Question

The simple interest on a sum of money is \( \frac{4}{9} \) of the principal. If the rate percent and time are numerically equal, then find the rate percent.

• A. \( 5\frac{2}{3}% \)
• B. \( 9\frac{2}{3}% \)
• C. \( 8\frac{2}{3}% \)
• D. \( 6\frac{2}{3}% \)

Hint:

Recall the formula for simple interest: \[ SI = \frac{P \times R \times T}{100} \] Use the given relationships to form an equation and solve for the required variable.

Answer 1

The Correct Option is D

Define the variables and the given information.
Let the principal sum of money be \( P \).
Let the rate of simple interest be \( R \) percent per annum.
Let the time period be \( T \) years.
The simple interest (SI) is given as \( \frac{4}{9} \) of the principal:
\[ SI = \frac{4}{9} P \] We are also given that the rate percent and time are numerically equal: \[ R = T \] Step 2: Use the formula for simple interest and substitute the given relationships.
The formula for simple interest is: \[ SI = \frac{P \times R \times T}{100} \] Substitute \( SI = \frac{4}{9} P \) and \( T = R \) into the formula: \[ \frac{4}{9} P = \frac{P \times R \times R}{100} \] \[ \frac{4}{9} P = \frac{P \times R^2}{100} \] Step 3: Solve the equation for \( R \).
We can cancel \( P \) from both sides of the equation (assuming \( P \neq 0 \)): \[ \frac{4}{9} = \frac{R^2}{100} \] Now, solve for \( R^2 \): \[ R^2 = \frac{4}{9} \times 100 \] \[ R^2 = \frac{400}{9} \] Take the square root of both sides to find \( R \) (since the rate cannot be negative, we consider the positive root): \[ R = \sqrt{\frac{400}{9}} \] \[ R = \frac{\sqrt{400}}{\sqrt{9}} \] \[ R = \frac{20}{3} \] Step 4: Express the rate as a mixed fraction.
\ Convert the improper fraction \( \frac{20}{3} \) to a mixed fraction: \[ \frac{20}{3} = 6 \frac{2}{3} \] So, the rate percent is \( 6 \frac{2}{3}% \).