Data sufficiency – ages: A father's age is 4 years greater than the mother's age and 28 years greater than the son's age. What is the father's age? Which of the following additional conditions should be used to calculate the age? A) The ratio mother/son is a non-prime integer. B) The ratio mother/son is a perfect square integer.

Introduction / Context:
This is a data sufficiency question based on ages and ratios. You are given linear relationships between the ages of a father, mother and son, and then two alternative conditions involving the ratio of the mother's age to the son's age. The task is not to compute the exact age but to decide whether each condition, alone or together, is enough to determine the father's age uniquely.

Given Data / Assumptions:

Let the father's age be F, the mother's age be M and the son's age be S (all in years).
F = M + 4 (father is 4 years older than mother).
F = S + 28 (father is 28 years older than son).
From these, M = F - 4 and S = F - 28, so M = S + 24.
Condition A: The ratio M / S is a non-prime integer (composite integer > 1).
Condition B: The ratio M / S is a perfect square integer (for example 4, 9, 16, ...).
We assume ages are positive integers.

Concept / Approach:
First, use the basic age relations to express M / S in terms of S. Then, enforce that this ratio be an integer and see which specific integer values satisfy condition A and condition B. Finally, check whether for each condition we get a unique father's age or multiple possibilities. If multiple father's ages remain, the information is not sufficient.

Step-by-Step Solution:
From M = S + 24, the ratio mother/son is M / S = (S + 24) / S = 1 + 24 / S. For M / S to be an integer, 24 / S must be an integer, so S must be a positive divisor of 24. The positive divisors of 24 are S = 1, 2, 3, 4, 6, 8, 12, 24. Compute M, the ratio M / S, and F for each S: S = 1 → M = 25, M / S = 25, F = 29
S = 2 → M = 26, M / S = 13, F = 30
S = 3 → M = 27, M / S = 9, F = 31
S = 4 → M = 28, M / S = 7, F = 32
S = 6 → M = 30, M / S = 5, F = 34
S = 8 → M = 32, M / S = 4, F = 36
S = 12 → M = 36, M / S = 3, F = 40
S = 24 → M = 48, M / S = 2, F = 52. Condition A (“non-prime integer”) is satisfied when the ratio is composite: 25, 9 and 4. These correspond to S = 1, 3 and 8, giving father's ages 29, 31 and 36 — three different possible ages. So A alone is not sufficient. Condition B (“perfect square integer”) is also satisfied by 25, 9 and 4, i.e., exactly the same three cases: father's ages 29, 31 and 36. So B alone is not sufficient. Even when A and B are combined (ratio is both non-prime and a perfect square), we are still left with the same three values (25, 9, 4), i.e., three possible father's ages. Thus, even together A and B cannot determine a unique age.

Verification / Alternative check:
The key is to see that both conditions describe the same set of ratios in this problem: the composite perfect squares 4, 9 and 25. Each one leads to a different, valid integer age for the son, and therefore to a different father's age. Because more than one age satisfies all the conditions simultaneously, the data are insufficient for a unique answer.

Why Other Options Are Wrong:
Options claiming that A alone or B alone is sufficient ignore the fact that multiple possible father's ages remain. Saying both A and B together are required is also incorrect: even together they fail to fix a single age. The information is not inconsistent; it simply leads to several valid solutions, so the inconsistency choice is inappropriate.

Common Pitfalls:
A common mistake is to assume that adding more conditions must eventually lead to a unique answer without actually checking all possibilities. Another pitfall is forgetting that “non-prime integer” includes perfect squares like 4, 9 and 25, so conditions A and B do not restrict the solution space as much as they might appear to at first glance.

Final Answer:
Even when both conditions are used together, the father's age is not uniquely determined. The correct choice is Even together A and B are not sufficient.