Step-by-Step Business Math and Statistics

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1 Step-by-Step Busness Math and Statstcs By Jn W. Cho Included n ths prevew: Copyrght Page Table of Contents Excerpt of Chapter For addtonal nformaton on adoptng ths book for your class, please contact us at x0 or va e-mal at nfo@cognella.com

2 Step-by-Step Busness Math and Statstcs Jn W. Cho DePaul Unversty

3 Copyrght 0 by Jn W. Cho. All rghts reserved. No part of ths publcaton may be reprnted, reproduced, transmtted, or utlzed n any form or by any electronc, mechancal, or other means, now known or hereafter nvented, ncludng photocopyng, mcroflmng, and recordng, or n any nformaton retreval system wthout the wrtten permsson of Unversty Readers, Inc. Frst publshed n the Unted States of Amerca n 0 by Unversty Readers, Inc. Trademark Notce: Product or corporate names may be trademarks or regstered trademarks, and are used only for dentfcaton and explanaton wthout ntent to nfrnge. Prnted n the Unted States of Amerca ISBN:

4 Contents Acknowledgments v Part. Busness Mathematcs Chapter. Algebra Revew Chapter. Calculus Revew Chapter. Optmzaton Methods 67 Chapter. Applcatons to Economcs 8 Part. Busness Statstcs Chapter. Introducton 08 Chapter. Data Collecton Methods Chapter. Data Presentaton Methods Chapter. Statstcal Descrptve Measures Chapter. Probablty Theory 7 Chapter 6. Dscrete Probablty Dstrbutons 79 Chapter 7. The Normal Probablty Dstrbuton 9 Chapter 8. The t-probablty Dstrbuton 8 Chapter 9. Samplng Dstrbutons 8 Chapter 0. Confdence Interval Constructon 9 Chapter. One-Sample Hypothess Testng 6 Chapter. Two-Sample Hypothess Testng Chapter. Smple Regresson Analyss Chapter. Multple Regresson Analyss 8 Chapter. The Ch-Square Test Appendx: Statstcal Tables 8 Subject Index 7

5 Acknowledgments I would lke to thank many professors who had used ths book n ther classes. Especally, Professors Bala Batava, Burhan Bner, Seth Epsten, Teresa Kler, Jn Man Lee, Norman Rosensten, and Cemel Selcuk had used prevous edtons of ths book n teachng GSB0 Appled Quanttatve Analyss at DePaul Unversty. Ther comments and feedbacks were very useful n makng ths edton more user-frendly. Also, I would lke to thank many current and past DePaul Unversty s Kellstadt Graduate School of Busness MBA students who studed busness mathematcs and statstcs usng the framework lad out n ths book. Ther comments and feedbacks were equally mportant and useful n makng ths book an excellent gude nto the often-challengng felds of mathematcs and statstcs. I hope and wsh that the knowledge ganed va ths book would help them succeed n ther busness endeavors. As s often the case wth equatons and numbers, I am sure ths book stll has some errors. If you fnd some, please let me know at jcho@depaul.edu. Best wshes to those who use ths book. Jn W. Cho, Ph.D. Kellstadt Graduate School of Busness DePaul Unversty Chcago, IL 6060 jcho@depaul.edu Acknowledgments v

6 Math. Chapter. Algebra Revew Part. Busness Mathematcs There are chapters n ths part of busness mathematcs: Algebra revew, calculus revew, optmzaton technques, and economc applcatons of algebra and calculus. A. The Number System Chapter. Algebra Revew The number system s comprsed of real numbers and magnary numbers. Real numbers are, n turn, grouped nto natural numbers, ntegers, ratonal numbers, and rratonal numbers.. Real Numbers = numbers that we encounter everyday durng a normal course of lfe the numbers that are real to us.. Natural numbers = the numbers that we often use to count tems countng trees, apples, bananas, etc.:,,,, a. odd numbers:,,, b. even numbers:,, 6,. Integers = whole numbers wthout a decmal pont: 0, +, +, +, +,. a. postve ntegers:,,,, b. negatve ntegers:,,,,. Ratonal numbers = numbers that can be expressed as a fracton of ntegers such as a/b (= a b) where both a and b are ntegers a. fnte decmal fractons: /, /, etc. b. (recurrng or perodc) nfnte decmal fractons: /, /9, etc. v. Irratonal Numbers = numbers that can NOT be expressed as a fracton of ntegers = nonrecurrng nfnte decmal fractons: a. n-th roots such as,, 7, etc. b. specal values such as (=p), or e (=exponental), etc. Chapter : Algebra Revew

