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Chapter 6

Chapter 6

Cost-Volume-ProfitRelationships

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Basics of Cost-Volume-Profit Analysis

Contribution Margin (CM) is the amountremaining from sales revenue after variable

expenses have been deducted.

Contribution Margin (CM) is the amountremaining from sales revenue after variable

expenses have been deducted.

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Basics of Cost-Volume-Profit Analysis

CM is used first to cover fixedexpenses. Any remaining CM

contributes to net operating income.

CM is used first to cover fixedexpenses. Any remaining CM

contributes to net operating income.

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Learning Objective

LO

LO

!o e"plain ho# changes inactivity affect contributionmargin an$ net operating

income%

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!he Contribution Approach &ales' variable e"penses' an$ contribution margin can

also be e"presse$ on a per unit basis% (f Racing sellsan a$$itional bicycle' )*++ a$$itional C, #ill be

generate$ to cover fi"e$ e"penses an$ profit%

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!he Contribution Approach

ach month' Racing must generate at least).+'+++ in total C, to brea/ even%

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(f Racing sells 0++ units0++ units in a month' it #illbe operating at the break-even point %

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(f Racing sells one more bi/e 10+ bi/es0+ bi/es2' netoperating income #ill increase by $200 %

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3e $o not nee$ to prepare an income statementto estimate profits at a particular sales volume%&imply multiply the number of units sol$ above

brea/-even by the contribution margin per unit%

(f Racing sells(f Racing sells

04+ bi/es' its04+ bi/es' its

net income #illnet income #illbe )6'+++%be )6'+++%

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Learning Objective

LO*LO*

!o prepare an$ interpret acost-volume-profit 1CVP2

graph%

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CVP Relationships in 5raphic orm

!he relationship among revenue' cost' profit an$ volume canbe e"presse$ graphically by preparing a CVP graph%

Racing $evelope$ contribution margin income statementsat 4++' 0++' an$ 7++ units sol$% 3e #ill use this

information to prepare the CVP graph%

Income

300 units

Income

00 units

Income

!00 units

"ales #!0$000% &00$000% &!0$000%

'ess variable expenses 0$000 #&0$000 #!0$000

Contribution margin *0$000% +0$000% #00$000%

'ess fixed expenses +0$000 +0$000 +0$000

,et operating income (&0$000)% -% &0$000%

Income

300 units

Income

00 units

Income

!00 units

"ales #!0$000% &00$000% &!0$000%

'ess variable expenses 0$000 #&0$000 #!0$000

Contribution margin *0$000% +0$000% #00$000%'ess fixed expenses +0$000 +0$000 +0$000

,et operating income (&0$000)% -% &0$000%

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CVP 5raph

nits

8 o

l l a r s

(n a CVP graph'(n a CVP graph' unit volumeunit volume isis

usually represente$ on theusually represente$ on the

hori/ontal () axishori/ontal () axis an$an$ dollarsdollars onon

thethe vertical (1) axisvertical (1) axis%%

(n a CVP graph'(n a CVP graph' unit volumeunit volume isis

usually represente$ on theusually represente$ on thehori/ontal () axishori/ontal () axis an$an$ dollarsdollars onon

thethe vertical (1) axisvertical (1) axis%%

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8 o

l l a r s

nits

CVP 5raph

2ixed xpenses

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CVP 5raph

8 o

l l a r s

nits

2ixed xpenses

4otal xpenses

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CVP 5raph

2ixed xpenses 8

o l l

a r s 4otal xpenses

4otal "ales

nits

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CVP 5raph

8 o

l l a r s

nits

5rea6-even point5rea6-even point

(00 units or %&00$000 in sales)(00 units or %&00$000 in sales)5rea6-even point5rea6-even point

(00 units or %&00$000 in sales)(00 units or %&00$000 in sales)

P r o f

i t A r e a

L o s s A r

e a

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Learning Objective

LO4LO4

!o use the contribution

margin ratio 1C, ratio2 tocompute changes in

contribution margin an$ netoperating income resulting

from changes in salesvolume%

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Contribution ,argin Ratio

!he contribution margin ratio is9

or Racing Bicycle Company the ratio is9

4otal CM4otal sales

CM 7atio 8

ach )%++ increase in sales results in atotal contribution margin increase of 0+:%

