Exp Functions & Investing

  1. Suppose you invest $700 in a high-yield savings account that earns 3% annual interest. How much will this investment be worth a year later if the interest is compounded annually? What if the interest is compounded monthly? Twice a month? Daily? Continually?
  2. Suppose you need $1000 four years in the future. You find a high-yield savings account that offers 3% interest and compounds your interest monthly. How much do you need to invest now so that your investment will be worth $1000 in four years?
  3. Suppose you need $1000 four years in the future, but only have $700 to invest now. What annual interest rate would you need to get this return on your investment? Assume the account compounds your interest monthly.
  4. Suppose you invest $700 in a high-yield savings account that earns 3% annual interest and compounds your interest monthly. In how many years will your investment be worth $1000?
  5. Suppose you have $123,456, and you choose to open a money-market account with your local credit union that guarantees 2.529% APY (Annual Percent Yield), where the account interest is compounded monthly.
    1. In order to write down an accurate formula for the account balance at any given time, we need the APR (Annual Percent Rate), not the APY. Recall that the APY is the literal percent that your balance will increase by in a year, whereas the APR is the raw interest rate prior to consideration of the compounding effect. Let \(r\) be the account’s APR and let \(R\) be the account’s APY. Find the account’s APR by solving for \(r\) in the following equation. Round the APR to the nearest tenth-of-a-percent. \[P\left(1+\frac{r}{12}\right)^{12} = P(1+R)\]
    2. Using the APR, write down a formula \(S(t)\) for the balance of the account after \(t\) years.
    3. What will the balance of your account be after five years? How much total interest did your money earn in those five years?
    4. What will the balance of your account be after one month, after you receive a single interest payment? What is the amount of this interest payment? (Typically the units of \(t\) are years. Be careful considering what \(t\) must be in this case.)
    5. How long before the balance of your account will reach $150,000?
    6. Suppose instead of compounding your interest monthly, the credit union compounded your interest continually. How much more interest would you be earning each year?

Puzzle

Missing Dollar Riddle

Three guests check into a hotel room. The manager says the bill is $30, so each guest pays $10. Later the manager realizes the bill should only have been $25. To rectify this, he gives the bellhop $5 as five one-dollar bills to return to the guests.

On the way to the guests’ room to refund the money, the bellhop realizes that he cannot equally divide the five one-dollar bills among the three guests. Since the guests are not aware of the total of the revised bill, the bellhop decides to just give each guest $1 back and keep $2 as a tip for himself, and proceeds to do so.

Since each guest got $1 back, each guest only paid $9, bringing the total paid to $27. The bellhop kept $2, which when added to the $27, comes to $29. So if the guests originally handed over $30, what happened to the remaining $1?