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Math  /  Algebra

QuestionSuppose you want to have $400,000\$ 400,000 for retirement in 20 years. Your account earns 4%4 \% interest. How much would you need to deposit in the account each month? \ \square$

Studdy Solution

STEP 1

What is this asking? How much money do we need to put away each month to have $400,000\$400,000 in 20 years, assuming a $4%\$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%\$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≈0.0033330.04 / 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⋅12=24020 \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⋅((1+r)n−1)rFV = P \cdot \frac{((1 + r)^n - 1)}{r} Where: FVFV is the **future value** (what we want to have at the end). PP is the **periodic payment** (what we need to deposit each month, and what we're trying to find). rr is the **periodic interest rate** (the monthly interest rate we calculated). nn is the **number of periods** (the total number of months we'll be saving).

STEP 6

We know that FV=$400,000FV = \$400,000, r≈0.003333r \approx 0.003333, and n=240n = 240.
Let's plug these values into the formula: 400000=P⋅((1+0.003333)240−1)0.003333400000 = 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≈2.21669(1 + 0.003333)^{240} \approx 2.21669 Now, subtract 1: 2.21669−1=1.216692.21669 - 1 = 1.21669 Divide this by the interest rate: 1.216690.003333≈365.02\frac{1.21669}{0.003333} \approx 365.02 So our equation now looks like this: 400000=P⋅365.02400000 = P \cdot 365.02

STEP 8

To find PP, we divide both sides of the equation by **365.02**: P=400000365.02≈1095.85P = \frac{400000}{365.02} \approx 1095.85 So, we need to deposit approximately $1095.85\$1095.85 each month.

STEP 9

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

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