Log and Root Homework
  1. If it takes you 25 years to turn $25,000 into $500,000, what must your APR be assuming annual compounding? monthly compounding?
  2. If you deposit $140 a month into a savings account with an APR of 7% over 20 years, what APR would you need to make a lump sum deposit of $2300 achieve the same amount in the account in the same amount of time?
  3. Solve for $y$
    1. $y=\log 1000$
    2. $\log y = 4$
    3. $7^{y} = 0.35$
    4. $3^y+1=12.6$
    5. $5=2(1.02)^{5y}$
    6. $\frac{5(10,000)^{2y}+11}{0.01}=25,000$
    For the following, give the answers in years and months to the nearest month (see example below).
  4. You have $1000 in an account with a 7% APR compounded annually. How long will it take so that you have $3000?
  5. You have $1000 in an account with a 7% APR compounded monthly. How long will it take for you to have $3000? Compare with your last answer.
  6. You can get an account with a 6% APR compounded monthly. How long will it take for your initial deposit to double?
  7. You are depositing $150 per month into a savings account with an APR of 4% compounded monthly. How long will it take to save $10,000?
  8. $^{*}$ You ran up a bill of $1400 on your credit card. The interest rate is 17% computed monthly. You can afford to pay $100 per month. How long will it take you to pay off your debt?
Note: If you are dealing with monthly payments, compounding, or deposits and you solve for "$y$" in the formula, the number will be in years. If you want the answer to the nearest month you must convert the decimal part to months.
Example: Suppose you have an initial deposit and you are asked to calculate how long it will take before your deposit will reach some value (monthly compounding). You solve and find that $y=12.357$. So you have 12.357 years, but in what month do we actually reach our goal?

$(0.357~\textrm{years})(\frac{12 ~ \textrm{months}}{1~\textrm{year}}) = 4.284~\textrm{months}$

So the answer is 12 years and 4.284 months. Since we are dealing with monthly compounding 12 years and 4 months is not long enough, so it will take 12 years and 5 months before your money reaches the value you want. You will have to make similar computations if the compounding period is weeks, quarters, days, etc.