Nasim is going to invest in an account paying an interest rate of 5.8% compounded continuously. How much would Nasim need to invest, to the nearest dollar, for the value of the account to reach $27,000 in 5 years?

To determine the initial investment required for Nasim's account to reach $\$27,000$ in $5$ years with an annual interest rate of $5.8\%$ compounded annually, we use the compound interest formula:
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
where:

$A = \$27,000$ is the future value of the investment.
$P$ is the initial principal (the amount to be invested).
$r = 5.8\% = 0.058$ is the annual interest rate.
$n = 1$ is the number of times the interest is compounded per year.
$t = 5$ is the time period in years.

Setting up the Equation for Principal $P$

We need to isolate the principal $P$ in the compound interest formula:
\[P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}\]
Substituting the given values into this expression:
\[P = \frac{27000}{\left(1 + \frac{0.058}{1}\right)^{1 \times 5}}\]
\[P = \frac{27000}{(1.058)^5}\]

Calculating the Initial Investment

First, we calculate the denominator:
\[(1.058)^5 \approx 1.3256565\]
Now, divide the future value by this factor:
\[P = \frac{27000}{1.3256565} \approx 20367.392\]
Rounding to the nearest dollar, we find that the initial investment required is approximately $\$20,367$.

Examples

Compound interest is a fundamental concept in finance that demonstrates how money can grow over time when interest is earned on both the initial principal and the accumulated interest from previous periods. For instance, if you are saving for a long-term goal like a down payment on a house or a retirement fund, understanding the power of compound interest helps you calculate exactly how much you need to set aside today to reach your financial target in the future. By using the formula $A = P(1 + r/n)^{nt}$, individuals can make informed decisions about their savings plans, ensuring they invest enough now to meet their financial milestones.