A man invested Rs. 1600 on compound interest for 2 years at the rate of R% per annum and received Rs. 2304 as the amount. If he invests the same sum at simple interest for the same period at the rate of (R − 8)% per annum, what will be the simple interest he earns?

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**Solution:**

1. Given:
   - Principal, *P* = Rs. 1600
   - Time, *T* = 2 years
   - Amount with compound interest, *A* = Rs. 2304
   - Rate of interest = *R*% per annum (compound interest)

2. Using the compound interest formula:
   $$A = P 	imes \left(1 + \frac{R}{100}ight)^T$$

Substitute the known values:
   $$2304 = 1600 	imes \left(1 + \frac{R}{100}ight)^2$$

3. Divide both sides by 1600:
   $$\left(1 + \frac{R}{100}ight)^2 = \frac{2304}{1600} = 1.44$$

4. Taking square root on both sides:
   $$1 + \frac{R}{100} = \sqrt{1.44} = 1.2$$

5. Solve for *R*:
   $$\frac{R}{100} = 1.2 - 1 = 0.2$$
   $$R = 20\%$$

6. Now, the simple interest rate is *(R - 8)% = (20 - 8)% = 12%*.

7. Calculate simple interest using:
   $$SI = \frac{P 	imes R 	imes T}{100}$$

Substitute values:
   $$SI = \frac{1600 	imes 12 	imes 2}{100} = \frac{1600 	imes 24}{100} = 384$$

**Answer:** Rs. 384

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