Direct Variation

Direct variation exists between two quantities when a change in one quantity brings a corresponding change in the other quantity.

Example:
Between cost and number of objects bought, there is direct variation.

If 2 pens cost Rs. 50, then what will 3 pens cost?

Before you answer, it is very obvious to note that more number of pens will involve more cost.

In other words, number of pens and cost are in direct variation with each other.

Whatever change in one quantity occurs, will bring a same change in the other too.

An increase in one quantity will bring about an increase in the other; and, a decrease in one will bring about decrease in the other.

In the above example on number of pens and price, more pens will induce greater cost, while fewer pens will involve less cost.

Another kind of variation is indirect variation, in which quantities will change inversely of each other.

i.e. an increase or decrease in one will cause an opposite effect – decrease or increase in the other quantity.

Example for indirect variation

Speed and time are two quantities which are in indirect variation.

One quantity varies inversely as the other.

In travelling a same distance, less speed will take greater time, while greater speed will involve less time.

Cross multiplication method for direct variation

Direct variation questions can be used using the method of cross multiplication.
Consider the above example:

If 2 pens cost Rs. 50, then how much will 3 pens cost?

To find, we can set up the following ratio between number of pens and price:

The ratio of 2 is 50 is same as the ratio of 3 to what?

i.e. 2: 50 = 3: ?

Now, from the law of means and extremes, we know

2 × ? = 3 × 50, i.e. (3 × 50)/2 = 75

Cross multiplication:

Since 2 pens cost Rs. 50, so each pen costs 50/2, which should be the same as x/3, i.e. the price of each of 3 pens.

So, 50/2 = x/3, Or x/3 = 50/2

This equation gives us the rule of cross multiplication