The simple interest on a sum of money is `(4)/(9)` times the principal and the rate of interest per annum is numerically equal to the number of years. Find the rate of interest per annum.

Options

A. `(10)/(3)%`
B. `(15)/(3)%`
C. `(20)/(3)%`
D. `(15)/(2)%`

Explanation

To solve the problem step by step, we will use the information provided in the question about simple interest, principal, rate, and time.

### Step 1: Understand the relationship between Simple Interest, Principal, Rate, and Time
The formula for Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \(SI\) = Simple Interest
- \(P\) = Principal
- \(R\) = Rate of interest per annum
- \(T\) = Time in years

### Step 2: Set up the equations based on the problem statement
According to the problem:
1. The Simple Interest is \(\frac{4}{9}\) times the Principal:
\[ SI = \frac{4}{9} P \]
2. The Rate of interest per annum is numerically equal to the number of years:
\[ R = T \]

### Step 3: Substitute SI in the formula
We can substitute the expression for Simple Interest into the formula:
\[ \frac{4}{9} P = \frac{P \times R \times T}{100} \]

### Step 4: Substitute \(R\) with \(T\)
Since \(R = T\), we can replace \(R\) in the equation:
\[ \frac{4}{9} P = \frac{P \times T \times T}{100} \]
This simplifies to:
\[ \frac{4}{9} P = \frac{P \times T^2}{100} \]

### Step 5: Cancel \(P\) from both sides
Assuming \(P \neq 0\), we can divide both sides by \(P\):
\[ \frac{4}{9} = \frac{T^2}{100} \]

### Step 6: Cross-multiply to solve for \(T^2\)
Cross-multiplying gives:
\[ 4 \times 100 = 9 \times T^2 \]
This simplifies to:
\[ 400 = 9T^2 \]

### Step 7: Solve for \(T^2\)
Now, divide both sides by 9:
\[ T^2 = \frac{400}{9} \]

### Step 8: Take the square root to find \(T\)
Taking the square root of both sides gives:
\[ T = \sqrt{\frac{400}{9}} = \frac{20}{3} \]

### Step 9: Since \(R = T\), find the Rate of Interest
Thus, the Rate of interest \(R\) is:
\[ R = \frac{20}{3} \text{ percent} \]

### Final Answer
The rate of interest per annum is \(\frac{20}{3}\) percent.

---