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2
MAT
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MAT
ICS 1 Life Mathematics
1.1 Introduction
1.2 Revision-Profit,LossandSimpleInterest
1.3
ApplicationofPercentage,ProfitandLoss,OverheadExpenses,DiscountandTax
1.4 CompoundInterest
1.5 DifferencebetweenS.I.andC.I.
1.6 FixedDepositsandRecurringDeposits
1.7 CompoundVariation
1.8 TimeandWork
1.1 Introduction
Every human being wants to reach the height of
‘WIN-WIN’situation throughouthis life.Toachieve
iteffectively,heallocateshistimetowork,toearnmorewealthandfame.
Fromstoneagetopresentworld,frommaterialexchangetomoneytransaction,forhisproduceandland,manhasbeenapplyingtheideaofratioandproportion.ThemonumentalbuildingsliketheTajMahalandtheTanjoreBrihadisvaraTemple,knownfortheiraestheticlooks,alsodemonstrateourancestors’knowledgeandskillofusingrightkindofratiotokeepthemstrongandwonderful.
Manyoftheexistingthingsintheworldareconnectedbycauseandeffectrelationshipasinrainandharvest,nutritionandhealth,incomeandexpenditure,etc.andhencecompoundvariationarises.
Inourefforttosurviveandambitiontogrow,weborrowordepositmoneyandcompensatetheprocesspreferablybymeansofcompoundinterest.
The government bears the responsibility of the sectors
likesecurity,health,educationandotheramenities.Todeliver these
toallcitizens,wepayvarioustaxesfromourincometothegovernment.
Thischaptercoversthetopicswhichareinterwoveninourlives.
Roger Bacon[1214-1294]
Englishphilosopher,Wonderfulteacheremphasisedonempiricalmethods.Hebecameamaster
at Oxford.
He stated: “Neglectofmathematics
worksinjurytoallknowledge”.Hesaid,“The
importanceofmathematics for a common man
tounderpinnedwheneverhevisitsbanks,shoppingmalls,railways,postoffices,insurancecompanies,ordealswithtransport,businesstransaction,importsandexports,tradeandcommerce”.
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1.2 Revision: Profit, Loss and Simple Interest
Wehavealreadylearntaboutprofitandlossandsimpleinterestinourpreviousclass.Letusrecallthefollowingresults:
ReSuLtS on PRofIt, LoSS and SImPLe InteReSt
(i) ProfitorGain = Sellingprice–Costprice
(ii) Loss = Costprice–Sellingprice
(iii) Profit% = 100.C.P.Profit
#
(iv) Loss% = 100C.P.Loss
#
(v)Simpleinterest(I) =100
Principal Time Rate# # Pnr100=
(vi) Amount = Principal+Interest
1.3 application of Percentage, Profit and Loss, overhead
expenses, discount and tax
1.3.1. application of Percentage
Wehavealreadylearntpercentagesinthepreviousclasses.Wepresenttheseideasasfollows:
(i) Twopercent = 2%=1002
(ii) 8%of600kg =1008 ×600=48kg
(ii) 125% =100125 =
45 =1
41
Now,welearntoapplypercentagesinsomeproblems.Example 1.1
Whatpercentis15paiseof2rupees70paise?Solution 2rupees70paise =
(2×100paise+70paise) = 200paise+70paise = 270paise
Requiredpercentage =
27015 100# =
950 =5 %
95 .
=21 =50%
=41 =25%
=43 =75%
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Example
1.2Findthetotalamountif12%ofitis`1080.SolutionLetthetotalamountbex
.Given : 12%ofthetotalamount = `1080 x
10012
# = 1080 x = ×
12108 1000 =`9000
` Thetotalamount = `9000.Example 1.3
72% of 25 students are good in Mathematics. How many are not
good inMathematics?
Solution PercentageofstudentsgoodinMathematics = 72%
NumberofstudentsgoodinMathematics = 72%of25students =
10072 25# = 18 students
NumberofstudentsnotgoodinMathematics = 25–18=7.Example 1.4
Findthenumberwhichis15%lessthan240.Solution 15%of240 =
10015 240# =36
` Therequirednumber = 240–36=204.Example 1.5
ThepriceofahouseisdecreasedfromRupeesFifteenlakhstoRupeesTwelvelakhs.Findthepercentageofdecrease.
