If the ratio of simple interest and principal is 8 ∶ 25/2 and rate of interest is equal to the time invested then find the time of investment?

Let’s denote the simple interest as \(SI\), the principal as \(P\), the rate of interest as \(R\) (in % per annum), and the time of investment as \(T\) (in years).

According to the given information, the ratio of simple interest to principal is \(8 : \frac{25}{2}\), which can be simplified to \(16 : 25\). So, we can write:

\[ \frac{SI}{P} = \frac{16}{25} \]

We also know that the rate of interest is equal to the time invested, so \(R = T\).

The formula for simple interest is:

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

Substituting \(R = T\) and rearranging the formula, we get:

\[ T^2 = \frac{100 \times SI}{P} \]

Using the ratio \(\frac{SI}{P} = \frac{16}{25}\), we can substitute for \(\frac{SI}{P}\) in the equation:

\[ T^2 = \frac{100 \times 16}{25} \]

Simplifying:

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

\[ T^2 = 64 \]

Taking the square root of both sides:

\[ T = 8 \text{ years} \]

Therefore, the time of investment is 8 years.