Question : A sum of money invested at 20% compound interest (compounded annually). It would fetch Rs. 723 more in 2 years if interest is compounded half-yearly. The sum is:
Option 1: Rs. 15,000
Option 2: Rs. 30,000
Option 3: Rs. 20,000
Option 4: Rs. 7,500
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Correct Answer: Rs. 30,000
Solution : We have, $R = 20\%$, and $T = 2$. Let the principal be Rs. $P$. When interest is compounded annually, $\text{Compound interest} = P[(1 + \frac{R}{100})^{t}-1]$ $CI=P[(1+\frac{20}{100})^2-1]$ $CI=P[(\frac{6}{5})^2-1]=\frac{11P}{25}$ When the interest is compounded half-yearly, $CI=P[(1+\frac{10}{100})^4-1]$ $CI=P[(\frac{11}{10})^2-1]$ $CI=P[(\frac{14641}{10000})-1]=\frac{4641P}{10000}$ From the question, $⇒\frac{4641P}{10000}-\frac{11P}{25}=723$ $⇒\frac{4641P-4400P}{10000}=723$ $⇒\frac{241P}{10000}=723$ $⇒P = 30000$ Hence, the correct answer is Rs. 30,000.
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