A sum of money becomes eight times of itself in 3 years at compound interest. The rate of interest per annum is

Answer

To solve the problem of finding the rate of interest per annum when a sum of money becomes eight times itself in 3 years at compound interest, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We know that the principal amount (P) becomes 8 times itself in 3 years. Therefore, if we let the principal amount be P, the amount after 3 years (A) will be:
\[ A = 8P \]

2. **Use the Compound Interest Formula**: The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( A \) = final amount
- \( P \) = principal amount
- \( r \) = rate of interest per annum
- \( n \) = number of years

In our case, we have:
\[ 8P = P \left(1 + \frac{r}{100}\right)^3 \]

3. **Simplify the Equation**: We can divide both sides of the equation by \( P \) (assuming \( P \neq 0 \)):
\[ 8 = \left(1 + \frac{r}{100}\right)^3 \]

4. **Take the Cube Root**: To find \( 1 + \frac{r}{100} \), we take the cube root of both sides:
\[ 1 + \frac{r}{100} = \sqrt[3]{8} \]
Since \( \sqrt[3]{8} = 2 \), we have:
\[ 1 + \frac{r}{100} = 2 \]

5. **Solve for \( r \)**: Now, we can isolate \( r \):
\[ \frac{r}{100} = 2 - 1 \]
\[ \frac{r}{100} = 1 \]
\[ r = 100 \]

6. **Conclusion**: The rate of interest per annum is:
\[ \boxed{100\%} \]