7 Math. Chapter. Algebra Revew v. Undefned fractons: a. any number that s dvded by a zero such as k/0 where k s any number b. a zero dvded by a zero = 0/0 c. an nfnty dvded by an nfnty = d. a zero dvded by an nfnty = 0 v. Defned fractons: a. a one dvded by a very small number 0 0 0,000,000, 000 a very large number such as a number that can approach b. a one dvded by a very large number /(a large number) = a small number 0 c. a scentfc noton the use of exponent.e+ =. x 0 =..E+6 =. x 0 6 =,,000.E =. x 0 - = E 6 =. x 0-6 = ,000,000 Smlarly, a caret (^) can be used as a sgn for an exponent: n = ^n 0 = ^0 Note: For example, E+6 means move the decmal pont 6 dgts to the rght of the orgnal decmal pont whereas E-6 means move the decmal pont 6 dgts to the left of the orgnal decmal pont. Step by Step Busness Math and Statstcs

8 Math. Chapter. Algebra Revew. Imagnary Numbers = numbers that are not easly encountered and recognzed on a normal course of lfe and thus, not real enough (or magnary) to an ndvdual. Often exsts as a mathematcal concepton. () = B. Rules of Algebra. a b b a + = +. ab ba x = x 6. aa for a 0 x - = 0 =. a( b c) ab ac x ( + ) = x + x. a ( a) a ( a) 0 + ( ) = (+) = = 0 6. ( a) b a( b) ab ( ) x = x ( ) 6 7. ( a)( b) ab ( ) x ( ) = x ( a b b) a ab ( + ) = + ()() + ( a b b) a ab ( ) = ()() + ( a b b)( a b) a ( + )( ) = a. ( a) /( b) a / b b ( ) /( ) /. a b a b a a b b () /( ). b ac b a c c. a c ad bc b d bd Chapter : Algebra Revew

9 Math. Chapter. Algebra Revew. b b ab a a c c c 6 6. a b c d a c a d ad b d b c bc 0 / a a a / 0. where a a /n = n a where a 0 / = ab = a * b = * a a 0.86 b b. a b a b a. b a b C. Propertes of Exponents Pay attenton to equvalent notatons It s very mportant that we know the followng propertes of exponents:. 0 Note that 0 0 = undefned. b b = ^ ( b) 0 0 = ^ ( 0). a b a b a b ab * ^ ( a b) * 7 = 8. a b a* b ab ab ( ) ^ ab * 6 ( ) Step by Step Busness Math and Statstcs

10 Math. Chapter. Algebra Revew. a b a ( ) = 096 b ab 6. ( Y) a a * Y a a Y a a Y a ( Y) * Y Y Y 7. n n / n / ( ) 8. p / q ( / q ) p ( ) p / q q p 0/ / 0 0 / 0 ( ) ( ) = = 8 / ( ) / 8 6 D. Lnear and Nonlnear Functons. Lnear Functons Lnear Functons have the general form of: Y = a + b where Y and are varables and a and b are constants. More specfcally, a s called an ntercept and b, a slope coeffcent. The most vsually dstngushable character of a lnear functon s that t s a straght lne. Note that +b means a postve slope and b means a negatve slope.. Nonlnear Functons There are many dfferent types of nonlnear functons such as polynomal, exponental, logarthmc, trgonometrc functons, etc. Only polynomal, exponental and logarthmc functons wll be brefly explaned below. ) The n-th degree polynomal functons have the followng general form: Chapter : Algebra Revew

11 Math. Chapter. Algebra Revew Y a b c d Or alternatvely expressed as:... p n q n n n Y q p... d c b a where a, b, c, d,, p and q are all constant numbers called coeffcents and n s the largest exponent value. Note that the n-th degree polynomal functon s named after the hghest value of n. For example, when n =, t s most often called a quadratc functon, nstead of a second-degree polynomal functon, and has the followng form: Y a b c When n =, t s called a thrd-degree polynomal functon or a cubc functon and has the followng form: Y a b c d ) Fndng the Roots of a Polynomal Functon Often, t s mportant and necessary to fnd roots of a polynomal functon, whch can be a challengng task. An n-th degree polynomal functon wll have n roots. Thus, a thrd degree polynomal functon wll have roots and a quadratc functon, two roots. These roots need not be always dfferent and n fact, can have the same value. Even though fndng roots to hgher-degree polynomal functons s dffcult, the task of fndng the roots of a quadratc equaton s manageable f one reles on ether the factorng method or the quadratc formula. If we are to fnd the roots to a quadratc functon of: a b c 0 we can fnd ther two roots by usng the followng quadratc formula:, b b ac a ) Examples: Fnd the roots, and, of the followng quadratc equatons: (a) 0 6 Step by Step Busness Math and Statstcs

