Math in Our World Section 8.2 Simple Interest

Learning Objectives Compute simple interest and future value. Compute principal, rate, or time. Compute interest using the Banker’s rule. Compute the true rate for a discounted loan.

Interest Interest (I ) is the fee charged for the use of money. Simple interest is a one-time percent of an amount of money.

Interest Principal (P) is the amount of money borrowed or placed into a savings account. Rate (r) is the percent of the principal paid for having money loaned, or earned for investing money. Unless indicated otherwise, rates are given as a percent for a term of 1 year. Time (t) or term is the length of time that the money is being borrowed or invested. When the rate is given as a percent per year, time has to be written in years. Future value (A) is the amount of the loan or investment plus the interest paid or earned.

Interest Formulas for Computing Simple Interest and Final Value 1. Interest = principal x rate x time: I = Prt 2. Future value principal interest: A = P + I or A = P (1 + rt)

EXAMPLE 1 Computing Simple Interest Find the simple interest on a loan of $3,600.00 for 3 years at a rate of 8% per year. SOLUTION Change the rate to a decimal and substitute into the formula I = Prt: 8% = 0.08 I = Prt = ($3,600.00)(0.08)(3) = $864.00 The interest on the loan is $864.00.

EXAMPLE 2 Finding Future Value Find the future value for the loan in Example 1. SOLUTION Recall that the Principal is $3,600.00 and the Interest on the loan is $864.00. Substitute into the formula A = P + I = $3,600.00 + $864.00 = $4,464.00 The total amount of money to be paid back is $4,464.00.

EXAMPLE 3 Computing Simple Interest for a Term in Months To meet payroll during a down period, United Ceramics Inc. needed to borrow $2,000.00 at 4% simple interest for 3 months. Find the interest. SOLUTION Change 3 months to years by dividing by 12, and change the rate to a decimal. Substitute in the formula I = Prt.

EXAMPLE 4 Computing Monthly Payments Admiral Chauffeur Services borrowed $600.00 at 9% simple interest for 1-1/2 years to repair a limousine. Find the interest, future value, and the monthly payment. SOLUTION Step 1 Find the interest.

EXAMPLE 4 Computing Monthly Payments SOLUTION Step 2 Find the future value of the loan. Step 3 Divide the future value of the loan by the number of months. Since 1-1/2 years = 18 months, divide $681.00 by 18 to get $37.83. The monthly payment is $37.83.

EXAMPLE 5 Computing Principal Phillips Health and Beauty Spa is replacing one of its workstations. The interest on a loan secured by the spa was $93.50. The money was borrowed at 5.5% simple interest for 2 years. Find the principal. SOLUTION The amount of the loan was $850.00.

EXAMPLE 6 Computing Interest Rate R & S Furnace Company invested $15,250.00 for 10 years and received $9,150.00 in simple interest. What was the rate that the investment paid?

EXAMPLE 6 Computing Interest Rate SOLUTION The interest paid on the investment was 6%.

EXAMPLE 7 Computing the Term of a Loan Fran and Rick borrowed $4,500.00 at 8-3/4% to put in a hot tub. They had to pay $2,756.25 interest. Find the term of the loan. SOLUTION The term of the loan was 7 years.

The Banker’s Rule The Banker’s rule treats every month like it has 30 days, so it uses 360 days in a year, instead of 365. They claim that the computations are easier to do. When a lending institution uses 360 days instead of 365, how does that affect the amount of interest? For example, on a $5,000 loan at 8% for 90 days, the interest would be

EXAMPLE 8 Using the Banker’s Rule Find the simple interest on a $1,800 loan at 6% for 120 days. Use the Banker’s rule. SOLUTION

Discounted Loans Sometimes the interest on a loan is paid upfront by deducting the amount of the interest from the amount the bank gives you. This type of loan is called a discounted loan. The interest that is deducted from the amount you receive is called the discount.

EXAMPLE 9 Finding the True Rate of a Discounted Loan A student obtained a 2-year $4,000 loan for college tuition. The rate was 9% simple interest and the loan was a discounted loan. (a) Find the discount. (b) Find the amount of money the student received. (c) Find the true interest rate.

EXAMPLE 9 Finding the True Rate of a Discounted Loan SOLUTION (a) The discount is the total interest for the loan. (b) The student received $4,000 - $720 = $3,280.

EXAMPLE 9 Finding the True Rate of a Discounted Loan SOLUTION (c) The true interest rate is calculated by finding the rate on a $3,280 loan with $720 interest. The true interest rate is approximately 10.98%.