Ages ratio now and after a fixed period: Samir and Saurabh are currently in the ratio 8 : 15 (respectively). After 9 years, their ages will be in the ratio 11 : 18. Find the difference between their present ages (in years).

Introduction / Context:
Problems on ages frequently use ratios at two different times to derive present ages. By translating the statement into linear equations, we can solve for a single scaling factor and then compute the required difference.

Given Data / Assumptions:

• Present ratio (Samir : Saurabh) = 8 : 15.
• After 9 years, ratio becomes 11 : 18.
• Let present ages be 8k and 15k.
• Both ages advance by the same 9 years in the future scenario.

Concept / Approach:
The core idea is that if present ages are in ratio a : b, then after t years they will be (a k + t) and (b k + t). Setting this over the target ratio yields a single equation in k. Once k is known, exact ages and differences follow immediately.

Step-by-Step Solution:
1) Assume Samir = 8k, Saurabh = 15k.2) After 9 years: (8k + 9) / (15k + 9) = 11 / 18.3) Cross-multiply: 18(8k + 9) = 11(15k + 9).4) Expand: 144k + 162 = 165k + 99.5) Rearrange: 63 = 21k ⇒ k = 3.6) Present ages: 8k = 24, 15k = 45. Difference = 45 − 24 = 21 years.

Verification / Alternative check:
Check the future ratio: 24 + 9 = 33 and 45 + 9 = 54. Then 33 : 54 simplifies by 3 to 11 : 18, matching the condition.

Why Other Options Are Wrong:

• 20/22/24/18 years do not reproduce the future ratio 11 : 18 when back-computed.

Common Pitfalls:
A frequent mistake is adding 9 to the ratio numbers instead of to the actual ages. Another error is simplifying ratios incorrectly before forming the equation.

Final Answer:
21 years