12 Math. Chapter. Algebra Revew Factorng Method : ( ) ( ) 0 Therefore, we fnd two roots as: = and =. Quadratc Formula : Note: a, b, and c, b b ac a ( ) ( ) ()() 9 8 =, (b) 6 0 Factorng Method: 6 ( 6) ( 6) ( 6) ( ) 0 Therefore, we fnd two dentcal roots (or double roots) as: Quadratc Formula: Note: a, b, and c 6, b b ac a () () ()(6) The factorng method often seems more convenent for people wth great experence wth algebra. That s, the easness comes wth experence. Those who lack algebrac skll may be better off usng the quadratc formula. In order to use the quadratc formula successfully, one must match up the values for, and c correctly. a, b Chapter : Algebra Revew 7

13 Math. Chapter. Algebra Revew = (c) 9Y 0 Factorng Method: 9Y ( Y ) ( Y ) 0 Therefore, we fnd two roots as: Y Y. Y and. Y Quadratc Formula : Note: a, b 0, and c 9Y, b b ac Y a (0) (0) ()( 9 ) 0 0 Y Y = Y.Y,.Y 8 8 E. Exponental and Logarthmc Functons. Exponental Functons An exponental functon has the form of Y a b where a and b are constant numbers. The smplest form of an exponental functon s Y b where b s called the base and s called an exponent or a growth factor. A unque case of an exponental functon s observed when the base of e s used. That s, Y e where e Because ths value of e s often dentfed wth natural phenomena, t s called the natural base. One must be very cognzant of the construct of ths quadratc equaton. Because we are to fnd the roots assocated wth, 9Y should be consdered as a constant term, lke c n the quadratc equaton. n Techncally, the expresson n approaches e as n ncreases. That s, as n approaches, e Step by Step Busness Math and Statstcs

14 Math. Chapter. Algebra Revew Examples> In order to be famlar wth how exponental functons work, please verfy the followng equaltes by usng a calculator. 6 a. e e e e b. (e ) (e ) e c. 0e e e e e Logarthmc Functons The logarthm of Y wth base b s denoted as log log Y f and only f b Y b b Y and s defned as: provded that b and Y are postve numbers wth b. The logarthm enables one to fnd the value of gven or. In both of these cases, we can easly fnd = due to the smple squarng process nvolved. However, fndng n s not easy. Ths s when knowng a logarthm comes n handy. Examples> Convert the followng logarthmc functons nto exponental functons: log 8 8 log 0 0 log log / = a. Specal Logarthms: A common logarthm and a natural logarthm. ) A Common Logarthm = a logarthm wth base 0 and often denoted wthout the base value. That s, log 0 log read as "a (common) logarthm of." ) A Natural Logarthm = a logarthm wth base e and often denoted as ln. Chapter : Algebra Revew 9

15 Math. Chapter. Algebra Revew That s, log ln read as "a natural logarthm of." e b. Propertes of Logarthms ) Product Property: log mn log m log n m ) Quotent Property: logb logb m logb n n n ) Power Property: log m n log m b b b b b Example > Usng the above propertes of logarthm, verfy the followng equalty or nequalty by usng a calculator. ) ln0 ln( 6) ln ln ) ln ln 0 ln 0 ln ) ln 0 0 ln ln 0 0 v) ln0 ln0 ln Example > Fnd n. (Ths soluton method s a bt advanced.) In order to fnd, () we can take a natural (or common) logarthm of both sdes as: ln ln () rewrte the above as: ln ln by usng the Power Property ln () solve for as: ln () use the calculator to fnd the value of as: ln ln Step by Step Busness Math and Statstcs

16 Math. Chapter. Algebra Revew Addtonal topcs of exponental and logarthmc functons are complcated and requre many addtonal hours of study. Because t s beyond our realm, no addtonal attempt to explore ths topc s made heren. F. Useful Mathematcal Operators. Summaton Operator = Sgma = n n n n n... = Sum s where goes from to n. Examples: Gven the followng data, verfy the summaton operaton. = = a b c d. ) ( ) ( ( 9 6) (6 7 ) e. ) ( ) ( ( 9 6) (6 7 ) Multplcaton Operator = p = n n n n... = Multply s where goes from to n. For detaled dscussons and examples on ths topc, please consult hgh school algebra books such as Algebra, by Larson, Boswell, Kanold, and Stff. ISBN=: Chapter : Algebra Revew

17 Step by Step Busness Math and Statstcs Math. Chapter. Algebra Revew Examples: Gven the followng data, verfy the multplcaton operaton. a b c. 8 6 d. ) ( ) ( ) (6 6) ( e. ) ( ) ( ) ( 6) ( f. ) ( ) ( ) (6 ) ( G. Multple-Choce Problems for Exponents, Logarthms, and Mathematcal Operators: Identfy all equvalent mathematcal expressons as correct answers.. ( + Y) = a. + Y + Y b. Y + Y c. + Y + Y d. + Y + Y e. none of the above = = 6

