If the simple interset on a certain sum of money afte `6 (1)/(4)` years is `(3)/(8)` of the principal, then find the rate of interest per annum.

Answer

To solve the problem, we will use the formula for Simple Interest (SI):

\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step-by-Step Solution:

1. **Define the Variables**:
Let the principal amount be \( P \).

2. **Identify the Simple Interest**:
According to the problem, the simple interest after \( 6 \frac{1}{4} \) years is \( \frac{3}{8} \) of the principal. Thus,
\[ SI = \frac{3}{8} P \]

3. **Convert Time to Decimal**:
The time period given is \( 6 \frac{1}{4} \) years, which can be converted to a decimal:
\[ 6 \frac{1}{4} = 6 + \frac{1}{4} = 6 + 0.25 = 6.25 \text{ years} \]

4. **Substitute Values into the SI Formula**:
Now we can substitute the values into the simple interest formula:
\[ \frac{3}{8} P = \frac{P \times R \times 6.25}{100} \]

5. **Cancel \( P \) from Both Sides**:
Assuming \( P \neq 0 \), we can cancel \( P \) from both sides:
\[ \frac{3}{8} = \frac{R \times 6.25}{100} \]

6. **Cross-Multiply to Solve for \( R \)**:
Cross-multiplying gives us:
\[ 3 \times 100 = 8 \times R \times 6.25 \]
Simplifying this:
\[ 300 = 50R \]

7. **Solve for \( R \)**:
Now, divide both sides by 50:
\[ R = \frac{300}{50} = 6 \]

8. **Conclusion**:
The rate of interest per annum is \( 6\% \).