In what time will the interest on a certain sum of money at 6% be `5/8` of itself ?

Answer

To solve the problem, we need to find the time (T) when the simple interest (SI) on a certain sum of money (P) at a rate of 6% per annum is equal to \( \frac{5}{8} \) of the principal amount (P).

### Step-by-Step Solution:

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

2. **Express the Simple Interest**:
According to the problem, the simple interest (SI) is given as:
\[ SI = \frac{5}{8} P \]

3. **Use the Simple Interest Formula**:
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( R \) is the rate of interest and \( T \) is the time in years.

4. **Substitute the Known Values**:
Here, the rate \( R = 6\% \). Substituting the values into the formula gives:
\[ \frac{5}{8} P = \frac{P \times 6 \times T}{100} \]

5. **Cancel the Principal (P)**:
Since \( P \) is common on both sides, we can cancel it out (assuming \( P \neq 0 \)):
\[ \frac{5}{8} = \frac{6 \times T}{100} \]

6. **Cross-Multiply to Solve for T**:
Cross-multiplying gives:
\[ 5 \times 100 = 6 \times T \times 8 \]
Simplifying this:
\[ 500 = 48T \]

7. **Solve for T**:
Now, divide both sides by 48:
\[ T = \frac{500}{48} \]
Simplifying this fraction:
\[ T = \frac{125}{12} \]

8. **Convert to Mixed Number**:
To convert \( \frac{125}{12} \) into a mixed number:
- Divide 125 by 12, which gives 10 with a remainder of 5.
- Thus, \( \frac{125}{12} = 10 \frac{5}{12} \).

### Final Answer:
The time period \( T \) is \( 10 \frac{5}{12} \) years.

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