Ages with present sum and past ratio: The present ages of A and B are in the ratio 3 : 5 and their sum is 56 years. What was the ratio of their ages 7 years ago?

Introduction / Context:
This is a classic ages and ratios problem. We are given a current ratio and total, and then asked for the ratio at a past time. The technique is to first compute actual present ages and then shift both by the same amount for the past scenario.

Given Data / Assumptions:

• Present ratio A : B = 3 : 5.
• A + B = 56 years.
• We need the ratio (A − 7) : (B − 7).

Concept / Approach:
Let A = 3k and B = 5k, so A + B = 8k = 56 ⇒ k = 7. Then find actual ages and subtract 7 years from both to get the past ages, and finally simplify that ratio if needed.

Step-by-Step Solution:
A = 3k = 21 years; B = 5k = 35 years.Seven years ago: A = 14, B = 28.Required ratio = 14 : 28 = 1 : 2.

Verification / Alternative check:
Check present sum: 21 + 35 = 56 (fits). The past ratio reduces perfectly to 1 : 2, which is consistent with equal subtraction from both ages.

Why Other Options Are Wrong:

• 3 : 4 or 2 : 3 do not match the computed 14 : 28.
• 3 : 5 repeats the present ratio, not the past one.

Common Pitfalls:

• Subtracting 7 only from the numerator term instead of both persons’ ages.
• Forgetting to compute exact ages before applying the 7-year shift.

Final Answer:
1 : 2