Question : On a certain sum of money lent out at 16% per annum, the difference between the compound interest for 1 year, payable half-yearly, and the simple interest for 1 year is Rs. 60. The sum is:
Option 1: Rs. 937.5
Option 2: Rs. 9,000
Option 3: Rs. 9,365
Option 4: Rs. 9,375
Correct Answer: Rs. 9,375
Solution :
Given: Rate = 16%
Time = 1 year
Difference between the compound interest for 1 year, payable half-yearly, and the simple interest for 1 year = Rs. 60
We know that,
Compound interest = $\text{Principal}×(1+\frac{\text{Rate}}{2×100})^{\text{2×Time}}-\text{Principal}$
⇒ Compound interest = $\text{Principal}×((1+\frac{8}{2×100})^{\text{2×1}}-1)$
= $\text{Principal}×(1.1664-1)$
= $0.1664 × \text{Principal}$ ................................................(1)
Now, Simple interest = $\frac{\text{Principal × Rate × Time}}{100}$
= $\frac{\text{Principal × 16 × 1}}{100}$
= $0.16 × \text{Principal}$ ................................................(2)
From equation (1) and (2), we get:
$0.1664 × \text{Principal} - 0.16 × \text{Principal} = 60$
⇒ $0.0064 × \text{Principal} = 60$
$\therefore\text{Principal} = \frac{60}{0.0064}= 9375$
Hence, the correct answer is Rs. 9,375.
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