The difference between the simple interest and the compound interest, compounded annually, on a certain sum of money for 2 years at 6% per annum is ₹78. Find the sum. [Give your answer to integral number without rounded off.]

Correct option is B

Given:

Rate of interest R = 6%

Time T = 2 years

The difference between Compound Interest and Simple Interest after 2 years is ₹78. 

Formula Used: 

Simple Interest (SI) formula:

$$\text{SI} = \frac{P \times R \times T}{100}$$​​

Compound Interest (CI) formula for annual compounding:​

$$\text{CI} = P \left(1 + \frac{R}{100}\right)^T - P$$ 

where P is the principal, R is the rate of interest, and T is the time period in years

Solution: 

Let PPP be the principal sum of money, which we need to find.

Rate of interest R =$6\%$

Time T=2 years

The difference between Compound Interest and Simple Interest after 2 years is ₹78.

For Simple Interest, using the formula: 

​$$\text{SI} = \frac{P \times R \times T}{100} = \frac{P \times 6 \times 2}{100} = \frac{12P}{100} = \frac{3P}{25}$$

For Compound Interest:

​$$\text{CI} = P \left(1 + \frac{R}{100}\right)^T - P = P \left(1 + \frac{6}{100}\right)^2 - P\\ \ \\ = P \left(1.06^2\right) - P = P \times 1.1236 - P = P \times (1.1236 - 1) = P \times 0.1236$$ 

We are told the difference between the Compound Interest and Simple Interest is ₹78:

CI − SI = 78 

Substitute the values of CI and SI:

​$$P \times 0.1236 - \frac{3P}{25} = 78$$ 

$$P×0.1236−P×0.12=78$$ 

$$P×(0.1236−0.12)=78$$ 

​$$P×0.0036=78$$ 

$$P = \frac{78}{0.0036} = 21666.67$$

Thus the sum of money (principal) is approximately ₹21,666.

$$P$$