To calculate how much Farah must invest today at age 19 to have a cool million dollars when she is 55, we need to know how many years she has until she retires. Since Farah wants to retire at age 55, she has 55-19 = <<55-19=36>>36 years until she retires.
We also need to know the interest rate that the account pays and the frequency at which the interest is compounded. In this case, the interest rate is 8% and the interest is compounded monthly.
To calculate how much Farah must invest, we can use the formula for the future value of an annuity, which is given by:
FV = PMT * (((1 + i)^n - 1) / i)
where FV is the future value that Farah wants to have, PMT is the amount that Farah must invest each period, i is the interest rate per period, and n is the total number of periods.
Since Farah wants to have a cool million dollars, we can set FV equal to $1,000,000. We also know that the interest rate is 8% and that the interest is compounded monthly, so we can set i equal to 8% / 12 = 0.006667. Finally, we know that Farah has 36 years until she retires, so we can set n equal to 36 * 12 = 432, since there are 12 months in a year.
Substituting these values into the formula above, we get:
FV = PMT * (((1 + i)^n - 1) / i)
FV = PMT * (((1 + 0.006667)^432 - 1) / 0.006667)
To solve for PMT, we can divide both sides of the equation by ((1 + 0.006667)^432 - 1) / 0.006667, which gives us:
PMT = FV / (((1 + i)^n - 1) / i)
PMT = $1,000,000 / (((1 + 0.006667)^432 - 1) / 0.006667)
Calculating this value gives us a result of PMT = $176.72, which is the amount that Farah must invest each month to have a cool million dollars when she is 55.
Now, let's consider Sam's situation. Sam also wants to have a cool million dollars when he is 55, but he is willing to put money into his account every year instead of every month. This means that the frequency at which the interest is compounded will be annual instead of monthly.
To calculate how much money Sam must put into his account each year, we can use the same formula as above but with a few changes. In particular, we need to set i equal to 8% / 1 = 0.08, since the interest is compounded annually, and we need to set n equal to 55-21 = <<55-21=34>>34, since Sam has 34 years until he retires.
Substituting these values into the formula above, we get:
FV = PMT * (((1 + i)^n - 1) / i)
FV = PMT * (((1 + 0.08)^34 - 1) / 0.08)
Solving for PMT gives us:
PMT = FV / (((1 + i)^n - 1) / i)
PMT = $1,000,000 / (((