of expressing the relationship between two or more values of the same type.

For example, if ‘A’ earns twice what ‘B’ earns, then we say that:

for every two dollars that ‘A’ earns, ‘B’ earns one, or

the ratio of A’s earnings to B’s earnings is 2 :1 

This ratio may be expressed as 4 : 2  or  6 : 3  or for that matter 200 : 100

While there is nothing wrong with the above representations of the ratio 2 : 1 , the smaller ratio 2 : 1 is preferred from a calculation point of view. 

Example:

If we had to express the ratio of the price of two cars, Swift, which costs ₹ 575,000 and Polo, which costs ₹ 500,000, then,

Ratio of price of Swift to price of Polo = 575000 : 500000

Since 1000 is a common multiple on both sides, we can remove this common multiple

∴ The ratio is 575 : 500

Here again, it is evident that 25 is a common multiple, therefore dividing both sides by 25 gives

23 : 20, the desired ratio.

Expressing a Ratio as a Fraction

A ratio can also be expressed as a fraction

For example, if the ratio of A's income to B's income is 2 : 1, we can say that

A’s income / B’s income =  2/1 

We can now cross multiply to get

A’s income × 1 =  B’s income × 2

Dividing a total in a given ratio

Any quantity can be divided in a given ratio as follows:

Suppose you have to divide an amount of $480 among 3 people A, B, and C in the ratio 2 : 3 : 7 respectively, then the division would be

A’s share   =  480 × 2/( 2+3+7)  =   480 × 2/12  =  80

B’s share   =  480 × 3/( 2+3+7)  =   480 × 3/12  =  120

C’s share   =  480 × 7/( 2+3+7)  =   480 × 7/12  =  280

Multiples to Ratio - How to convert

Sometimes, the relationship between quantities may be expressed as a relationship of multiples instead of ratios. 

For example,   

Three times A’s share = Four times B’s share

Is this the same as saying that the ratio of  A : B = 3 : 4?

No, it is not the same thing. What we are saying here is that

3A  = 4B

∴  A/B  =  4/3

∴   A  :  B  =  4  :  3  

For example, if we are given that

2A = 3B = 4C = 5D

then what is the ratio of A, B and C?

Solution :    

Ratio  A  :  B  :  C  :  D  =   1/2  :  1/3  :  1/4  :  1/5   

=   30/60 : 20/60 : 15/60 : 12/60    

[since 60 is LCM of the denominators 2, 3, 4 and 5, we write each fraction with denominator 60] 

Now 60 cancels throughout and we get

=  30  :  20  :  15  :  12

Linking more than one pair of ratios 

For example,

If a : b = 2 : 3 and b : c = 4 : 7, then what is a : b : c ?

The common link between the ratios   a : b and b : c  is ‘b’. 

However, it does not have the same value in both the ratios, so we cannot link the ratios as they are. What we need to do in order to link the ratios is have a common value for ‘b’. 

We obtain this as follows:

Multiply the first ratio by 4 (the value of ‘b’ in the second ratio) 

and 

Multiply the second ratio by 3 (the value of ‘b’ in the first ratio).

∴ a : b = 2 : 3   or  2 × 4 : 3 × 4  =  8 : 12

Similarly, b : c = 4 : 7   or  4 × 3 : 7 × 3  =  12 : 21

Now ‘b’ has the same value in both the ratios, so we can link them up as 

    a : b : c = 8 : 12 : 21