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Indirect (Inverse) Variation

In the case of a direct variation, we observed that the two quantities increase or decrease together.
But there are situations in which an increase (or decrease) in one quantity results in a decrease
(or increase) in the other. Such variation is called inverse proportion or inverse variation.

Let us understand with an example.
Suppose you went to buy a few pens for `100. Think about the rate per pen and the number of
pens. If one pen is `10 then you can take 10 pens, if one pen is `5 then you can take 20 pens, and
if one pen is `25 then you can take only 4 pens and so on.

Rate of pen (in `) (x) 10 5 25 2 20 12.50
Number of pens (y) 10 20 4 50 5 8
Note that:

• The higher the rate per pen, the less is the number of pens.

• The lower the rate per pen, the more is the number of pens.
Let us consider another situation.
If 24 labourers work 30 days to make a certain number of bricks, then 48 labourers will complete
the target in 15 days, 12 labourers will complete the target in 60 days, 72 labourers will complete

the target in 10 days and so on.

Number of labourers (x) 24 48 12 72 30 90 Quick Check
Number of days (y) 30 15 60 10 24 8
Can you tell something about the
In the above two cases, if one quantity (x) increases, the other changes in these two quantities
quantity (y) decreases and if one quantity (x) decreases, the — the number of labourers and
other quantity (y) increases. the number of days?

Here, the ratio x : y is not constant in all cases. Let us see if x × y is constant.
In the first case, 10 × 10 = 100 = 5 × 20 = 25 × 4 = 2 × 50 = 20 × 5 = 12.50 × 8

Clearly, xy = 100, a constant.
In the second case, 24 × 30 = 720 = 48 × 15 = 12 × 60 = 72 × 10 = 30 × 24 = 90 × 8. Clearly, xy = 720, a
constant.
x y
So, when x and y are in inverse proportion, we can write x × y = x × y or 1 = 2 . That is, when
2
1
1
2
x 2 y 1
two quantities x and y are in inverse proportion (or vary inversely), we write it as x ∝ 1 , which
y
gets translated to xy = k.
Thus, we can say that x and y are in inverse proportion (variation) if xy = k, where k is the constant
of inverse proportion (variation).
Example 9: Check whether x and y are in inverse variation to each other in each of the following
tables:

(a) x 4 8 15 10 30 36
y 60 30 20 24 12 5

285 Direct and Indirect Variations