8 09%+0$000

%&00$000

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Or' in terms of units' the contribution marginratio is9

or Racing Bicycle Company the ratio is9%&00%!00 8 09

nit CM

nit selling priceCM 7atio 8

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00 5i6es !00 5i6es

"ales &00$000% &!0$000%

'ess variable expenses #&0$000 #!0$000

Contribution margin +0$000 #00$000 'ess fixed expenses +0$000 +0$000

,et operating income -% &0$000%

00 5i6es !00 5i6es

"ales &00$000% &!0$000%

'ess variable expenses #&0$000 #!0$000

Contribution margin +0$000 #00$000 'ess fixed expenses +0$000 +0$000



A )7+'+++ increase in sales revenue A )7+'+++ increase in sales revenueresults in a )*+'+++ increase in C,%results in a )*+'+++ increase in C,%

1)7+'+++ ; 0+< = )*+'+++21)7+'+++ ; 0+< = )*+'+++2

A )7+'+++ increase in sales revenue A )7+'+++ increase in sales revenueresults in a )*+'+++ increase in C,%results in a )*+'+++ increase in C,%

1)7+'+++ ; 0+< = )*+'+++21)7+'+++ ; 0+< = )*+'+++2

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>uic/ Chec/

Coffee ?latch is an espresso stan$ in a $o#nto#noffice buil$ing% !he average selling price of a cupof coffee is )%0@ an$ the average variablee"pense per cup is )+%46% !he average fi"e$

e"pense per month is )'4++% On average' *'++cups are sol$ each month% 3hat is the C, Ratiofor Coffee ?latch

a% %4@

b% +%7.c% +%*0*

$% 0%4@



a% %4@

b% +%7.c% +%*0*

$% 0%4@

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a% %4@

b% +%7.c% +%*0*

$% 0%4@



a% %4@

b% +%7.c% +%*0*

$% 0%4@

>uic/ Chec/

nit contribution marginnit selling price

CM 7atio 8

8(%#.-%0.3*)

%#.

8%#.#3%#.

8 0.:!+

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Learning Objective

LO0LO0

!o sho# the effects on

contribution margin ofchanges in variable costs'

fi"e$ costs' selling price' an$volume%

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Changes in i"e$ Costs an$ &ales Volume

3hat is the profit impact if Racing canincrease unit sales from 7++ to 70+ by

increasing the monthly a$vertising

bu$get by )+'+++

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Changes in i"e$ Costs an$ &ales Volume%+0$000 ; %#0$000 advertising 8 %0$000%+0$000 ; %#0$000 advertising 8 %0$000%+0$000 ; %#0$000 advertising 8 %0$000%+0$000 ; %#0$000 advertising 8 %0$000

"ales"ales increased increased by %&0$000$ but net operatingby %&0$000$ but net operating

incomeincome decreased decreased by %&$000by %&$000%%

"ales"ales increased increased by %&0$000$ but net operatingby %&0$000$ but net operating

incomeincome decreased decreased by %&$000by %&$000%%

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Changes in i"e$ Costs an$ &ales Volume

!he &hortcut &olution

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Change in Variable Costs an$ &ales Volume

3hat is the profit impact if Racing canuse higher uality ra# materials' thus'

increasing variable costs per unit by )+'

to generate an increase in unit salesfrom 7++ to 7.+

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Change in Variable Costs an$ &ales Volume

!+0 units!+0 units

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Change in i"e$ Cost' &ales Price an$ Volume

3hat is the profit impact if Racing 12 cutsits selling price )*+ per unit' 1*2 increases

its a$vertising bu$get by )7'+++ per

month' an$ 142 increases unit sales from7++ to 67+ units per month

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"ales"ales increaseincrease by %*&$000$ fixed costs increase byby %*&$000$ fixed costs increase by

%#!$000$ and net operating income%#!$000$ and net operating income increasesincreases by %&$000by %&$000%%

"ales"ales increaseincrease by %*&$000$ fixed costs increase byby %*&$000$ fixed costs increase by

%#!$000$ and net operating income%#!$000$ and net operating income increasesincreases by %&$000by %&$000%%

Change in i"e$ Cost' &ales Price an$ Volume

Change in Variable Cost i"e$ Cost an$

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Change in Variable Cost' i"e$ Cost an$&ales Volume

