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Direct Variation

If two quantities are associated in such a way that an increase in one quantity leads to a corresponding
increase in the other and vice versa, then such a variation is called a direct variation.
Let us think about the fee collection from a class. If one student pays `1000, then two students will
pay `2000, three students will pay `3000 and so on.

Number of students 1 2 3 5 20 36
Fee collection (in `) 1000 2000 3000 5000 20000 36000
Notice that:

• More the number of students, more is the fee collection.
• Less the number of students, less is the fee collection.

Let us consider another situation. If Shally uses 600 g flour to make 72 pieces of cookies, then she
will use 300 g flour to make 36 pieces of cookies, 100 g flour to make 12 pieces of cookies, and so on.

Quantity of flour (in g) 600 300 100 500 200 1000

Number of cookies prepared 72 36 12 60 24 120
In the above two cases, if one quantity (x) increases, the other
quantity (y) also increases, and if one quantity (x) decreases, Quick Check
the other quantity (y) also decreases. Moreover, the ratio x : Can you tell something about the
y is constant. In the first example, it is 1 : 1000, while 25 : 3 in changes in two quantities — the
the second example. This type of relationship between two quantity of flour and the number
quantities is called direct proportion or direct variation, and of cookies prepared?
we can define direct variation in mathematical form as follows:
x x
When x and y are in direct variation, we can write 1 = 2 . That is when two quantities x and y are
y 1 y 2
in direct proportion (or vary directly), we write it as x ∝ y, which gets translated to x = ky.

x
Thus, we say that x and y are in direct variation if = k, where k is the constant of direct variation.
y
Example 1: Check whether x and y are in direct variation to each other in each of the following
tables.

(a) x 4 12 8 10 30 36 (b) x 6 12 15 20 14 24
y 10 30 20 25 75 90 y 18 20 30 50 21 80

x
Solution: We find the ratios for the values of x and the corresponding values of y and compare
them. y
4 2 12 2 8 2 10 2 30 2 36 2
(a) We have, = , = , = , = , = , =
10 5 30 5 20 5 25 5 75 5 90 5
4 12 8 10 30 36 2
That is, = = = = = = = k
10 30 20 25 75 90 5
2
Thus, the ratio of the corresponding values of x and y is constant and equal to .
5

2
So, x and y are in direct variation with the constant of variation that is equal to .
5

Mathematics-8 280