Two persons with a fixed present difference and future ratio: The difference between the present ages of P and Q is 4 years. After 5 years, the ratio P : Q will be 9 : 8. Find the present age of P.

Introduction / Context:
We combine a fixed present difference with a future ratio, leading to a single variable equation. Sometimes the computed value does not align with any listed numeric option; in such cases, “None of these” is correct if the arithmetic is sound.

Given Data / Assumptions:

• P − Q = 4 (assume P is older; if not, algebra reveals consistency).
• After 5 years: (P + 5) / (Q + 5) = 9 / 8.

Concept / Approach:
Express P as Q + 4, substitute into the future ratio, and solve for Q. Recover P and compare with options.

Step-by-Step Solution:
1) (Q + 4 + 5) / (Q + 5) = 9 / 8 ⇒ (Q + 9) / (Q + 5) = 9 / 8.2) Cross-multiply: 8(Q + 9) = 9(Q + 5).3) 8Q + 72 = 9Q + 45 ⇒ 27 = Q.4) Then P = Q + 4 = 31.

Verification / Alternative check:
Future ages: P = 36, Q = 32 ⇒ 36 : 32 = 9 : 8 (true). Hence P = 31 is exact.

Why Other Options Are Wrong:

• 24/30/32 years are not equal to 31 years; the only valid choice is None of these.

Common Pitfalls:
Students sometimes misread “difference is 4” as “ratio difference” or add 5 to the ratio instead of ages. Maintain proper algebraic structure.

Final Answer:
None of these (P = 31 years)