Question 1137709: Joan quit her job at GM when she was 35 years old. Her friend, who is a financial advisor, recommended that she leave her 401(k) retirement savings in GM’s retirement plan rather than withdrawing or transferring the money to a new plan. Her friend said that, on average, Joan could expect a 6% increase per year if she left her money in the plan for many years, based on the past performance of GM’s plan. Joan will not be able to add more money to the GM 401(k) account, but can open a new one at her new job.
A. Create a model for the amount of money Joan will have in her 401(k) after any number of years. Use A for the accumulated amount of money, t for the number of years, and P for the amount of money in the plan when she quit, which is called the principal.
B. What percent of the principal will Joan have if she leaves the money in the account for the 32 years until she retires? What percent increase does this represent? Round each answer to the nearest hundredth of a percent.
C. If Joan's original principal was $20,300 how much would be in the account when she retires?
D. Approximately how many years will it take the principal to double? How long to quadruple?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula is A = P * (1 + r) ^ T
r is the interest rate per year.
P is the amount invested at the time she leaves her position at GM.
A is the future value after T years.
if she leaves the money in her account for 32 years, the formula becomes:
A = P * (1 + .06) ^ 32
that gets her A = P * 6.453386682.
the increase is A - P which is equal to 6.453386882 * P - P which is equal to P * (6.453386882 - 1) which is equal to 5.453386882 * P.
that is an increase of 545.3386882 percent.
if her principal was $20,300, then A would be equal to 6.453386882 * 20,300 = 131,003.7496.
if her principal was $20,300, then the increase would be 5.453386882 * 20,300 = 110,703.7496, which, when added to the principal, becomes 110,703.7496 + 20,300 = 131,003.7496.
if you go back to the original formula of A = P * (1 + .06) ^ 32 and replace P with 20,300, you will see tht A = 20,300 * (1 + .06) ^ 32 = 131,003.7496.
to find out how long it will take for her money to double, you would do the following.
let A = 2 * 20,300 and let P = 20,300.
the formula becomes 40,600 = 20,300 * (1 + .06) ^ T
divide both sides of this formula by 20,300 and it becomes 40,600 / 20,300 = (1 + .06) ^ T
simplify to get 2 = (1 + .06) ^ T
take the log of both sides of this equation to get:
log(2) = log((1 + .06) ^ T)
since log((1 + .06) ^ T) is equal to T * log(1 + .06), your equation becomes:
log(2) = T * log(1 + .06)
divide both sides of this equaiton by log(1 + .06) to get:
log(2) / log(1 + .06) = T
solve for T to get T = log(2) / log(1.06) = 11.89566105 years.
confirm by replacing T in the original equation to get:
40,600 = 20,300 * (1 + .06) ^ 11.89566105 which becomes 40,600 = 40,,600, confirming the solution is correct.
to quadruple the money, the formula becomes 4 * 20,300 = 20,300 * 1.06 ^ T
divide both sides of this equation by 20,300 to get 4 = 1.06 ^ T
take the log of both sides of this equation to get log(4) = log(1.06 ^ T)
the equaiton becomes log(4) = T * log(1.06)
solve for T to get T = log(4) / log(1.06) = 23.79132209.
confirm by replacing T in the original equation to get:
4 * 20,300 = 20,300 * 1.06 ^ 23.79132209 which becomes 4 * 20,300 = 81,200.
since 4 * 20,300 = 81,200, the solution is confirmed to be correct.
the answers to your questions, as best i can determine, are:
A. Create a model for the amount of money Joan will have in her 401(k) after any number of years. Use A for the accumulated amount of money, t for the number of years, and P for the amount of money in the plan when she quit, which is called the principal.
A = P * 1.06 ^ T
B. What percent of the principal will Joan have if she leaves the money in the account for the 32 years until she retires? What percent increase does this represent? Round each answer to the nearest hundredth of a percent.
A = P * 1.06 ^ 32 becomes A = P * 6.45338... * 100 = 645.338% = 645.34% rounded to a hundredth of a percent.
the increase is 6.45338... - 1 = 5.45338... * 100 = 545.338% = 545.34% rounded to a hundredth of a percent.
C. If Joan's original principal was $20,300 how much would be in the account when she retires?
A = P * 1.06 ^ 32 becomes A = 20,300 * 1.06 ^ 32 which becomes A = 131,003.7496.
D. Approximately how many years will it take the principal to double? How long to quadruple?
it will take approximately 11.89566105 years for the principal to double.
it will take approximately 23.79132209 years for the principal to quadruple.
note that P * (1 + .06) ^ T simplifies to P * 1.06 ^ T.
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