WGUl D771l Objectivel Assessmentl (Latestl
2026/l 2027l Update)l Quantitativel Literacyl
Guide|l Questionsl &l Answersl |l Gradel A|l
100%l Correctl (Verifiedl Solutions)
Linearl Models
Growthl Models
Populationsl ofl people,l animals,l andl itemsl arel growingl alll aroundl us.l Byl
understandingl howl thingsl grow,l wel l canl betterl understandl whatl tol expectl inl thel
future.l Inl thisl chapter,l wel l focusl onl time-dependentl change.
Linearl (Algebraic)l Growth
Marcol isl al collectorl ofl antiquel sodal bottles.l Hisl collectionl currently
containsl 437l bottles.l Everyl year,l hel budgetsl enoughl moneyl tol buyl 32l newl bottles.l
Canl wel l determinel howl manyl bottlesl hel willl havel inl 5l years,l andl howl longl itl
willl takel forl l hisl collectionl tol reachl 1,000l bottles?l Whilel bothl ofl thesel questionsl
youl couldl probablyl solvel withoutl anl equationl orl l formall mathematics,l wel l arel
goingl tol formalizel ourl approachl tol thisl probleml tol providel al meansl tol answerl
morel complicatedl questions.l Suppose
thatl Pnl representsl thel number,l orl l population,l ofl bottlesl Marcol has
afterl nl years.l Sol l P0l wouldl representl thel numberl ofl bottlesl now,l P1l wouldl
representl thel numberl ofl bottlesl afterl 1l year,l P2l wouldl representl thel numberl ofl
bottlesl afterl 2l years,l andl sol on.l Wel couldl describel howl Marco’sl bottlel collectionl isl
changingl using:
P0=437
Pn=Pn−1+32
Thisl isl calledl al recursivel relationship.l Al recursivel relationshipl isl al formulal whichl
relatesl thel nextl valuel inl al sequencel tol thel previousl values.l Here,l thel numberl ofl
bottlesl inl yearl nl canl bel foundl byl addingl 32l tol thel numberl ofl bottlesl inl thel
previousl year,l Pn−1.l Usingl thisl relationship,l wel l couldl calculate:
P1=P0+32=437+32=469
P2=P1+32=469+32=501
P3=P2+32=501+32=533
P4=P3+32=533+32=565
P5=P4+32=565+32=597
,Wel havel answeredl thel questionl ofl howl manyl bottlesl Marcol willl havel inl 5l years.l
However,l solvingl howl longl itl willl takel forl l hisl collectionl to
reachl 1,000l bottlesl wouldl requirel al lotl morel calculations.l Whilel recursivel
relationshipsl arel excellentl forl l describingl simplyl andl cleanlyl howl al quantityl isl
changing,l theyl arel notl convenientl forl l makingl predictionsl orl l solvingl problemsl thatl
stretchl farl intol thel future.l Forl l that,l al closedl orl l explicitl forml forl l thel
relationshipl isl preferred.l Anl explicitl equationl allowsl usl to
calculatel Pnl directly,l withoutl needingl tol knowl Pn−1.l Whilel youl mayl already
bel ablel tol guessl thel explicitl equation,l letl usl derivel itl froml thel recursivel formula.l
Wel canl dol sol byl selectivelyl notl simplifyingl asl wel l go:
P1=437+32=437+1(32)l P2=P1+32=437+32+32=437+2(32)l
P3=P2+32=(437+2(32))+32=437+3(32)l P4=P3+32=(437+3(32))+32=437+4(32)
Youl canl probablyl seel thel patternl now,l andl generalizel that
Pn=437+n(32)=437+32n
Usingl thisl equation,l wel l canl calculatel howl manyl bottlesl he’lll havel afterl 5l years:
P5=437+32(5)=437+160=597
Wel canl nowl alsol solvel forl l whenl thel collectionl willl reachl 1,000l bottlesl byl
substitutingl inl 1,000l forl l Pnl andl solvingl forl l n
1,000=437+32n
563=32nl n=56332=17.59
Sol l Marcol willl reachl 1,000l bottlesl inl 18l years.
Inl thel previousl example,l Marco’sl collectionl grewl byl thel samel numberl ofl bottlesl
everyl year.l Thisl constantl changel isl thel definingl characteristicl ofl linearl growth.l
Plottingl thel valuesl wel l calculatedl forl l Marco’sl collection,l wel canl seel thel valuesl
forml al straightl line,l thel shapel ofl linearl growth.
