A, B and C together receive an amount of Rs 8400 and distribute it among themselves in the ratio 6 : 8 : 7 respectively. They save in the ratio 3 : 2 : 4 respectively and B saves Rs 400. What is the ratio of the expenditures of A, B and C respectively?

Introduction / Context:
This problem studies income, savings and expenditure using ratios. Three people share a total amount, then they save in a given ratio, and we know the actual saving of one of them. From this information we need to compute their expenditures and finally express those expenditures as a ratio. It combines basic ratio operations with the idea that expenditure equals income minus saving.

Given Data / Assumptions:
Total amount received by A, B and C together is Rs 8400. Income ratio (shares received) A : B : C = 6 : 8 : 7. Savings ratio A : B : C = 3 : 2 : 4. B saves Rs 400. Expenditure for each person = income share minus saving of that person.

Concept / Approach:
We first convert the income ratio into actual incomes by using the total Rs 8400. Next we use the savings ratio along with B actual saving to compute the savings of A and C. Once incomes and savings are known for each person, expenditure is simply income minus saving. The last step is expressing the three expenditure amounts as a simplified ratio. This follows the standard approach of using ratios to convert between totals, shares, savings and expenditures.

Step-by-Step Solution:
Step 1: Income ratio A : B : C is 6 : 8 : 7. Step 2: Sum of ratio parts = 6 + 8 + 7 = 21. Step 3: Value of one income part = 8400 / 21 = Rs 400. Step 4: Income of A = 6 * 400 = Rs 2400. Step 5: Income of B = 8 * 400 = Rs 3200. Step 6: Income of C = 7 * 400 = Rs 2800. Step 7: Savings are in ratio 3 : 2 : 4 for A : B : C. Step 8: Let savings of A, B and C be 3s, 2s and 4s respectively. We are told that B saves Rs 400, so 2s = 400. Step 9: From 2s = 400 we get s = 200. Step 10: Saving of A = 3 * 200 = Rs 600. Step 11: Saving of B = 2 * 200 = Rs 400. Step 12: Saving of C = 4 * 200 = Rs 800. Step 13: Expenditure of A = 2400 - 600 = Rs 1800. Step 14: Expenditure of B = 3200 - 400 = Rs 2800. Step 15: Expenditure of C = 2800 - 800 = Rs 2000. Step 16: Expenditure ratio A : B : C = 1800 : 2800 : 2000. Step 17: Divide all by 200 to simplify, giving 9 : 14 : 10.

Verification / Alternative check:
We can verify that incomes minus savings equal expenditures and everything balances. Total savings = 600 + 400 + 800 = Rs 1800. Total expenditure = 1800 + 2800 + 2000 = Rs 6600. Sum of total savings and total expenditure is 1800 + 6600 = Rs 8400, which matches the original total received. Also, the savings are indeed in the ratio 3 : 2 : 4 and B saving of 400 matches the given condition. Hence the expenditure ratio 9 : 14 : 10 is consistent.

Why Other Options Are Wrong:
The ratio 6 : 8 : 7 is the income ratio, not the expenditure ratio, so option 6 : 8 : 7 is incorrect. The ratio 8 : 6 : 7 does not match the expenditure values and simply reorders the income ratio. The ratio 12 : 7 : 9 does not correspond to 1800, 2800 and 2000 in any simple scaling. Only 9 : 14 : 10 matches the correctly derived expenditure amounts.

Common Pitfalls:
Learners sometimes confuse income and savings ratios and may try to divide the total Rs 8400 directly according to the savings ratio. Another common oversight is to assume that B saving is equal to one of the income parts rather than a separate quantity. Clearly distinguishing income, saving and expenditure, and working stepwise from income to saving to expenditure, prevents such mistakes.

Final Answer:
The ratio of the expenditures of A, B and C is 9 : 14 : 10.