At a certain rate, a sum becomes$\dfrac{729}{625}$ of itself in one year when interest is compounded half yearly. What will be the compound interest

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At a certain rate, a sum becomes$\dfrac{729}{625}$ of itself in one year when interest is compounded half yearly. What will be the compound interest on Rs. 10,000 for $2\dfrac{1}{2}$ years at the same rate of interest, if interest is compounded annually (correct to the nearest rupee)?
1. Rs.4,693
2. Rs.4,352
3. Rs.14,352
4. Rs.4,532

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Correct Answer - Option 4 : Rs.4,532

Given:

Amount = 729/625 × Sum

Formula:

Let P = principal, R = rate of interest and N = time period

Compound interest (calculated annually) = P(1 + R/100)n - P

Amount(compounded half yearly) = P(1 + (R/2)/100)2n

Let P = Principal, R = R% and N = $a\dfrac{1}{b}$ years

Compound interest = P(1 + R/100)a{1 + (1/b × R/100)} - P

Calculation:

Time = 1 year

⇒ Amount = P(1 + R/200)2

⇒ 729P/625 = P(1 + R/200)2

⇒ (27/25)2 = (1 + R/200)2

⇒ 27/25 = 1 + R/200

⇒ R/200 = 2/25

⇒ R = 16%

∴ Required compound interest = [10000(1 + 16/100)2{1 + (16/2)/100}] - 10000 = Rs.4532.48 ≈ Rs.4532