3hat is the profit impact if Racing 12 paysa )7 sales commission per bi/e sol$'

instea$ of paying salespersons flat salaries

that currently total )6'+++ per month' an$1*2 increases unit sales from 7++ to 77

bi/es

Change in Variable Cost i"e$ Cost an$

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Change in Variable Cost' i"e$ Cost an$&ales Volume

"ales"ales increaseincrease by %3:$!00$ variable costsby %3:$!00$ variable costs increaseincrease byby

%3#$#&!$ but fixed expenses%3#$#&!$ but fixed expenses decreasedecrease by %*$000by %*$000%%

"ales"ales increaseincrease by %3:$!00$ variable costsby %3:$!00$ variable costs increaseincrease byby

%3#$#&!$ but fixed expenses%3#$#&!$ but fixed expenses decreasedecrease by %*$000by %*$000%%

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Change in Regular &ales Price

(f Racing has an opportunity to sell 7+bi/es to a #holesaler #ithout $isturbing

sales to other customers or fi"e$

e"penses' #hat price #oul$ it uote to the#holesaler if it #ants to increase monthly

profits by )4'+++

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Change in Regular &ales Price

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Learning Objective

LO7LO7

!o compute the brea/evenpoint in unit sales an$ $ollar

sales%

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Brea/-ven Analysis

Brea/-even analysis can beBrea/-even analysis can be

approache$ in t#o #ays9approache$ in t#o #ays9

%% uation metho$uation metho$*%*% Contribution margin metho$Contribution margin metho$

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uation ,etho$

>rofits 8 ("ales ? @ariable expenses) ? 2ixed expenses

"ales 8 @ariable expenses ; 2ixed expenses ; >rofits

7

At the brea6-even pointAt the brea6-even point

profits eBual /eroprofits eBual /ero

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Brea/-ven Analysis

Dere is the information from Racing Bicycle Company9

4otal >er .nit >ercent"ales (!00 bi6es) &!0$000% !00% #009

'ess variable expenses #!0$000 300 *09

Contribution margin #00$000% &00% 09

'ess fixed expenses +0$000

,et operating income &0$000%

4otal >er .nit >ercent"ales (!00 bi6es) &!0$000% !00% #009

'ess variable expenses #!0$000 300 *09

Contribution margin #00$000% &00% 09


,et operating income &0$000%

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uation ,etho$

%!00 8 %300 ; %+0$000 ; %0

Dhere 8 ,umber of bi6es sold %!00 8 nit selling price %300 8 nit variable expense %+0$000 8 4otal fixed expense

3e calculate the brea/-even point as follo#s9

"ales 8 @ariable expenses ; 2ixed expenses ; >rofits"ales 8 @ariable expenses ; 2ixed expenses ; >rofits

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uation ,etho$

%!00 8 %300 ; %+0$000 ; %0%&00 8 %+0$000 8 %+0$000 E %&00 per bi6e

8 400 bikes400 bikes

3e calculate the brea/-even point as follo#s9


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uation ,etho$

!he euation can be mo$ifie$ to calculatethe brea/-even point in sales $ollars%


8 0.*0 ; %+0$000 ; 8 0.*0 ; %+0$000 ; %0%0

Dhere 8 4otal sales dollars

0.*0 8 @ariable expenses as a 9 of sales%+0$000 8 4otal fixed expenses

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uation ,etho$

8 0.*0 ; %+0$000 ; %0 0.0 8 %+0$000

8 %+0$000 E 0.0 8 $200,000 $200,000


!he euation can be mo$ifie$ to calculatethe brea/-even point in sales $ollars%

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Contribution ,argin ,etho$

!he contribution margin metho$ has t#o/ey euations%

2ixed expensesnit contribution margin

85rea6-even point

in units sold

2ixed expenses CM ratio8

5rea6-even point in

total sales dollars

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Contribution ,argin ,etho$

LetEs use the contribution margin metho$ tocalculate the brea/-even point in total

sales $ollars at Racing%

2ixed expenses CM ratio

85rea6-even point intotal sales dollars

).+'+++).+'+++0+

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>uic/ Chec/


e"pense per month is )'4++% On average *'++cups are sol$ each month% 3hat is the brea/-evensales in units

a% .* cups

b% 4'6 cups

c% '*++ cups

$% '7+ cups



a% .* cups

b% 4'6 cups

c% '*++ cups

$% '7+ cups

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a% .* cups

b% 4'6 cupsc% '*++ cups

$% '7+ cups



a% .* cups

b% 4'6 cupsc% '*++ cups

$% '7+ cups

>uic/ Chec/

2ixed expensesnit CM

5rea6-even 8

%#$300%#.=cup - %0.3*=cup

%#$300%#.#3=cup

8 1,150 cups

8

8

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>uic/ Chec/


e"pense per month is )'4++% On average *'++cups are sol$ each month% 3hat is the brea/-evensales in $ollars

a% )'4++

b% )'7

c% )'..

