K men agree to contribute a total gift of Rs. L to a trust. If three men drop out and do not contribute, by how much does the individual contribution of each remaining man increase in order to keep the total gift amount unchanged?

Introduction / Context:
This problem tests algebraic reasoning with variables and proportional sharing of a fixed amount. A group of K men plan to donate a fixed total amount Rs. L. When some men drop out, the remaining men must share the same total, so each individual share increases. The question asks specifically for the increase per man, not the new share itself.

Given Data / Assumptions:
Number of men originally = K.Total intended gift to the trust = L rupees.Three men drop out, so remaining men = K − 3.The total contribution L remains the same.We assume K is greater than 3 and all contributions are equal within each scenario.

Concept / Approach:
The key idea is equal sharing of a fixed sum. Initially, each of the K men pays L / K. After three men drop out, each of the remaining K − 3 men pays L / (K − 3). The required answer is the difference between the new share and the old share. We work symbolically with K and L throughout and simplify the algebraic expression.

Step-by-Step Solution:
Step 1: Original contribution per man = L / K.Step 2: New contribution per man after three men drop out = L / (K − 3).Step 3: Increase per man = new share − old share.Step 4: Increase per man = L / (K − 3) − L / K.Step 5: Take a common denominator K(K − 3): increase = L[K − (K − 3)] / [K(K − 3)].Step 6: Simplify the numerator: K − (K − 3) = 3.Step 7: Therefore increase per man = 3L / [K(K − 3)].

Verification / Alternative check:
Choose simple numerical values for checking. Let K = 5 and L = 1000. Initially, each man gives 1000 / 5 = 200. If three men drop out, remaining men = 2 and each must give 1000 / 2 = 500. Increase per man is 500 − 200 = 300. Our formula gives 3L / [K(K − 3)] = 3 * 1000 / [5 * 2] = 300, which matches perfectly.

Why Other Options Are Wrong:
The expression L / (K − 3) is the new contribution per man, not the increase, so it ignores the original share. The expression L / K is just the old share. The term 3L / K does not involve K − 3 at all, so it cannot correctly reflect the effect of three men leaving. The term L / (K * (K − 3)) is dimensionally too small and does not come from the algebraic difference of the two shares.

Common Pitfalls:
Many learners mistakenly mark L / (K − 3) as the answer because they misread the question as asking for the new contribution rather than the extra amount. Others forget to subtract the old share from the new one when computing the increase. Careful reading and stepwise algebraic simplification avoid these issues.

Final Answer:
The increase in contribution for each remaining man is 3L / (K * (K − 3)).