The simple interest on a sum of money is `(1)/(9)` th of the principal, and the number of years is equal to the rate per cent per annum. Find the rate per cent.

Options

A. `3(1)/(3)%`
B. `3%`
C. `10%`
D. None of these

Answer

To solve the problem step by step, we will follow the given information and apply the formula for simple interest.

### Step 1: Understand the given information
We know that:
- The simple interest (SI) is \( \frac{1}{9} \) of the principal (P).
- The rate of interest (R) is equal to the time (T) in years.

### Step 2: Set up the equation
From the information given, we can express the simple interest as:
\[ SI = \frac{1}{9} P \]
Since the rate (R) and time (T) are equal, we can denote both as R.

### Step 3: Use the formula for simple interest
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting T with R, we get:
\[ SI = \frac{P \times R \times R}{100} = \frac{P \times R^2}{100} \]

### Step 4: Set the two expressions for SI equal to each other
Now we can set the two expressions for SI equal to each other:
\[ \frac{1}{9} P = \frac{P \times R^2}{100} \]

### Step 5: Simplify the equation
We can cancel P from both sides (assuming P ≠ 0):
\[ \frac{1}{9} = \frac{R^2}{100} \]

### Step 6: Cross-multiply to solve for R^2
Cross-multiplying gives us:
\[ 100 = 9R^2 \]

### Step 7: Solve for R^2
Now, we can solve for R^2:
\[ R^2 = \frac{100}{9} \]

### Step 8: Take the square root to find R
Taking the square root of both sides, we find:
\[ R = \sqrt{\frac{100}{9}} = \frac{10}{3} \]

### Step 9: Convert R to a percentage
To convert \( \frac{10}{3} \) to a mixed number:
\[ R = 3 \frac{1}{3} \text{ percent} \]

### Final Answer
Thus, the rate of interest is:
\[ \text{Rate} = 3 \frac{1}{3} \text{ percent} \]

---