Questions: As part of your retirement plan, you have decided to deposit 6000 at the beginning of each year into an account paying 5% interest compounded annually. (Round your answers to the nearest cent.) (a) How much (in ) would the account be worth after 10 years? (b) How much (in ) would the account be worth after 20 years? (c) When you retire in 30 years, what will be the total worth (in ) of the account? (d) If you found a bank that paid 6% interest compounded annually rather than 5%, how much (in ) would you have in the account after 30 years? (e) Use the future value of an annuity due formula to calculate how much (in ) you would have in the account after 30 years if the bank in part (d) switched from annual compounding to monthly compounding and you deposited 500 at the beginning of each month instead of 6000 at the beginning of each year.

As part of your retirement plan, you have decided to deposit 6000 at the beginning of each year into an account paying 5% interest compounded annually. (Round your answers to the nearest cent.)
(a) How much (in ) would the account be worth after 10 years?

(b) How much (in ) would the account be worth after 20 years?

(c) When you retire in 30 years, what will be the total worth (in ) of the account?

(d) If you found a bank that paid 6% interest compounded annually rather than 5%, how much (in ) would you have in the account after 30 years?

(e) Use the future value of an annuity due formula to calculate how much (in ) you would have in the account after 30 years if the bank in part (d) switched from annual compounding to monthly compounding and you deposited 500 at the beginning of each month instead of 6000 at the beginning of each year.

Transcript text: As part of your retirement plan, you have decided to deposit $6000 at the beginning of each year into an account paying 5% interest compounded annually. (Round your answers to the nearest cent.) (a) How much (in $) would the account be worth after 10 years? (b) How much (in $) would the account be worth after 20 years? (c) When you retire in 30 years, what will be the total worth (in $) of the account? (d) If you found a bank that paid 6% interest compounded annually rather than 5%, how much (in $) would you have in the account after 30 years? (e) Use the future value of an annuity due formula to calculate how much (in $) you would have in the account after 30 years if the bank in part (d) switched from annual compounding to monthly compounding and you deposited $500 at the beginning of each month instead of $6000 at the beginning of each year.

Solution

Solution Steps

To solve this problem, we will use the future value of an annuity due formula, which accounts for regular deposits made at the beginning of each period. The formula for the future value of an annuity due is:

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

where:

$ P $ is the annual deposit,
$ r $ is the annual interest rate,
$ n $ is the number of years.

We will calculate the future value for each scenario given in the questions.

Step 1: Future Value After 10 Years

To calculate the future value of the account after 10 years with annual deposits of $ P = 6000 $ and an interest rate of $ r = 0.05 $, we use the future value of an annuity due formula:

\[ FV_{10} = 6000 \times \left( \frac{(1 + 0.05)^{10} - 1}{0.05} \right) \times (1 + 0.05) \]

Calculating this gives:

\[ FV_{10} \approx 79240.72 \]

Step 2: Future Value After 20 Years

Next, we calculate the future value after 20 years using the same formula:

\[ FV_{20} = 6000 \times \left( \frac{(1 + 0.05)^{20} - 1}{0.05} \right) \times (1 + 0.05) \]

This results in:

\[ FV_{20} \approx 208315.51 \]

Step 3: Future Value After 30 Years

Finally, we calculate the future value after 30 years:

\[ FV_{30} = 6000 \times \left( \frac{(1 + 0.05)^{30} - 1}{0.05} \right) \times (1 + 0.05) \]

The calculation yields: