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

Options

A. `3%`
B. `1/3%`
C. `3(1)/(3)%`
D. `(3)/(10)%`

Answer

To solve the problem step by step, we will use the formula for simple interest and the relationships given in the question.

### Step 1: Understand the given information
The problem states that the simple interest (SI) on a sum of money is \( \frac{1}{9} \) of the principal (P). It also mentions that the number of years (T) is equal to the rate of interest (R) per cent per annum.

### Step 2: Set up the equations
From the information provided:
- Simple Interest (SI) = \( \frac{1}{9} P \)
- The relationship between SI, P, R, and T is given by the formula:
\[ SI = \frac{P \times R \times T}{100} \]

Since T = R, we can replace T in the formula:
\[ SI = \frac{P \times R \times R}{100} = \frac{P \times R^2}{100} \]

### Step 3: Substitute the value of SI
Now, substituting the value of SI from Step 1 into the equation from Step 2:
\[ \frac{1}{9} P = \frac{P \times R^2}{100} \]

### Step 4: Cancel out P
Assuming P is not zero, we can cancel P from both sides:
\[ \frac{1}{9} = \frac{R^2}{100} \]

### Step 5: Cross-multiply to solve for R^2
Cross-multiplying gives us:
\[ 100 = 9R^2 \]

### Step 6: Solve for R^2
Now, divide both sides by 9:
\[ R^2 = \frac{100}{9} \]

### Step 7: Take the square root to find R
Taking the square root of both sides, we get:
\[ R = \sqrt{\frac{100}{9}} = \frac{\sqrt{100}}{\sqrt{9}} = \frac{10}{3} \]

### Step 8: Convert R to percentage
To express R as a percentage, we convert \( \frac{10}{3} \) into a mixed fraction:
\[ \frac{10}{3} = 3 \frac{1}{3} \]

Thus, the rate of interest per annum is \( 3 \frac{1}{3} \% \).

### Final Answer
The rate per annum is \( 3 \frac{1}{3} \% \).

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