Question

Ms. Sheela creates a fund of 100,000 to provide scholarships to needy children. The scholarship is provided at the beginning of each year, and the fund earns an interest of r% annually. If the scholarship amount is 8,000, find r.

• A. \(8\frac{1}{2}\%\)
• B. \(16\frac{8}{23}\%\)
• C. \(17\frac{8}{25}\%\)
• D. \(8\frac{8}{5}\%\)

Hint:

When dealing with perpetual withdrawals, it’s essential to balance the interest earned with the withdrawal amount. The formula \( \text{Interest Earned} = \frac{r}{100} \times \text{Fund} \) allows us to find the interest rate needed to match the annual withdrawals. It’s important to solve the equation step by step, ensuring to simplify fractions and percentages correctly. This will help ensure that the fund continues to sustain itself without running out of money.

Answer 1

The Correct Option is B

The total fund remains 1,00,000, earning an interest of \( r\% \) annually, while 8,000 is withdrawn at the start of each year. The fund must sustain itself indefinitely, which means the interest earned should exactly match the withdrawn amount.

The formula for perpetual withdrawals is:

Interest Earned Annually = Withdrawal Amount.

The interest earned is given by:

Interest Earned = \( \frac{r}{100} \times \text{Fund} \).

Substitute the known values:

\( \frac{r}{100} \times 1,00,000 = 8,000 \).

Simplify:

\( r = \frac{8,000 \times 100}{1,00,000} \).

\( r = 8 \frac{16}{23} \% \).

Thus, \( r = 8 \frac{16}{23} \% \).

Answer 2

The total fund remains 1,00,000, earning an interest of \( r\% \) annually, while 8,000 is withdrawn at the start of each year. The fund must sustain itself indefinitely, which means the interest earned should exactly match the withdrawn amount.

Step 1: Formula for perpetual withdrawals:

The formula for perpetual withdrawals specifies that the amount of interest earned annually must be equal to the withdrawal amount to ensure the fund sustains itself indefinitely. This is expressed as:

Interest Earned Annually = Withdrawal Amount.

Step 2: Expression for the interest earned:

The interest earned annually is calculated by applying the interest rate \( r\% \) to the total fund of 1,00,000. This is given by the formula:

Interest Earned = \( \frac{r}{100} \times \text{Fund} \).

Step 3: Substitute the known values:

Substituting the values into the formula, where the fund is 1,00,000 and the withdrawal amount is 8,000, we get:

\( \frac{r}{100} \times 1,00,000 = 8,000 \).

Step 4: Solve for \( r \):

Simplifying the equation:

\( r = \frac{8,000 \times 100}{1,00,000} \).

Step 5: Final calculation:

Simplifying further:

\( r = 8 \frac{16}{23} \% \).

Conclusion: Therefore, the interest rate required to ensure the fund sustains itself indefinitely is \( r = 8 \frac{16}{23} \% \).