Simple interest on a certain sum is $$\frac{54}{75}$$ of the sum. What is the rate of interest per annum if the value of the rate of interest per annum is half the time period (in years) of investment?

Let's assume the initial sum is '75y'.

Simple interest on a certain sum is $$\frac{54}{75}$$ of the sum.

Simple interest = $$\frac{54}{75}\times75y$$ = 54y

Let's assume the time period of investment is '2z'.

If the value of the rate of interest per annum is half the time period (in years) of investment.

rate of interest per annum = $$\frac{2z}{2}$$ = z

We know that Simple interest = $$\frac{sum\ of\ money\times\ rate\times\ time}{100}$$

$$54y=\frac{75y\times\ z\times\ 2z}{100}$$

$$18=\frac{\ z\times z}{2}$$

$$36 = z^2$$

rate of interest per annum = z = 6%