18 Math. Chapter. Algebra Revew. ( Y) = a. + Y + Y b. ( Y) ( Y) c. Y + Y d. Y + Y e. only (b) and (c) of the above. ( + Y) = a. + 6Y + 9Y b. + Y + 9Y c. + 6Y + Y d. + 9Y e. none of the above. ( Y) = a. 9Y b. + 6Y + Y c. Y + 9Y d. + 9Y e. none of the above. ( )(6 0 ) = a. +0 b. 0 c. 8 /0 d. e. none of the above 6. ( 6 Y )(Y )( Y ) = a. 7 Y 9 b. 7 Y 8 c. 7 0 Y 0 d. 7Y 8 e. only (b) and (d) of the above 7. ( + Y) = Chapter : Algebra Revew

19 Math. Chapter. Algebra Revew a. ( + Y + Y ) b Y + Y c. + Y + Y d. all of the above e. none of the above 8. 6 = a. c. b. d. e. none of the above 9. ( )/(6 0 ) = a. c b. 7 d. only (a) and (c) of the above e. all of the above Y a. c. Y = Y Y b. 9 Y Y d. only (a) and (b) of the above e. all of the above. Y Y = a. d. Y Y b. e. Y Y c. Y Step by Step Busness Math and Statstcs

20 Math. Chapter. Algebra Revew.. ( 6 ) (8Y ) 8 Y = a. Y b. Y c. Y Y d. e. none of the above. ( ) = a. 6 b. 8 6 c. 8 d. 6 6 e. ( ). [( Y ) ] = a. 9 8 Y 7 b. 9 6 Y c. 8 6 Y 0 d. 8 6 Y e. 8 8 Y 7. ( Y ) /( Y ) = a. Y - b. Y c. /Y d. e. / Usng the followng data, answer Problems 6 0. = 6 7 = a. 6 b. 0 c. 9 d. 00 e. none of the above Chapter : Algebra Revew

21 Math. Chapter. Algebra Revew 6 7. a. 6 b. 80 c. 0 d. 78 e. none of the above 8. a. b. 0 c. d. 0 e. none of the above 7 9. a. 9 b. 8 c. 676 d. 6 e. none of the above a. 0 b. 8 c. 89 d. 67 e. 8. Fnd the value of n 909. a. 0 b. c. 0 d. e. none of the above. Identfy the correct relatonshp(s) shown below: a. log0 log0 b. c. ln ln ln d. lny ln lny e. none of the above s correct. Answers to Exercse Problems for Exponents and Mathematcal Operators. ( + Y) = a.* + Y + Y because ( + Y) ( + Y) = + Y + Y + Y = + Y + Y 6 Step by Step Busness Math and Statstcs

22 Math. Chapter. Algebra Revew. ( Y) = e.* only (b) and (c) of the above because ( Y) ( Y) = Y Y + Y = Y + Y. ( + Y) = b.* + Y + 9Y because ( + Y) ( + Y) = + 6Y + 6Y + 9Y = + Y + 9Y. ( Y) = c.* Y + 9Y because ( Y) = ( Y) ( Y) = 6Y 6Y + 9Y = Y + 9Y. ( )(6 0 ) = d.* because ()(6) +0 = +0 = 6. ( 6 Y )(Y )( Y ) = a.* 7 Y 9 because ()()() 6++ Y ++ = 7 Y 9 7. ( + Y) = d.* all of the above because ( + Y + Y ) = Y + Y = + Y + Y 6 8. = b.* because 6 9. ( )/(6 0 ) = e.* all of the above because (/6) -0 = (/) -7 7 = Y d.* Y = only (a) and (b) of the above because Chapter : Algebra Revew 7

23 Math. Chapter. Algebra Revew Y Y 0 Y 0 Y Y. Y Y = a.* Y because 0.. Y Y Y Y. 0. Y.. ( 6 ) (8Y ) 8 Y = a.* Y because (6 ) 8. (8Y ) Y (6) 8. (8) Y Y = (8) 8 () Y. Y. () Y Y Y. ( ) = b.* 8 6 because ( ) ( ) ( ) = () ++ = x = 8 6. [( Y ) ] = d.* 8 6 Y because [ x Y x ] = x 8x Y 6x = 8 6 Y. ( Y ) /( Y ) = c.* /Y because ( Y ) ( Y ) - = [( ) 8 Y 6 ][() - -8 Y -8 ]= Y - = /Y c.* 9 because d.* 78 because Step by Step Busness Math and Statstcs

24 Math. Chapter. Algebra Revew e.* 7 none of the above because ( ) ( ) ( 0 ) (67 6) c.* 676 because 7 ( ) ( ) 6 7 ( 6 ) (67 6) e.* 8 because ( ) ( ) ( ). Fnd the value of n 909. c.* ( ) (67 ) (6 ) In order to fnd, () we can take a natural (or common) logarthm of both sdes as: ln ln909 () rewrte the above as: ln ln 909 by usng the Power Property ln909 () solve for as: ln () use the calculator to fnd the value of as: ln ln Identfy the correct relatonshp(s) shown below: e.* none of the above s correct. Note that Chapter : Algebra Revew 9

