The sum of the present ages of P and Q is 41 years. The age of P 2 years from now will equal the age of R 1 year ago. The age of P 4 years from now will equal the age of Q 1 year ago, and the ratio of the present ages of P and S is 3 : 4. What is the difference between the present ages of R and S?

Introduction / Context:
This multi person age problem links four people with several time shifted equalities and a given ratio. We must find the difference between the present ages of R and S. To solve it, we first determine P and Q using the sum of their ages and one time based relation, then find R and S through the remaining equations and the ratio. It is a good exercise in handling multiple constraints consistently.

Given Data / Assumptions:

• P + Q = 41 years at present.
• P 2 years hence equals R 1 year ago.
• P 4 years hence equals Q 1 year ago.
• The ratio of the present ages of P and S is 3 : 4.
• We must find the difference between the present ages of R and S.

Concept / Approach:
Let the present ages of P, Q, R and S be P, Q, R and S years. From P + Q = 41 and P 4 years hence equal to Q 1 year ago, we get P + 4 = Q - 1, hence Q in terms of P. Substituting into the sum gives P. From P 2 years hence equals R 1 year ago, we find R. The ratio P : S = 3 : 4 gives S in terms of P. Finally, we compute the difference between R and S.

Step-by-Step Solution:
Given P + Q = 41. (Equation 1) P 4 years hence equals Q 1 year ago: P + 4 = Q - 1. So Q = P + 5. (Equation 2) Substitute Equation 2 into Equation 1: P + (P + 5) = 41. This gives 2P + 5 = 41, so 2P = 36 and P = 18 years. Then Q = P + 5 = 18 + 5 = 23 years. P 2 years hence equals R 1 year ago: P + 2 = R - 1. So R = P + 3 = 18 + 3 = 21 years. The ratio P : S is 3 : 4, so 18 : S = 3 : 4. From 18 / S = 3 / 4, we get S = 18 * 4 / 3 = 24 years. Difference between present ages of R and S = 24 - 21 = 3 years.

Verification / Alternative check:
Check each condition. P + Q = 18 + 23 = 41, correct. Four years from now P will be 22, and one year ago Q was 22, matching P + 4 = Q - 1. Two years from now P will be 20, and one year ago R was 20, matching P + 2 = R - 1. The ratio P : S is 18 : 24, which simplifies to 3 : 4. All constraints hold, so the computed ages and the difference are consistent.

Why Other Options Are Wrong:
Differences of 2, 4, 5 or 6 years would not arise from a consistent set of ages that satisfy all the given equations. Any attempt to adjust one age to change the difference breaks at least one of the time based equalities or the ratio condition. Only a difference of 3 years preserves all relationships simultaneously.

Common Pitfalls:
It is easy to confuse which age is advanced or reduced in each statement, such as mixing up P 4 years hence with Q 4 years hence. Another frequent error is to misapply the ratio P : S and invert it when solving. Writing each equation explicitly and substituting step by step prevents algebraic and logical confusion.

Final Answer:
Thus, the difference between the present ages of R and S is 3 years.