A sum of money at 8% Simple Interest was placed in a deposit for 7 years. The matured
amount was then invested in a scheme offering 10% Compound Interest compounded annually, thereby procuring an interest of Rs.1638 in 2 years. What was the original sum?

Correct option is A

Given:

Let the original sum be  P . The interest rate for Simple Interest is 8%, and it was deposited for 7 years. The matured amount was then invested at 10% Compound Interest compounded annually for 2 years, procuring an interest of Rs.1638.

Formula Used:

The formula for Simple Interest is:

$$A = P + P \times \frac{r \times t}{100}$$​ where  A  is the amount,  P  is the principal,  r  is the rate of interest, and  t  is the time.

The formula for Compound Interest is:

$$A = P(1 + \frac{r}{100})^t$$​ where  A  is the amount,  P  is the principal, r  is the rate of interest, and  t  is the time.

Solution:

First, calculate the amount after 7 years of Simple Interest:

​$$A = P + P \times \frac{8 \times 7}{100} = P(1 + \frac{56}{100}) = P \times 1.56$$​​

Now, this amount was invested at 10% Compound Interest for 2 years, and the interest earned is Rs. 1638.

Let the amount after Simple Interest be  A . Using the Compound Interest formula:

​$$A(1 + \frac{10}{100})^2 - A = 1638$$​​

This simplifies to:

​$$A \times 1.21 - A = 1638$$

$$0.21A = 1638$$​​​

Therefore,  A = $$\frac{1638}{0.21} = 7800$$​​

Since  $$A = P \times 1.56$$ we now have:

​$$7800 = P \times 1.56$$​​

Thus, $$P = \frac{7800}{1.56} = 5000$$​​

Therefore, the original sum was Rs. 5000.