2013/04/121WORKING WITH INTERESTX-Kit TextbookChapter 4CONTENTSimpleInterestCompoundInterestTime LinesTERMINOLOGYTerm ExplanationInvestment When you save moneyLoan When you borrow moneyDebt When you owe moneyInterest • The price you pay for borrowingmoney.• Earning for saving money.• Interest is added to the originalloan or investment.SIMPLEINTERESTJim wants to study graphic design ata college and needs R10 000 for hisstudies. He goes to a bank toborrow the money. The bank ask 5%interest on study loans. At the bankthe following is discussed:TERMINOLOGYTerm ExplanationPresent Value (PV) • The amount of money we borrow or save• The value today/now• The principal part of the loan• Amount before adding interest• Example R10 000Interest Rate (𝒊) • The rate is used to calculate the amount ofinterest• Given as percentages• Change to decimal fractions• Example 𝟓% = 𝟎, 𝟎𝟓Interest (I) • The amount we pay (borrow) or get (save)TERMINOLOGYTerm ExplanationTerm of period (𝒏) • The length of time over which we borrow orsave money• Due date for paying back the a loan• Maturity date for an investment• Measure term in years, half-years, months,weeks or days• The longer the term, the greater theamount of interestFuture Value (FV) • The value of money at the end of the term• The sum of the principal and interest
2013/04/122𝑰 = 𝑷𝑽 × 𝒊 × 𝒏• Simple interest is interest that is calculated on theprincipal amount for the length of time for which it isborrowed.• Simple interest is due at the end of the term• We calculate simple interest by multiplying the presentvalue by the interest rate by the term• Always use the same length of time to measure the rate ofinterest and the term, for example:If the interest rate is per year and the term is for 8 months,show the term as a fraction of a year ( 𝟖𝟏𝟐)ExampleCalculate the simple interest that Jim has to pay thebank if he borrows R10 000 for 1 year. The interest rateis 5% per annum.𝑰 = 𝑷𝑽 × 𝒊 × 𝒏• 𝐼 = the interest (in rands)• 𝑃𝑉= the present value (the amount borrowed/saved)• 𝑖 = rate of interest (a percentage)• 𝑛 = the term or timeExampleWhat if Jim borrows the moneyfor 2 years?CALCULATINGFUTUREVALUE(FV)Future Value = Present Value + Interest𝑭𝑽 = 𝑷𝑽 + 𝑰𝑭𝑽 = 𝑷𝑽 + 𝑷𝑽 × 𝒊 × 𝒏𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊𝒏ExampleJim wants to know how muchhe will have to pay in interest ona 3-year loan of R10 000. Jimalso wants to know how muchmoney he will have paid in totalby the end of the loan period?CALCULATINGPRESENTVALUE(PV)𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊𝒏𝑷𝑽 =𝑭𝑽𝟏 + 𝒊𝒏
2013/04/123ExampleJim is still at school and can makesome money as a waiter during theholidays. Instead of borrowing themoney, Jim wants to save for hisstudies. He wants to know how muchmoney he must save now, at 5%simple interest to have R10 000 in 3years’ time?CALCULATINGINTERESTRATE (𝒊)𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊𝒏Jim made a lot of money in tips working as awaiter over the summer holidays. He wants toput R8 500 in the bank to save for his post-matric studies. He has calculated the futurevalue. He will need R10 000 for his studies in 3years’ time. How much interest will he have toearn on his principal of R8 500? What is therequired interest rate?CALCULATINGTHE TERM(𝒏)𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊𝒏Jim has R8 000 and he can bank it at aninterest rate of 5%. He still thinks that hewould need R10 000 for his studies. Jimwants to know how long it will take thepresent value of R8 000 to grow to a futurevalue of R10 000 at a simple interest rateof 5%?CALCULATINGSIMPLEINTERESTFOR AFRACTIONOF THETERMJim invested R8 000 for 7 months before heplans to begin college. What interest will Jimreceive if he banks R8 000 for 7 months at4.5% simple interest per annum?You are asked to calculate interest (amount inrands), not interest rate (a percentage)ExampleGrace receives R750 on her birthday. Shedecides to invest the money for 8 monthsat 6% per annum.1. Find the simple interest.2. Find the future value.ExampleTo have a future value of R1 050 after 7months, how much money must Jim saveat 8% simple interest per annum?(Remember to change the time to thesame unit as the interest rate.)
