21.1 Arithmetic Growth and Simple Interest
When you open a savings account, your primary concerns are the safety and growth of your savings. Suppose you deposit $100 in an account that pays interest at a rate of 10% annually. Assuming that you leave the money alone, how much is in the account after 1 year? 2 years? Before we can do any calculations, we need to convert our interest rate from a percent to a decimal. Percent means "per hundred," so 10% means 10/100 or 0.10
(The easiest way to convert from a % to a decimal is to move the decimal point 2 places to the left.)
Convert the following percents to decimals.
a) 5%
b) 3.2%
c) 25%
d) 89.9%
e) 0.4% In some problems, we also need to convert the unit of time from months to years. Since there are 12 months in a year, divide the number of months by 12 to get the number of years.
Give the number of years (or fraction of a year) represented by each of the following:
a) 1 month
b) 6 months
c) 36 months
d) 54 months
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Now, back to our problem:
Suppose you deposit $100 in an account that pays interest at a rate of 10% annually. Assuming that you leave the money alone, how much is in the account after 1 year? 2 years? With simple interest, interest is paid only on the original balance, no matter how much interest has accumulated. With simple interest, for a principal of $100 and a 10% interest rate, you receive $10 interest at the end of the first year; so at the beginning of the second year, the account contains $110. But at the end of the 2nd year, you again receive only $10 (the interest on your original deposit).
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Simple interest is interest that is paid on the original principal only, not on any accumulated interest.
Simple interest is often used for the following transactions:
1) private loans between individuals, because it is easy to calculate;
2) commercial loans for less than one year­­not just because it is easy to calculate, but also because for low interest rates, simple interest differs very little from compound interest;
3) financing of corporations and the government through bonds. A bond is a loan with repayment at the end of a fixed term and simple interest in the meantime, paid usually annually or semiannually.
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The formulas for simple interest are below.
For a principal P and an annual rate of interest r, the interest I earned in t years is
I = Prt
and the total amount A accumulated in the account is
A = P + I = P + Prt Back to our problem again:
If you deposit $100 in an account that pays interest at a rate of 10% annually, how much interest is earned in 5 years? 20 years?
How much money has accumulated after 5 years?
20 years?
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Do the following calculations and tell what they represent in the context of simple interest:
a) 5000(.03)(12)
b) 400(.01)(7)
Example: Simple Interest on a Student Loan
Suppose you have exhausted the amount that you can borrow under federal loan programs and need a private direct student loan for $10,000. PNC Financial Services offers an interest­only repayment option, under which you make monthly interest payments while you are in school and pay on the principal only after graduation. Under this plan, PNC earns simple interest from you while you are in school.
If they charge 3.21% interest, how much will you pay each month on your $10,000 loan?
If you are in school for 24 months, how much total interest will you pay?
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Ex: If one of your ancestors had bought a $100 bond back in 1850 that paid 5% simple interest, how much interest would it have earned by now?
What would the total worth of the bond be (principal plus interest)?
Ex: Suppose your parents loan you $8000 to buy a car, and ask for you to pay simple interest at a rate of 3.1% each month until you can pay them back the original loan.
a) If it takes you 6 months to pay them back, how much interest will you have paid them?
b)What if it takes you 36 months?
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Ex: Suppose you buy a 20­year U.S.Treasury bond for $5000 that pays 3.8% simple interest every year for 20 years. At the end of 20 years, how much total interest will you have earned?
We frequently observe the kind of growth corresponding to simple interest, called arithmetic growth or linear growth, in other contexts.
Arithmetic growth (also called linear growth) is growth by a constant amount in each time period.
For example, the population of medical doctors in the U.S. grows arithmetically because the medical schools graduate the same number of doctors each year (and the number of doctors dying is also fairly constant).
If you graph arithmetic growth over time, the graph will be a straight line, hence the name linear growth.
Homework #19: pp. 794 ­ 795: 1 ­ 4, plus handout
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