Question

A certain sum of money becomes 13,824 after 3 years and 32,768 after 6 years if the interest is compounded annually. What is the rate of interest per annum?

• A. 6.66%
• B. 3.33%
• C. 16.66
• D. 33.33%

Answer 1

The Correct Option is D

Finding the Rate of Interest Using the Compound Interest Formula

Step 1: Define the Compound Interest Formula 

The compound interest formula is:

\[ A = P \left(1 + \frac{r}{100} \right)^t \]

Step 2: Given Data

• \( A_2 = 32,768 \) (Amount after 6 years)
• \( A_1 = 13,824 \) (Amount after 3 years)

Step 3: Find the Interest Rate

Using the compound interest formula for two different time periods:

\[ \frac{A_2}{A_1} = \left(1 + \frac{r}{100} \right)^{6-3} \]

Substituting the values:

\[ \frac{32,768}{13,824} = \left(1 + \frac{r}{100} \right)^3 \]

Step 4: Solve for \( r \)

Calculating the ratio:

\[ 2.37 = \left(1 + \frac{r}{100} \right)^3 \]

Taking the cube root on both sides:

\[ 1 + \frac{r}{100} = 1.333 \]

Solving for \( r \):

\[ r = (1.333 - 1) \times 100 = 33.33\% \]

Final Answer:

Thus, the rate of interest is 33.33% (Option D).