Second Hand Software needs to accumulate $$ \$ 12,000 $$ in 3 years to meet future needs. What sinking fund payment would they need to make at the end of each year, at $$ 6 \% $$ interest compounded annually? Select one: a. $$ \$ 4,110.34 $$ b. $$ \$ 3,769.32 $$ c. $$ \$ 4,392.65 $$ d. $$ \$ 4,634.15 $$

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UpStudy AI · Accepted Answer

To find the sinking fund payment that Second Hand Software needs to make at the end of each year to accumulate $12,000 in 3 years at 6% interest compounded annually, we can use the formula for the future value of a sinking fund:

$$ FV = PMT \times \frac{(1 + r)^n - 1}{r} $$

Where:
- $$ FV $$ is the future value of the sinking fund, which is $12,000 in this case.
- $$ PMT $$ is the periodic payment (the sinking fund payment).
- $$ r $$ is the annual interest rate, which is 6% or 0.06 in decimal form.
- $$ n $$ is the number of periods, which is 3 years in this case.

We can rearrange the formula to solve for $$ PMT $$:

$$ PMT = \frac{FV \times r}{(1 + r)^n - 1} $$

Substitute the given values into the formula:

$$ PMT = \frac{12,000 \times 0.06}{(1 + 0.06)^3 - 1} $$

Now, we can calculate the sinking fund payment.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{12000\times 0.06}{\left(1+0.06\right)^{3}-1}$$
- step1: Add the numbers:
  $$\frac{12000\times 0.06}{1.06^{3}-1}$$
- step2: Convert the expressions:
  $$\frac{12000\times 0.06}{\left(\frac{53}{50}\right)^{3}-1}$$
- step3: Multiply the numbers:
  $$\frac{720}{\left(\frac{53}{50}\right)^{3}-1}$$
- step4: Subtract the numbers:
  $$\frac{720}{\frac{53^{3}-50^{3}}{50^{3}}}$$
- step5: Multiply by the reciprocal:
  $$720\times \frac{50^{3}}{53^{3}-50^{3}}$$
- step6: Multiply:
  $$\frac{720\times 50^{3}}{53^{3}-50^{3}}$$
The sinking fund payment that Second Hand Software needs to make at the end of each year to accumulate $12,000 in 3 years at 6% interest compounded annually is approximately $3,769.32.

Therefore, the correct answer is:
b. $3,769.32
The sinking fund payment needed is approximately \$3,769.32 per year.

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