25 Math. Chapter. Algebra Revew log 0 log 0 log b. () a. 0 c. d. ln ln( ) ln ln lny ln ln lny Y H. Graphs In economcs and other busness dscplnes, graphs and tables are often used to descrbe a relatonshp between two varables and Y. s often represented on a horzontal axs and Y, a vertcal axs.. A Postve-Slopng Lne and a Negatve-Slopng Lne For example, a functon of Y = + 0., as plotted below, has an ntercept of and a postve slope of +0.. Therefore, t rses to the rght (and declnes to the left) and thus, s characterzed as a postve slopng or upward slopng lne. It shows a pattern where as ncreases (decreases), Y ncreases (decreases). Ths relatonshp s also known as a drect relatonshp. 0 y y= x - -0 On the other hand, a functon of Y = 0. as plotted below, has an ntercept of and a negatve slope of 0.. Therefore, t declnes to the rght (and rses to the left) and thus, s characterzed as a negatve slopng or downward slopng lne. It shows a pattern where as ncreases (decreases), Y decreases (ncreases). That s, because and Y move n an opposte drecton, t s also known as an ndrect or nverse relatonshp. 0 Step by Step Busness Math and Statstcs

26 Math. Chapter. Algebra Revew y=-0. 0 y -0-0 x -. Shfts n the Lnes -0 Often, the lne can move up or down as the value of the ntercept changes, whle mantanng the same slope value. When the followng two equatons are plotted n addton to the orgnal one we plotted above, we can see how the two lnes dffer from the orgnal one by ther respectve ntercept values: Orgnal Lne: Y = + 0. New Lne #: Y = New Lne #: Y = + 0. The mddle lne The top lne The bottom lne Y 0 Y=6+0. Y=+0. Decrease Increase Y= Note : Note : As the ntercept term ncreases from to 6, the mddle lne moves up to become the top lne. Ths upward shft n the lne ndcates that the value of has decreased whle the Y value was held constant (or unchanged). Thus, the upward shft s the same as a shft to the left and ndcates a decrease n gven the unchanged (or same) value of Y. As the ntercept term decreases from to, the mddle lne moves down to become the bottom lne. Ths downward shft n the lne ndcates that the value of has ncreased whle the Y value was held Chapter : Algebra Revew

27 Math. Chapter. Algebra Revew constant (or unchanged). Thus, the downward shft s the same as a shft to the rght and ndcates an ncrease n gven the unchanged (or same) value of Y. Note : Ths observaton s often utlzed n the demand and supply analyss of economcs as a shft n the curve. A leftward shft s a "decrease" and a rghtward shft s an "ncrease.". Changes n the Slope When the value of a slope changes, holdng the ntercept unchanged, we wll note that the lne wll rotate around the ntercept as the center. Let s plot two new lnes n addton to the orgnal lne as follows: Orgnal Lne: Y = + 0. New Lne #: Y = + New Lne #: Y = + 0 = New Lne #: Y = 0. The orgnal (=mddle) lne The top lne The flat lne The bottom lne y 0 y=+ Steep slope y=+0. y= Flat Slope -0-0 x - y=-0. Negatve Slope -0 Note that the steepness (or flatness) of the slope as the value of the slope changes. Lkewse, note the relatonshp among a flat, a postve, and a negatve slope. I. Applcatons: Compound Interest Step by Step Busness Math and Statstcs

28 Math. Chapter. Algebra Revew. The Concept of Perodc Interest Rates Assume that the annual percentage rate (APR) s ( r 00 )%. That s, f an APR s 0%, then r = 0.. Also, defne FV = future value, PV = present value, and t = number of years to a maturty. ) Annual compoundng for t years ) Semannual compoundng for t years ) Quarterly compoundng for t years v) Monthly compoundng for t years v) Weekly compoundng for t years v) Daly compoundng for t years v) Contnuous compoundng for t years 6 t FV PV ( r) r t FV PV ( ) r t FV PV ( ) r t FV PV ( ) r t FV PV ( ) r FV PV ( ) 6 rt FV PV e 6t Examples> Assume that $00 s deposted at an annual percentage rate (APR) of % for year. ) Annual compoundng one -year depost nterest calculaton FV PV ( r) t $00 ( 0.) $.00 ) Semannual compoundng two ½-year deposts nterest calculatons n year FV r PV ( ) t 0. $00 ( ) $00 ( 0.06) $.6 ) Quarterly compoundng four ¼-year deposts nterest calculatons n year FV r PV ( ) t 0. $00 ( ) $00 ( 0.0) $. 6 Do you remember that ths s an exponental functon wth a natural base of e? Chapter : Algebra Revew

