The rate of interset per annum at which the total simple interset of a certain capital for 1 year is equal to the total simple interset of the same capital at the rate of 5% per annum for 2 years, is

Options

A. `(5)/(2)` %
B. 0.1
C. 0.25
D. 0.125

Answer

To solve the problem, we need to find the rate of interest per annum (denoted as R) at which the total simple interest for a capital (P) for 1 year is equal to the total simple interest of the same capital at the rate of 5% per annum for 2 years.

### Step-by-Step Solution:

1. **Understand the Simple Interest Formula**:
The formula for calculating simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount (capital)
- \( R \) = Rate of interest per annum
- \( T \) = Time period in years

2. **Calculate Simple Interest for the First Case**:
For the first case, where the time period is 1 year and the rate is \( R \):
\[ SI_1 = \frac{P \times R \times 1}{100} = \frac{P \times R}{100} \]

3. **Calculate Simple Interest for the Second Case**:
For the second case, where the time period is 2 years and the rate is 5%:
\[ SI_2 = \frac{P \times 5 \times 2}{100} = \frac{P \times 10}{100} \]

4. **Set the Two Simple Interests Equal**:
According to the problem, the simple interest for both cases is equal:
\[ SI_1 = SI_2 \]
Substituting the expressions we derived:
\[ \frac{P \times R}{100} = \frac{P \times 10}{100} \]

5. **Eliminate \( P \) from Both Sides**:
Since \( P \) is common in both sides and is not zero, we can cancel it out:
\[ \frac{R}{100} = \frac{10}{100} \]

6. **Solve for \( R \)**:
Multiplying both sides by 100 gives:
\[ R = 10 \]

7. **Final Answer**:
The rate of interest per annum \( R \) is:
\[ R = 10\% \]