Seven years ago, the ages of A and B were in the ratio 4 : 5. Seven years hence, they will be in the ratio 5 : 6. What is B’s present age?

Difficulty: Medium

56 yr
63 yr
70 yr
77 yr
None of these

Correct Answer: 77 yr

Explanation:

Introduction / Context:
Two time-shifted ratios yield two linear equations in present ages. Solve by elimination for exact present ages.

Given Data / Assumptions:

(A − 7) : (B − 7) = 4 : 5
(A + 7) : (B + 7) = 5 : 6

Concept / Approach:
Convert ratios to equations and solve the system for A and B.

Step-by-Step Solution:

5A − 35 = 4B − 28 ⇒ 5A − 4B = 7 … (1)6A + 42 = 5B + 35 ⇒ 6A − 5B = −7 … (2)Multiply (1) by 5: 25A − 20B = 35Multiply (2) by 4: 24A − 20B = −28Subtract ⇒ A = 63; then from (1): 5·63 − 4B = 7 ⇒ 315 − 4B = 7 ⇒ B = 77

Verification / Alternative check:
7 years ago: 56 : 70 = 4 : 5; 7 years hence: 70 : 84 = 5 : 6, both correct.

Why Other Options Are Wrong:
56, 63, 70 do not satisfy both time-shifted ratios simultaneously.

Common Pitfalls:
Applying the 7-year shift to only one person or inverting the ratios inadvertently.

Seven years ago, the ages of A and B were in the ratio 4 : 5. Seven years hence, they will be in the ratio 5 : 6. What is B’s present age?

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