Question : An amount was lent for two years at 20% per annum, compounded annually. Had the compounding been done half-yearly, the interest would have increased by Rs. 241. What was the amount (in Rs.) lent?
Option 1: Rs. 10,000
Option 2: Rs. 12,000
Option 3: Rs. 20,000
Option 4: Rs. 24,000
Correct Answer: Rs. 10,000
Solution :
Given: An amount was lent for two years at 20% per annum, compounded annually. The difference in interest is Rs. 241.
We know the formula, $A = P(1 + \frac{r}{n})^{nt}$.
For compounding annually $(n = 1)$,
$A = P(1 + \frac{0.2}{1})^{1×2}$
⇒ $A = P(1.44)$
Now, we need to find the interest earned when compounding annually.
Interest annually = $A – P$
⇒ $P(1.44) – P = 0.44P$
The interest when compounding half-yearly $(n=2)$.
Interest half-yearly = $P(1 + \frac{r}{n})^{nt} – P$.
⇒ $P(1 + \frac{0.2}{2})^{2×2} – P = P(1.4641) – P= 0.4641P$
Interest half-yearly – Interest annually = Rs. 241
⇒ $(0.4641P) – 0.44P = 241$
⇒ $0.0241P = 241$
⇒ $P = \frac{241}{0.0241} = 10,000$
So, the amount lent (Principal amount) was Rs. 10,000.
Hence, the correct answer is Rs. 10,000.
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