29 Math. Chapter. Algebra Revew v) Monthly compoundng twelve /-year deposts nterest calculatons n year FV r PV ( ) t 0. $00 ( ) $00 ( 0.0) $.68 v) Weekly compoundng ffty-two /-year deposts nterest calculatons n year FV r PV ( ) t 0. $00 ( ) $00 ( ) $.7 v) Daly compoundng 6 /6-year deposts 6 nterest calculatons n year FV r PV ( ) 6 6t $00 ( 0. ) 6 6 $00 ( ) 6 $.7 v) Contnuous compoundng for year contnuous nterest calculatons FV PV e rt $00 e 0. $00 e 0. $.7 Examples> Calculate the annual rate of return (ROR) based on the varous compoundng schemes shown above. ) For annual compoundng, ROR = P P 0 00 =0. % 00 0 P ) For sem-annual compoundng, ROR = P P =0.6.6% 00 0 P ) For monthly compoundng, Step by Step Busness Math and Statstcs

30 Math. Chapter. Algebra Revew ROR = P P = % 00 0 P Note: The rate of return on an annual bass s known as the Annual Percentage Yeld (APY). Even though APR may be the same, APY wll ncrease as the frequency of compoundng ncreases because an nterest s earned on an nterest more frequently.. Annuty Calculaton Annuty Formulas: FV n A[( ) ] PV n A[( ) ] n ( ) Examples where A= the fxed annuty amount; n = the number of perods; and = a perodc nterest rate. Of course, FV = the future (or fnal or termnal) value and PV = the present (or current) value. ) If you obtan a 0 year mortgage loan of $00,000 at an annual percentage rate (APR) of 6%, what would be your monthly payment? Answer: 00, A[( ) ( ) x0 ] x0 Therefore, A=$99. ) If you nvest $,000 a month n an account that s guaranteed to yeld a 0% rate of return per year for 0 years (wth a monthly compoundng), what wll be the balance at the end of the 0-year perod? Answer: Chapter : Algebra Revew

31 Math. Chapter. Algebra Revew FV 0. $,000 [( ) ] $,60,87.9 ) If you are guaranteed of a 0% rate of return for 0 years, how much should you save and nvest each month to accumulate $ mllon at the end of the 0-year perod? Answer: $,000, A[( ) ] Therefore, A=$.8 v) Suppose that you have saved up $00,000 for your retrement. You expect that you can contnuously earn 0% each year for your $00,000. If you know that you are gong to lve for addtonal years from the date of your retrement and that the balance of your retrement fund wll be zero at the end of the -year perod, how much can you wthdraw to spend each month? Answer: $00, A[( ) ] ( ) Therefore, A=$,07.6 v) Assume the same stuaton as Problem above, except that now you have to ncorporate an annual nflaton rate of %. What wll be the possble monthly wthdrawal, net of nflaton? Answer: $00, A[( ) ] ( ) 80 6 Step by Step Busness Math and Statstcs

32 Math. Chapter. Algebra Revew Therefore, A=$898.8 Note: Combnng Answers to Problems (v) and (v) above, t means that you wll be actually wthdrawng $,07.6 per month but ts purchasng power wll be equvalent to $ Ths s because nflaton only erodes the purchasng power; t does not reduce the actual amount receved. If one goes through a professonal fnancal plannng, the fnancal planner wll expand on ths smple assumpton to a more complex and realstc scenaro. v) Assumng only annual compoundng, how long wll t take to double your nvestment f you earn 0% per year? Answer 7 : A ( 0.). x x A Now, take the natural logarthm of both sdes as follows: x ln. ln ln. ln ln 7.7years ln. v) Assumng monthly compoundng, how long wll t take to double your nvestment f you earn 0% per year? Answer: 0. A ( ) x.008 ln.008 ln x ln.008 A x ln ln ln months 6.96years 7 When ether the natural logarthm or the common logarthm s taken, the exponent,, as n ths case, wll become a coeffcent as shown heren. Then, use the calculator wth a ln functon to complete the calculaton. Chapter : Algebra Revew 7

33 Math. Chapter. Algebra Revew v) Assume that you have a 0-year, $00,000 mortgage loan at an annual percentage rate (APR) of 6%. How long wll t take you to pay off ths loan f you pay off $,000 a month? Answer: Use the nformaton on Answers to Problem as follows: 0.06,000 [( ) ] 00, ( ) Therefore, ,000 ( ),000 [( ) ] ( ) 000 ( ) (.00) 000 ln.00 ln ln 8.97months.8years ln.00 J. Inequaltes. If a > 0 and b > 0, then (a+b) > 0 and ab > 0 If a=7 and b=, then (7+) > 0 and (7)() > 0. If a > b, then (a b) > 0 If a=7 and b=, then (7 ) > 0. If a > b, then (a+c) > (b+c) for all c If a=7 and b=, then (7+c) > (+c) 7 >. If a > b and c > 0, then ac > bc If a=7 and b= and c=, then (7)() > ()() > 8 Step by Step Busness Math and Statstcs

