Math Snap

STEP 1

What is this asking?
How much money do we need to put away each month to have $\$400,000$ in 20 years, assuming a $\$4\%$ yearly interest rate?
Watch out!
The interest rate is yearly, but we're making monthly deposits, so we need to be careful with our units!

STEP 2

1. Convert yearly terms to monthly terms.
2. Calculate the monthly deposit.

STEP 3

We're given a yearly interest rate of $\$4\%$.
To find the monthly interest rate, we divide the yearly rate by 12, the number of months in a year.
So, our monthly interest rate is $0.04 / 12 \approx 0.003333$.

STEP 4

We're saving for 20 years.
Since there are 12 months in a year, the total number of months we'll be saving is $20 \cdot 12 = 240$ months.

STEP 5

We'll use the Future Value of an Ordinary Annuity formula, which is perfect for figuring out how much we need to deposit regularly to reach a specific goal.
The formula is:
$$FV = P \cdot \frac{((1 + r)^n - 1)}{r}$$ Where:
$FV$ is the future value (what we want to have at the end).
$P$ is the periodic payment (what we need to deposit each month, and what we're trying to find).
$r$ is the periodic interest rate (the monthly interest rate we calculated).
$n$ is the number of periods (the total number of months we'll be saving).

STEP 6

We know that $FV = \$400,000$, $r \approx 0.003333$, and $n = 240$.
Let's plug these values into the formula:
$$400000 = P \cdot \frac{((1 + 0.003333)^{240} - 1)}{0.003333}$$

STEP 7

First, let's simplify the expression inside the parentheses:
$$(1 + 0.003333)^{240} \approx 2.21669$$ Now, subtract 1:
$$2.21669 - 1 = 1.21669$$ Divide this by the interest rate:
$$\frac{1.21669}{0.003333} \approx 365.02$$ So our equation now looks like this:
$$400000 = P \cdot 365.02$$

STEP 8

To find $P$, we divide both sides of the equation by 365.02:
$$P = \frac{400000}{365.02} \approx 1095.85$$ So, we need to deposit approximately $\$1095.85$ each month.

We need to deposit approximately $\$1095.85$ each month to reach our goal of $\$400,000$ in 20 years.