you want to save in order to buy a car, in 5 years, without taking out a loan. you determine that you’ll…

you want to save in order to buy a car, in 5 years, without taking out a loan. you determine that you’ll need $34,000.00 for the purchase. if you deposit money into an ordinary annuity that yields 5.38% interest compounded monthly, how much will you need to deposit each month? use this formula: $pmt=\frac{fv\times\frac{r}{n}}{(1 + \frac{r}{n})^{nt}-1}$ enter the dollar amount rounded to the nearest cent. your answer:

you want to save in order to buy a car, in 5 years, without taking out a loan. you determine that you’ll need $34,000.00 for the purchase. if you deposit money into an ordinary annuity that yields 5.38% interest compounded monthly, how much will you need to deposit each month? use this formula: $pmt=\frac{fv\times\frac{r}{n}}{(1 + \frac{r}{n})^{nt}-1}$ enter the dollar amount rounded to the nearest cent. your answer:

Answer

Explanation:

Step1: Identify the values

$FV = 34000$, $r=0.0538$, $n = 12$ (month - compounding), $t = 5$

Step2: Calculate the exponent

$nt=12\times5 = 60$

Step3: Calculate the denominator part

$(1+\frac{r}{n})^{nt}-1=(1+\frac{0.0538}{12})^{60}-1$ Let $x=\frac{0.0538}{12}\approx0.0044833$, then $(1 + 0.0044833)^{60}-1$. Using the formula $a^b=e^{b\ln(a)}$, $(1 + 0.0044833)^{60}=e^{60\ln(1.0044833)}$. $\ln(1.0044833)\approx0.004473$, $60\ln(1.0044833)\approx60\times0.004473 = 0.26838$, $e^{0.26838}\approx1.30714$. So $(1 + 0.0044833)^{60}-1\approx1.30714 - 1=0.30714$

Step4: Calculate the numerator part

$FV\times\frac{r}{n}=34000\times\frac{0.0538}{12}=34000\times0.0044833\approx152.4322$

Step5: Calculate the payment amount

$pmt=\frac{FV\times\frac{r}{n}}{(1+\frac{r}{n})^{nt}-1}=\frac{152.4322}{0.30714}\approx496.29$

Answer:

$496.29$