Riya has an amount of Rs $6X$ and she invested $\frac{1}{3}$rd of the amount in two different schemes A and B in the ratio of 1 : 7 respectively. From the remaining amount, she paid Rs 2100 to her maid and invested the remaining Rs $P$ in the post office. If $X : P = 3 : 5$, then which of the following statement/s is or are correct?

I. Amount invested in scheme B is more than the amount invested in post office.

II. Amount spent on maid is more than the amount spent on scheme A.

III. Amount invested in scheme B is equal to amount spent on maid.

Question

Riya has an amount of Rs $6X$ and she invested $\frac{1}{3}$rd of the amount in two different schemes A and B in the ratio of 1 : 7 respectively. From the remaining amount, she paid Rs 2100 to her maid and invested the remaining Rs $P$ in the post office. If $X : P = 3 : 5$, then which of the following statement/s is or are correct?

I. Amount invested in scheme B is more than the amount invested in post office.

II. Amount spent on maid is more than the amount spent on scheme A.

III. Amount invested in scheme B is equal to amount spent on maid.

Accepted Answer

**Solution:**

1. Total amount Riya has = Rs $6X$

2. Amount invested in schemes A and B together = $\frac{1}{3} 	imes 6X = 2X$

3. The ratio of investment in A and B is 1 : 7.
   - Total parts = 1 + 7 = 8 parts
   - Amount in A = $\frac{1}{8} 	imes 2X = \frac{2X}{8} = \frac{X}{4}$
   - Amount in B = $\frac{7}{8} 	imes 2X = \frac{14X}{8} = \frac{7X}{4}$

4. Remaining amount after investing in A and B = $6X - 2X = 4X$

5. From this remaining amount, Rs 2100 is paid to maid and Rs $P$ is invested in post office.
   - So, $4X = 2100 + P$

6. Given ratio: $X : P = 3 : 5$
   - Let $X = 3k$ and $P = 5k$

7. Substitute $X$ and $P$ in equation $4X = 2100 + P$:
   $$4 	imes 3k = 2100 + 5k$$
   $$12k = 2100 + 5k$$
   $$12k - 5k = 2100$$
   $$7k = 2100$$
   $$k = 300$$

8. Calculate $X$ and $P$:
   - $X = 3k = 3 	imes 300 = 900$
   - $P = 5k = 5 	imes 300 = 1500$

9. Now calculate amounts:
   - Amount in A = $\frac{X}{4} = \frac{900}{4} = 225$
   - Amount in B = $\frac{7X}{4} = \frac{7 	imes 900}{4} = \frac{6300}{4} = 1575$
   - Amount paid to maid = 2100
   - Amount invested in post office = $P = 1500$

10. Evaluate each statement:
    - I. Amount in scheme B (1575) > Amount in post office (1500) → **True**
    - II. Amount spent on maid (2100) > Amount spent on scheme A (225) → **True**
    - III. Amount in scheme B (1575) = Amount spent on maid (2100) → **False**

**Answer:** Both I and II are correct, so option (b) Both I and II.

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