One year ago, the ratio of mother’s age to daughter’s age was 4 : 1. Four years from now, the ratio will be 3 : 1. What was the ratio of their ages five years ago?

Introduction / Context:
Age-ratio problems rely on expressing ages at different reference times from a common base. Start with last year’s ratio, solve for the base parameter, then compute ages at any other time point as required.

Given Data / Assumptions:

• One year ago: Mother : Daughter = 4 : 1 ⇒ (M − 1) = 4k, (D − 1) = k
• Four years from now: (M + 4) : (D + 4) = 3 : 1

Concept / Approach:
Express M and D in terms of k from the first condition, substitute into the future ratio, and solve for k. Then compute ages five years ago and form the ratio.

Step-by-Step Solution:
From (M − 1, D − 1) = (4k, k) ⇒ M = 4k + 1, D = k + 1Future ratio: (M + 4)/(D + 4) = 3/1(4k + 1 + 4)/(k + 1 + 4) = 3 ⇒ (4k + 5)/(k + 5) = 34k + 5 = 3k + 15 ⇒ k = 10Ages 5 years ago: M − 5 = (4k + 1 − 5) = 4k − 4 = 36; D − 5 = (k + 1 − 5) = k − 4 = 6 ⇒ ratio 36 : 6 = 6 : 1

Verification / Alternative check:
Current ages: M = 41, D = 11. Last year 40 : 10 = 4 : 1; in 4 years 45 : 15 = 3 : 1. Both match.

Why Other Options Are Wrong:
They don’t match the back-calculated ages five years ago for k = 10.

Common Pitfalls:
Forgetting to shift the ages correctly when moving between “ago” and “after” time frames; mixing which ratio applies to which time.

Final Answer:
6 : 1