Ratio
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 
Worked out examples
EXAMPLE 1
If three numbers are in the ratio 1 3 : 5 and
their sum is 108, then the largest number is _______.
Solution
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
EXAMPLE 2
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. .
Solution
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. .
EXAMPLE 3
If 
, find 
.
Solution
Dividing
both numerator and denominator of by y, it becomes 
As 
.
EXAMPLE 4
Two numbers are in the ratio 9 : 7. If 14 is
subtracted from each, the new ratio is 7 : 5. Find the numbers. .
Solution
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. .
EXAMPLE 5
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. .
Solution
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. .
