Question

The difference between the Compound Interest (compounded annually) and the Simple Interest on a certain sum at 10% per annum for 2 years is $15. Find the principal sum.

Hint:

The difference between Compound Interest and Simple Interest for 2 years is given by the formula \( \frac{P \times R^2}{100^2} \).

Answer 1

Step 1: Formula for Compound Interest (CI) and Simple Interest (SI). 
The formulas for SI and CI are as follows: \[ \text{SI} = \frac{P \times R \times T}{100} \] \[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P \] where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time. 
Step 2: Using the information provided. 
We are given:
- Rate \( R = 10% \)
- Time \( T = 2 \, \text{years} \)
- Difference between CI and SI after 2 years = $15
The formula for the difference between CI and SI is: \[ \text{CI} - \text{SI} = P \times \frac{R^2}{100^2} \] Substitute the values \( R = 10 \) and the difference = 15: \[ 15 = P \times \frac{10^2}{100^2} \] \[ 15 = P \times \frac{100}{10000} \] \[ 15 = \frac{P}{100} \] \[ P = 1000 \] 
Step 3: Conclusion. 
The principal sum is $1000.