Question : An amount was lent for one year at 18% per annum, compounding annually. Had the compounding been done half-yearly, the interest would have increased by 324. What was the amount (in Rs.) lent?
Option 1: Rs. 32,000
Option 2: Rs. 36,000
Option 3: Rs. 40,000
Option 4: Rs. 72,000
Correct Answer: Rs. 40,000
Solution :
Given: An amount was lent for one year at 18% per annum, compounding annually.
We know the formula, $P(1 + \frac{r}{n})^{nt} = A$.
For yearly compounding (n = 1),
$P(1 + \frac{0.18}{1}) = A$
⇒ $P(1.18) = A$
The interest each year is given as $A - P$.
⇒ $ P(1.18) -P = 0.18P$
The interest is at (n = 2) when compounded half–yearly
Half– yearly interest = $P(1 + \frac{r}{n})^{nt}-P$.
⇒ $P(1 + \frac{0.18}{2})^{2}-P= P(1.09)^2 -P$
⇒ $1.1881P-P=0.1881P$
According to the question,
$(0.1881P) - 0.18P = 324$
⇒ $0.0081P = \frac{324}{0.0081}$
⇒ $P=Rs. 40,000$
Thus, Rs. 40,000 was the loaned money (Principal amount).
Hence, the correct answer is Rs. 40,000.
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