Problems on Ages — “A : B = 11 : 13 now. After 7 years the ratio will be 20 : 23. What is the difference in their present ages?”

Introduction / Context:Present and future ratios lead to a pair of linear equations in a single multiplier. Once the multiplier is determined, individual ages and their difference follow immediately.

Given Data / Assumptions:

• Now: A = 11x, B = 13x.
• After 7 years: (11x + 7) : (13x + 7) = 20 : 23.

Concept / Approach:Cross-multiply the future ratio and solve for x. Compute A and B, then take the difference |A − B|.

Step-by-Step Solution:

Verification / Alternative check:Future ages: 40 and 46 ⇒ 20 : 23 ✓.

Why Other Options Are Wrong:4/5/7/8 years do not reconcile with x = 3 and the given future ratio.

Common Pitfalls:Attempting to equate termwise (11 to 20, 13 to 23) instead of using proper cross-multiplication; ignoring that both gain the same 7 years.

Final Answer:6 years