A sum becomes 16 times itself when compounded half yearly for 2 years. It becomes 27 times itself in $X$ years when compounded annually at the same rate. Find the difference between the compound interest (compounded annually) and simple interest received when Rs 15,000 is invested for $X$ years at 10\% per annum.

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Accepted Answer

**Solution:**

1. Let the principal be $P$ and the rate of interest be $r\%$ per annum.

2. Given that the sum becomes 16 times in 2 years when compounded half yearly.

- Number of half years = $2 \times 2 = 4$
   - Compound Interest formula for half yearly compounding:
     $$
     P \times \left(1 + \frac{r}{2 \times 100}\right)^4 = 16P
     $$
   - Dividing both sides by $P$:
     $$
     \left(1 + \frac{r}{200}\right)^4 = 16
     $$

3. Taking the fourth root on both sides:
   $$
   1 + \frac{r}{200} = \sqrt[4]{16} = 2
   $$

4. Solving for $r$:
   $$
   \frac{r}{200} = 2 - 1 = 1 \implies r = 200\% \text{ per annum}
   $$

5. Now, the sum becomes 27 times in $X$ years when compounded annually at the same rate $r = 200\%$.

- Using compound interest formula:
     $$
     P \times (1 + \frac{r}{100})^X = 27P
     $$
   - Dividing both sides by $P$:
     $$
     (1 + 2)^X = 27
     $$
   - Simplify:
     $$
     3^X = 27
     $$

6. Since $27 = 3^3$, we get:
   $$
   X = 3 \text{ years}
   $$

7. Now, calculate the difference between compound interest (CI) and simple interest (SI) for Rs 15,000 at 10\% per annum for 3 years.

- Simple Interest:
     $$
     SI = \frac{P \times r \times t}{100} = \frac{15000 \times 10 \times 3}{100} = 4500
     $$

- Compound Interest:
     $$
     CI = P \times \left(1 + \frac{r}{100}\right)^t - P = 15000 \times (1.1)^3 - 15000
     $$

- Calculate $ (1.1)^3 = 1.331 $

- So,
     $$
     CI = 15000 \times 1.331 - 15000 = 19965 - 15000 = 4965
     $$

8. Difference between CI and SI:
   $$
   4965 - 4500 = 465
   $$

**Answer:** Rs 465

**Note:** The options given include Rs 465, but the correct answer is Rs 465. However, the question states the correct answer as Rs 475 in option (a). This discrepancy arises because the rate calculated from the first part is 200\%, but the question states 10\% for the final calculation. So, we must use 10\% for the last calculation as given.

Recalculating difference with 10\% rate for 3 years:

- SI = $\frac{15000 \times 10 \times 3}{100} = 4500$

- CI = $15000 \times (1.1)^3 - 15000 = 15000 \times 1.331 - 15000 = 19965 - 15000 = 4965$

- Difference = $4965 - 4500 = 465$

Hence, the correct answer is Rs 465 (option b).

**Final Answer:** Rs 465

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