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4x − 3 4 × 12 − 3 48 − 3 45 3
Verification: LHS = = = = =
2x + 6 2 × 12 + 6 24 + 6 30 2
3
And RHS = ⇒ LHS = RHS, hence verified.
2
Example 19: The denominator of a rational number is greater than its numerator by 3. If numerator
3
is increased by 7 and denominator is decreased by 1, the new number becomes . Find the original
number. 2

Solution: Let the numerator of a rational number be x.
Then, the denominator of the rational number = x + 3
When the numerator is increased by 7, then the new numerator = x + 7

When denominator is decreased by 1, then new denominator = x + 3 – 1 = x + 2.
3
New number formed = 2
According to the question,
(x + 7 ) 3
=
(x + 2 ) 2
⇒ 2(x + 7) = 3(x + 2) ⇒ 2x + 14 = 3x + 6

3 x – 2x = 14 – 6 ⇒ x = 8
x 8 8
Thus, the original number is = = .
8
x ( + 3 ) ( + 3 ) 11
Example 20: The ages of Ritu and Reema are in the ratio 2 : 3. After 6 years, the ratio of their ages
will be 8 : 11. Find their present ages.
Solution: Let the present ages of Ritu and Reema be 2x and 3x respectively.

After 6 years, Ritu’s age = (2x + 6) years and Reema’s age = (3x + 6) years.
According to the question,
2x + 6 8 Quick Check
=
3x + 6 11 The ages of Mohan and
By cross-multiplication, Ram are in the ratio 5 : 7.
11(2x + 6) = 8(3x + 6) ⇒ 22x + 66 = 24x + 48 Four years later, their ages
⇒ 66 – 48 = 24x – 22x ⇒ 18 = 2x will be in the ratio 3 : 4.
Find their present ages.
⇒ x = 9
Thus, Ritu’s present age = 2x = 2 × 9 = 18 years
Reema’s present age = 3x = 3 × 9 = 27 years

Practice Time 2C

1. Solve the following equations.

2x − 3 − 1 4x − 3 1 3x + 1 − 6 9x + 1
( a) = ( b) + = 0 ( c) = ( d) = 2
5 3 2 5 x − 2 11 3x + 5
4x − 5 2 2x − 9 − 1 x + 1 3 4x
( e) = ( f) = ( g) = ( h) = 3
2x 7 3x + 4 16 2 x + 3 8 2x + 7

47 Linear Equations in One Variable