Question

how much would you need to deposit in an account now in order to have $4000 in the account in 5 years? assume the account earns 7% interest compounded monthly. round to the nearest penny if needed.

Explanation:

Step1: Recall the compound - interest formula for present value

The formula for compound interest is $A = P(1+\frac{r}{n})^{nt}$, where:

• $A$ is the future value of the investment/loan, including interest.
• $P$ is the principal investment amount (the initial deposit or loan amount).
• $r$ is the annual interest rate (in decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the time the money is invested or borrowed for, in years.

We want to find $P$, so we can rearrange the formula to $P=\frac{A}{(1 + \frac{r}{n})^{nt}}$.

Step2: Identify the values of the variables

• We know that $A=\$4000$ (the future value we want to have).
• The annual interest rate $r = 7\%=0.07$.
• Since the interest is compounded monthly, $n = 12$ (because there are 12 months in a year).
• The time $t = 5$ years.

Step3: Substitute the values into the formula

First, calculate the exponent $nt$: $nt=12\times5 = 60$.

Then, calculate the value of $(1+\frac{r}{n})$: $\frac{r}{n}=\frac{0.07}{12}$, and $1+\frac{0.07}{12}=1+\frac{7}{1200}=\frac{1200 + 7}{1200}=\frac{1207}{1200}\approx1.005833$.

Next, calculate $(1+\frac{r}{n})^{nt}=(1.005833)^{60}$. We can use a calculator to find that $(1.005833)^{60}\approx1.416018$.

Finally, calculate $P$: $P=\frac{4000}{1.416018}\approx2824.41$.

Answer:

2824.41