In how much time will the simple interest on certain sum of money at `12(1)/(2)%` per annum be `(7)/(4)` of itself ?

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI) and the information provided in the question.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Rate of interest (R) = \(12\frac{1}{2}\%\) per annum = \(12.5\%\) = \(\frac{25}{2}\%\)
- Simple Interest (SI) is given as \(\frac{7}{4}\) of itself.

2. **Set Up the Simple Interest Formula**:
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \(P\) = Principal amount
- \(R\) = Rate of interest
- \(T\) = Time in years

3. **Express SI in Terms of P**:
According to the problem, the Simple Interest is \(\frac{7}{4}\) of itself. Therefore, we can express this as:
\[ SI = \frac{7}{4} P \]

4. **Substitute SI into the Formula**:
Now substitute SI into the formula:
\[ \frac{7}{4} P = \frac{P \times \frac{25}{2} \times T}{100} \]

5. **Cancel P from Both Sides**:
Since \(P\) is common on both sides, we can cancel it out (assuming \(P \neq 0\)):
\[ \frac{7}{4} = \frac{\frac{25}{2} \times T}{100} \]

6. **Simplify the Right Side**:
To simplify, multiply both sides by 100:
\[ 100 \times \frac{7}{4} = \frac{25}{2} \times T \]
This gives:
\[ 1750 = \frac{25}{2} \times T \]

7. **Solve for T**:
Now, multiply both sides by 2 to eliminate the fraction:
\[ 3500 = 25T \]
Now divide both sides by 25:
\[ T = \frac{3500}{25} = 140 \]

8. **Final Calculation**:
To find \(T\) in years:
\[ T = 14 \text{ years} \]

### Final Answer:
The time will be **14 years**.