Solution Originalprice = `15,00,000 Changeinprice = `12,00,000
Decreaseinprice = 15,00,000–12,00,000=3,00,000 ̀
Percentageofdecrease= 100
1500000300000
# =20%Remember
Percentageofincrease = Original amountIncrease in amount
×100
Percentageofdecrease = Original amountDecrease in amount
×100
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exeRcISe 1.1
1. Choosethecorrectanswer.
(i)
Thereare5orangesinabasketof25fruits.Thepercentageoforangesis___
(A)5% (B)25% (C)10% (D)20%
(ii) 252 =_______%.
(iii) 15%ofthetotalnumberofbiscuitsinabottleis30.
Thetotalnumberofbiscuitsis_______.
(A)100 (B)200 (C)150 (D)300
(iv) Thepriceofascooterwas`
34,000lastyear.Ithasincreasedby25%thisyear.Thentheincreaseinpriceis_______.
(A) `6,500 (B)`8,500 (C)`8,000 (D)`7,000
(v) A man saves
`3,000permonthfromhistotalsalaryof`20,000.Thepercentageofhissavingsis_______.
(A)15% (B)5% (C)10% (D)20%
2. (i) 20%ofthetotalquantityofoilis40litres.
Findthetotalquantityofoilinlitres.
(ii) 25%ofajourneycovers5,000km.Howlongisthewholejourney? (iii)
3.5%ofanamountis` 54.25. Find the amount. (iv)
60%ofthetotaltimeis30minutes.Findthetotaltime. (v)
4%salestaxonthesaleofanarticleis`2.Whatistheamountofsale?
3.
Meenuspends`2000fromhersalaryforrecreationwhichis5%ofhersalary.Whatishersalary?
4.
25%ofthetotalmangoeswhicharerottenis1,250.Findthetotalnumberofmangoesinthebasket.Also,findthenumberofgoodmangoes.
15sweetsaredividedbetweenSharathandBharath,sothattheyget20%and80%ofthemrespectively.Findthenumberofsweetsgotbyeach.
MyGrandmasays,inherchildhood,goldwas`100pergram.Readanewspaper
toknow thepriceofgoldandnotedownthepriceon
thefirstofeverymonth.Findthepercentageofincrease every month.
(A)25 (B)4 (C)8 (D)15
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5.
ThemarksobtainedbyRaniinhertwelfthstandardexamsaretabulatedbelow.Expressthesemarksaspercentages.
Subjects maximum marks marks obtainedPercentage of
marks (out of 100)(i)English 200 180(ii)Tamil 200 188(iii)
Mathematics 200 195(iv) Physics 150 132(v) Chemistry 150
142(vi)Biology 150 140
6.
Aschoolcricketteamplayed20matchesagainstanotherschool.Thefirstschoolwon25%ofthem.Howmanymatchesdidthefirstschoolwin?
7.
Rahimdeposited`10,000inacompanywhichpays18%simpleinterestp.a.Findtheinteresthegetsforaperiodof5years.
8.
Themarkedpriceofatoyis`1,200.Theshopkeepergaveadiscountof15%.Whatisthesellingpriceofthetoy?
9.
Inaninterviewforacomputerfirm1,500applicantswereinterviewed.If12%ofthemwereselected,howmanyapplicantswereselected?Alsofindthenumberofapplicantswhowerenotselected.
10.
Analloyconsistsof30%copperand40%zincandtheremainingisnickel.Findtheamountofnickelin20kilogramsofthealloy.
11.
PandianandThamaraicontestedfortheelectiontothePanchayatcommitteefromtheirvillage.Pandiansecured11,484voteswhichwas44%ofthetotalvotes.Thamaraisecured36%ofthevotes.Calculate(i)thenumberofvotescastinthevillageand(ii)thenumberofvoterswhodidnotvoteforboththecontestants.
12.
Amanspends40%ofhisincomeforfood,15%forclothesand20%forhouserentandsavestherest.Whatisthepercentageofhissaving?Ifhisincomeis
`34,400,findtheamountofhissavings.
13.
Jyothikasecured35marksoutof50inEnglishand27marksoutof30inMathematics.Inwhichsubjectdidshegetmoremarksandhowmuch?
