Rs. $(2000 + x)$ is invested at 10\% compound interest for two years. The interest received is Rs. 525. The interest received when extit{m} times the amount extit{x} is invested again at 20\% compound interest for the same time period is $(y + 1)$ times Rs. 880. Which of the following option(s) is/are incorrect?

(i) $m = 4(y + 1)$

(ii) $x = ext{Rs. } 500$

(iii) $x = ext{Rs. } 1500$

Question

Rs. $(2000 + x)$ is invested at 10\% compound interest for two years. The interest received is Rs. 525. The interest received when 	extit{m} times the amount 	extit{x} is invested again at 20\% compound interest for the same time period is $(y + 1)$ times Rs. 880. Which of the following option(s) is/are incorrect?

(i) $m = 4(y + 1)$

(ii) $x = 	ext{Rs. } 500$

(iii) $x = 	ext{Rs. } 1500$

Accepted Answer

**Solution:**

1. Given: Principal = Rs. $(2000 + x)$, rate = 10\%, time = 2 years, compound interest (CI) = Rs. 525.

2. Compound Interest formula for 2 years:
   $$	ext{CI} = P 	imes \left(1 + \frac{r}{100}ight)^2 - P$$

Substitute values:
   $$525 = (2000 + x) 	imes \left(1 + \frac{10}{100}ight)^2 - (2000 + x)$$

3. Calculate the growth factor:
   $$\left(1 + \frac{10}{100}ight)^2 = (1.1)^2 = 1.21$$

4. So,
   $$525 = (2000 + x) 	imes 1.21 - (2000 + x) = (2000 + x)(1.21 - 1) = (2000 + x)(0.21)$$

5. Solve for $(2000 + x)$:
   $$2000 + x = \frac{525}{0.21} = 2500$$

6. Therefore,
   $$x = 2500 - 2000 = 500$$

7. Now, the amount invested again is $m 	imes x = m 	imes 500$ at 20\% CI for 2 years.

8. Calculate the interest on this amount:
   $$	ext{CI} = m 	imes 500 	imes \left(1 + \frac{20}{100}ight)^2 - m 	imes 500 = m 	imes 500 	imes (1.44 - 1) = m 	imes 500 	imes 0.44 = 220m$$

9. Given this interest is $(y + 1)$ times Rs. 880:
   $$220m = (y + 1) 	imes 880$$

10. Simplify:
    $$220m = 880(y + 1) \implies m = 4(y + 1)$$

11. Check the options:
    - (i) $m = 4(y + 1)$ is correct.
    - (ii) $x = 500$ is correct.
    - (iii) $x = 1500$ is incorrect because we found $x = 500$.

12. The question asks which options are incorrect.
    - (i) is correct, so not incorrect.
    - (ii) is correct, so not incorrect.
    - (iii) is incorrect.

13. Therefore, only option (iii) is incorrect.

**Answer:** (a) Both (i) and (ii) are incorrect is not true since (i) and (ii) are correct.

Re-examining the question: It asks which options are incorrect.
- (i) is correct
- (ii) is correct
- (iii) is incorrect

Hence, only (iii) is incorrect.

So the correct choice for incorrect options is (d) Only (iii).

**Correction:** The correct answer is (d) Only (iii).

**Final Answer:** (d) Only (iii)

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