If the simple interest on a certain sum of money is `4//25th` of the sum and the rate per cent equal to the number of years, then find the rate of interest per annum.

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI) and the relationships given in the question.

### Step 1: Understand the given information
We know that the simple interest (SI) is given as \( \frac{4}{25} \) of the principal (P). We also know that the rate of interest (R) is equal to the number of years (T).

### Step 2: Set up the equation for Simple Interest
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
From the question, we can substitute \( SI \) with \( \frac{4}{25} P \):
\[ \frac{4}{25} P = \frac{P \times R \times R}{100} \]
Here, we replace \( T \) with \( R \) since the rate is equal to the number of years.

### Step 3: Simplify the equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{4}{25} = \frac{R^2}{100} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 4 \times 100 = 25 \times R^2 \]
This simplifies to:
\[ 400 = 25 R^2 \]

### Step 5: Solve for \( R^2 \)
Now, divide both sides by 25:
\[ R^2 = \frac{400}{25} \]
Calculating the right side gives:
\[ R^2 = 16 \]

### Step 6: Take the square root
Taking the square root of both sides yields:
\[ R = \sqrt{16} \]
Thus:
\[ R = 4 \]

### Conclusion
The rate of interest per annum is \( 4\% \).

---