The simple interset on a sum of money is `(16)/(25)` of the principal. The number of year is equal to the rate of interset per annum?

Options

A. 4
B. 16
C. 8
D. 12

Answer

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

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

**Step 2: Define the variables.**
Let the Principal amount be \( P \).
According to the question, the Simple Interest is \( \frac{16}{25} \) of the Principal:
\[ \text{SI} = \frac{16}{25} P \]

**Step 3: Set the relationship between Time and Rate.**
The question states that the number of years (T) is equal to the rate of interest (R). Therefore, we can write:
\[ T = R \]

**Step 4: Substitute the values into the Simple Interest formula.**
Now we can substitute SI, P, R, and T into the formula:
\[ \frac{16}{25} P = \frac{P \times R \times R}{100} \]
Since \( T = R \), we replace T with R.

**Step 5: Simplify the equation.**
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ \frac{16}{25} = \frac{R^2}{100} \]

**Step 6: Cross-multiply to solve for R.**
Cross-multiplying gives us:
\[ 16 \times 100 = 25 \times R^2 \]
\[ 1600 = 25 R^2 \]

**Step 7: Divide both sides by 25.**
\[ R^2 = \frac{1600}{25} \]
\[ R^2 = 64 \]

**Step 8: Take the square root of both sides.**
\[ R = \sqrt{64} \]
\[ R = 8 \]

Thus, the rate of interest per annum is \( R = 8\% \).

### Final Answer:
The rate of interest per annum is **8%**.