UNIT 1: LINEAR EQUATIONS AND THEIR INTERPRETATIVE APPLICATIONS IN BUSINESS Contents
1.0 Aims and Objectives 1.1 Introduction 1.2 Linear equations 1.2.1 Developing equation of a line 1.2.2 Special formats 1. Application of linear equations of linear 1..1 Linear cost!output relations"ips 1..2 #rea$!even anal%sis 1.& 'odel e(amination questions 1.0AIMS AND OBJECTIVES
After reading t"e c"apter students must be able a ble to) •
define algebraic e(pression* equation + linear equation define
•
e(plain t"e different ,a%s of formulating or developing equations of a e(plain
line
•
understand t"e brea$even point and its application
•
define t"e cost output relation s"ip
•
e(plain t"e different cost elements
1.1 INTRODUCTION
'at"ematics* old and ne,l% created* coupled ,it" innovative applications of t"e rapidl% evolving electronic computer and directed to,ard management problems* resulted in a ne, field of stud% called quantitative met"ods* ,"ic" "as become part of t"e curriculum of colleg colleges es of bus busine iness. ss. -"e import importance ance of quantit quan titati ative ve approa app roac"e c"es management ent quanti qua ntitat tative ive approac" appr oac"es ess to managem problems is no, ,idel% accepted and a course in mat"ematics* ,it" management applica applicatio tions ns is includ included ed in t"e core core of sub subjec jects ts studie studied d b% almost almost all manage ma nagemen ment managem man agement entt
1
students. -"is manual develops mat"ematics in t"e applied conte(t required for an understanding of t"e quantitative approac" to management problems. 1.2 Line! E"#tions Equation: !! A mat"ematical statement ,"ic" indicates t,o algebraic e(pressions are
equal (am ple) / / 2 (ample) Algebraic expressions) ! A mat"ematical statement indicating t"at numerical quantities
are lin$ed b% mat"ematical operations. (am ple) 2 (ample) Linear equations: - are equations ,it" a variable + aa constant constant ,it" ,it" degree degree one. one.
!
Are equations ,"ose terms 3t"e parts separated b% * !* signs4
!
Are a constant* or a constant times one variable to t"e first po,er
(am ple) 2 5 / (ample) 6 ! t"e degree 3t"e po,er4 of t"e variables is 1 ! t"e constant or t"e fi(ed value is 6 ! t"e terms of t"e equation are 2 and / separated b% 5 sign 7o,ever 2 / 6 isn8t a linear equation* because / is a constant times t"e product of 2 variables. 9 :o 2 terms* terms* :o ;/ terms* and no / terms are allo,ed. allo,ed. !
Linear equations are equations ,"ose slope is is constant constant t"roug"out t"e line.
!
-"e general notion of a linear equation is e(pressed in a form / m( b ,"ere m slope* b t"e /! intercept* / dependent variable and independent variable.
If / represents -otal
Y −Y ∆Y rise ( fall ) = = ∆ X run X − X 2 2
1
if 1 ≠ 2 if
1
Slope measures t"e steepness of a line. -"e larger t"e slope t"e more steep 3steeper4 t"e line is* bot" in value and in absolute value.
2
/
/
m undefined
ive slope m0 !ve slope
!
A line t"at is parallel to t"e !a(is is t"e gentlest gentlest of all lines i.e. m 0
!
A line t"at is parallel to t"e /!a(is is t"e steepest of all lines i.e. m undefined or infinite.
