Mathematical Explorations January 24

Would you rather?

Simple vs. compound interest If you invest $1000 and get $100 in interest every year, this is called simple interest. This is linear growth: B = 1000 + 100t, where t is the number of years the investment has been in the bank. $1000 is the y-intercept, or the starting point of the function. $100/year is the slope, or the rate of increase of your investment. It’s constant.

Compound interest If you invest $1000 and get 8% interest every year, this is called compound interest. After one year, the new balance is 1.08*$1, 000 = $1, 080. After two, it is 1.08 * $1, 080 = $1, 166.40. The expression is B = 1000*(1.08)t. $1000 is the y-intercept, or the starting point of your investment. 1.08 is the growth factor of your investment. Here the rate of change is not constant, but the percent change is.

Compounding more often If you invest $1000 and get 8% interest every year, but it is compounded quarterly, this means that you get 2% interest added on the previous month’s amount every quarter. After one year, the new balance is $1, 000(1.02)4 = $1,082.43. The difference is not that big, but will it add up? The expression for the amount of money in the account after t years is $1000(1.02)4t What if you are compounding monthly? Daily? All the time?

Depreciation Suppose you buy a new car for $20,000. If the car is a business expense you might choose linear depreciation for tax purposes. Suppose the value decreases by $1,800 each year. The equation that determines the value V as a function of the age A is V = 20, 000 - 1, 800A. But a more realistic way to model the value of the car is to assume that the percentage decrease is the same each year. Suppose it's 13%. Then each year its value is 87% of what it was the year before. The corresponding equation is V = 20, 000 * 0.87A.

Cars Let’s investigate this claim: While different cars depreciate at different rates, it's a good rule of thumb to assume that a new car will lose approximately 20 percent of its value in the first year and 15 percent each year after that until, after 10 years, it's worth around 10 percent of what it originally cost. Assume a car cost $30,000 when it was new. How much will it cost after 10 years?

Summary When there is exponential growth, then the growth factor is 1 + percent growth rate expressed as decimal When there is exponential decay, or depreciation, then the decay factor is 1 – percent growth rate expressed as decimal

Doubling time How long will it take to double your money? Answering this question exactly requires the use of logarithms, so instead we will use some guess and check and graphing (use logarithms if you remember how!) Suppose you start with $1000 and interest is 8%. When will you have $2000? Do some guessing and checking on your calculator or computer.

You will have $1469. 33 after 5 years, and $2158. 92 after 10 years You will have $1469.33 after 5 years, and $2158.92 after 10 years. It turns out that after 9 years you will have $1999.00, so we will say it will take 9 years. What if you started with $500? Experiment again

After 5 years you will have $734. 66 After 5 years you will have $734.66. After 10 years you will have $1079.46. After 9 years, you will have $999.50. Really? What if you start with $8500? Same thing: 9 years. Try it. In fact, the doubling time does not depend on the initial investment, just the interest rate. Try this: divide 70 by the interest rate.

70/8=8.75, which is a really good estimate for the doubling time. This is the rule of 70: 70/interest rate is a good estimate for doubling time. Unfortunately, interest rates in the U.S. are currently very low. My savings account has an interest rate of 0.01%. A bigger bank might give you 0.03%. My investment will double in 7000 years, alas.

Half-life Half-life is used a lot in science, in particular in a process called carbon-dating. Briefly, since we know the half-life of Carbon-14, which is radioactive and therefore decays exponentially, based on its amount in, say, a fossil, we can estimate the age of that fossil. Half-life is the amount of time it takes for an exponentially decaying function to reach half of its initial value. This will not be too important to us. Note that the rule of 70 still applies.

Challenge What if we know the doubling time but not the growth rate? Some of you need this for your projects. More generally, consider this example: According to one of your book problems, the average cost of private colleges in 1999 was 15,518 and was 27,293 in 2010. Assuming the growth was exponential, how much did it grow each year? Use your calculator, logarithms, Excel, or Desmos to solve this problem in any way you know how. Then I will show you how you can solve it in Desmos.

Loans All loans grow exponentially, but because we make regular payments, the amount of our loan changes after every payment. Tomorrow we will look at student loans and mortgages.

Your work for today We are going to look at incarceration rates in the U.S. We will use regressions for this assignment. Work as a group and send me your work when you are done. Please write detailed responses rather than just one brief sentence. All instructions are on the written handout.