College Math

Question

Mitch and Bill are the same age. When Mitch is 25 years old, he begins depositing $100 every month into a long-term savings plan. He consistently makes these deposits for 10 years, at which point he is forced to stop making deposits of any kind. However, he leaves his money in this same account for the next 40 years (where it continues to earn interest that is compounded monthly). Bill, on the other hand, doesn’t start saving until he is 35 years old. At that point in his life (age 35) and for the next 40 years after that, he makes consistent monthly deposits of $100. Assume that both accounts earn interest at an annual percentage rate of 7% and interest in both accounts is compounded monthly. (Be sure to label your two answers with (a) and (b) so they are easy to read.)
a. How much money does Mitch and Bill have in their individual accounts by age 75?
b. Compare how much their individual principal contributions were and write a brief paragraph summarizing your take-away message you get from this parable.

Answer 1

We know that Mitch put 100/month in his account, for 10 years, or 10yr.*12mon.=120 months. So that means Mitch has put 100(dollars/month)*120(months)=$12,000.00, which is a very nice amount of cash to have to put somewhere to earn interest! Mitch has left all this money in that account same account, which we're told gives 7% interest (compounded monthly) for 40 years and we know at that time Mitch is 25+40=65 years old, at that time.

The formula of monthly compound interest is: CI = P(1 + (r/12) )^12t - P where, P is the principal amount, r is the interest rate in decimal form(0.07), and t is the time. We know what these variables are from the work we've just done, so plug them in this formula to get the first number we need to answer for Part a) Mitch. But, Mitch is only 65, and we are asked how much his account has in it at age 75. Since we're not told anything about what Mitch has done with this money from age 65-75, If Mitch is smart, I think he would just leave it in that account for the next 10 years too. So, we just need to use that formula again and change the time from 40 years to 50 years. That will give us the amount of money Mitch has when he is 75, **But I would also be sure to include that assumption in your answer for clarity, and something close to my wording should suffice. So that's the answer for Part a) Mitch.

Let see how Bill is doing: Bill is 35 when he starts to save $100/month, but like Mitch, he puts it in an account that yields 7% interest, and it's also compounded monthly, so we can use the same formula as before, we just need to use Bill's numbers this time(note how much lower Bill's principlr is & how that affects his end amount!), now you have the answer Form Part a) Bill.

For the last part of the problem, you need to compare the 2 principles & how much money each person ended with. Then write that info up in a paragraph, and be sure to include the 'lesson' you learned from this problem. I can't (& shouldn't) do it for you. Just use your common sense & you'll be fine.