Question

Reagan has a savings account that earns 1.87% interest, compounded quarterly. If she needs $20,857 for a down payment on a house in 9 years, how much money will she need to invest in the account today?
A. 10,704.95
B. 17,633.17
C. 19,355.32
D. 20,508.99

Answer 1

To solve this, we are going to use the compounded interest formula: $A=P(1+ \frac{r}{n} )^{nt}$
where 
$A$ is the final amount after $t$ years
$P$ is the initial investment 
$r$ is the interest rate in decimal form 
$n$ is the number of times the interest is compounded per year
$t$ 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 $A=20857$ and $t=9$. To convert the interest rate to decimal form, we are going to divide the rate by 100%
$r= \frac{1.87}{100} =0.0187$
Since the interest is compounded quarterly, it is compounded 4 times per year; therefore, $n=4$. 
Lets replace the values in our formula to find $P$:
$A=P(1+ \frac{r}{n} )^{nt}$
$20857=P(1+ \frac{0.0187}{4} )^{(4)(9)}$
$P= \frac{20857}{(1+ \frac{0.0187}{4} )^{(4)(9)}}$
$P= \frac{20857}{(1+ \frac{0.0187}{4} )^{36}}$
$P=17633.17$

We can conclude that the correct answer is: B. 17,633.17