Question

How much money should be invested in an account that earns 7% interest compounded quarterly to yield $10,000 in 5 years. (Round to the nearest cent)

Answer 1

In general, the compound interest formula is

$A=P(1+\frac{r}{n})^{nt}$

In our case, since it is compounded quarterly, n=3 (3 periods of 4 months in a year), and

$n=4,r=7\%=0.07,A=10000,t=5$

Thus, solving for P,

$\begin{gathered} \Rightarrow10000=P(1+\frac{0.07}{4})^{4\cdot5} \\ \Rightarrow P=\frac{10000}{(1+\frac{0.07}{4})^{4\cdot5}}=\frac{10000}{(1+\frac{0.07}{4})^{20}} \\ \Rightarrow P\approx7068.25 \end{gathered}$

Thus, the answer is approximately $7068.25

A completely different situation is when we fix

$A=P+10000,r=0.07,n=4$$\begin{gathered} \Rightarrow P+10000=P(1+\frac{0.07}{4})^{4\cdot5} \\ \Rightarrow P-P(1+\frac{0.07}{4})^{4\cdot5}=-10000 \end{gathered}$