$% )4'*@



a% )'4++

b% )'7

c% )'..

$% )4'*@

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a% )'4++

b% )'7c% )'..

$% )4'*@



a% )'4++

b% )'7c% )'..

$% )4'*@

>uic/ Chec/

2ixed expenses

CM 7atio

5rea6-even

sales%#$3000.:!+

8 $1,715

8

8

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Learning Objective

LO6LO6

!o $etermine the level ofsales nee$e$ to achieve a

$esire$ target profit%

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!arget Profit Analysis

!he euation an$ contribution marginmetho$s can be use$ to $etermine the

sales volume nee$e$ to achieve a target

profit%

&uppose Racing Bicycle Company

#ants to /no# ho# many bi/es mustbe sol$ to earn a profit of )++'+++%

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!he CVP uation ,etho$


%!00 8 %300 ; %+0$000 ; %#00$000

%&00 8 %#+0$000

8 900 bikes

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!he Contribution ,argin Approach

!he contribution margin metho$ can be use$to $etermine that @++ bi/es must be sol$ to

earn the target profit of )++'+++%

2ixed expenses ; 4arget profit nit contribution margin

8nit sales to attain

the target profit

%+0$000 ; %#00$000 %&00=bi6e

8 900 bikes

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>uic/ Chec/


e"pense per month is )'4++% Do# many cups ofcoffee #oul$ have to be sol$ to attain target profitsof )*'7++ per month

a% 4'464 cups

b% *'** cups

c% '7+ cups

$% 0'*++ cups



a% 4'464 cups

b% *'** cupsc% '7+ cups

$% 0'*++ cups

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a% 4'464 cups


$% 0'*++ cups



a% 4'464 cups


$% 0'*++ cups

>uic/ Chec/ 2ixed expenses ; 4arget profit

nit CM

nit salesto attain

target profit

8 3$3*3 cups

8%3$+00%#.#3

%#$300 ; %&$!00%#. - %0.3*

8

8

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Learning Objective

LOLO

!o compute the margin ofsafety an$ e"plain its

significance%

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!he ,argin of &afety

!he margin of safety is the e"cess ofbu$gete$ 1or actual2 sales over the brea/-

even volume of sales%

Margin of safety 8 4otal sales - 5rea6-even salesMargin of safety 8 4otal sales - 5rea6-even sales

LetEs loo/ at Racing Bicycle Company an$

$etermine the margin of safety%

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(f #e assume that Racing Bicycle Company has actualsales of )*7+'+++' given that #e have alrea$y$etermine$ the brea/-even sales to be )*++'+++'

the margin of safety is )7+'+++ as sho#n

5rea6-even

sales

00 units

Actual sales

!00 units

"ales &00$000% &!0$000%

'ess variable expenses #&0$000 #!0$000 Contribution margin +0$000 #00$000

'ess fixed expenses +0$000 +0$000


5rea6-even

sales

00 units

Actual sales

!00 units

"ales &00$000% &!0$000%




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!he margin of safety can be e"presse$ as&09 of sales%

1)7+'+++ F )*7+'+++2

5rea6-even

sales

00 units

Actual sales

!00 units

"ales &00$000% &!0$000%




5rea6-even

sales

00 units

Actual sales

!00 units

"ales &00$000% &!0$000%




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!he margin of safety can be e"presse$ interms of the number of units sol$% !hemargin of safety at Racing is )7+'+++'

an$ each bi/e sells for )7++%

,argin of ,argin of

&afety in units&afety in units== = ++ bi/es= ++ bi/es

)7+'+++)7+'+++

)7++)7++

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>uic/ Chec/

Coffee ?latch is an espresso stan$ in a$o#nto#n office buil$ing% !he average sellingprice of a cup of coffee is )%0@ an$ theaverage variable e"pense per cup is )+%46% !he

average fi"e$ e"pense per month is )'4++% Onaverage *'++ cups are sol$ each month% 3hatis the margin of safety

