gloria took out a 10 - year loan for $50,000 ...
gloria took out a 10 - year loan for $50,000 at an apr of 5%, compounded monthly. what will her balance be after she has made exactly half of her monthly payments?\n\na. $31,819.65\nb. $18,180.35\nc. $21,897.57\nd. $28,102.43
Answer
# Explanation:
## Step1: Calculate the monthly - interest rate and number of payments
The annual percentage rate (APR) is $r = 5\%=0.05$. The monthly - interest rate $i=\frac{r}{12}=\frac{0.05}{12}$. The loan is for $n = 10$ years, so the total number of monthly payments is $N=10\times12 = 120$ payments. After making half of the payments, the number of remaining payments $k = 60$. The loan amount $P = 50000$.
## Step2: Use the present - value of an ordinary annuity formula for the remaining balance
The formula for the remaining balance $B$ of a loan is $B = P\times\frac{(1 + i)^{N}-(1 + i)^{m}}{(1 + i)^{N}-1}$, where $m$ is the number of payments made and $N$ is the total number of payments. Here, $m = 60$ and $N = 120$.
First, calculate $(1 + i)=(1+\frac{0.05}{12})$.
Let $x=(1+\frac{0.05}{12})$.
$(1 + i)^{N}=x^{120}$, $(1 + i)^{m}=x^{60}$.
$B = 50000\times\frac{x^{120}-x^{60}}{x^{120}-1}$.
$x=(1+\frac{0.05}{12})\approx1.004167$.
$x^{60}\approx1.283359$, $x^{120}\approx1.647009$.
$B = 50000\times\frac{1.647009 - 1.283359}{1.647009-1}$
$B = 50000\times\frac{0.36365}{0.647009}$
$B\approx50000\times0.562049$
$B\approx28102.43$
# Answer:
D. $\$28,102.43$