-"e slope of a line is defined as t"e c"ange!ta$ing place along t"e vertical a(is relative to t"e corresponding c"ange ta$ing place along t"e "ori=ontal a(is* or t"e c"ange in t"e t"e value of / relative to a one!unit c"ange in t"e value of . 1.2.1 Developing equation of a line
-"ere are at least t"ree ,a%s of developing t"e equation of a line. -"ese are) 1. -"e slope!intercept form 2. -"e slope!point form . -,o!point form 1. T$e s%o&e'inte!(e&t )o!*
-"is ,a% of developing t"e equation of a line involves t"e use of t"e slope + t"e intercept to formulate t"e equation. Often t"e slope + t"e /!intercept for a specific linear function are obtained directl% from t"e description of t"e situation ,e ,is" to model. (am ple > 1 (ample ?iven Slope 10 /!intercept 20* t"en Slope!intercept form) t"e equation of a line ,it" slope m and /!intercept b is / m( b / 10 20
Interpretative Exercises
>2 Suppose t"e @i(ed cost 3setup cost4 for producing product be br. 2000. After setup it costs br. 10 per produced. If t"e total cost is represented b% /) 1. rite t"e equation of t"is relations"ip in slope!intercept form. 2. State t"e slope of t"e line + + interpret t"e number . State t"e /!intercept of t"e line + interpret t"e number number >. A sales man "as a fi(ed salar% of br. 200 a ,ee$ In additionB "e receives a sales commission t"at is 20C of "is total volume of sales. State t"e relations"ip bet,een t"e sales man8s total ,ee$l% salar% + "is sales for t"e ,ee$. Ans,er Ans,er / / 0.20 200 2. T$e s%o&e &oint )o!*
-"e equation of a non!vertical line* L* of slope* m* t"at passes t"roug" t"e point 3 1* / 14 is ) defined b% t"e formula / 5 /1 m 3 5 14 / 5 /1 m 3 5 14 (am ple >1 (ample ?iven* slop & and oint 31* 24
⇒
/ 5 2 & 3 5 14 / 5 2 & ! & / & 5 2
>2 A sales man earns a ,ee$l% basic salar% plus a sales commission of 20C of "is total sales. "en "is total ,ee$l% sales total br. 1000* "is total salar% for t"e ,ee$ is &00. derive t"e formula describing t"e relations"ip bet,een total salar% and sales. Ans,er / / 0.2 200 > If t"e relations"ip bet,een -otal
-,o points completel% determine a straig"t line + of course* t"e% determine t"e slope of t"e line. 7ence ,e can first compute t"e slope* t"en use t"is value of m toget"er ,it"
&
eit"er point in t"e point!slope form / 5 /1 m 3 5 14 to generate t"e equation of a line. #% "aving t,o coordinate of a line ,e can determine t"e equation of t"e line. ⇒ #% -,o point form of linear equation)
3/ 5 /14
Y 2 −Y 1 ( X − X 1 ) − X X 2 1
(am ple >1 given 31* 104 + 3F* 04 (ample @irst slope
0 −10 F −1
/ 5 /1 m 3 5 14
=
−10
⇒
G
= −2
** t"en
/ 5 10 !2 3 5 14 / 5 10 !2 2 / !2 12
>2 A salesman "as a basic salar% +* in addition* receives a commission ,"ic" is a fi(ed percentage of "is sales volume. "en "is ,ee$l% sales are #r. #r. 1000* "is total salar% is br. &00. "en "is ,ee$l% sales are G00.00* "is total salar% is br. 00. Determine "is basic salar% + "is commission percentage + e(press t"e relations"ip bet,een sales + salar% in equation form. Ans,er) / / 0.2 200 > A printer costs a price of birr 1*&00 for printing 100 copies of a report + br. 000 for printing G00 copies. copies. Assuming a linear relations"ip ,"at ,ould be t"e price price for printing printing 00 copiesH Ans,er) / & 1000
"en t"e equation of a line is to be determined from t,o given points* it is a good idea to compare corresponding coordinates because if t"e / values are t"e same t"e line is "ori=ontal + if t"e values are t"e same t"e line is vertical (am ple) 1 (ample)
?iven t"e t"e points 3* 3* F4 + + 3E* 3E* F4 t"e line line t"roug" t"roug" t"em t"em is is "ori=ontal "ori=ontal because because bot" /!coordinates are t"e same i.e. F bot"
G
-"e equation of t"e line -"e line becomes becomes / F*,"ic" is different from t"e form / m( b If t"e !coordinates of t"e t,o different points are equal (am ple 3G* (ample 3G* 24 + 3G* 124 t"e line t"roug" t"em is vertical* + its equations is G i.e. is equal to a constant. If ,e proceed proceed to to appl% appl% t"e point slope procedure* ,e ,ould obtain. Slope 3'4
12
−10
G −G
=
2 0
=
+ if m ∞ infinite infinite t"e line is vertical + t"e form of
t"e equation is) constant b $arallel # perpen%icular lines & an%
-,o lines are parallel if t"e t,o lines "ave t"e same slope* + t,o lines are perpendicular if t"e product of t"eir slope is 51 or t"e slope of one is t"e negative reciprocal of t"e slope of t"e ot"er. 7o,ever* for vertical + "ori=ontal lines. 3-"e% are perpendicular to eac" ot"er4* t"is rule of ' 1 3t"e 3t"e first slope4 times ' 2 3t"e second slope4 equals 51 doesn8t "old true. i.e. '1 ( '2 ≠ !1 !1 ple) / / 2 5 10 + / 2 1& are parallel because t"eir slope are equal i.e. 2 (am (ample) / ;2 10 + / !2; 100 are perpendicular to eac" ot"er because t"e multiplication result of t"e t,o slope are 51 i.e.
2
x
−
2
1
=−
c Lines t'roug' t'e origin
An% equation in t"e variables + / t"at "as no constant term ot"er t"an =ero ,ill "ave a grap" t"at passes t"roug" t"e origin. Or* a line ,"ic" passes t"roug" t"e origin "as an ! intercept of 30* 04 i.e. bot" and / intercepts are =ero.
