Usig Math to Uderstad Our World Project 5 Buildig Up Savigs Ad Debt Note: You will have to had i aswers to all umbered questios i the Project Descriptio See the What to Had I sheet for additioal materials to submit 1 Itroductio Durig the Jauary Workshop, oh so log ago, we talked about compoud iterest Recall that if you deposit a pricipal P ito a savigs accout with a iterest rate of r% compouded times a year, the amout of moey you have after the first compoudig period is P r P = P 1 r ) After two compoudig periods it is P 1 r ) P 1 r ) r ; that is, the previous balace, P 1 r ), plus the iterest o that balace, P 1 r ) r We could also write this as P 1 r ) 1 r ) = P 1 r ) What is the balace after k compoudig periods? Now suppose, as is ofte the case, that each compoudig period we make a deposit of amout v to the accout v for value) This is ofte how retiremet fuds work - you start with a iitial ivestmet ad the have a certai amout deducted from your paycheck each moth to add to it Of course, the deposits really may ot happe as ofte as the compoudig, but the math is easier if they do, so let s go with that for ow So suppose we start with a iitial ivestmet P, a iterest rate of r, ad suppose we add v to the accout at the ed of each compoudig period The at the ed of the first compoudig period we have a balace of P r P v = P 1 r ) v the pricipal plus the iterest plus v) It s a little bit easier if we let s stad for 1 r The we would write our first balace as P s v 1
1 Show that at the ed of the secod compoudig period, the balace is P s v)s v Multiply this out so that you have o paretheses remaiig Be sure to show all the steps What is the balace after the third compoudig period? Agai multiply it out Be sure to show all the steps Do you see a patter startig to emerge? Ca you use this patter to predict the balace after 10 compoudig periods? After 100 compoudig periods you do t have to write it all out)? After k compoudig periods? Fidig A Usable Formula The formula you got above is ot very usable I mea, if you wated to compute your balace at the ed of 40 years, it would be a real pai But there is a way to simplify this! 1 The Geometric Series The sum 1 z z z z k is part of what is kow as the geometric series, which is very famous i mathematics The etire geometric series is 1 z z z z k z k1 where we just keep goig o ad o forever, so you have a sum of a ifiite umber of thigs You might thik that such a sum would be ifiite, but, as log as z < 1, it s ot! For example, use your calculator to help you decide what the geometric series would be if z = 1/; that is, what is 1 1 1 ) ) 1 ) 4 1? The sum of 1 z z z z k is sometimes called the fiite geometric series There is actually a very simple formula for the sum of the geometric series ad for the sum of a fiite geometric series The formula is tricky to fid, especially for the ifiite series, but let s see if we ca figure it out for the fiite series First multiply the etire fiite series by z 1); ie, cosider 1 z z z z k) z 1) Multiply it out; that is, multiply each term i the series by z ad the multiply each term by 1 Is there some cacellig you ca do? What happes? Ca you use this to get a simple formula for the fiite geometric series itself?
