Ten years ago A was half as old as B. The ratio of their present ages is 3 : 4. What is the sum of their present ages in years?

Introduction / Context:
This question involves both a past age relationship and a present age ratio between two people, A and B. It checks your ability to use linear equations to connect these conditions and then find the required sum of their present ages. Such problems are common in competitive exams under the topic problems on ages.

Given Data / Assumptions:

• Ten years ago, A was half as old as B.
• The ratio of their present ages is 3 : 4.
• We need to find the sum of their present ages.
• All ages are in years and taken as positive.

Concept / Approach:
The problem combines information from two different times. We start by expressing the present ages of A and B in terms of a common variable using the given ratio. Then we relate these present ages back to the condition that held ten years ago. From this, we obtain a simple equation in one variable and solve it to find the ages and their sum.

Step-by-Step Solution:
Let present age of A be 3 * x years.Let present age of B be 4 * x years.Ten years ago, A age was 3 * x - 10 and B age was 4 * x - 10.The condition says that ten years ago A was half as old as B, so 3 * x - 10 = (1 / 2) * (4 * x - 10).Multiply both sides by 2 to clear the fraction: 2 * (3 * x - 10) = 4 * x - 10.This gives 6 * x - 20 = 4 * x - 10.Rearrange: 6 * x - 4 * x = -10 + 20, so 2 * x = 10.Thus x = 5.Present age of A = 3 * 5 = 15 years, present age of B = 4 * 5 = 20 years.Sum of their present ages = 15 + 20 = 35 years.

Verification / Alternative check:
Check the past condition: Ten years ago, A was 15 - 10 = 5 years old, B was 20 - 10 = 10 years old.Now verify the relationship: 5 is half of 10, so the condition is satisfied.Check the present ratio: 15 : 20 simplifies to 3 : 4, confirming consistency with the given ratio.

Why Other Options Are Wrong:
15 years: This is only the age of A, not the sum of ages.25 years: Does not match the sum of any pair of ages that satisfies the given conditions.45 years: This would imply significantly larger ages and would not satisfy the past relationship.

Common Pitfalls:
Some students forget to subtract 10 from both ages when considering the situation ten years ago.Another mistake is treating A was half of B as A = 2 * B instead of A = B / 2.Algebraic mistakes during rearrangement of 6 * x - 20 = 4 * x - 10 can also lead to incorrect values of x.

Final Answer:
The sum of the present ages of A and B is 35 years.