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# John decides to start saving money for a new car. He knows he can invest money into an account which will earn a 6.5% APR, compounded weekly, and would like to have saved $10,000 after 5 years. a. How much money will he need to invest into the account now so that he has$10,000 after 5 years? b. Determine the APY (Annual Percent Yield) for the account. c. Determine the 5-year percent change for the account.

Solution

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$A$

There is the formula:

\begin{aligned} A&=p\left(1+\dfrac{r}{n}\right)^{nt}\\ \end{aligned}

He wants to save $\10000$, so $A=10000$ After $5$ years means $t=5$ $6,5\%$ APR compounded weekly means $r=0,065$ and $n=\dfrac{365}{7}=52$

Let us find $A$:

\begin{aligned} A&=p\left(1+\dfrac{r}{n}\right)^{nt}\\ \\ 10000&=p\left(1+\dfrac{0,065}{52}\right)^{52\cdot5}\\ \\ p&=\dfrac{10000}{\left(1+\dfrac{0,065}{52}\right)^{52\cdot5}}\\ \\ &\boxed{p=7226,74} \end{aligned}

Thus, he will need to invest $\7226,74$ into the account now so that he has $\10000$ after $5$ years.

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