The simple interest on a sum of moneyis 1/9 of the principal. Find the rate percent if time and rate are numerically equal.

Options

A. `6(2)/(3)%`
B. `3(1)/(3)`
C. `2(2)/(3)`
D. `13(1)/(3)%`

Answer

To solve the problem, we need to find the rate percent given that the simple interest (SI) on a sum of money is 1/9 of the principal (P), and that time (T) and rate (R) are numerically equal.

### Step-by-step Solution:

1. **Understand the Relationship**:
- We know that Simple Interest (SI) is given by the formula:
\[ SI = \frac{P \times R \times T}{100} \]
- According to the problem, SI is \( \frac{1}{9} \) of the principal (P):
\[ SI = \frac{P}{9} \]

2. **Set Up the Equation**:
- Since SI is equal to \( \frac{P}{9} \), we can equate the two expressions:
\[ \frac{P}{9} = \frac{P \times R \times T}{100} \]

3. **Substituting T with R**:
- The problem states that time (T) and rate (R) are numerically equal, so we can substitute T with R:
\[ T = R \]
- This gives us:
\[ \frac{P}{9} = \frac{P \times R \times R}{100} \]
- Simplifying this, we have:
\[ \frac{P}{9} = \frac{P \times R^2}{100} \]

4. **Canceling P**:
- Assuming \( P \neq 0 \), we can divide both sides by P:
\[ \frac{1}{9} = \frac{R^2}{100} \]

5. **Cross-Multiplying**:
- Cross-multiplying gives us:
\[ 100 = 9R^2 \]

6. **Solving for R**:
- Rearranging the equation:
\[ R^2 = \frac{100}{9} \]
- Taking the square root of both sides:
\[ R = \sqrt{\frac{100}{9}} = \frac{10}{3} \approx 3.33 \]

7. **Finding the Rate Percent**:
- Since R is the rate percent, we can express it as:
\[ R \approx 3.33\% \]

### Final Answer:
The rate percent is approximately **3.33%**.