When you open a savings account, your primary concerns are the safety and the growth of the “population” of your savings. We assume the safety and concentrate on measuring the growth.
EXAMPLE 1 Simple Interest
Interest is the money earned on a savings account or other fund. Suppose that you deposit $1000 in an account that “pays interest at a rate of 10% annually.” (This is an unrealistic rate, particularly in this era of very low interest rates. We use it solely because it makes the calculations easy.) Assuming that you make no other deposits or withdrawals, how much is in the account after 5 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, or
So the balance at the beginning of the second year is . We express an interest rate either as a percentage or as a decimal fraction. “Percent” means “per 100,” so you can think of the symbol “%” as standing for “1 per 100” or . So to convert from a percentage to a decimal fraction, divide the percentage by 100 by moving the decimal point two places to the left. An interest rate of 10% is 10/100 or 0.10; an interest rate as a decimal number (such as 0.10) is 100r% (10%). (Caution: A common error in using the formulas in this chapter is to forget to express the percentage as a decimal; for example, for , don’t substitute 5 for the in the formula, but instead substitute 0.05.)
With simple interest, interest is paid only on the original balance, no matter how much interest has accumulated. At the end of the first year, the account will contain $1100. But at the end of the second year, you again receive only $100; so at the beginning of the third year, the account contains $1200. In fact, at the end of each year, you receive just $100 in interest, amounting to a final balance of $1500 at the end of the fifth year.
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What will be the balance after 10 years?
Algebra Review Appendix
Fractions, Percents, and Percentages
Simple Interest DEFINITION
Simple interest is interest that is paid on the original principal only, not on any accumulated interest.
The formulas for simple interest are themselves simple.
Simple Interest RULE
For a principal and an annual rate of interest , the interest earned in years is
and the total amount accumulated in the account is
You may find this method for interest rather strange if you are used to a different system-compound interest, which we will consider shortly. However, simple interest is often used for the following transactions:
EXAMPLE 2 Simple Interest on a Student Loan
Let’s suppose that you have exhausted the amount that you can borrow under federal loan programs and need a private direct student loan for $10,000. The lowest fixed interest rate from PNC Bank, Pittsburgh, Pennsylvania, in July 201 5 was 6.49%. There is an interest-only repayment option, under which you make monthly interest payments while you are in school and pay toward the principal only after graduation. Under this plan, PNC earns simple interest from you while you are in school.
How much monthly interest would you pay for such a $10,000 loan? The principal is , the annual interest rate is per year, and the number of years is year. The interest for one month would be
(Actually, the interest rate might not be 6.49% but could be as much as 12.99%, since it would depend on the creditworthiness of you and any cosigner.)
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If you opt not to make the monthly interest payments, then when you are supposed to begin to repay the loan, the accumulated simple interest is capitalized, that is, added to the principal of the loan. If you are supposed to begin repayment after 51 months, what would be the principal of the loan at that time, assuming that the interest rate stays the same throughout?
We frequently observe the kind of growth corresponding to simple interest, called arithmetic growth or linear growth, in other contexts.
Arithmetic Growth DEFINITION
Arithmetic growth (pronounced with accent on the “met” syllable) (also called linear growth) is growth by a constant amount in each time period.
For example, the population of active medical doctors in the United States grows arithmetically because the medical schools graduate the same number of doctors each year and the numbers of doctors dying and retiring are also fairly constant but smaller. The concept of linear growth has appeared already in this book in the discussions of linear programming (Chapter 4) and linear regression (Chapter 6).