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:

Instead of recalling the CI minus SI shortcut formula, try writing both the Simple Interest and the Compound Interest as separate fractions of the principal for the given rate and time, then subtract one from the other. The unknown principal will cancel down to a single simple equation.

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: \[ 15 = P \times \frac{10^2}{100^2} \] \[ 15 = P \times \frac{100}{10000} \] \[ 15 = \frac{P}{100} \] \[ P = 1500 \] 
Step 3: Conclusion. 
The principal sum is $1500.

Answer 2

Concept:

• Instead of quoting the CI minus SI shortcut formula, the Simple Interest and Compound Interest can each be written directly in terms of the principal, and then subtracted, avoiding the need to remember a separate formula.
• Both SI and CI can be expressed as a fraction of the principal, so the difference also comes out as a fraction of the principal.

Step 1: Write the Simple Interest for 2 years as a fraction of the principal.
$\text{SI} = \dfrac{P \times 10 \times 2}{100} = 0.2P$

Step 2: Write the Compound Interest for 2 years as a fraction of the principal.
$\text{CI} = P(1.1)^2 - P = P(1.21 - 1) = 0.21P$

Step 3: Subtract to get the difference as a fraction of the principal.
$\text{CI} - \text{SI} = 0.21P - 0.2P = 0.01P$

Step 4: Substitute the given difference and solve for P.
$0.01P = 15$
$P = \dfrac{15}{0.01} = 1500$

Final Answer: The principal sum is $1500.