a% 4'*7+ cups

b% @7+ cups

c% '7+ cups

$% *'++ cups

Coffee ?latch is an espresso stan$ in a$o#nto#n office buil$ing% !he average sellingprice of a cup of coffee is )%0@ an$ theaverage variable e"pense per cup is )+%46% !he

average fi"e$ e"pense per month is )'4++% Onaverage *'++ cups are sol$ each month% 3hatis the margin of safety

a% 4'*7+ cups

b% @7+ cups

c% '7+ cups

$% *'++ cups

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Coffee ?latch is an espresso stan$ in a$o#nto#n office buil$ing% !he average sellingprice of a cup of coffee is )%0@ an$ theaverage variable e"pense per cup is )+%46%

!he average fi"e$ e"pense per month is)'4++% On average *'++ cups are sol$ eachmonth% 3hat is the margin of safety

a% 4'*7+ cups

b% @7+ cups

c% '7+ cups

$% *'++ cups


!he average fi"e$ e"pense per month is)'4++% On average *'++ cups are sol$ eachmonth% 3hat is the margin of safety

a% 4'*7+ cups

b% @7+ cups

c% '7+ cups

$% *'++ cups

>uic/ Chec/

Margin of safety 8 4otal sales ? 5rea6-even sales

8 950 cups

8 &$#00 cups ? #$#!0 cups

or !0 cups

&$#00 cupsMargin of safety

percentage8 8 45%

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Cost &tructure an$ Profit &tability

Cost structure refers to the relative proportionof fi"e$ an$ variable costs in an organiGation%

,anagers often have some latitu$e in

$etermining their organiGationEs cost structure%

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Cost &tructure an$ Profit &tability

!here are a$vantages an$ $isa$vantages to highfi"e$ cost 1or lo# variable cost2 an$ lo# fi"e$ cost

1or high variable cost2 structures%

An a$vantage of a high fi"e$ An a$vantage of a high fi"e$cost structure is that incomecost structure is that income#ill be higher in goo$ years#ill be higher in goo$ years

compare$ to companiescompare$ to companies

#ith lo#er proportion of #ith lo#er proportion of

fi"e$ costs%fi"e$ costs%

An a$vantage of a high fi"e$ An a$vantage of a high fi"e$cost structure is that incomecost structure is that income#ill be higher in goo$ years#ill be higher in goo$ years

compare$ to companiescompare$ to companies



A $isa$vantage of a high fi"e$ A $isa$vantage of a high fi"e$

cost structure is that incomecost structure is that income#ill be lo#er in ba$ years#ill be lo#er in ba$ yearscompare$ to companiescompare$ to companies#ith lo#er proportion of #ith lo#er proportion of


A $isa$vantage of a high fi"e$ A $isa$vantage of a high fi"e$

cost structure is that incomecost structure is that income#ill be lo#er in ba$ years#ill be lo#er in ba$ yearscompare$ to companiescompare$ to companies



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Learning Objective

LO.LO.

!o compute the $egree of

operating leverage at aparticular level of sales an$

e"plain ho# it can be use$ topre$ict changes in net

operating income%

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Operating Leverage

Contribution margin,et operating income

Fegree of operating leverage

=

A measure of hoG sensitive net operatingincome is to percentage changes in sales.

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Operating Leverage

Actual sales

!00 5i6es

"ales &!0$000%'ess variable expenses #!0$000

Contribution margin #00$000


,et income &0$000%

Actual sales

!00 5i6es

"ales &!0$000%'ess variable expenses #!0$000

Contribution margin #00$000


,et income &0$000%

%#00$000

%&0$000

8 5 5

At Racing' the $egree of operating leverage is7%

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Operating Leverage

3ith an operating leverage of 7' if Racing3ith an operating leverage of 7' if Racingincreases its sales by +

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Operating Leverage

10% increase in sales from$250,000 to $275,000 . . .

10% increase in sales from$250,000 to $275,000 . . .

. . . results in a 50% increase in

income from $20,000 to $30,000.

. . . results in a 50% increase in

income from $20,000 to $30,000.

