Two-time-ratio system: One year ago, A : B = 3 : 4. One year hence, A : B = 5 : 6. What is B’s present age?

Introduction / Context:
Two ratios at different times generate a 2×2 linear system in present ages. Solve systematically and check whether the resulting integer matches any listed options; if not, select “None of these.”

Given Data / Assumptions:

• (A − 1) : (B − 1) = 3 : 4
• (A + 1) : (B + 1) = 5 : 6

Concept / Approach:
Convert ratios to linear equations by cross-multiplication and then solve simultaneously for A and B.

Step-by-Step Solution:

Verification / Alternative check:
Check both ratios: (4−1):(5−1)=3:4 and (4+1):(5+1)=5:6 ✓.

Why Other Options Are Wrong:
8, 4, and 6 do not equal the computed B = 5; hence “None of these.”

Common Pitfalls:
Sign errors when shifting ages by ±1; misapplying ratios to present instead of past/future ages.

Final Answer:
None of these (B’s present age is 5 years).