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Let’s explore a simple but important concept – Variation.

Two quantities A and B are said to be varying with each other if there exists some relationship between A and B such that the change in A and B is uniform and guided by some rule.

For example, if A = k * B or if A = k/B, where k is a constant, then A and B are said to be varying with each other.

In this case, A = k * B is an example of Direct variation.

What is Direct Variation?

Let’s say there are two quantities A and B such that if B increases in a certain ratio, A also increases in the same ratio and if B decreases in a certain ratio, A also decreases in the same ratio. Such a case is an example of Direct Variation.

This is denoted as A varies directly as B. If A varies directly as B, then A = k * B, where k is a constant – the constant of proportionality.

Remember a simple relation always: Suppose X varies directly with Y. Also, when X = X1, Y = Y1 and when X = X2, Y = Y2. In such a case,

X1/Y1 = X2/Y2

or

X1/X2 = Y1/Y2

Thus, in the above example:

42 / 6 = 56 / 8

or

42 / 56 = 6 / 8

What is Inverse Variation?

Let’s say there are two quantities A and B such that if B increases in a certain ratio, A decreases in the same ratio and if B decreases in a certain ratio, then A increases in the same ratio. Such a case is an example of Inverse Variation.

If A varies inversely with B then A = k/ B, where k is a constant – the constant of proportionality.

Remember a simple relation always: Suppose X varies inversely with Y. Also, when X = X1, Y = Y1 and when X = X2, Y = Y2. In such a case,

X1 * Y1 = X2 *Y2

or

X1/X2 = Y2/Y1

Thus, in the above example,

6 * 8 = 12 * 4

or

6 / 12 = 4 / 8.

Let us now look at Applications of Ratio and Proportion.

For example;

In a mixture of 60 litres, the ratio of milk and water is 2: 1. Find out the quantity of water that should be added to this to change the ratio to 1: 2?

Explanation:

Quantity of milk

= 60 * 2/3 litres

= 40 litres

Quantity of water in the mixture

= 60 – 40 litres

= 20 litres

Now, new ratio = 1: 2 This means that in the new mixture, the quantity of water should be double the quantity of milk.

Thus, quantity of water = 2 * quantity of milk = 2 * 40 = 80 litres.

But, we already have 20 litres of water in the original mixture.

Hence, the quantity of water to be added is 80 – 20 = 60 litres

Now let us look at another simple example.

A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1,000 more than D, what is B’s share?

Explanation:

Let the shares of A, B, C and D be Rs. 5x, 2x, 4x and 3x respectively.

Then, 4x – 3x = 1000

i.e. x = 1000

Therefore, B’s share = Rs 2x = Rs 2 * 1000 = Rs 2,000