A scholarship will be set up to provide an annual prize of $400 to the best mathematics student in a school. The scholarship is paid for by investing an amount of money into a perpetuity, paying interest of 3.4% per annum, compounding annually. The amount that needs to be invested to provide this scholarship is closest to:

Question

BlackTom AI · Accepted Answer

Using the perpetuity present-value formula PV = C / r, with C = $400 (annual payment) and r = 3.4% = 0.034 (annual interest rate).

1) Relevant concepts:
- A perpetuity pays a fixed amount C forever. Its present value (the lump sum that must be invested today to produce C each year indefinitely) is PV = C / r, where r is the annual effective interest rate.
- Because interest is compounded annually and payments are annual, we use r = 0.034 directly.

2) Step-by-step calculation:
- Convert the percentage to a decimal: 3.4% = 0.034.
- Compute PV = 400 / 0.034.
- To show intermediate arithmetic: multiply numerator and denominator by 1000 to avoid decimals: PV = (400 × 1000) / (0.034 × 1000) = 400000 / 34.
- Divide: 400000 ÷ 34 = 11764.705882352941... .

3) Rounding and final selection:
- The exact value is approximately $11,764.705882. Rounded to the nearest cent would be $11,764.71. Among the given choices, $11,764.70 (option d) is the closest available option (difference of $0.0059 from the exact rounded cent).

4) Why the correct answer is correct:
- The perpetuity formula directly yields this lump sum: investing about $11,764.71 at 3.4% annually produces $400 of interest each year indefinitely. Option d ($11,764.70) is effectively the correct present value rounded to the available choices.

5) Why other options are incorrect:
- a. $11,764.50 is slightly off (difference ≈ $0.21) and not the closest.
- b. $400 is just the annual payment, not the required capital.
- c. $800 is unrelated (perhaps a misunderstanding of doubling) and far too small.

Conclusion: The amount to invest is approximately $11,764.71, so the closest provided choice is d. $11,764.70.

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