MA 111 Worksheet 3.2Interest1. Suppose we have $1000, and we want to calculate its value after a 5% increase. Thereare two (equivalent) ways we have learned to do this:Way #1: $1000 +5100· $1000Way #2: $1000 · 1.05We will be using the second way because it is simpler . . . just a single multiplication.Note the specific type of question we usually are interested in: “How much moneywill I have after some period of time?” We are not only interested in the increase, butrather what the total amount will be.2. Annual percentage rateThis is the interest rate being charged/paid on a sum of money during one year’s time.It is often abbreviated “APR.”For the following examples, I will use an APR of R = 5% which, as a decimal, equalsr =5100= 0.05. I will use R when expressing the rate as a percent, and r whenexpressing the rate as a decimal, where r =R100.3. PrincipalThe principal is the amount of money that is earning interest.For the following examples, I will use a starting principal of $1000.4. Simple interest over one yearExample. 1.05 · $1000 = $1050. After one year of simple interest, we have $1050.5. Simple interest over three yearsFor “simple interest,” we just use the APR and the starting principal to figure interestearned.Example. Interest from first year: 0.05 · $1000 = $50.Interest from second year: 0.05 · $1000 = $50.Interest from third year: 0.05 · $1000 = $50.Total interest: $150.Final amount: $1150, which equals $1000 + (3 · 0.05 · $1000) or $1000(1 + 3 · 0.05).Formula. If principal P is invested at simple annual interest rate R% per year for tyears, the final amount A is given by:A = P (1 + tr),where r =R100.16. Compound interest over three years, compounding annuallyFor “compound interest,” the interest that has already been earned is added to theprincipal. That is, we will earn interest on the interest we’ve already accrued.Example. Interest from first year: 0.05 · $1000 = $50.Principal after first year: $1050.Interest from second year: 0.05 · $1050 = $52.50.Principal after sec ond year: $1102.50.Interest from third year: 0.05 · $1102.50 = $55.13.Principal after third year: $1157.63.Shorter: We can combine the two steps (interest and revised principal) into one step:After the first year: $1000 · 1.05 = $1050.After the second year: $1050 · 1.05 = $1102.50.After the third year: $1102.50 · 1.05 = $1157.63.Even shorter: We can combine these three steps into one:After three years: $1000 · 1.05 · 1.05 · 1.05 = $1157.63.Shortest!: We can use an exponent to indicate the number of years (that is, the numberof factors of 1.05):After three years: $1000 · (1.05)3= $1157.63.Formula. If principal P is invested at annual interest rate R% compounded everyyear for t years, the final amount A is given by:A = P (1 + r)t,where r =R100.7. Compound interest over twenty years, compounding annuallyExample. $1000 · (1.05)20= $2653.30.8. Compound interest over one year, compounding monthlyWe first have to find the monthly interest rate. To do this, divide the APR r by 12.Example. After one month: $1000 ·1 +0.0512= $1004.17.After twelve months: $1000 ·1 +0.051212= $1051.16.9. Compound interest over t years, compounding monthlyFormula. If principal P is invested at annual interest rate R% compounded everymonth for t years, the final amount A is given by:A = P1 +r1212t,where r =R100.Example. After three years: $1000 ·1 +0.051236= $1161.47.210. Compound interest over t years, compounding dailyFormula. If principal P is invested at annual interest rate R% compounded every dayfor t years, the final amount A is given by:A = P1 +r365365t,where r =R100.Example. After three years: $1000 ·1 +0.053651095= $1161.82.11. Compound interest over t years, compounding continuouslyWe might consider compounding more often. How much does this help us?Example. Compound hourly. After three years we have $1000(1+.058760)3·8760= $1161.833745.Example. Compound every minute. After three years we have $1000(1+.05525600)3·525600=$1161.834234.It is possible to imagine, in the limit, compounding continuously, all the time. It turnsout that a simple formula involving the number e results:Formula. If principal P is invested at annual interest rate R% compounded continu-ously for t years, the final amount A is given by:A = P ert,where r =R100.On my calculator, the “e” button is below the “π” button.Example. After three years: $1000 · e0.15= $1161.834243.12. Reviewing the above results, we see there is a big jump in the final amount when wewent from simple to compound interest (compounding annually). There is another bigjump when we compound monthly instead of annually.Beyond that, there is not a lot to be gained from using shorter time periods for com-pounding. In practice, most banks will use monthly compounding for everything (butmay use daily compounding; e.g., for outstanding credit card balances).313. Some Problems to Try. SHOW YOUR WORK.(a) What is the final amount if you invest $5000 for 30 years at 4.2% simple interest?(b) What is the final amount if you invest $5000 for 30 years at 4.2% interest com-pounded annually?(c) What is the final amount if you invest $5000 for 30 years at 4.2% interest com-pounded quarterly (four times a year)?(d) What is the final amount if you invest $5000 for 30 years at 4.2% interest com-pounded continuously?(e) You want to buy a $25,000 car in 5 years. How much should you invest today at5.3% interest compounded monthly to achieve that amount?(f) Suppose you invest $1000 for one year at 4% interest compounded monthly. Whatinterest rate R, compounded annually, would yield the same final amount? Thisis called the Annual Percent Yield (APY).(g) If you invest $10,000 today at 3.5% interest compounded monthly, how manyyears will it take to double your