Apply the Compound Interest Formula for monthly ...

In the real world, interest is often compounded more than once a year. In many cases, it is compounded monthly, which means that the interest is added back to ... Home Math Writing LanguageSkills Tutoring CalculatingInterestandExcelFunctions: ApplytheCompoundInterestFormulaformonthlyCompoundingInterest Intherealworld,interestisoftencompoundedmorethanonceayear.Inmanycases,itiscompoundedmonthly,whichmeansthattheinterestisaddedbacktotheprincipaleachmonth. Inordertocalculatecompoundingmorethanonetimeayear,weusethefollowingformula: \({\text{A}}={\text{P}}(1+\frac{\text{r}}{\text{n}})^{\text{nt}}\) A=Amount(endingamount) P=Principal(beginningamount) r=Rate(annually)asadecimal t=Timeinyears n=Numberofcompoundingperiodsperyear Thefollowingvideowillexplainalittlemoreabouteachoftheformulaswehavelearnedtocalculateinterestsofarandhowtheyarerelated. VideoSource(05:01mins)|Transcript Thefollowingvideodemonstrateshowtodothecompoundinterestcalculationusingtheorderofoperations.Italsodemonstrateshowtoenterthenumbersintoacalculatorinordertoavoidroundingerrors. HowtoAvoidRoundingErrors AvoidroundingerrorsbyNOTroundinguntilthefinalanswer.Dothisbyfollowingtheorderofoperationsandusingallthedigitsinthecalculatorfromeachpreviousstep. VideoSource(07:57mins)|Transcript PracticeProblems Ifyouinvest$1000inanaccountthatpays9%interestannually,compoundedmonthly,whatisthetotalamountofmoneythatyouwouldhaveattheendofoneyear?(Hint:Inthiscase,n=12becauseitiscompoundingmonthly,butt=1becausewearecalculatingfor1year.) Supposethatyouinvest$2000inanaccountthatpays4%interestannually,compoundedmonthly.Howmuchmoneywouldyouhaveintheaccountafterthreeyears?(Hint:n=12andt=3.) Lookattheanswertoquestion2.Howmuchinterestwasearnedoverthethreeyears? Supposethatyouinvest$2000inanaccountthatpays8%interestannually,compoundedmonthly.Howmuchmoneywouldyouhaveintheaccountafterthreeyears? Lookattheanswertoquestion4.Howmuchinterestwasearnedoverthethreeyears? Comparetheanswerstoquestions3and5.Whentheinterestrateisdoubledfrom4%to8%,Whathappenstotheamountofcompoundinterestearnedafterthreeyears? Exactlythesame Lessthandoubled Exactlydoubled Morethandoubled ViewSolutions Solutions \(1,000(1+\frac{0.09}{12})^{(12\times1)}=1,000(1.0075)^{12}=\$1,093.81\) $2254.54 $254.54 $2540.47(WrittenSolution) Inthissituation,wearefindingtheendingbalanceofanaccountthatpays8%annualinterestcompoundedmonthly,sowewillusetheformula: \({\text{A}}={\text{P}}\left(1+\frac{\text{r}}{\text{n}}\right)^{\text{nt}}\) Tofindtheamountintheaccountattheendofthreeyearsweneedthefollowing: P=Principal(beginningamount)=2000 r=Rate(annually)asadecimal=0.08 t=timeinyears=3 n=numberofcompoundingperiodsperyear=12(sincetheaccountiscompoundedmonthly. Weplugeachoftheaboveintoourformula: \(\text{A}={\color{MediumVioletRed}2000}\left(1+\frac{{\color{DodgerBlue}0.08}}{{\color{DarkOrange}12}}\right)^{{\color{DarkOrange}12}\cdot{\color{MediumSeaGreen}3}}\) Thendivide0.08by12toget: \(\text{A}=2000(1+{\color{DodgerBlue}{0.0066667}})^{12\cdot3}\) Thenadd1to0.006667: \({\text{A}}=2000({\color{DodgerBlue}{1.0066667}})^{12\cdot3}\) Thenmultiply12by3: \({\text{A}}=2000(1.0066667)^{\color{DodgerBlue}{36}}\) Thenraise1.0066667tothethirty-sixthpower: \(\text{A}=2000\left({\color{DodgerBlue}1.270237}\right)\) Thenmultiply2000by1.270237: A=2540.47 Sotheaccountwillhave$2,540.47attheendof3years $540.47(WrittenSolution) Westartedwith$2,000andattheendof3yearswehad$2,540.47.Thedifferencebetweenthetwois$540.47sothatishowmuchweearnedininterest. D.Morethandoubled.Ifwedoubletheinterestearnedat4%weget$254.54×2=$509.08.At8%,weearn$540.47ininterest.Theinterestat8%ismorethandoubletheamountearnedat4%.