Two statements are given below. Based on these statements, answer the following question:

What is the respective ratio of savings of V and D?

I. Income of V is 4\% less than that of S, and the ratio of expenditure of V to S is 7 : 8. D spends $ \frac{3}{5} $ of his income.

II. S saves Rs. 14,000 and V saves Rs. 14,800. Income of D is Rs. 2,000 more than that of S.

Which of the statements is/are sufficient to answer the question?

Question

Two statements are given below. Based on these statements, answer the following question:

What is the respective ratio of savings of V and D?

I. Income of V is 4\% less than that of S, and the ratio of expenditure of V to S is 7 : 8. D spends $ \frac{3}{5} $ of his income.

II. S saves Rs. 14,000 and V saves Rs. 14,800. Income of D is Rs. 2,000 more than that of S.

Which of the statements is/are sufficient to answer the question?

Accepted Answer

**Solution:**

1. From statement I:
   - Let income of S be $25x$.
   - Then income of V is 4\% less than S, so $24x = 25x \times \frac{96}{100}$.
   - Let expenditure of V be $7y$, so expenditure of S is $8y$.
   - D spends $\frac{3}{5}$ of his income.

2. From statement II:
   - Saving of S = Rs. 14,000.
   - Saving of V = Rs. 14,800.
   - Income of D is Rs. 2,000 more than that of S.

3. Using both statements together:
   - Savings of S = Income - Expenditure = $25x - 8y = 14,000$.
   - Savings of V = Income - Expenditure = $24x - 7y = 14,800$.

4. From the ratio of expenditures given in statement I, we have:
   $$
   \frac{8y}{7y} = \frac{25x - 14,000}{24x - 14,800}
   $$

5. Cross-multiplying and simplifying:
   $$
   8(24x - 14,800) = 7(25x - 14,000) \ 
   192x - 118,400 = 175x - 98,000 \ 
   192x - 175x = 118,400 - 98,000 \ 
   17x = 20,400 \ 
   x = 1,200
   $$

6. Income of D = Income of S + 2,000 = $25 \times 1,200 + 2,000 = 32,000$.

7. Since D spends $\frac{3}{5}$ of his income, savings of D = $\frac{2}{5} \times 32,000 = 12,800$.

8. Therefore, the ratio of savings of V to D is:
   $$
   14,800 : 12,800 = 37 : 32
   $$

**Answer:** Both statements I and II together are sufficient to find the ratio.

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