Find the final amount (rounded to the nearest dollar) in this retirement account, in which the rate of return on the account and the regular contribution change over time.$700 per month invested at 6%, compounded monthly, for 25 years; then $3000 per month invested at 7%, compounded monthly, for 25 years.
Question
Find the final amount (rounded to the nearest dollar) in this retirement account, in which the rate of return on the account and the regular contribution change over time.3000 per month invested at 7%, compounded monthly, for 25 years.
Solution
To solve this problem, we need to calculate the future value of two different annuities (regular investments) and then add them together.
- First, calculate the future value of the $700 per month investment at 6% for 25 years.
The formula for the future value of an annuity is:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
where: P = monthly investment = $700 r = annual interest rate = 6% = 0.06 n = number of times interest is compounded per year = 12 (monthly) t = number of years = 25
Substituting these values into the formula, we get:
FV = 700 * [(1 + 0.06/12)^(12*25) - 1] / (0.06/12)
This gives us a future value of approximately $1,348,807 for the first 25 years.
- Next, calculate the future value of the $3000 per month investment at 7% for the next 25 years.
Using the same formula, but with P = $3000, r = 7% = 0.07, n = 12, and t = 25, we get:
FV = 3000 * [(1 + 0.07/12)^(12*25) - 1] / (0.07/12)
This gives us a future value of approximately $3,982,484 for the next 25 years.
- Finally, add the two amounts together to get the total amount in the retirement account after 50 years:
3,982,484 = $5,331,291
So, the final amount in the retirement account, rounded to the nearest dollar, is approximately $5,331,291.
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