Question : The difference between the compound interest (compounding half-yearly) for 1 year and the simple interest for 1 year on a certain sum lent out at the annual rate of 10% is Rs, 25. What is the sum?
Option 1: Rs. 12000
Option 2: Rs. 6000
Option 3: Rs. 8000
Option 4: Rs. 10000
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Correct Answer: Rs. 10000
Solution :
Given,
Difference between compound interest and simple interest = Rs. 25,
Annual interest rate = 10%
We know,
Compound Interest for 1 year = $P[(1 + \frac{r}{100})^n- 1]$, where $r$ is the annual interest rate and $n$ is the number of compounding periods.
Simple Interest for 1 year = $\frac{P × r × t}{100}$, where $r$ is the annual interest rate and $t$ is the time in years.
Let the principal sum be $P$.
The given difference is:
⇒ $P[(1 + \frac{5}{100})^2 - 1] - \frac{P × 10 × 1}{100} = 25$
⇒ $P[(1+0.05)^2-1]-\frac{P}{10}=25$
⇒ $P[(1.05)^2-1]-\frac{P}{10}=25$
⇒ $P[1.1025-1]-0.1P=25$
⇒ $0.1025P-0.1P=25$
⇒ $0.0025P=25$
⇒ $P = \text{Rs.}\ 10000$
Hence, the correct answer is Rs. 10000.
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