F
1.+ A&&%i(tion o) %ine! e"#tions 1.(.1 linear cost output relations'ips 5 <* @<* -<* A<* '<* -J* Π)
-J;-<
profit
Or -
region
-J
-J K -<
Loss region
-< -< -@<
- -J ! -<
A
@
# -< 7 ? -<
K 5 3< @<4 @<
K 5 K.< ! -@<
-@< #
K 3 5 <4 ! @< <
D
?3:o of units4 "ere K units product + units sold in revenue -< -otal
Interpretation of t'e grap':
1. -"e vertical distance bet,een A#* @<* ?D is t"e same because @i(ed
6
G. As production increases* -otal ariable
AJ Q
=
P . Q Q
⇒ AJ AJ in linear functions
1.2.2 B!e-een An%/sis
#rea$!even point is t"e point at ,"ic" t"ere is no loss or profit to t"e compan%. It can be e(pressed as eit"er in terms of production quantit% or revenue level depending on "o, t"e compan% states its cost equation. 'anufacturing companies usuall% state t"eir cost equation in terms of quantit% 3because t"e% produce and sell4 ,"ere as retail business state t"eir cost equation in terms terms of of revenue 3because t"e% purc"ase and sell4 Cse 1: Mn#)(t#!in Co*&nies
-J -<
i.e -J 5 -< 0
K < @< K 5 <.K @< K 3 5 <4 @<
,"ere Ke #rea$even Kuantit% @< @i(ed cost unit selling price
E
FC
Ke
P
< unit variable cost
VC
−
-J -<;-J
-<
K FC
Ke P −VC (am ple >1 A (ample >1 A manufacturing
-J K
-< GK 10*000
-J 10K
rofit3Π4 -J 5 -< 10K 5 3GK 10*0004 GK 5 10*000
b4 At brea$ even point -J -< 10K GK 10*000 GK 5 10*000 0 Ke
10*000 G
Ke 2000 units i.e. #rea$even Kuantit% is 2000 units Sales volume 2000 10 20*000 br. N
2G000
20000
1G000
10000
G000
Π
-J 10K
-< GK 10000
-< GK -Π GK ! 10000
-<
-@<
0 1000
2000
000 &000
G000
K 3no. of units produced + sold4
!G000 ! !10000 !
Interpretation) "en a co. produces + sells 2000 units of output* t"ere ,ill not be an% loss or gain 3no profit* no loss4 T$e e))e(t o) ($nin one !i%e -ee&in ot$e! (onstnt Cse 1 ' i3e4 (ost
Assume for t"e above problem @< is decreased b% #r. G000*
Ke ⇒ Ke
G000 G
1000 units
-J 10K -"erefore*
@< ↓ → Ke Ke ↓
@< + Ke "ave direct relations"ip
@< ↑ → Ke Ke ↑ Cse 2'Unit !i%e (ost
Assume for t"e above problem < decreased b% 1 br.
Ke ⇒ Ke
10000 F
=
1*FF6 units
-J 10K
10
-"erefore*
< ↓ → Ke Ke ↓
< + Ke "ave direct relations"ip
< ↑ → Ke Ke ↑
Cse +' Se%%in &!i(e
Assume for t"e above problem selling price is decreased b% br. 1*
-< GK 10*000
10000 &
=
2G00 units
-J NK -"erefore ↓ → Ke Ke ↑
rice and brea$even point "ave indirect relations"ip
↑ → Ke Ke ↓ In t"e above e(ample a compan% "as t"e follo,ing options 3to minimi=e its brea$even point and ma(imi=e profit4. !
decreasing @<
!
decreasing unit <
!
increasing t"e unit selling price
And if t"e organi=ation is bet,een option 2 + * it is preferable to decrease t"e unit variable cost because if ,e increase t"e selling price* t"e organi=ation 'a% loose its customers + also decreasing t"e @< is preferable. -J -< -@<
Ke
K
11
in4in t$e "#ntit/ %ee% ,$i($ ino%es &!o)it o! %oss FC + 0
# P −V ** an% K is is related related to to t"e t"e cost cost ** profit!!!! profit!!!! Π -J 5 -c
"ere) # brea$even point
K 5 3<.K @<4 Π
rofit Π
Π 3.K 5 <.K4 5 @<
-J -otal revenue
Π K 3 5 <4 5 @<
-< -otal cost
± Π+ FC
P −VC
=
for any qty level
Q
=
FC ±Π P −VC
K Kuantit% ∀< nit variable cost
(am ple >1 (ample >1 @or t"e above manufacturing co. if it ,ants to ma$e a profit of 2G000 br. "at s"ould be t"e quantit% levelH -J 10K
K
FC ±Π
,"en t"ere is Π* t"e quantit% produced + ,"en
P − V
sold "ave to be greater t"an t"e #rea$even quantit% 10*000 +2G000
-< GK 10*000
Π 2G*000
6000 units
10 −G
KH If it e(pects a loss of br. G000 ,"at ,ill be t"e quantit% level. K
FC ±VC P −VC
=
10*000 −G000 10 −G
9 ,"en t"ere is is loss* t"e qt% produced + sold s"ould be less t"an t"e #K
Cse 2 Me!($n4isin 5Reti% B#siness
#rea$even Jevenue #K Assume a bus. @irm ,it" product A "as t"e follo,ing cost cost + revenue items. items. ariable cost of A 100 br. Selling price 1G0 br.
