A sum invested at 10% p.a. for 2 years becomes ₹ 3,267, when the interest is compounded annually.
What will be the simple interest on the same sum at the same rate in 2$\frac{1}{2}$ years ?

Question

Accepted Answer

To determine the simple interest on the same sum at the same rate for 2.5 years, we first need to find the principal amount using the information given about compound interest.

### Step 1: Calculate the Principal Amount

The formula for compound interest is:

$$ A = P \left(1 + \frac{r}{100}\right)^n $$

Where:
- $ A $ is the amount after $ n $ years,
- $ P $ is the principal amount,
- $ r $ is the annual interest rate,
- $ n $ is the number of years.

Given:
- $ A = ₹ 3,267 $
- $ r = 10\% $
- $ n = 2 $

Substitute the given values into the formula:

$$ 3,267 = P \left(1 + \frac{10}{100}\right)^2 $$

$$ 3,267 = P \left(1.1\right)^2 $$

$$ 3,267 = P \times 1.21 $$

$$ P = \frac{3,267}{1.21} $$

$$ P = ₹ 2,700 $$

### Step 2: Calculate the Simple Interest

Now that we have the principal amount, we can calculate the simple interest for 2.5 years at the same rate of 10%.

The formula for simple interest is:

$$ SI = \frac{P \times r \times t}{100} $$

Where:
- $ SI $ is the simple interest,
- $ P = ₹ 2,700 $,
- $ r = 10\% $,
- $ t = 2.5 $ years.

Substitute the values into the formula:

$$ SI = \frac{2,700 \times 10 \times 2.5}{100} $$

$$ SI = \frac{67,500}{100} $$

$$ SI = ₹ 675 $$

### Conclusion

The simple interest on the same sum at the same rate for 2.5 years is ₹ 675, which corresponds to Option 3.

### Explanation of Incorrect Options

1. **Option 1: ₹ 625** - This amount is less than the calculated simple interest of ₹ 675. It does not account for the full 2.5 years at the 10% rate.

2. **Option 2: ₹ 650** - This amount is also less than the calculated simple interest of ₹ 675. It does not accurately reflect the interest accrued over 2.5 years.

3. **Option 4: ₹ 700** - This amount exceeds the calculated simple interest of ₹ 675. It overestimates the interest for the given period and rate.

Therefore, Option 3 is the correct answer, as it accurately represents the simple interest calculated for the given conditions.