In how much time would the simple interest on a certain sum be `0.125` times the principal at `10 %` per annum?

Options

A. `1` ` 1/4 years`
B. `1` ` 1/3 years `
C. `2` ` 1/4 years`
D. `2` ` 3/4 years `

Answer

To solve the problem step by step, we need to find the time \( t \) when the simple interest (SI) is \( 0.125 \) times the principal (P) at a rate of \( 10\% \) per annum.

### Step-by-Step Solution:

1. **Identify the variables:**
- Let the principal \( P = x \).
- The simple interest \( SI = 0.125 \times P = 0.125x \).
- The rate \( R = 10\% \).

2. **Use the formula for Simple Interest:**
The formula for simple interest is given by:
\[ SI = \frac{P \times R \times t}{100} \]

3. **Substitute the known values into the formula:**
We can substitute \( SI \), \( P \), and \( R \) into the formula:
\[ 0.125x = \frac{x \times 10 \times t}{100} \]

4. **Simplify the equation:**
We can simplify the equation by canceling \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 0.125 = \frac{10t}{100} \]

5. **Solve for \( t \):**
Rearranging the equation gives:
\[ 0.125 = 0.1t \]
Now, divide both sides by \( 0.1 \):
\[ t = \frac{0.125}{0.1} = 1.25 \]

6. **Convert the time into years:**
Since \( t = 1.25 \) years, we can also express this as:
\[ t = 1 \frac{1}{4} \text{ years} \text{ or } \frac{5}{4} \text{ years} \]

### Final Answer:
The time required for the simple interest to be \( 0.125 \) times the principal at \( 10\% \) per annum is \( 1.25 \) years or \( 1 \frac{1}{4} \) years.

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