Joe needs $$ \$ 12,000 $$ in 4 years. What amount can he deposit at the end of each month at $$ 5 \% $$ annual rate of interest compounded monthly, so he will have his $$ \$ 12,000 $$ ? Round to the nearest cent. A. $$ \$ 226.35 $$ B. $$ \$ 314.78 $$ C. $$ \$ 325.16 $$ D. $$ \$ 287.62 $$

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UpStudy AI · Accepted Answer

To determine how much Joe needs to deposit at the end of each month to accumulate $$ \$12,000 $$ in 4 years with an annual interest rate of $$ 5\% $$ compounded monthly, we can use the future value of an ordinary annuity formula:

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

Where:
- $$ FV = \$12,000 $$ (future value)
- $$ r = \frac{0.05}{12} = 0.0041667 $$ (monthly interest rate)
- $$ n = 4 \times 12 = 48 $$ (total number of payments)
- $$ PMT $$ is the monthly payment we need to find.

Rearranging the formula to solve for $$ PMT $$:

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

Plugging in the values:

$$
PMT = \frac{12000 \times 0.0041667}{(1 + 0.0041667)^{48} - 1}
$$

First, calculate $$ (1 + 0.0041667)^{48} $$:

$$
(1.0041667)^{48} \approx 1.22089
$$

Then,

$$
PMT = \frac{12000 \times 0.0041667}{1.22089 - 1} = \frac{50}{0.22089} \approx 226.42
$$

Rounding to the nearest cent, the monthly deposit amount is approximately **\$226.42**.

Among the provided options, the closest value is:

**Answer:** A. $$ \$226.35 $$
Joe needs to deposit approximately \$226.35 at the end of each month to reach \$12,000 in 4 years with a 5% annual interest rate compounded monthly.

Joe needs in 4 years. What amount can he deposit at the end of each month at annual rate of interest compounded monthly, so he will have his ? Round to the nearest cent.
A.
B.
C.
D.

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