A sum of money becomes 7/6 of itself in 3 years at a certain rate of simple interest. The rate per annum is.

Options

A. `6(6)/(9)%`
B. `5(5)/(9)%`
C. `4(4)/(9)%`
D. `11(1)/(9)%`

Answer

To find the rate of simple interest per annum when a sum of money becomes \( \frac{7}{6} \) of itself in 3 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Amount (Principal)**:
Let the initial amount (Principal) be \( S \).

2. **Determine the Total Amount After 3 Years**:
According to the problem, after 3 years, the amount becomes \( \frac{7}{6} \) of itself. Thus, the total amount \( A \) after 3 years is:
\[ A = \frac{7}{6} S \]

3. **Calculate the Simple Interest**:
The simple interest (SI) can be calculated using the formula:
\[ \text{SI} = A - P \]
Substituting the values we have:
\[ \text{SI} = \frac{7}{6} S - S \]
To combine these fractions, we can express \( S \) as \( \frac{6}{6} S \):
\[ \text{SI} = \frac{7}{6} S - \frac{6}{6} S = \frac{1}{6} S \]

4. **Use the Simple Interest Formula**:
The formula for simple interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Here, \( P = S \), \( R \) is the rate we need to find, and \( T = 3 \) years. Substituting these values into the formula gives:
\[ \frac{1}{6} S = \frac{S \times R \times 3}{100} \]

5. **Eliminate \( S \) from Both Sides**:
Since \( S \) is common on both sides, we can cancel it out (assuming \( S \neq 0 \)):
\[ \frac{1}{6} = \frac{R \times 3}{100} \]

6. **Cross Multiply to Solve for \( R \)**:
Cross multiplying gives:
\[ 100 = 3R \times 6 \]
Simplifying this:
\[ 100 = 18R \]

7. **Calculate the Rate \( R \)**:
To find \( R \), divide both sides by 18:
\[ R = \frac{100}{18} = \frac{50}{9} \]

8. **Convert to Decimal (if needed)**:
To convert \( \frac{50}{9} \) to a decimal:
\[ R \approx 5.56\% \]

### Final Answer:
The rate of simple interest per annum is \( \frac{50}{9}\% \) or approximately \( 5.56\% \).

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