14.
Aworkerreceives`11,250asbonus,whichis15%ofhisannualsalary.Whatishismonthlysalary?
15.
Thepriceofasuitisincreasedfrom`2,100to`2,520.Findthepercentageofincrease.
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1.3.2 applications of Profit and Loss
Inthissection,welearntosolveproblemsonapplicationsofprofitandloss.
(i) Illustration of the formula for
S.P.Considerthefollowingsituation:Rajesh
buysapenfor`80andsellsittohisfriend.
Ifhewantstomakeaprofitof5%,canyousaythepriceforwhichhewouldhavesold?
Rajesh bought the pen for ` 80 which is the
CostPrice(C.P.).Whenhesold,hemakesaprofitof5%whichiscalculatedontheC.P.
` Profit = 5%ofC.P.=1005 ×80=` 4
Sincethereisaprofit,S.P.>C.P. S.P. = c.P. + Profit = 80+4=`
84.` ThepriceforwhichRajeshwouldhavesold=`
84.Thesameproblemcanbedoneusingtheformula.
Sellingprice(S.P.) = 100100 Profit% C.P#+^ h .
=100
100 580#
+^ h =100105 80# =` 84.
1. 40%=100%–_____% 2.
If25%ofstudentsinaclasscometoschoolbywalk,65%ofstudents
come by bicycle and the remaining percentage by school bus,
whatpercentageofstudentscomebyschoolbus?
3.
Inaparticularclass,30%ofthestudentstakeHindi,50%ofthemtakeTamilandtheremainingstudentstakeFrenchastheirsecondlanguage.WhatisthepercentageofstudentswhotakeFrenchastheirsecondlanguage?
4.
Inacity,30%arefemales,40%aremalesandtheremainingarechildren.Whatisthepercentageofthechildren?
AmuthabuyssilksareesfromtwodifferentmerchantsGanesanandGovindan.Ganesanweaves200gmofsilverthreadwith100gmofbronzethreadwhereasGovindanweaves300gmofsilverthreadwith200gmofbronzethreadforthesarees.Calculatethepercentageofsilverthreadineachandfindwhogivesabetterquality.[Note:Morethesilverthreadbetterthequality.]
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(ii) Illustration of the formula for
c.P.Considerthefollowingsituation:Supposeashopkeepersellsawristwatchfor`540
makingaprofitof5%,thenwhatwouldhavebeenthecostofthewatch?
Theshopkeeperhadsoldthewatchataprofitof5%ontheC.P.SinceC.P.isnotknown,letustakeitas`100.
Profitof5%ismadeontheC.P. ` Profit = 5%ofC.P. =
1005 ×100
Weknow, S.P. = c.P. + Profit = 100+5 =
`105.Here,ifS.P.is`105,thenC.P.is`100.WhenS.P.ofthewatchis`540,C.P.
=
105540 100# = ` 514.29
`
Thewatchwouldhavecost`514.29fortheshopkeeper.Theaboveproblemcanalsobesolvedbyusingtheformulamethod.
c.P. = %100100profit S.P.#+c m
=100 5100+
×540
=105100 ×540
= ` 514.29.
WenowsummarizetheformulaetocalculateS.P.andC.P.asfollows:
1. Whenthereisaprofit 1. Whenthereisaloss
(i)C.P.= 100 profit%100 S.P.#
+c m (ii)C.P.= 100 loss%
100 S.P.#-
` j
2.Whenthereisaprofit 2. Whenthereisaloss,
(i)S.P.= %100
100 profit C.P.#+c m (ii)S.P.= %100
100 loss C.P.#-` j
=` 5.
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Revathiusedtheformulamethod: Loss = 15%. S.P. = `13,600
C.P. = 100 Loss%100-
×S.P.
=100 15
100-
×13600
=85100 ×13600
= `16,000
Example 1.6
HameedbuysacolourT.Vsetfor`15,200andsellsitatalossof20%.Whatis
thesellingpriceoftheT.Vset?SolutionRaghulusedthismethod:Lossis20%oftheC.P.
= 1520010020
#
= `3040
S.P. = C.P.–Loss
= 15,200–3,040
= `12,160
BothRaghulandRoshancameoutwiththesameanswerthatthesellingpriceoftheT.V.setis`12,160.