2013/04/124ExampleMary banked R500. After 180 days theamount in her account is R507. What wasthe interest rate that she received?ExampleJoe banked R500 at a bank that gave him3% interest per annum. When he lookedat his balance (the amount in the bank) itwas R520. How long had Joe’s moneybeen in the bank?TIMELINESBeginningof TermTimePeriodInterestRateEnd ofTermExampleWhen Jim started high school, Jim’s motherbegan to save for his post-matric studies. Shecalculated that 5 years from then, when Jimleaves school, he will need at least R20 000 tostudy for 2 years. Jim’s mother has a savingsaccount at a bank that pays 4,5% interest peryear. How much would she have to save todayin order to have R20 000 in 5 years’ time? Showyour calculations on a time line.ExampleA man is offered R200 000 cash now for hishouse or R202 000 after 6 months. Whichis the better offer if the current interestrate is 5% per year? Present allinformation on a time line.COMPOUNDINTEREST–Interestupon Interest•Compound interest is interest paid on theoriginal investment as well as on theinterest that you have earned previously.•Simple interest is only earned on theoriginal principal.
2013/04/125R100investedat 10% annuallyYear Simple Interest Compound Interest1 R100 + R10 = R110 R100 + R10 = R1102 R110 + R10 = R120 R110 + R11 = R1213 R120 + R10 = R130 R121 + R12,10 = R133,104 R130 + R10 = R140 R133,10 + R13,31 = R146,415 R140 + R10 = R150 R146,41 + R14,64 = R161,0510 R190 + R10 = R200 R235,79 + R23,58 = R259,3750 R590 + R10 = R600 R10 671,90 + R1 067,19 = R11 739,09WORKINGWITH COMPOUNDINTEREST𝑭𝑽 = 𝑷𝑽 𝟏 + 𝒊 𝒏• 𝐹𝑉 = Future Value• 𝑃𝑉 = Present Value (Principal)• 𝑖 = Annual Interest Rate• 𝑛 = Number of years/period•We can change the subject of the formulaCALCULATINGFUTUREVALUEJim receives R1 000 on his birthday anddecides to save it. He can get an interestrate of 4% at the bank. Interest iscompounded annually (yearly). Jim wantsto know how much his investment will beworth at the end of 3 years.EXAMPLEIn 10 years’ time, you want to have yourown business. You believe that it is betterto start small than not to start saving at all.You invest R5 000 now and leave it for 10years. Interest is compounded at 8%annually.CALCULATINGPRESENTVALUEBefore you start your own business, you’dlike to travel. In 7 years’ time, you thinkyou would need at least R85 000 to seesome of the world. How much moneywould you need to invest now? Theinterest rate is 18% p.a. compoundedannually.CALCULATINGINTERESTRATEA friend wants to borrow R3 000 from you.She says that she will give you R4 000 backafter 3 years. But, you know about compoundinterest. You know that if you put your R3 000in a bank for 3 years, you will earn compoundinterest at a rate of 8% p.a. What is theinterest rate if you lend your friend themoney?
2013/04/126CALCULATINGTHE TERM/PERIODYour younger brother earns R500 workingin your family’s shop during the schoolholidays. He wants to buy a bicycle so thathe can make more money by deliveringpizzas. His dream bicycle cost R700. Howlong will he have to save if he can earn 8%interest compounded annually?DIFFERENTTIMEPERIODS• Interest compounded annually means that the interest is addedto the capital amount ones a year at the end of the year.• To handle different time periods the compound interestformula is changed:𝑭𝑽 = 𝑷𝑽 𝟏 +𝒊𝒎𝒏×𝒎• 𝐹𝑉 = Future Value after 𝑛 × 𝑚 periods of compounding• 𝑃𝑉 = Present Value• 𝑖 = Compound Interest Rate• 𝑚 = Number of compounding periods during one year• 𝑛 = Number of years that the investment is heldEXAMPLELet’s see how much R1 000 would beworth if we invested it for 10 years at 5%interest, and interest is compounded:1. Annually2. Semi-annually3. Quarterly4. Monthly5. DailyThe Future Value increases asthe compounding periodsincrease.MORE EXAMPLESFind the future value of R200 invested for 7years at 7,5% per annum and compoundedannually.MORE EXAMPLESYou have won a scratch-and-wincompetition held by your bank. You canchoose one of the following prizes:•R10 000 now, or•R18 000 at the end of 5 yearsThe interest rate is 12% compoundedannually. Which prize do you choose?
2013/04/127MORE EXAMPLESFor R2 000 to become R3 000 in 10 years,work out what the annual compoundinterest rate should be.MORE EXAMPLESHow long will it take R2 000 to growto R3 000 at 8% interest compoundedannually?MORE EXAMPLESJim saves R500 at an interest rate of 6%.What is his investment worth after 5 yearsif interest is compounded:1. Yearly2. Quarterly3. Monthly4. Weekly5. Daily