Nature's Way wants to build a $175,000 greenhouse in 5 years. The company sets up a sinking fund with payments made quarterly. Find the payment into this fund if the money earns 5% interest compounded quarterly.

Question

BlackTom AI · Accepted Answer

We start by restating what the problem asks: Nature's Way wants to accumulate 175,000 in a sinking fund over 5 years with quarterly payments, and the fund earns 5% interest compounded quarterly.

First, determine the periodic terms:
- Quarterly interest rate i = 5% per year / 4 = 0.05/4 = 0.0125 per quarter.
- Total number of payments (periods) n = 5 years × 4 quarters per year = 20 payments.

Next, use the sinking fund accumulation formula. The future value of an ordinary annuity (regular payments at the end of each period) is:
FV = PMT × [((1 + i)^n − 1) / i]
We know FV = 175,000, i = 0.0125, n = 20, so we solve for PMT:
PMT = FV × i / ((1 + i)^n − 1)

Now compute the bracket term and the PMT step by step. Calculate (1 + i)^n = (1.0125)^20. Using incremental multiplication (or a calculator), (1.0125)^20 ≈ 1.2820372. Therefore ((1 + i)^n − 1) ≈ 0.2820372.

Then the denominator is 0.2820372, and the numerator is FV × i = 175,000 × 0.0125 = 2,187.50.

Thus PMT ≈ 2,187.50 / 0.2820372 ≈ 7,756 to 7,758, depending on rounding choices in intermediate steps. A precise calculation gives about PMT ≈ 7,757.

With this result in mind, examine the options:
- Option "$8,751": This value is higher than the precise calculation of roughly 7,757, by about 1,000. If someone used a rounded bracket value or a different compounding interpretation, they might land here, but it is noticeably above the mathematically derived amount.
- Option "$7,757": This aligns closely with the computed figure from the formula using i = 0.0125 and n = 20. The small difference, if any, would come from rounding at intermediate steps; this option is consistent with the standard sinking fund calculation.
- Option "$3,562": This is far too small for accumulating 175,000 in 20 quarters at the given rate; it would leave a large shortfall.
- Option "$15,679": This is almost double the required payment, which would dramatically overshoot the target and is not consistent with the formula unless i or n were misapplied.

To summarize the reasoning, the correct approach is to apply the sinking fund future value formula with the quarterly rate 0.0125 and 20 periods, solve for PMT, and compare with the given options. The math indicates a payment near 7,757 per quarter, making the $7,757 option the one that best matches the calculation, while the other options deviate meaningfully from the computed value.

Nature's Way wants to build a $175,000 greenhouse in 5 years. The company sets up a sinking fund with payments made quarterly. Find the payment into this fund if the money earns 5% interest compounded quarterly.单项选择题

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