 ## Proportion | What is proportion?

Proportion is the equality of two or more than two ratios. Four numbers a, b, c and d are said to be in proportion if the ratio of a and b is equal to the ratio of c and d. This is written as a:b::c:d and read as ‘a is to b is as c is to d’. First term and fourth term are called extremes while second and third terms are called means.
In proportion, the product of extremes is always equal to the product of means. That is a×d = b×c.

### Types of Proportions

There are 2 types of proportions; they are:
a.    Direct proportion
b.    Indirect proportion

#### Direct proportion

If two quantities are so related that the increase or decrease in one causes the corresponding increase or decrease in the other, they are said to be direct proportion. Let us take an example of no. of copies and their cost:

 No. of Copies Total Price (Rs.) 1 50 3 150

Ratio of copies = 1/3
Ratio of cost = 50/150 = 1/3

Here, the ratio of copies = ratio of cost i.e if number of copies increases their cost also increase. So they are said to be in direct proportion.

#### Indirect proportion

If two quantities are so related that the decrease or increase in one causes the corresponding increase or decrease in the other is called indirect (inverse) proportion. Let us take an example of no. of men and their working days:

 No. of men No. of days 1 16 2 8

Ratio of men = 1/2
Inverse ratio of days = 8/16 = 1/2

Here, the ratio of men = inverse ratio of days i.e if number of men increases their working days decreases. So they are said to be in indirect proportion.

### Workout Examples

Example 1: Find whether 5, 7, 20 and 28 are in proportion or not.

Solution: Here,
Product of extremes = 5 × 28 = 140
Product of means = 7 × 20 = 140

Since, product of extremes = product of means, the terms 5, 7, 20 and 28 are in proportion.

Example 2: Find the value of x if 5, x, 10 and 12 are proportional.

Solution: Here,
Product of extremes = 5 × 12 = 60
Product of means = x × 10 = 10x

Since, the terms are proportional,

product of means = product of extremes
i.e.       10x = 60
or,        x = 60/10
or,        x = 6

Example 3: If the terms 4, 5 and 8 are three terms of proportion, find the fourth proportion.

Solution: Let x be the fourth proportion.
Now, 4, 5, 8 and x are in proportion.
4/5 = 8/x
or,        4x = 40
or,        x = 40/4
or,        x = 10

The fourth proportion is 10.

Example 4: If the cost of 18 kg of rice is Rs. 720, find the cost of 5 kg of rice.

Solution: Let x be the cost of 5 kg of rice.

 Rice (kg) Cost (Rs.) 18 720 5 x

Since quantity of rice and their cost are in direct proportion,
18/5 = 720/x
or,        18x = 3600
or,        x = 3600/18
or,        x = 200

Cost of 5 kg of rice is Rs. 200.

Example 5: A group of 400 men has provision for 90 days. For how many men would the provision last for 60 days?

Solution: Let the provision last for 60 days for x men.

 No. of Men No. of Days 400 90 x 60

Since no. of men and days are in indirect proportion,
400/x = 60/90
or,        60x = 400 × 90
or,        x = 400 × 90/60
or,        x = 600

The required number of men is 600.

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