Page 135 - Maths Skills - 8

Direct and Inverse Proportions 133

INTRODUCTION
(i) When the speed of a vehicle is increased, the time taken to cover the same distance is decreased.
(ii) The more you purchase a particular product, the more you have to pay.

(iii) For constructing a building, if the number of workers is increased, then the number of days taken to
complete the construction is decreased.

By analyzing we find the following:
● In (i): When one quantity (speed) is increased, the other quantity (time) is decreased.
● In (ii): When one quantity (product) is increased, the other quantity (cost) is increased.
● In (iii): When one quantity (workers) is increased, the other quantity (days) is decreased.

Thus for any two related quantities, we see that if one quantity is changed, then its corresponding quantity is also
changed. This gives rise to the concept of Proportion. In this chapter, we shall learn about direct and inverse
Proportion. And, we shall also apply this concept in solving problems on
(i) Time and work. (ii) Time and distance.

TYPES OF PROPORTIONS

On the basis of relation between two quantities, proportions are of two types:

Direct Proportions
Two quantities are said to be in direct proportion if the increase (or decrease) in the measure of one result in the
corresponding increase (or decrease) in the measure of the other. For example,

(a) If the rate of inflation rises, the cost of the food articles also rises. So the rate of inflation and the cost of
food articles are in direct proportion.
(b) Consider a table and the force required to move it. If the weight of the table is increased, more force is
required to move it. Similarly, if the weight of the table is reduced, less force is required to move it. So,
the weight of the table and the force required to move it are in direct proportion.

Inverse Proportion
Two quantities are said to be in inverse proportion if the increase (or decrease) in the measure of one result in the
corresponding decrease (or increase) in the measure of the other. For example,
(a) If the pressure on a gas at constant temperature is increased, it results in a decrease in the volume and vice
versa. So the pressure and the volume of a gas, at constant temperature, are in inverse variation.
(b) 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 proportion.

DIRECT PROPORTION
Suppose the cost of a pencil is ` 2. Then, the cost of 2, 3, 4 and 5 pencils can be calculated using unitary method.
Thus, the cost of 2 pencils = 2 × 2 = ` 4; 3 pencils = 3 × 2 = ` 6; 4 pencils = 4 × 2 = ` 8; and 5 pencils = 5 × 2 = ` 10.
We may put the number of pencils (p) and their costs (q) in a table as under:

Number of pencils (p) 1 2 3 4 5

Cost in ` (q) 2 4 6 8 10