To solve this, we are going to use the compounded interest formula: [tex]A=P(1+ \frac{r}{n} )^{nt}[/tex]
where
[tex]A[/tex] is the final amount after [tex]t[/tex] years
[tex]P[/tex] is the initial investment
[tex]r[/tex] is the interest rate in decimal form
[tex]n[/tex] is the number of times the interest is compounded per year
[tex]t[/tex] is the time in years
We know from our problems that she needs $20,857 for a down payment on a house in 9 years, so [tex]A=20857[/tex] and [tex]t=9[/tex]. To convert the interest rate to decimal form, we are going to divide the rate by 100%
[tex]r= \frac{1.87}{100} =0.0187[/tex]
Since the interest is compounded quarterly, it is compounded 4 times per year; therefore, [tex]n=4[/tex].
Lets replace the values in our formula to find [tex]P[/tex]:
[tex]A=P(1+ \frac{r}{n} )^{nt}[/tex]
[tex]20857=P(1+ \frac{0.0187}{4} )^{(4)(9)} [/tex]
[tex]P= \frac{20857}{(1+ \frac{0.0187}{4} )^{(4)(9)}} [/tex]
[tex]P= \frac{20857}{(1+ \frac{0.0187}{4} )^{36}} [/tex]
[tex]P=17633.17[/tex]
We can conclude that the correct answer is: B. 17,633.17
