Question

At a certain simple rate of interest, a given sum amounts to Rs 13920 in 3 years, and to Rs 18960 in 6 years and 6 months. If the same given sum had been invested for 2 years at the same rate as before but with interest compounded every 6 months, then the total interest earned, in rupees, would have been nearest to:

• A. \(3096\)
• B. \(3221\)
• C. \(3180\)
• D. \(3150\)

Hint:

When you know the amounts at two different times under simple interest, the difference of amounts directly gives you the interest for the extra period. From that, you can easily find the yearly interest, then the rate and principal, and finally plug those into a compound interest calculation.

Answer 1

The Correct Option is B

Step 1: As before, \(P = 9600\) and the half-yearly rate is \(7.5\% = 0.075\), compounded over 4 half-years, so we need \((1.075)^4\).
Step 2: Expand \((1 + 0.075)^4\) term by term using the binomial theorem:
\[ 1 + 4(0.075) + 6(0.075)^2 + 4(0.075)^3 + (0.075)^4 = 1 + 0.3 + 0.033750 + 0.001688 + 0.0000316 \approx 1.335470. \]
Step 3: So \(A = 9600 \times 1.335470 \approx 12820.5\), and the interest earned is
\[ 12820.5 - 9600 \approx 3220.5. \]
\[ \boxed{\text{Interest} \approx \text{Rs } 3221} \]