Direct and inverse proportion:

Let us consider some examples to understand direct and inverse proportion:

1. More you ride your bicycle the more distance you will cover.
2. More the packets of biscuits more the number of biscuits.
3. If there are many shops of the same kind then less will be the crowd.
4. If there is more advertisement in newspaper then less will be the news.

From the above examples, it is clear that change in one quantity leads to change in the other quantity.

Types of proportions:

There are two proportions i.e. direct and inverse proportion.

1. Direct proportion:

• We can say two quantities x and y in direct proportion if they increase (decrease) together in the same manner.
• Also, two quantities x and y in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. That is if \(\frac{x}{y}\) = k [k is a positive number], then x and y are said to vary directly.
• In such a case if y₁, y₂ are the values of y corresponding to the values x₁ , x₂ of x respectively then \(\frac{x_{1}}{y_{1}}\) = \(\frac{x_{2}}{y_{2}}\)
• If x is directly proportional to y, we can write it as x ∝ y.

Variables increasing (or decreasing) together need not always be in direct proportion. For example:

• In human beings, physical changes occur with time but not necessarily in a predetermined ratio.
• Among individuals when we talk about changes in weight and height are not in any known proportion.
• In natural phenomena such as there is no direct relationship or ratio between the height of a tree and the number of leaves growing on its branches.

2. Inverse proportion:

• We can say two quantities x and y in direct proportion if they increase (decrease) together in the same manner.
• Also, two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy = k, then x and y are said to vary inversely.
• In such a case if y₁, y₂ are the values of y corresponding to the values x₁ , x₂ of x respectively then \(\frac{x_{1}}{x_{2}}\) = \(\frac{y_{2}}{y_{1}}\)
• If x is inversly proportional to y, we can write it as x ∝ \(\frac{1}{y}\).

Direct and inverse proportion examples:

1. An electric pole, 14 metres high, casts a shadow of 10 metres. Find the height of a tree that casts a shadow of 15 metres under similar conditions.

Solution:

More the height of an object, the more would be the length of its shadow.
Thus, x1 : x2 = y1 : y2
14 : x = 10 : 15
10 × x = 15 × 14
thus, x = 21

2. 6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if we use only 5 pipes of the same type?
   Solution:

Number of pipes	6	5
Time (in minutes)	80	x

This is a case of inverse proportion as less number of pipes, it will require more time to fill the tank.
Hence, 80 × 6 = x × 5  as [x₁y₁ = x₂ y₂]
or, \(\frac{80×6}{5}\)
x = 96
Thus, time taken to fill the tank by 5 pipes is 96 minutes or 1 hour 36 minutes.

Also, read:

Ratio and Proportion: Learn, Take Notes, Revise, Examples