Question

A sum amounts to Rs.~9680 in 2 years and to Rs.~10648 in 3 years respectively at compound interest. What will be the amount if the same sum is invested for \(1\dfrac{2}{5}\) years at the same rate of compound interest?

• A. Rs.~9025
• B. Rs.~9152
• C. Rs.~9215
• D. Rs.~9251

Hint:

When dealing with compound interest for fractional years, use the power form \(A = P(1 + r/100)^t\) where \(t\) can be fractional. Find \(r\) first from the difference in consecutive years’ amounts.

Answer 1

The Correct Option is B

Step 1: Find the annual rate of interest.
From the problem: Amount after 2 years = \(9680\) Amount after 3 years = \(10648\) The amount for the 3rd year alone is: \[ \frac{10648}{9680} = 1 + \frac{r}{100}. \] Simplify: \[ \frac{10648}{9680} = 1.10 \quad \Rightarrow \quad \frac{r}{100} = 0.10 \quad \Rightarrow \quad r = 10%. \] Step 2: Find the principal (initial sum).
Using \(A = P(1 + r/100)^t\): \[ 9680 = P (1.1)^2. \] So: \[ P = \frac{9680}{1.21} = 8000. \] Step 3: Find the amount for \(1\dfrac{2}{5}\) years.
\(1\dfrac{2}{5}\) years = \(1.4\) years = \(1 + \frac{2}{5}\) years. For 1 year: \[ \text{Amount after 1 year} = P(1.1) = 8000 \times 1.1 = 8800. \] Now apply compound interest for the remaining \(0.4\) years (which is \(\frac{2}{5}\) year). For fraction of a year at compound interest: \[ \text{Multiplier} = (1.1)^{0.4}. \] Step 4: Compute \((1.1)^{0.4\).}
\((1.1)^{0.4} \approx 1.039772\). So: \[ \text{Final amount} = 8000 \times (1.1)^{1.4} \approx 8000 \times 1.14396 = 9151.68 \approx 9152. \] \[ \boxed{\text{Rs.~9152}} \]