Ten years ago, A was exactly half of B in age. Their present-age ratio is 3 : 4. What is the sum of their present ages?

Difficulty: Medium

8 years
20 years
35 years
45 years
None of these

Correct Answer: 35 years

Explanation:

Introduction / Context:
This age problem combines a past-time relation with a present-time ratio. We convert both statements into equations with a single parameter and compute the present ages, then sum them.

Given Data / Assumptions:

10 years ago: A was half of B.
Present ratio: A : B = 3 : 4.
Find the sum of present ages.

Concept / Approach:
Let A = 3k and B = 4k at present. Use the “10 years ago” condition (A − 10) = 0.5 * (B − 10) to solve for k and then compute A + B.

Step-by-Step Solution:

Let present ages be A = 3k and B = 4k.Ten years ago: 3k − 10 = (1/2)(4k − 10) = 2k − 5.Therefore 3k − 10 = 2k − 5 ⇒ k = 5.Present ages: A = 15, B = 20. Sum = 35.

Verification / Alternative check:

Ten years ago: A = 5, B = 10; indeed A was half of B.

Why Other Options Are Wrong:

8, 20, 45: Do not match the equations simultaneously.“None of these” is unnecessary because 35 is valid.

Common Pitfalls:

Applying the present ratio to the past directly without subtracting 10.Using A : B = 4 : 3 by mistake (order matters).

Ten years ago, A was exactly half of B in age. Their present-age ratio is 3 : 4. What is the sum of their present ages?

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