Question

select the correct calculation and payment. gabe is getting married in three years and wants to save $20,000 for his wedding. he opens an account earning 6% interest, compounded monthly. find the periodic monthly deposit (payment) needed to achieve a balance of $20,000 after three years. table with formulas and p values: ( p = \frac{20,000 cdot \frac{0.06}{12}}{left(1 + \frac{0.06}{12}
ight)^{12 cdot 3} - 1} ) (p=209), ( p = \frac{20,000 cdot \frac{0.06}{12}}{left(1 + \frac{0.06}{1}
ight)^{1 cdot 3} - 1} ) (p=6282), ( p = \frac{20,000 cdot \frac{0.06}{12}}{left(1 + \frac{0.06}{12}
ight)^{12 cdot 3} - 1} ) (p=508), ( p = \frac{20,000 cdot \frac{0.06}{1}}{left(1 + \frac{0.06}{1}
ight)^{1 cdot 3} - 1} ) (p=3876); buttons: reset, next

Explanation:

Step1: Identify the formula for future value of an ordinary annuity

The formula for the future value \( FV \) of an ordinary annuity is \( FV = P \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}} \), where \( P \) is the periodic payment, \( r \) is the annual interest rate (in decimal), \( n \) is the number of compounding periods per year, and \( t \) is the number of years. We need to solve for \( P \), so rearranging the formula gives \( P=\frac{FV\times\frac{r}{n}}{(1 + \frac{r}{n})^{nt}-1} \).

Here, \( FV = 20000 \), \( r = 0.06 \) (6% annual interest), \( n = 12 \) (compounded monthly), and \( t = 3 \) years.

Step2: Calculate the values of \( \frac{r}{n} \) and \( nt \)

First, calculate \( \frac{r}{n}=\frac{0.06}{12}=0.005 \).

Then, calculate \( nt = 12\times3 = 36 \).

Step3: Substitute into the formula for \( P \)

Substitute \( FV = 20000 \), \( \frac{r}{n}=0.005 \), and \( nt = 36 \) into the formula \( P=\frac{20000\times0.005}{(1 + 0.005)^{36}-1} \).

First, calculate the denominator: \( (1 + 0.005)^{36}\approx1.196680525 \), so \( (1 + 0.005)^{36}-1\approx0.196680525 \).

Then, calculate the numerator: \( 20000\times0.005 = 100 \).

Now, \( P=\frac{100}{0.196680525}\approx508.47 \approx 508 \) (when rounded to the nearest whole number).

Looking at the options, the formula \( P=\frac{20000\cdot\frac{0.06}{12}}{(1 + \frac{0.06}{12})^{12\cdot3}-1} \) (which simplifies to the calculation we did) gives \( P = 508 \).

Answer:

The correct calculation is \( P=\frac{20000\cdot\frac{0.06}{12}}{(1 + \frac{0.06}{12})^{12\cdot3}-1} \) with \( P = 508 \) (the top - right cell in the table with the formula and \( P = 508 \)).