The simple interest on a sum of money is `(8)/(25)` of the sum. If the number of years is numerically half of the rate percent per annum, then find the rate percent per annum.

Answer

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

### Step 1: Understand the problem
We know that the simple interest (SI) is given as \( \frac{8}{25} \) of the principal amount (P). We also know that the number of years (T) is numerically half of the rate percent per annum (R).

### Step 2: Set up the equations
Let:
- Principal (P) = P
- Rate of interest per annum = R%
- Time (T) = \( \frac{R}{2} \) years (since T is half of R)

According to the formula for simple interest:
\[ SI = \frac{P \times R \times T}{100} \]

### Step 3: Substitute the values
From the problem, we know that:
\[ SI = \frac{8}{25} P \]
Substituting the values into the SI formula:
\[ \frac{8}{25} P = \frac{P \times R \times \frac{R}{2}}{100} \]

### Step 4: Simplify the equation
We can cancel P from both sides (assuming P ≠ 0):
\[ \frac{8}{25} = \frac{R \times \frac{R}{2}}{100} \]
This simplifies to:
\[ \frac{8}{25} = \frac{R^2}{200} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 8 \times 200 = 25 \times R^2 \]
\[ 1600 = 25R^2 \]

### Step 6: Solve for R²
Now, divide both sides by 25:
\[ R^2 = \frac{1600}{25} \]
\[ R^2 = 64 \]

### Step 7: Find R
Taking the square root of both sides:
\[ R = \sqrt{64} \]
\[ R = 8 \]
Since the rate of interest cannot be negative, we take the positive value.

### Conclusion
The rate percent per annum is **8%**.

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