Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part or parts, one quantity is of the other. The ratio of two quantities “a” and “b” is represented as a : b and read as “a is to b”. “a” is called antecedent, “b” is the consequent. Since the ratio expresses the number of times one quantity contains the other, it’s an abstract quantity. .

A ratio a : b can also be expressed as a/b. So if two items are in the ratio 2 : 3, we can say that their ratio is 2/3. If two terms are in the ratio 2, it means that they are in the ratio of 2/1, i.e., 2 : 1.

“A ratio is said to be a ratio of greater inequality or lesser inequality or of equality according as antecedent is greater than, less than or equal to consequent”. From this we find that a ratio of greater inequality is diminished and a ratio of lesser inequality is increased by adding same quantity to both terms. .

i.e., in a : b .

if a < b then (a + x) : (b + x) > a : b and if a > b then (a + x) : (b + x) < a : b

If ………….., then each of these ratios is equal to

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Worked out examples **

If three numbers are in the ratio 1 3 : 5 and their sum is 108, then the largest number is _______.

Let the three numbers be x, 3x and 5x. .

Given, x + 3x + 5x = 108

&⇒ x = 12

\ The largest number is 5x = 5 x 12 = 60

Find the numbers which are in the ratio 3 : 2 : 4 such that the sum of first and second added to the difference of third and second is 21. .

Let the numbers be a, b and c. .

Given that a, b and c are the ratio 3 : 2 : 4. .

a : b : c = 3 : 2 : 4

Let, a = 3x, b = 2x and c =4x

Given, (a + b) + (c - b) = 21

&⇒ a + b + c – b = 21

&⇒ a + c = 21

&⇒ 3x + 4x = 21

&⇒ 7x = 21

&⇒ x = 3

a, b, c are 3x, 2x, 4x. .

\ a, b, c are 9, 6, 12. .

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If , find .

Dividing both numerator and denominator of by y, it becomes

As .

Two numbers are in the ratio 9 : 7. If 14 is subtracted from each, the new ratio is 7 : 5. Find the numbers. .

Let the numbers be 9x and 7x. .

5 (9x - 14) = 7 (7x - 14)

&⇒ 45x - 70 = 49x - 98

&⇒ 98 - 70 = 49x - 45x

&⇒ 28 = 4x

&⇒ 7 = x. .

Hence the numbers are 9x = 63 and 7x = 49. .

The ratio of the incomes of Chetan and Dinesh is 3 : 4. The ratio of their expenditures is 5 : 7. If both save Rs.2000, find the incomes of both. .

Let the incomes of Chetan and Dinesh be 3x and 4x respectively. Let the expenditures of Chetan and Dinesh be 5y and 7y respectively. Savings is defined as (Income) — (Expenditure). Hence the savings of Chetan and Dinesh are 3x - 5y and 4x - 7y respectively. .

3x - 5y = 2000 → (1)

4x - 7y = 2000 → (2)

Multiplying (1) by 7 and (2) by 5 and subtracting the resultant equation (2) from resultant equation (1), we get, x = 4000

The incomes of Chetan and Dinesh are 3x = Rs.12000 and 4x = Rs.16000 respectively. .