A man invested a certain sum in scheme A at a simple interest rate of 15% per annum for 2 years and earned a total simple interest of Rs. 7800. He then increased his investment by Rs. *X* and invested the new amount in scheme B, which offers compound interest at 10% per annum for 2 years, earning Rs. 6720 as compound interest. Find the value of *X* (in Rs.).

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**Solution:**

1. Let the initial amount invested in scheme A be *P*.

2. Given that the simple interest (SI) for 2 years at 15% p.a. is Rs. 7800.

Using the formula for simple interest:
   $$SI = \frac{P 	imes R 	imes T}{100}$$
   Substitute the values:
   $$7800 = \frac{P 	imes 15 	imes 2}{100}$$

3. Simplify to find *P*:
   $$7800 = \frac{30P}{100}$$
   $$7800 = 0.3P$$
   $$P = \frac{7800}{0.3} = 26000$$

4. The man increases the amount by Rs. *X* and invests in scheme B at compound interest (CI) of 10% p.a. for 2 years.

So, the new principal is:
   $$P + X = 26000 + X$$

5. The compound interest earned is Rs. 6720.

Compound interest formula:
   $$CI = P \left(1 + \frac{R}{100}ight)^T - P$$

Substitute the values:
   $$6720 = (26000 + X) \left(1 + \frac{10}{100}ight)^2 - (26000 + X)$$

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

6. Rewrite the equation:
   $$6720 = (26000 + X)(1.21) - (26000 + X)$$
   $$6720 = (26000 + X)(1.21 - 1)$$
   $$6720 = (26000 + X)(0.21)$$

7. Solve for *X*:
   $$26000 + X = \frac{6720}{0.21} = 32000$$
   $$X = 32000 - 26000 = 6000$$

**Answer:** Rs. 6000

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