34 Math. Chapter. Algebra Revew. If a > b and c < 0, then ac < bc If a=7 and b= and c=, then (7)( ) < ()( ) < K. Absolute Values and Intervals. f 0 and f 0 Examples> a. + = + = and = ( ) = + = b. +0 = 0 and 0 = ( 0) = 0. If n, then n n Examples> a. If <, then < < b. If <, then < < + < < + < < 7 c. If + < 0, then 0 < + < 0 < < 6 7 < <. If n, then n f > 0 or n f < 0 Note that when a negatve number s multpled to both sdes of the nequalty sgn, the drecton of the nequalty sgn reverses. Examples> a. If >, then, > or > < b. If >, then, ( ) > > 8 or ( ) > ( ) < < Chapter : Algebra Revew 9

35 Math. Chapter. Algebra Revew c. If 6 >, then, (6 ) > > 6 < or (6 ) > (6 ) < < 8 > 6 L. A System of Lnear Equatons n Two Unknowns Gven the followng system of lnear equatons, solve for and Y. Y Y. Soluton Method : The Substtuton Method () Rearrange the bottom equaton for as follows: Y Y () Substtute ths nto the top equaton as follows: (Y ) Y 8Y 6 Y () Substtute ths Y nto any of the above equaton for value: Y () Verfy f the values of and Y satsfy the system of equatons: Y Y () Verfcaton completed and solutons found.. Soluton Method : The Elmnaton Method () Match up the varables as follows: 0 Step by Step Busness Math and Statstcs

36 Math. Chapter. Algebra Revew Y Y () Multply ether of the two equatons to fnd a common coeffcent. (Y s chosen to be elmnated and thus, the top equaton s multpled by as follows:) Y 6 Y 6 () Subtract the bottom equaton from the adjusted top equaton n () above and obtan: 6 Y 6 Y or 6 Y 6 Y 6 ( ) Y Y 6 or 6 Y Y () Substtute ths nto any of the above equaton for Y value: 6 Y 6 6Y 6 Y 68 Y () Verfy f the values of and Y satsfy the system of equatons: Y Y (6) Verfcaton completed and solutons found.. An Example Suppose that you have $0 wth whch you can buy apples (A) and oranges (R). Also, assume that your bag can hold only tems such as apples, or oranges, or some combnaton of apples and oranges. If the apple prce s $ and the orange prce s $0.0, how many apples and oranges can you buy wth your $0 and carry them home n your bag? Chapter : Algebra Revew

37 Math. Chapter. Algebra Revew Answer: The Substtuton Method: () dentfy relevant nformaton: Budget Condton: A + 0.R = 0 Bag-Sze Condton: A + R = () convert the Bag-Sze Condton as: A = R () substtute A = R nto the Budget Condton as: ( R) + 0.R = 0 0.R = R= () substtute R= nto () above and fnd: A = = 8 () verfy the answer of A=8 and R= by pluggng them nto the above two condtons as: Budget condton: () = 0 Bag-Sze condton: 8 + = Because both condtons are met, the answer s A=8 and R=. The Elmnaton Method: () dentfy relevant nformaton: Budget Condton: A + 0.R = 0 Bag-Sze Condton: A + R = () subtract the bottom equaton from the top: 0.R = R= () plug ths R= nto ether one of the two condtons above: A + 0.() = 0 A=8 Step by Step Busness Math and Statstcs

38 Math. Chapter. Algebra Revew Or A + () = A=8 () verfy the answer of A=8 and R= by pluggng them nto the above two condtons as: Budget condton: () = 0 Bag-Sze condton: 8 + = Because both condtons are met, the answer s A=8 and R=.. Solve the followng smultaneous equatons by usng both the substtuton and elmnaton methods: a. 0 + Y = 80 0Y 9 = 0 b. + 7 = Y Y + 7 = Answer: =0 and Y=0 Answer: = and Y= c. (/) (/)Y = 7. Y = 0 Answer: =0 and Y=0 Note that there s no way of tellng whch soluton method the substtuton or the elmnaton s superor to the other. Even though the elmnaton method s often preferred, t s the experence and preference of the solver that wll decde whch method would be used. M. Examples of Algebra Problems. For your charty organzaton, you had served 00 customers who bought ether one hot dog at $.0 or one hamburger at $.0, but never the two together. If your total sales of hot dogs (HD) and hamburgers (HB) were $60 for the day, how many hot dogs and hamburgers dd you sell?. You are offered an dentcal sales manager job. However, Company A offers you a base salary of $0,000 plus a year-end bonus of % of the gross sales you make for that year. Company B, on the other hand, offers a base salary of $,000 plus a year-end bonus of % of the gross sales you make for that year. a. Whch company would you work for? Chapter : Algebra Revew