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>uic/ Chec/


!he average fi"e$ e"pense per month is)'4++% On average *'++ cups are sol$ eachmonth on% 3hat is the operating leverage

a% *%*

b% +%07c% +%40

$% *%@*


!he average fi"e$ e"pense per month is)'4++% On average *'++ cups are sol$ eachmonth on% 3hat is the operating leverage

a% *%*

b% +%07c% +%40

$% *%@*

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!he average fi"e$ e"pense per month is)'4++% On average*'++ cups are sol$ eachmonth% 3hat is the operating leverage

a% *%*

b% +%07c% +%40

$% *%@*


!he average fi"e$ e"pense per month is)'4++% On average*'++ cups are sol$ eachmonth% 3hat is the operating leverage

a% *%*

b% +%07c% +%40

$% *%@*

>uic/ Chec/

Contribution margin,et operating income

peratingleverage 8

%&$3:3%#$0:38 8 2.21

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>uic/ Chec/

At Coffee ?latch the average selling price of a At Coffee ?latch the average selling price of acup of coffee is )%0@' the average variablecup of coffee is )%0@' the average variablee"pense per cup is )+%46' an$ the average fi"e$e"pense per cup is )+%46' an$ the average fi"e$e"pense per month is )'4++% On average *'++e"pense per month is )'4++% On average *'++

cups are sol$ each month%cups are sol$ each month%(f sales increase by *+

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At Coffee ?latch the average selling price of a At Coffee ?latch the average selling price of acup of coffee is )%0@' the average variablecup of coffee is )%0@' the average variablee"pense per cup is )+%46' an$ the average fi"e$e"pense per cup is )+%46' an$ the average fi"e$e"pense per month is )'4++% On average *'++e"pense per month is )'4++% On average *'++

cups are sol$ each month%cups are sol$ each month%(f sales increase by *+

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Verify (ncrease in Profit

Actualsales

Increasedsales

2,100 cups 2,520 cups

"ales 3$#&% 3$:!!%

'ess @ariable expenses :!* 0: Contribution margin &$3:3 &$++

'ess 2ixed expenses #$300 #$300

,et operating income #$0:3% #$!+%9 change in sales &0.09

9 change in net operating income .&9

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&tructuring &ales Commissions

Companies generally compensatesalespeople by paying them either a

commission base$ on sales or a salary plus a

sales commission% Commissions base$ onsales $ollars can lea$ to lo#er profits in a

company%

LetEs loo/ at an e"ample%LetEs loo/ at an e"ample%

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Pipeline Hnlimite$ pro$uces t#o types of surfboar$s'the IR an$ the !urbo% !he IR sells for )++ an$generates a contribution margin per unit of )*7% !he!urbo sells for )7+ an$ earns a contribution margin

per unit of ).%

!he sales force at Pipeline Hnlimite$ is

compensate$ base$ on sales commissions%

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(f you #ere on the sales force at Pipeline' you #oul$push har$ to sell the !urbo even though the IR

earns a higher contribution margin per unit%

!o eliminate this type of conflict' commissions cancommissions can

be based on contribution marginbe based on contribution margin rather than onselling price alone%

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Learning Objective

LO@LO@

!o compute the brea/-even

point for a multipro$uctcompany an$ e"plain the

effects of shifts in the salesmi" on contribution margin

an$ the brea/-even point%

C f &

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!he Concept of &ales ,i"

&ales mi" is the relative proportion in #hich acompanyEs pro$ucts are sol$%

8ifferent pro$ucts have $ifferent sellingprices' cost structures' an$ contribution

margins%

LetEs assume Racing Bicycle Company sells

bi/es an$ carts an$ that the sales mi"bet#een the t#o pro$ucts remains the same%

, l i $ b / l i

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,ulti-pro$uct brea/-even analysis

Racing Bicycle Co% provi$es the follo#ing information9

%&*!$000

%!!0$000

8 +.&9 (rounded)

, lti $ t b / l i

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,ulti-pro$uct brea/-even analysis2ixed expenses

CM 7atio

5rea6-even

sales%#:0$000

+.&9

8 $352,697

8

=

? A ti f CVP A l i

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?ey Assumptions of CVP Analysis

&elling price is constant%&elling price is constant%Costs are linear%Costs are linear%

(n multi-pro$uct companies' the(n multi-pro$uct companies' the

sales mi" is constant%sales mi" is constant%

(n manufacturing companies'(n manufacturing companies'

inventories $o not change 1unitsinventories $o not change 1unitspro$uce$ = units sol$2%pro$uce$ = units sol$2%

$ f Ch t 6

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n$ of Chapter 6

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