Ratio and Proportion is simple to understand (despite the long and confusing name). It is also quite important since it regularly features in various examinations like Banking entrance exams, UPSC CSAT, SSC CGL and many more tests.

For example, Suppose that the salaries of your friends Dante and Nero are in the ratio of 5:9. The sum of their salaries is Rs 35,000. Find out their individual salaries.

The answer to this question is quite simple.

Ratios essentially compare two quantities and give us the relation between them.

As we know that the ratio of their salaries is 5:9.

So, let us assume that Dante’s salary is 5x and Nero’s salary is 9x.

Now, this means that 5x + 9x = 35,000

14x = 35,000

or x = 2,500

Therefore, Dante’s salary = 5x = 5 * 2,500 = Rs 12,500

and, Nero’s salary = 9x = 9 * 2,500 = Rs 22,500

Thus, ratios are used to compare values. So, a ratio is a quantity which expresses the relationship between two similar quantities or two numbers of the same kind (e.g., objects, persons, students, number of books, units of identical dimension, etc.)

Note: Using ratios, we can only compare two similar quantities.

If the ratio of two quantities A and B is expressed as x: y, it means that A/B = x/y.

Types of Ratio:

Compounded Ratio of two ratios a/b and c/d is ac/bd,

Duplicate ratio of a : b is a2 : b2

Triplicate ratio of a : b is a3 : b3

Sub-duplicate ratio of a : b is √a : √b

Sub-triplicate ratio of a : b is ∛a : ∛b

Reciprocal ratio of a : b is b : a

Proportion:

Proportions are related to ratios.

When two ratios are equal, then the four quantities involved in the two ratios are said to be proportional.

Thus, if a/b = c/d, then a, b, c and d are said to be proportional.

This is represented as a: b :: c:d and is read as “a is to b as c is to d”.

For example: 2/3 is equal to 8/12. So, we write it as 2/3 = 8/12. Thus, we say that 2, 3, 8, 12 are proportional and write it as 2 : 3 :: 8 : 12.

When 2, 3, 8, 12 are in proportion, then 2 and 12 are known as ‘Extremes’ and 3 and 8 are known as ‘Means’.

Remember this interesting formula always:

Product of Extremes = Product of Means

i.e. A * D = B * C

For example, in this case:

2 * 12 = 3 * 8

Fourth proportional:

If a:b :: c:x, x is called the fourth proportional of a, b, c.

We have or, .

Thus, fourth proportional of a, b, c is .

Third Proportional:

If a:b:: b:x, x is called the third proportional of a, b.

We have

Thus, third proportional of a, b is .

Mean Proportional:

If a:x:: x:b, x is called the mean or second proportional of a, b.

We have

Mean proportional of a and b is .

We also say that a, x, b are in continued proportion.

Componendo and Dividendo :

If , then

(i) (Componendo)

(ii) (Dividendo)

(iii) (Componendo and Dividendo)

(iv) .