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 $1000 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? The $1000 is the principal, the initial balance of the account. At the end of one year, interest is added. The amount of interest (in this case) is 10% of the principal. 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% 1
Now, back to our problem: Suppose you deposit $1000 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? The $1000 is the principal, the initial balance of the account. At the end of one year, interest is added. The amount of interest is 10% of the principal. 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 $1000 and a 10% interest rate, you receive $100 interest at the end of the first year; so at the beginning of the second year, the account contains $1100. But at the end of the 2nd year, you again receive only $100 (the interest on your original deposit). 2
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 year not 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. 3
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 = P(1 + rt) Back to our problem again: If you deposit $1000 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 is in the account after 5 years? 20 years? 4
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 interest only 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 earn 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? 5
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? 6
Ex: Suppose you buy a 20 year 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? 7
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). Homework: pp. 794 795: 1 4 8