Page 153 - Start Up Mathematics_8 (Non CCE)

9 Direct and Inverse Variation

Two quantities are said to be in variation, if the change in the measure of one results in a corresponding change
in the measure of the other. In other words, two quantities are in variation if any increase or decrease in one
quantity results in increase or decrease in the other.
For example, if we increase the number of articles to be purchased, the total cost will also increase. Consider
another example of the speed of a moving train and the time it takes to cover a distance. If the speed increases,
the time taken to cover the distance decreases. So, we conclude that the speed and time taken to cover a distance
by a moving train are in variation.

Types of Variation
Direct Variation: Two quantities are said to be in direct variation if the increase (or decrease) in the measure
of one results in the corresponding increase (or decrease) in the measure of the other.
For example, consider a rock and the force required to move it. If the weight of the rock is increased, more
force is required to move it. Similarly, if the weight of the rock is reduced, less force is required to move it.
So, the weight of the rock and the force required to move it are in direct variation.
Inverse Variation: Two quantities are said to be in inverse variation if the increase (or decrease) in the measure
of one results in the corresponding decrease (or increase) in the measure of the other.
For example, in a road project, assuming that all the workers work at the same rate, if we increase the number
of the workers, the time taken to complete the project will decrease. Similarly, if the number of workers are
reduced, the time taken to complete the project will increase. Thus, the number of workers and the time taken
to complete the project are in inverse variation.

Direct variation
x
If two quantities x and y vary in such a manner that is constant and positive, then x and y are said to be in
y x
direct variation. In other words, x and y are in direct variation if is always constant. This constant is the
constant of variation (say k). x y

y = k or x = ky
If two quantities x and y vary directly and x and y are their values at one point, then
1
1
x
1 = k, a constant ...(1)
y 1
If x and y are the values which x and y assume at second point, then
2
2
x
2 = k, a constant ...(2)
y 2
Equating (1) and (2), we get x 1 = x 2
y
fi x y = x y 1 y 2
1 2
2 1
x y
fi 1 = 1
x 2 y 2
fi x : x : : y : y 2
2
1
1
fi x : x = y : y 2
1
2
1