1 Show your formula ad its step by step derivatio The fial form for your formula should be 1 z z z k = a ice expressio Use your formula to compute the followig 1 1 ) 1 1 ) What is 1 1 1 ) Verify this with your calculator ) 1 ) 4 1 ) 4 1 ) 16 1 ) 10 1? 4 This is ot immediately applicable to our project, but ca you make a guess as to the formula for the ifiite geometric series 1 z z z z k z k1 i the case whe z < 1? Why does z have to be less tha 1?) Try out your formula for 1 1 ) ) 4 1 1 1 ) How does this compare to your aswer i umber? Are the aswers exactly the same? a little bit differet? a lot differet? Explai The ifiite geometric series is oly tagetially related to our problem of computig balace, but it is so famous that I could t resist showig it to you Ad it s kid of cool that you ca add up a ifiite umber of umbers ad get a fiite aswer Now you ca aswer a famous puzzle that Archimedes wodered about Suppose a frog starts at poit A ad hops 1 uit to the right The she hops half a uit to the right, the she hops a fourth of a uit to the right Each hop is half the distace of the previous hop How far does she get i the ed or would she get if she lived forever)? OK, so ow we will stick to the fiite geometric series for the rest of the project Puttig it all Together 1 Use your formula for the fiite geometric series to write simple formulas for the expressios i problem from Part 1 the Itroductio) Suppose you start a accout with a iitial deposit of $1000 ad add $100 a moth where the iterest is compouded mothly at a iterest rate of 8% quite obtaiable from a stock portfolio, or at least it used to be) How much moey will you have after 0 years? How much moey have you put ito the accout over that time? What if you had t made the mothly deposits, but just made the iitial deposit ad let the iterest accumulate? That is, i this case, how much moey would you have after 0 years ad how much moey would you have made? Try ot to roud too much Whe you have large expoets, a little roudig ca make a big differece
Suppose you bega the savigs pla above whe you were 5 agai payig i $100 a moth) What would your balace be whe you retired at age 65? How much moey would you have paid i durig that time? 4 Suppose you bega the savigs pla above for your child at birth What would the balace be whe your child retired at age 65? How much moey would you have paid i? 5 I each of the scearios -4, how much would you have to deposit each moth to have a millio dollars i the ed? You ca read about these ad other formulas by goig to Start by clickig o compoud iterest Payig Out wwwmoeychimpcom Oce you reach retiremet age you ofte stop makig paymets ito your retiremet accout ad start withdrawig moey at regular itervals to support yourself At the same time, of course, you ll cotiue to ear iterest o whatever remais i the accout It s importat to figure out how much you ca take out of the accout ad still have eough to last for the rest of your life Suppose you start with a balace of R whe you retire, ad your iterest rate is still r, ad you take out a amout w at the begiig of each compoudig period 1 Fid a formula that shows how much moey you have left i the accout after k compoudig periods Your work will be very similar to what you did above Start by writig dow the balace after oe compoudig period, two compoudig periods, three compoudig periods Multiply thigs out ad see if you ca spot a patter You should get a sum z z z z i your formula which you ca simplify as you did above You might fid it useful to thik of z z z z as z1 z z z 1 )) If you start with $600,000 ad you take out $48,000 a year ad the iterest rate is 6%, compouded quarterly, how log will your retiremet savigs last? If you eed it to last 0 years, how much ca you take out each quarter? each year? 4 Credit Card Debt Now let s talk about credit card debt Suppose you curretly have a balace of P dollars o your credit card ad you pay off w dollars at the begiig of each moth Suppose the iterest rate is r, usually compouded mothly 1 Fid a formula that tells you the balace o your card after m moths provided you do t charge aythig else to the card i the meatime This will be very similar to what you did above 4
If your credit card balace is $10,000 ad your iterest rate is 14% compouded mothly) ad you pay off $00 a moth, how log will it take you to pay it off provided you do t charge aythig else i the meatime)? How much moey will you pay i total? What if you pay oly $100 a moth? What if you cotiue to charge to your card, say at a rate of $50 a moth? PS if your balace is $10,000 ad the iterest rate is 14% compouded mothly, how much iterest does the credit card compay charge you the first moth?) 5 The Natioal Debt It s quite complicated to discuss atioal debt correctly The govermet borrows from may may differet leders, each with a differet iterest rate But you ca imagie how quickly debt ca grow whe the govermet does ot pay off at least the accrued iterest each moth You ca check the curret atioal debt at wwwbrilligcom/debt_clock/ 6 Usig Up Oil Reserves I the Jauary Workshop we leared that world oil cosumptio is curretly growig at a rate of % per year I 000, world oil cosumptio was about 7,740,000,000 barrels The total amout believed to be i the earth is about 107 billio barrels 1 At this rate, how log will it be util we ru out of oil? 7 How May Grais Of Rice? Remember the Chiese emperor who was so take with the game of chess that he offered the ivetor aythig he wated? The ivetor asked for oe grai of rice o the first square of a chessboard, two o the secod, 4 o the third, 8 o the 4th, ad so o, doublig the amout each square A chess board has 64 squares 1 Ca you use the ideas i this project to develop a simple formula that will tell us how may grais of rice the emperor of Chia would have had to have give the ivetor of chess show your work)? What if he had oly had to go up to square istead? Albert Eistei called compoud iterest the 8th Woder of the World 5