39 Math. Chapter. Algebra Revew b. If you can acheve a total sale of $,000,000 for ether A or B, whch company would you work for?. A ftness club offers two aerobcs classes. In Class A, 0 people are currently attendng and attendance s growng people per month. In Class B, 0 people are regularly attendng and growng at a rate of people per month. Predct when the number of people n each class wll be the same.. Everybody knows that Dr. Cho s the best nstructor at DePaul. When a student n GSB 0 asked about the mdterm exam, he sad the followng: a. The mdterm exam wll have a total of 00 ponts and contan problems. Each problem s worth ether ponts or ponts. Now, you have to fgure out how many problems of each value there are n the mdterm exam. b. The mdterm exam wll have a total of 08 ponts and there are twce as many -pont problems than -pont problems. Each problem s worth ether ponts or ponts. Now, you have to fgure out how many problems of each value there are n the mdterm exam.. Your boss asked you to prepare a company party for 0 employees wth a budget of $00. You have a choce of orderng a steak dnner at $0 per person or a chcken dnner at $ per person. (All tps are ncluded n the prce of the meal.) a. How many steak dnners and chcken dnners can you order for the party by usng up the budget? b. How many steak dnners and chcken dnners can you order for the party f the budget ncreases to $0? 6. Your father just receved a notce from the Socal Securty Admnstraton sayng the followng: If you retre at age 6, your monthly socal securty payment wll be $00. If you retre at age 66, your monthly socal securty payment wll be $00. a. Your father s askng you to help decde whch opton to take. What would you tell hm? Do not consder the tme value of money. (Hnt: Step by Step Busness Math and Statstcs

40 Math. Chapter. Algebra Revew Calculate the age at whch the socal securty ncome receved wll be the same.) b. The Socal Securty Admnstraton has gven your father one more opton as: If you retre at age 70, your monthly socal securty payment wll be $800. What would you now tell hm? Do not consder the tme value of money. (Hnt: Calculate the age at whch the socal securty ncome receved wll be the same.) Answers to Above Examples of Algebra Problems. Quantty Condton: HD + HB = 00 Sales Condton:.0 HD +.0 HB = 60 Solvng these two equatons smultaneously, we fnd HD* = 00 and HB* = 00.a. We have to dentfy the break-even sales (S) for both companes as follows: Therefore, Compensaton from A = $0, S Compensaton from B = $, S Compensaton from A = Compensaton from B $0, S = $, S S* = $600,000 Concluson: If you thnk you can sell more than $600,000, you had better work for B. Otherwse, work for A..b. Snce you can sell more than $600,000, such as $ mllon, work for B and possbly realze a total compensaton of $,000 (=$, x ($ mllon)). If you work for A, you would receve $0,000 (=$0, x ($ mllon)).. Attendance n A = 0 + (Months) Attendance n B = 0 + (Months) Attendance n A = Attendance n B Therefore, 0 + (Months) = 0 + (Months) Chapter : Algebra Revew

41 Math. Chapter. Algebra Revew Months* =.a. Total Ponts: + Y = 00 Number of Problems: + Y = where = the number of pont problems and Y = the number of pont problems. Therefore, *= and Y*=0.b. Total Ponts: + Y = 08 Number of Problems: = Y where = the number of pont problems and Y = the number of pont problems. Therefore, *=9 and Y*=8.a. Total Number of Employees: S + C = 0 Budget: 0S + C = 00 where S = number of steak dnner and C = chcken dnner. Therefore, C*=0 and S*=0.b. Total Number of Employees: S + C = 0 Budget: 0S + C = 0 where S = number of steak dnner and C = chcken dnner. Therefore, C*=0 and S*=0 6.a. Total Recept between 6 and = ( 6)*00* Total Recept between 66 and = ( 66)*00* Total Recept between 6 and = Total Recept between 66 and That s, ( 6)*00* = ( 66)*00* Therefore, * = 76 That s, f your father can lve longer than 76 of age, he should start recevng the socal securty payment at 66 of age. Otherwse, he should retre at 6. 6.b. If the retrement decson s between 6 vs. 70: 6 Step by Step Busness Math and Statstcs

42 Math. Chapter. Algebra Revew ( 6)*00* = ( 70)*800* Therefore, * = 79. That s, f your father can lve longer than 79. of age, he should retre at 70 of age. Otherwse, he should retre at 6. If the retrement decson s between 66 vs. 70: ( 66)*00* = ( 70)*800* Therefore, * = 8 That s, f your father can lve longer than 8 of age, he should retre at 70 of age. Otherwse, he should retre at 66. Chapter : Algebra Revew 7