The simple interest on a sum of money is `9//49` of the principal. The number of years are equal to the rate of interest per annum. What is the rate (in percentage) of interest per annum?

Options

A. 36/7
B. 24/7
C. 15/7
D. 30/7

Answer

To solve the problem step by step, we will use the information provided about simple interest (SI), principal (P), rate of interest (R), and time (T).

### Step 1: Understand the relationship between SI, P, R, and T
The formula for simple interest 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 known values
According to the problem:
- The simple interest (SI) is \( \frac{9}{49} \) of the principal (P).
- The time (T) is equal to the rate (R).

Let's assume the principal (P) is 49. Therefore:
\[ SI = \frac{9}{49} \times 49 = 9 \]
So, we have:
- SI = 9
- P = 49

### Step 3: Substitute the known values into the SI formula
Since T = R, we can substitute T with R in the SI formula:
\[ 9 = \frac{49 \times R \times R}{100} \]

### Step 4: Rearrange the equation to solve for R
We can rearrange the equation:
\[ 9 = \frac{49R^2}{100} \]
Multiplying both sides by 100:
\[ 900 = 49R^2 \]
Now, divide both sides by 49:
\[ R^2 = \frac{900}{49} \]

### Step 5: Take the square root to find R
Taking the square root of both sides:
\[ R = \sqrt{\frac{900}{49}} \]
This simplifies to:
\[ R = \frac{\sqrt{900}}{\sqrt{49}} = \frac{30}{7} \]

### Step 6: Convert R to a percentage
To find the rate in percentage, we multiply by 100:
\[ R = \frac{30}{7} \times 100 = \frac{3000}{7} \approx 428.57\% \]

### Final Answer
The rate of interest per annum is approximately \( 428.57\% \).

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