Question

A sum amounts to Rs 9,680 in 2 years and Rs 10,648 in 3 years at compound interest. Find the rate of interest per annum.

• A. 8%
• B. 10%
• C. 12%
• D. 15%

Hint:

When compound interest amounts are given for consecutive years, you can find the rate of interest directly without finding the principal.
Simply find the difference between the two amounts, which is the interest earned in that year:
\[ \text{Interest} = 10648 - 9680 = 968 \].
The rate of interest is this interest divided by the amount of the previous year, multiplied by 100:
\[ R = \frac{968}{9680} \times 100 = 10% \].
This saves significant time during the examination.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks us to find the annual compound interest rate when a certain principal sum grows to two different amounts in consecutive years (2 years and 3 years).

Step 2: Key Formula or Approach:
Under compound interest, the amount at the end of any year acts as the principal for the succeeding year.
Therefore, the interest earned in the 3rd year is simply the interest calculated on the amount accumulated at the end of the 2nd year.
The formula for the compound interest amount is:
\[ A = P\left(1 + \frac{R}{100}\right)^n \]

Step 3: Detailed Explanation:
1. Let the principal sum be \(P\) and the annual rate of interest be \(R\)%.
2. The amount at the end of 2 years (\(A_2\)) is given as Rs 9,680.
Using the formula:
\[ 9680 = P\left(1 + \frac{R}{100}\right)^2 \quad \text{--- (Equation 1)} \]
3. The amount at the end of 3 years (\(A_3\)) is given as Rs 10,648.
Using the formula:
\[ 10648 = P\left(1 + \frac{R}{100}\right)^3 \quad \text{--- (Equation 2)} \]
4. To eliminate the principal variable \(P\), we divide Equation 2 by Equation 1:
\[ \frac{10648}{9680} = \frac{P\left(1 + \frac{R}{100}\right)^3}{P\left(1 + \frac{R}{100}\right)^2} \]
5. This simplifies directly to:
\[ \frac{10648}{9680} = 1 + \frac{R}{100} \].
6. Let us calculate the value of the fraction on the left-hand side:
\[ \frac{10648}{9680} = 1.1 \].
7. Substitute this back into our simplified equation:
\[ 1.1 = 1 + \frac{R}{100} \].
8. Subtracting 1 from both sides gives:
\[ \frac{R}{100} = 0.1 \implies R = 10% \].

Step 4: Final Answer:
The rate of interest per annum is 10%, which is Option (B).