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

Options

A. 8
B. `6(1)/(4)`
C. 5
D. 4

Answer

To solve the problem, we will use the formula for Simple Interest (SI):

\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount (the sum of money)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

Given that the Simple Interest is \( \frac{8}{25} \) of the principal amount, we can express this as:

\[ SI = \frac{8}{25} P \]

We are also given that the number of years \( T \) is numerically half of the rate percent \( R \). This can be expressed as:

\[ T = \frac{R}{2} \]

Now, substituting these values into the Simple Interest formula:

\[ \frac{8}{25} P = \frac{P \times R \times T}{100} \]

Substituting \( T \) with \( \frac{R}{2} \):

\[ \frac{8}{25} P = \frac{P \times R \times \frac{R}{2}}{100} \]

Now, we can cancel \( P \) from both sides (assuming \( P \neq 0 \)):

\[ \frac{8}{25} = \frac{R \times \frac{R}{2}}{100} \]

This simplifies to:

\[ \frac{8}{25} = \frac{R^2}{200} \]

Now, we can cross-multiply to eliminate the fraction:

\[ 8 \times 200 = 25 \times R^2 \]

Calculating the left side:

\[ 1600 = 25R^2 \]

Now, divide both sides by 25 to isolate \( R^2 \):

\[ R^2 = \frac{1600}{25} \]

Calculating the right side:

\[ R^2 = 64 \]

Now, take the square root of both sides to find \( R \):

\[ R = \sqrt{64} \]

Calculating the square root:

\[ R = 8 \]

Thus, the rate percent per annum is:

\[ \boxed{8\%} \]