Shortcut Trick
Present ratio = 4 : 3 = 8 : 6 (to equalise the age difference of 2 parts).
After 4 years, ratio = 9 : 7. The increase is 1 part (9 − 8 = 1), which equals 4 years.
Present ages: Mr. X = 8 × 4 = 32 years, Wife = 6 × 4 = 24 years (Difference = 8 years).
At marriage, ratio = 5 : 3. Difference = 2 parts = 8 years, so 1 part = 4 years.
Age of Mr. X at marriage = 5 × 4 = 20 years. Years since marriage = 32 − 20 = 12 years.
∴ The correct answer is 12 years.
Alternate Method
Given:
Present ratio of age of Mr. X and his wife = 4 : 3
Ratio after 4 years = 9 : 7
Ratio at the time of marriage = 5 : 3
At Marriage
Ratio = 5 : 3
Ages = 20 & 12
Present
Ratio = 4 : 3
Ages = 32 & 24
+4 Years
Ratio = 9 : 7
Ages = 36 & 28
12 Years Ago
+4 Years
Calculation:
Let the present ages of Mr. X and his wife be 4x and 3x respectively.
After 4 years, their ages will be (4x + 4) and (3x + 4).
⇒ (4x + 4) ÷ (3x + 4) = 9 ÷ 7
⇒ 7 × (4x + 4) = 9 × (3x + 4)
⇒ 28x + 28 = 27x + 36
⇒ 28x − 27x = 36 − 28
⇒ x = 8
⇒ Present age of Mr. X = 4 × 8 = 32 years
⇒ Present age of his wife = 3 × 8 = 24 years
Let them be married y years ago.
⇒ Ages at the time of marriage were (32 − y) and (24 − y).
According to the given ratio at marriage:
⇒ (32 − y) ÷ (24 − y) = 5 ÷ 3
⇒ 3 × (32 − y) = 5 × (24 − y)
⇒ 96 − 3y = 120 − 5y
⇒ 5y − 3y = 120 − 96
⇒ 2y = 24
⇒ y = 12
∴ The correct answer is 12 years.
Additional Information
Constant Age Difference Rule
The absolute difference between the ages of two individuals always remains constant regardless of the time period.
Ratio Equalisation Method
When solving age ratios across different times, multiply the given ratios by appropriate integers so that the difference between the numerator and denominator is the same in both scenarios.
Cross-Multiplication Trick
For ratio changes from a : b to c : d over time t, setting up an equation and using the cross-product difference directly yields the exact multiplier for the ratio parts.