Figurel 3.7.1:l Marco'sl Bottlel Collection
,Linearl Growth
Ifl al quantityl startsl atl sizel P0l andl growsl byl dl everyl timel period,l thenl thel
quantityl afterl nl timel periodsl canl bel determinedl usingl eitherl ofl thesel relations:
Recursivel form:l Pn=Pn−1+d
Explicitl form:l Pn=P0+dn
Inl thisl equation,l dl representsl thel commonl difference,l thel amountl thatl thel populationl
changesl eachl timel nl increasesl byl 1.
Connectionl tol Priorl Learningl –l Slopel andl Intercept
Youl mayl recognizel thel commonl difference,l d,l inl ourl linearl equationl asl slope.l Inl
fact,l thel entirel explicitl equationl shouldl lookl l familiar,l itl isl thel samel asl thel linearl
equation,l y=mx+b.
Inl thel standardl algebraicl equation,l y=mx+b,l bl wasl thel y-intercept,l or
thel yl valuel whenl xl wasl zero.l Inl thel forml ofl thel equationl wel l arel using,l wel
arel usingl P0l tol representl thatl initiall amount.
Example
Thel populationl ofl elkl inl al nationall forestl wasl measuredl to
bel 12,000l inl 2018,l andl wasl measuredl againl tol bel 15,000l inl 2022.l Ifl thel
populationl continuesl tol growl linearlyl atl thisl rate,l whatl willl thel elkl populationl bel
inl 2029?
, Tol begin,l wel l needl tol definel howl wel l arel goingl tol measurel n.l Rememberl thatl
P0l isl thel populationl whenl n=0,l sol wel l probablyl dol notl wantl tol literallyl usel thel
yearl 0.l Sincel wel l alreadyl knowl thel populationl inl 2018,l letl us
definel n=0l tol bel thel yearl 2018.l Thenl P0=12,000.
Nextl wel l needl tol findl d.l Rememberl dl isl thel growthl perl timel period,l inl thisl
casel growthl perl year.l Betweenl thel twol measurements,l thel populationl grewl byl
15,000−12,000=3,000,l butl itl tookl 2022−2018=4l yearsl tol growl thatl much.l Tol findl thel
growthl perl year,l wel l can
divide:l 3000l elkl /l 4l yearsl =750l elkl perl year.
Wel canl nowl writel ourl equationl inl whicheverl forml isl preferred.l Recursivel form:
P0=12,000
Pn=Pn−1+750
Explicitl form:l Pn=12,000+750n
Tol answerl thel question,l wel l needl tol firstl notel thatl thel yearl 2029l will
bel n=11,l sincel 2029l isl 11l yearsl afterl 2018.l Thel explicitl forml willl bel easierl tol
usel forl l thisl calculation:
P11=12,000+750(11)=20,250l elk
Whenl goodl modelsl gol bad
Whenl usingl mathematicall modelsl tol predictl futurel behavior,l itl isl importantl tol keepl
inl mindl thatl veryl fewl trendsl willl continuel indefinitely.
Supposel al four-year-oldl boyl isl currentlyl 39l inchesl tall,l andl youl arel toldl tol expectl
himl tol growl 2.5l inchesl al year.
Wel canl setl upl al growthl model,l withl n=0l correspondingl tol 4l yearsl old.l Recursivel
form:
P0=39
Pn=Pn−1+2.5
Explicitl form:
Pn=39+2.5n
Sol l atl 6l yearsl old,l wel l wouldl expectl himl tol be
P2=39+2.5(2)=44l inchesl tall
Mostl mathematicall modelsl willl breakl downl eventually.l Certainly,l wel l shouldl notl
expectl thisl boyl tol continuel tol growl atl thel samel ratel alll hisl life.l Ifl hel did,l atl
agel 50l hel wouldl bel P46=39+2.5(46)=154l inchesl talll =12.8l feetl tall!
Whenl usingl anyl mathematicall model,l wel l havel tol considerl whichl inputsl arel
reasonablel tol use.l Wheneverl wel l extrapolate,l orl l makel predictionsl intol thel future,l
wel l arel assumingl thel modell willl continuel tol bel valid.
Mostl everydayl numbersl involvel units.l Al unitl relatesl al valuel tol al specificl quantityl
ofl al thing,l asl inl 90l minutes:l 90l isl al value,l whilel 90l minutesl isl al specificl
quantityl ofl al thingl (minutes).l Unitsl oftenl arel objectsl likel peoplel orl cars,l orl l
measurementsl likel inchesl orl l pounds.
