If the amount is `3(3)/8` times the sum after 3 years at compound interest compounded annually, then the rate of interest per annum is

Options

A. `25%`
B. `50%`
C. `16(2)/3%`
D. `33(1)/3%`

Answer

To solve the problem step by step, we will use the formula for compound interest and the information given in the question.

### Step 1: Understand the given information
We are given that the amount after 3 years is \( \frac{27}{8} \) times the principal. We need to find the rate of interest per annum.

### Step 2: Assume the principal
Let us assume the principal amount (P) is \( 8x \).

### Step 3: Calculate the amount using the given ratio
According to the problem, the amount (A) after 3 years is:
\[ A = \frac{27}{8} \times P = \frac{27}{8} \times 8x = 27x \]

### Step 4: Use the compound interest formula
The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( A \) is the amount after time \( n \),
- \( P \) is the principal,
- \( r \) is the rate of interest,
- \( n \) is the number of years.

Substituting the values we have:
\[ 27x = 8x \left(1 + \frac{r}{100}\right)^3 \]

### Step 5: Simplify the equation
Dividing both sides by \( 8x \):
\[ \frac{27x}{8x} = \left(1 + \frac{r}{100}\right)^3 \]
This simplifies to:
\[ \frac{27}{8} = \left(1 + \frac{r}{100}\right)^3 \]

### Step 6: Take the cube root of both sides
To solve for \( 1 + \frac{r}{100} \), we take the cube root:
\[ 1 + \frac{r}{100} = \sqrt[3]{\frac{27}{8}} \]

### Step 7: Calculate the cube root
Calculating the cube root:
\[ \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} \]

### Step 8: Solve for \( r \)
Now we have:
\[ 1 + \frac{r}{100} = \frac{3}{2} \]
Subtracting 1 from both sides:
\[ \frac{r}{100} = \frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2} \]
Multiplying both sides by 100:
\[ r = 100 \times \frac{1}{2} = 50 \]

### Final Answer
Thus, the rate of interest per annum is \( 50\% \).

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