Rs 6500 is to be divided among A, B and C so that after A spends 90% of his share, B spends 75% of her share and C spends 60% of his share, their remaining savings are in the ratio 3 : 5 : 6. What are the respective shares of A, B and C?

Introduction / Context:
This problem involves reverse working from final savings back to original shares when different spending percentages are applied. It combines ratio, percentage and linear equations in a practical money distribution context, which is common in aptitude exams.

Given Data / Assumptions:

• Total amount to be divided = Rs 6500.
• A spends 90% of his share, so he saves 10%.
• B spends 75% of her share, so she saves 25%.
• C spends 60% of his share, so he saves 40%.
• Their savings are in the ratio 3 : 5 : 6.
• We must find the original shares of A, B and C.

Concept / Approach:
Let the savings of A, B and C be 3k, 5k and 6k for some k. Relate these savings back to their original shares by using the percentage of saving for each person. Then use the fact that the sum of the original shares equals 6500 to determine k and hence each share. This is a standard method for such reverse percentage problems.

Step-by-Step Solution:
Step 1: Let savings of A, B and C be 3k, 5k and 6k. Step 2: A saves 10% of his share. So 0.10 * share of A = 3k, giving share of A = 3k / 0.10 = 30k. Step 3: B saves 25% of her share. So 0.25 * share of B = 5k, giving share of B = 5k / 0.25 = 20k. Step 4: C saves 40% of his share. So 0.40 * share of C = 6k, giving share of C = 6k / 0.40 = 15k. Step 5: Total of all shares = share of A + share of B + share of C = 30k + 20k + 15k = 65k. Step 6: It is given that the total is Rs 6500, so 65k = 6500. Step 7: Therefore k = 6500 / 65 = 100. Step 8: Share of A = 30k = 30 * 100 = Rs 3000. Step 9: Share of B = 20k = 20 * 100 = Rs 2000. Step 10: Share of C = 15k = 15 * 100 = Rs 1500.

Verification / Alternative check:
Compute savings from these shares. A saves 10% of 3000 = 300. B saves 25% of 2000 = 500. C saves 40% of 1500 = 600. Savings ratio is 300 : 500 : 600. Dividing by 100 gives 3 : 5 : 6, exactly as required. Sum of shares 3000 + 2000 + 1500 is 6500, matching the total amount. Thisdouble check confirms the shares are correct.

Why Other Options Are Wrong:
Other combinations such as A = 1000, B = 2000, C = 1500 do not satisfy both the total Rs 6500 and the required savings ratio when the spending percentages are applied. At least one of the constraints will fail for each incorrect option, especially the precise 3 : 5 : 6 savings ratio.

Common Pitfalls:
Some learners confuse spending percentage with saving percentage, using 90% as the saving for A instead of 10%. Another common mistake is to allocate the 3 : 5 : 6 ratio directly to the shares instead of to the savings. It is important to read carefully that the ratio applies to savings after different spending percentages, not to the original amounts themselves.

Final Answer:
The shares are A = Rs 3000, B = Rs 2000 and C = Rs 1500.