2026/l 2027l Update)l Quantitativel Literacyl
Guide|l Questionsl &l Answersl |l Gradel A|l
100%l Correctl (Verifiedl Solutions)
Linearl Models
Growthl Models
Populationsl ofl people,l animals,l andl itemsl arel growingl alll aroundl us.l Byl
understandingl howl thingsl grow,l wel l canl betterl understandl whatl tol expectl inl thel
future.l Inl thisl chapter,l wel l focusl onl time-dependentl change.
Linearl (Algebraic)l Growth
Marcol isl al collectorl ofl antiquel sodal bottles.l Hisl collectionl currently
containsl 437l bottles.l Everyl year,l hel budgetsl enoughl moneyl tol buyl 32l newl bottles.l
Canl wel l determinel howl manyl bottlesl hel willl havel inl 5l years,l andl howl longl itl
willl takel forl l hisl collectionl tol reachl 1,000l bottles?l Whilel bothl ofl thesel questionsl
youl couldl probablyl solvel withoutl anl equationl orl l formall mathematics,l wel l arel
goingl tol formalizel ourl approachl tol thisl probleml tol providel al meansl tol answerl
morel complicatedl questions.l Suppose
thatl Pnl representsl thel number,l orl l population,l ofl bottlesl Marcol has
afterl nl years.l Sol l P0l wouldl representl thel numberl ofl bottlesl now,l P1l wouldl
representl thel numberl ofl bottlesl afterl 1l year,l P2l wouldl representl thel numberl ofl
bottlesl afterl 2l years,l andl sol on.l Wel couldl describel howl Marco’sl bottlel collectionl isl
changingl using:
P0=437
Pn=Pn−1+32
Thisl isl calledl al recursivel relationship.l Al recursivel relationshipl isl al formulal whichl
relatesl thel nextl valuel inl al sequencel tol thel previousl values.l Here,l thel numberl ofl
bottlesl inl yearl nl canl bel foundl byl addingl 32l tol thel numberl ofl bottlesl inl thel
previousl year,l Pn−1.l Usingl thisl relationship,l wel l couldl calculate:
P1=P0+32=437+32=469
P2=P1+32=469+32=501
P3=P2+32=501+32=533
P4=P3+32=533+32=565
P5=P4+32=565+32=597
,Wel havel answeredl thel questionl ofl howl manyl bottlesl Marcol willl havel inl 5l years.l
However,l solvingl howl longl itl willl takel forl l hisl collectionl to
reachl 1,000l bottlesl wouldl requirel al lotl morel calculations.l Whilel recursivel
relationshipsl arel excellentl forl l describingl simplyl andl cleanlyl howl al quantityl isl
changing,l theyl arel notl convenientl forl l makingl predictionsl orl l solvingl problemsl thatl
stretchl farl intol thel future.l Forl l that,l al closedl orl l explicitl forml forl l thel
relationshipl isl preferred.l Anl explicitl equationl allowsl usl to
calculatel Pnl directly,l withoutl needingl tol knowl Pn−1.l Whilel youl mayl already
bel ablel tol guessl thel explicitl equation,l letl usl derivel itl froml thel recursivel formula.l
Wel canl dol sol byl selectivelyl notl simplifyingl asl wel l go:
P1=437+32=437+1(32)l P2=P1+32=437+32+32=437+2(32)l
P3=P2+32=(437+2(32))+32=437+3(32)l P4=P3+32=(437+3(32))+32=437+4(32)
Youl canl probablyl seel thel patternl now,l andl generalizel that
Pn=437+n(32)=437+32n
Usingl thisl equation,l wel l canl calculatel howl manyl bottlesl he’lll havel afterl 5l years:
P5=437+32(5)=437+160=597
Wel canl nowl alsol solvel forl l whenl thel collectionl willl reachl 1,000l bottlesl byl
substitutingl inl 1,000l forl l Pnl andl solvingl forl l n
1,000=437+32n
563=32nl n=56332=17.59
Sol l Marcol willl reachl 1,000l bottlesl inl 18l years.
Inl thel previousl example,l Marco’sl collectionl grewl byl thel samel numberl ofl bottlesl
everyl year.l Thisl constantl changel isl thel definingl characteristicl ofl linearl growth.l
Plottingl thel valuesl wel l calculatedl forl l Marco’sl collection,l wel canl seel thel valuesl
forml al straightl line,l thel shapel ofl linearl growth.
