Means – possible value with AM = 25 and GM = 7: Two positive numbers A and B have arithmetic mean 25 and geometric mean 7. Which of the following could be A?

Introduction / Context:
Given AM and GM of two positive numbers, we can reconstruct them up to order by solving a quadratic with known sum and product. The question asks which listed value can be one of the numbers.

Given Data / Assumptions:

• (A + B)/2 = 25 ⇒ A + B = 50
• √(AB) = 7 ⇒ AB = 49
• A, B > 0

Concept / Approach:
Let t be a number satisfying t^2 − (A + B)t + AB = 0 ⇒ t^2 − 50t + 49 = 0. The roots are the possible values for A and B (order-free).

Step-by-Step Solution:
Discriminant Δ = 50^2 − 4*49 = 2500 − 196 = 2304.√Δ = 48 ⇒ t = (50 ± 48)/2 ⇒ t ∈ {49, 1}.Thus {A, B} = {49, 1}; one valid value for A is 49.

Verification / Alternative check:
AM = (49 + 1)/2 = 25, GM = √(49*1) = 7 — both conditions satisfied.

Why Other Options Are Wrong:
10, 20, 25 do not complete a pair with sum 50 and product 49; 1 would also be valid but is not given in the original options set for selection of A (included here only to clarify the pair).

Common Pitfalls:
Misusing AM/GM inequalities or assuming A = B (which would give 25, not 7 as GM).

Final Answer:
49