MAT 142 College Mathematics Module #4FinanceTerri L. Miller & Elizabeth E. K. Jones Spring 2009revised April 13, 20091. Simple InterestInterest is the money earned (profit) on a savings account or investment. Principalor present value is the amount of money invested, sometimes referred to as the initialamount.Simple interest is when the money earned is computed as a percentage of the principalper year. The interest rate is the decimal equivalent of the percentage that will be earned.Simple Interest Formula:I = P rtwhere I is the interest, P is the principal, r is the rate, and t is the time in years.Example 1. Calculate the interest for a deposit of $850 into an account paying 3.5% annualsimple interest if the money is in the account for 7 months.Solution: We are given P = 850 and r = 0.035, since there are 12 months in a year and themoney will be in the account for 7 months, t = 7/12. So the interest will be:I = P rt =⇒ I = 850(0.035)712≈ $7.35.We round our answer to two decimal places since this is money.At the end of the time, the total amount, principal and interest, is called the future valueor maturity value. There are two ways to compute this value.Future Value for Simple Interest Formula:F V = P + I or F V = P (1 + rt)where I is the interest, P is the principal, r is the rate, and t is the time in years.Example 2. What is the future value of a savings account earning 312% simple interest, ifthe present value is $538 and the money is in the account for 7 months?Solution:(1) The first method is to compute the interest and then add that to the principal. Weare given P = 538, r = 0.035, and t =712.I = 538(0.035)712= 10.98 =⇒ F V = 538 + 10.98 = $548.98.(2) The second method is to compute the future value directly.F V = 538(1 + 0.035112= $548.98.Simple interest is used as the basis for other types of interest. The most common applicationof simple interest is called an add-on loan. An add-on loan is a loan in which the futurevalue of the loan is calculated and then payments are determined by dividing this by thenumber of payments to be made. The following example demonstrates this type of loan.Example 3. The Perez family buys a bedroom set at Mor Furniture for $3,700. They put$500 down and finance the rest through the store at 9.8% add-on interest. If they agree tomake 36 monthly payments, find the size of each payment.Solution: This is a simple interest problem. The interest is determined at the simple interestrate and added on to the amount of the loan. Our first step in the problem is to determinethe amount of the loan. Since $500 is used as a down payment, the amount of the loan willbe $3700 − $500 = $3200. Now we need to figure out what formula we will need to use.Remember that the amount of the loan plus the interest is the future value of the loan. Thuswe will need to use the future value simple interest formula of F V = P (1 + rt) . In ourproblem P is the amount of the loan or $3200. The interest rate is 9.8%. We need this tobe changed from a percent to a decimal. To do this, we divide 9.8 by 100 to get 0.098. t isthe time in years. Since the Perez family will be paying for 36 months, t will be 3.F V = 3200(1 + 0.098 ∗ 3).Now we will plug all this in to our calculator to getF V = 4140.80.We now know how much the Perez family will pay in total for the bedroom set. We stillhave to figure out how much this will cost them in monthly payments. To do this, we willneed to divide the future value by the number of month. This will give uspymt =4140.8036= 115.02222222.Since we are talking about an amount of money, we must have only dollars and cents (twoplaces to the right of the decimal). In this class we will round using standard rounding. Thiswill make the payment amount $115.02.2. Compound InterestInterest is compounded when the interest eaned for a specified time period is added intothe account and then it also earns interest. Here is a simple example of how it works.2c 2009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller & Elizabeth E. K. JonesExample 4. George makes a deposit of $50 into an account that earns 24% per year com-pounded monthly. He will leave the money in the account for 3 months. How much will hehave at the end of three months.Solution: Since the interest is 24% per year, at the end of the first month, George will get50(0.24)(1/12) = $1.00 in interest (simple interest formula) deposited into the account. Thisincreases his balamce to $51 At the end of the second month, he will get 51(0.24)(1/12) =$1.02 in interest deposited into the account. His balance increases again to $52.02. At theend of the third month, he will have another 52.02(0.24)(1/12) = $1.04 deposited into hisaccount, so that when he draws out his money he will get $53.06.It should be pretty clear that we do not want to build this up one month at a time wheneverwe need to find the amount of an account earning compound interest. Hence. we use aformula but first we need to know another term.The compounding period is the length of time that elapses before a new interest is de-posited into the account. In our Example 4, the compounding period is one month ormonthly.Compound Interest Formula:F V = P1 +rn(nt)where F V is the future value, P is the principal, r is the rate, t is the time in yearsand n is the number of compounding periods per year.Example 5. When Jacob was born, his grandparents deposited $10,000 into a special ac-count for Jacobs college education. The account earned 614% interest compounded daily.(1) How much will be in the account when Jacob is eighteen?(2) If, on becoming eighteen, Jacob arranges for the monthly interest to be sent to him,how much would he receive each 30-day month?Solution:(1) The first part of this problem is a basic future value of a compound interest accountquestion. For this, we will need the future value formula for compound interestF V = P1 +rn(nt). For our problem P is 10000; r is 0.0625 (obtained by calculating(6 + 1 ÷ 4) ÷ 100); n is 365 (number of days in a year); and t is 18 (number of yearsfrom Jacob’s birth to age 18). We now plug these numbers into the formula to findthe future value.F V = 100001 +0.0625365(365·18)= 30799.20215.Since we are talking about an amount of money, we must have only dollars and cents(two places to the right of the decimal). In this class we will round using standardrounding. This will make the future value $30799.20.3c 2009 ASU School of
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