Questions: If you save 490 per month for retirement in an account, how much will you have after 37 years?

Transcript text: If you save $\$ 490$ per month for retirement in an account, how much will you have after 37 years?

Solution

Solution Steps

To solve this problem, we need to calculate the future value of a series of monthly savings over a period of 37 years. This can be done using the future value of an annuity formula. The formula for the future value of an annuity is:

\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \]

where:

$ P $ is the monthly savings amount ($490)
$ r $ is the monthly interest rate (annual rate divided by 12)
$ n $ is the total number of payments (37 years times 12 months per year)

Assuming an annual interest rate of 5%, we can calculate the future value.

Solution Approach

Define the monthly savings amount, annual interest rate, and the number of years.
Convert the annual interest rate to a monthly interest rate.
Calculate the total number of payments.
Use the future value of an annuity formula to find the total amount saved.

Step 1: Define Variables

Let $ P = 490 $ (monthly savings), $ r = 0.05 $ (annual interest rate), and $ t = 37 $ (years).

Step 2: Convert Annual Interest Rate

Convert the annual interest rate to a monthly interest rate: \[ r_{monthly} = \frac{r}{12} = \frac{0.05}{12} \approx 0.0041667 \]

Step 3: Calculate Total Payments

Calculate the total number of payments over 37 years: \[ n = t \times 12 = 37 \times 12 = 444 \]

Step 4: Calculate Future Value

Using the future value of an annuity formula: \[ FV = P \times \left( \frac{(1 + r_{monthly})^n - 1}{r_{monthly}} \right) \] Substituting the values: \[ FV = 490 \times \left( \frac{(1 + 0.0041667)^{444} - 1}{0.0041667} \right) \approx 627445.69 \]