12
'ar$up Selling price 5 ariable cost 1G0 5 100 G0 i. as a function of cost* t"e mar$up is G0;100 G0C ii. as a function of retail price* t"e mar$up is G0;1G0 . C it is also called margin. 'argin
-"e cost of goods sold 100C ! . C FF.FC ≈ F6C F6C -"e Selling price
?iven ot"er selling e(pense 1Cof t"e selling selling price price i.e. i.e. 0.01 0.01 So* t"e -< equation becomes) / 0.FE @< "ere) is sales revenue / is total cost Out of 100C selling price FEC is t"e variable cost of goods purc"ased + sold sold (am ple Suppose Suppose a retail business sale its commodities at a margin of 2GC on all items (ample purc"ased + sold. 'oreover t"e compan% uses GC commission as selling e(pense + br. 12000 as a @i(ed
'argin
2GC
6GC
Let represents selling price / total cost @< 12000
GC
e #rea$even revenue
E0C
∴/ 0.E 12000 #rea$ even revenue is obtained b% ma$ing sales revenue + cost equals At brea$even point -< -J
/ m( b
i.e. / t"en* 0.E 12000
unit variable cost ⇒
FC 1 −m
or
FC ±Π 1 −m
where m =
VC P
=
TVC TR
!0.2 !12000
1
"en t"e co. receives br. F0*000 as sales revenue* ⇒ "en
F0*000 F0*000 br.
t"ere ,ill be no loss or profit. -"e #rea$even revenue revenue 3#J
FC ±Π
1 −m
4 met"od is useful* because ,e can use a
single formula for different goods so far as t"e compan% uses t"e same amount of profit margin for all goods. 7o,ever* in #rea$even quantit% met"od or #K
FC ±Π P − V
it is it
not possible and "ence ,e "ave to use different formula for different items. (am ple >1 (ample
It is estimated t"at t"at sales in t"e coming period ,ill ,ill be br. F000 + t"at @<
,ill be br. 1000 + variable costs br. br. F00* develop t"e total cost equation + t"e brea$even revenue. brea$even Ans,er) / Ans,er) /
F00 F000
1000 0.F 1000 "ere / -otal
#J e
1000 1 −0.F
=
1000 0.&
= 2G00 br .
At t"e sales volume of br. 2G00* t"e compan% brea$s even. 9 "en t"e brea$even revenue equation is for more t"an one item it is impossible to find t"e brea$even quantit%. It is onl% possible for one item b% Ke e; "ere e #rea$ even revenue selling price Ke #rea$even quantit% -o c"ange t"e brea$even revenue equation in to #rea$even quantit% . e "ave to multiple price b% t"e coefficient of . li$e,ise* to c"ange in to brea$even revenue from #rea$ even quantit%* ,e "ave to to divide t"e unit < b% price. 1.6 Mo4e% e3*intion "#estions
1. / compan%8s cost function for t"e ne(t four four mont"s mont"s is is < G00*000 GK
1&
a4 @ind t"e # dollar volume of sales if t"e selling price is br. F ; unit b4 "at ,ould be t"e compan%8s cost if it decides to s"utdo,n operations for t"e ne(t four mont"s c4 If* because of stri$e* t"e most t"e compan% can produce is br. 100*000 units* s"ould it s"utdo,nH "% or ,"% notH 2. In its first %ear* PAbol #una
Selling price br. 100
-< br. 1*G00*000
-@< br. G0*000
Re"#i!e4:
1. Develop Jevenue* cost + profit profit functions functions for for t"e t"e co. in terms of quantit%. 2. @ind t"e #rea$even point in in terms terms of of quantit% quantit% .
@ind t"e equation relating -otal
ii.
@ind t"e profit if sales are #r. F0*000
iii.
@ind t"e brea$even revenue
iv.
If profit is #r. 1G*000 ,"at s"ould be t"e t"e revenue revenue levelH levelH
1G
v.
If %ou "ave an% one item at a price of #r. 1G;unit "o, do %ou convert t"e cost equation in terms of revenue in to a cost equation in terms of quantit%H
1F