Figurel 3.7.1:l Marco'sl Bottlel Collection
,Linearl Growth
Ifl al quantityl startsl atl sizel P0l andl growsl byl dl everyl timel period,l thenl thel
quantityl afterl nl timel periodsl canl bel determinedl usingl eitherl ofl thesel relations:
Recursivel form:l Pn=Pn−1+d
Explicitl form:l Pn=P0+dn
Inl thisl equation,l dl representsl thel commonl difference,l thel amountl thatl thel populationl
changesl eachl timel nl increasesl byl 1.
Connectionl tol Priorl Learningl –l Slopel andl Intercept
Youl mayl recognizel thel commonl difference,l d,l inl ourl linearl equationl asl slope.l Inl
fact,l thel entirel explicitl equationl shouldl lookl l familiar,l itl isl thel samel asl thel linearl
equation,l y=mx+b.
Inl thel standardl algebraicl equation,l y=mx+b,l bl wasl thel y-intercept,l or
thel yl valuel whenl xl wasl zero.l Inl thel forml ofl thel equationl wel l arel using,l wel
arel usingl P0l tol representl thatl initiall amount.
Example
Thel populationl ofl elkl inl al nationall forestl wasl measuredl to
bel 12,000l inl 2018,l andl wasl measuredl againl tol bel 15,000l inl 2022.l Ifl thel
populationl continuesl tol growl linearlyl atl thisl rate,l whatl willl thel elkl populationl bel
inl 2029?
, Tol begin,l wel l needl tol definel howl wel l arel goingl tol measurel n.l Rememberl thatl
P0l isl thel populationl whenl n=0,l sol wel l probablyl dol notl wantl tol literallyl usel thel
yearl 0.l Sincel wel l alreadyl knowl thel populationl inl 2018,l letl us
definel n=0l tol bel thel yearl 2018.l Thenl P0=12,000.
Nextl wel l needl tol findl d.l Rememberl dl isl thel growthl perl timel period,l inl thisl
casel growthl perl year.l Betweenl thel twol measurements,l thel populationl grewl byl
15,000−12,000=3,000,l butl itl tookl 2022−2018=4l yearsl tol growl thatl much.l Tol findl thel
growthl perl year,l wel l can
divide:l 3000l elkl /l 4l yearsl =750l elkl perl year.
Wel canl nowl writel ourl equationl inl whicheverl forml isl preferred.l Recursivel form:
P0=12,000
Pn=Pn−1+750
Explicitl form:l Pn=12,000+750n
Tol answerl thel question,l wel l needl tol firstl notel thatl thel yearl 2029l will
bel n=11,l sincel 2029l isl 11l yearsl afterl 2018.l Thel explicitl forml willl bel easierl tol
usel forl l thisl calculation:
P11=12,000+750(11)=20,250l elk
Whenl goodl modelsl gol bad
Whenl usingl mathematicall modelsl tol predictl futurel behavior,l itl isl importantl tol keepl
inl mindl thatl veryl fewl trendsl willl continuel indefinitely.
Supposel al four-year-oldl boyl isl currentlyl 39l inchesl tall,l andl youl arel toldl tol expectl
himl tol growl 2.5l inchesl al year.
Wel canl setl upl al growthl model,l withl n=0l correspondingl tol 4l yearsl old.l Recursivel
form:
P0=39
Pn=Pn−1+2.5
Explicitl form:
Pn=39+2.5n
Sol l atl 6l yearsl old,l wel l wouldl expectl himl tol be
P2=39+2.5(2)=44l inchesl tall
Mostl mathematicall modelsl willl breakl downl eventually.l Certainly,l wel l shouldl notl
expectl thisl boyl tol continuel tol growl atl thel samel ratel alll hisl life.l Ifl hel did,l atl
agel 50l hel wouldl bel P46=39+2.5(46)=154l inchesl talll =12.8l feetl tall!
Whenl usingl anyl mathematicall model,l wel l havel tol considerl whichl inputsl arel
reasonablel tol use.l Wheneverl wel l extrapolate,l orl l makel predictionsl intol thel future,l
wel l arel assumingl thel modell willl continuel tol bel valid.
Mostl everydayl numbersl involvel units.l Al unitl relatesl al valuel tol al specificl quantityl
ofl al thing,l asl inl 90l minutes:l 90l isl al value,l whilel 90l minutesl isl al specificl
quantityl ofl al thingl (minutes).l Unitsl oftenl arel objectsl likel peoplel orl cars,l orl l
measurementsl likel inchesl orl l pounds.
