The simple interest on a sum of money is 4/9 of the principal. What will be the annual rate of interest and time (in years), if both of these are numerically equal?

The correct option is 6 2/3 % and 6 2/3 Years.

Let us break down the explanation and mathematical derivation step-by-step:

Step 1: Identify the variables and given conditions
Let the principal sum of money be represented by $P$.
Let the annual rate of interest be $R$% and the time period be $T$ years.
The problem states that the simple interest (SI) is equal to $\frac{4}{9}$ of the principal. Thus:

$\mathrm{SI}=\frac{4}{9}P$

We are also given that the numerical value of the rate of interest ($R$) is equal to the numerical value of the time ($T$). Let this common value be $x$:

$R=T=x$

Step 2: Set up the simple interest formula
The standard formula for calculating simple interest is:

$\mathrm{SI}=\frac{P\times R\times T}{100}$

Substituting the values we identified in Step 1:

$\frac{4}{9}P=\frac{P\times x\times x}{100}$

Step 3: Solve for the unknown value $x$
Since the principal $P$ is on both sides of the equation and is non-zero, we can divide both sides by $P$ to simplify the equation:

$\frac{4}{9}=\frac{x^2}{100}$

Next, multiply both sides by 100 to isolate the $x^2$ term:

$x^2=\frac{4\times 100}{9}$

$x^2=\frac{400}{9}$

Now, find the positive square root on both sides to determine the value of $x$ (since rate and time must be positive numbers):

$x=\sqrt{\frac{400}{9}}$

$x=\frac{20}{3}$

Step 4: Convert to mixed fraction and state the final answer
Converting the improper fraction $\frac{20}{3}$ to a mixed fraction:

$x=6\frac{2}{3}$

Therefore, the annual rate of interest is $6\frac{2}{3}\%$ and the time period is $6\frac{2}{3}$ Years.