Example 1.7
Ascootyissoldfor`13,600andfetchesalossof15%.Findthecostpriceof
the scooty.Deviusedthismethod:Lossof15%means,if C.P. is
`100,Loss=` 15Therefore,S.P.wouldbe (100–15) = `
85IfS.P.is`85,C.P.is`100WhenS.P.is`13,600,then C.P. =
85100 13600#
= `16,000Hencethecostpriceofthescootyis`16,000.
Items cost price in ` Profit/LossSelling
price in `WashingMachine 16,000 9%Profit
MicrowaveOven 13,500 12%LossWoodenShelf 13%Loss 6,786
Sofaset 12½%Profit 7,000Air Conditioner 32,400 7%Profit
Roshanusedtheformulamethod: C.P. = `15,200 Loss = 20% S.P. =
100
100 Loss%- × C.P.
= 15200100
100 20#-
= 1520010080
#
= `12,160
OR
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Example 1.8
Thecostpriceof11pensisequaltothesellingpriceof10pens.Findtheloss
orgainpercent.SolutionLetS.P.ofeachpenbex. S.P.of10pens = `10x
S.P.of11pens = ` 11xGiven: C.P.of11pens =
S.P.of10pens=`10xHere,S.P.>C.P. ` Profit = S.P.–C.P. = 11x–10x=x
Profit% = C.P.
Profit ×100=x
x10
×100=10%.Example 1.9
Amansellstwowristwatchesat`594each.Ononehegains10%andontheotherheloses10%.Findhisgainorlosspercentonthewhole.
SolutionGiven :S.P.ofthefirstwristwatch =`594,Gain%=10%
` C.P.ofthefirstwristwatch =%100 profit
100 S.P.#+
= 594100 10
100#
+^ h
= 594110100
# =`540.
Similarly,C.P.ofthesecondwatchonwhichheloses10%is
= 100 Loss%100 S.P.#-^ h
= 594100 10100
#-^ h = 90
100 594# =` 660.
TosaywhethertherewasanoverallProfitorLoss,weneedtofindthecombinedC.P.andS.P.
TotalC.P.ofthetwowatches = 540+660=`1,200.
TotalS.P.ofthetwowatches = 594+594=`1,188. NetLoss = 1,200–1,188=`
12.
Loss% = 100C.P.Loss
#
=120012 100# =1%.
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1.3.3. application of overhead expenses
Maya went with her father to purchase an Aircooler.Theybought it
for` 18,000.The shopwhereinthey bought was not closer to their
residence. Sothey had to arrange a vehicle to take the air cooler
totheir residence. They paid conveyance charges of ` 500. Hence the
C.P. of the air cooler is not only` 18,000 but it also includes the
conveyance charges (Transportation charges)
`500whichiscalledasoverhead expenses .
Now, C.P.oftheaircooler = Realcost+Conveyancecharges =
18,000+500=`18,500Consideranothersituation,whereKishore’sfatherbuysanoldMaruticarfrom
aChennaidealerfor`2,75,000andspends`25,000forpaintingthecar.Andthenhetransportsthecartohisnativevillageforwhichhespendsagain`2,000.Canyouanswerthefollowingquestions:
(i) Whatisthetheoverallcostpriceofthecar? (ii)
Whatistherealcostpriceofthecar? (iii)
Whataretheoverheadexpensesreferredhere?In the
aboveexamplethepaintingchargesandthetransportationchargesare
theoverheadexpenses.
\Costpriceofthecar = Realcostprice+Overheadexpenses
= 2,75,000+(25,000+2,000) =
2,75,000+27,000=`3,02,000.Thus,wecometotheconclusionthat,
Sometimeswhen an article is bought or sold, some additional
expenses
occurwhilebuyingorbeforesellingit.Theseexpenseshavetobeincludedinthecostprice.Theseexpensesarereferredtoasoverhead
expenses.Thesemayincludeexpenseslikeamountspentonrepairs,labourcharges,transportation,etc.,
Example 1.10Raju bought amotorcycle for` 36,000 and then bought
some extra fittings
tomakeitperfectandgoodlooking.Hesoldthebikeataprofitof10%andhegot
`44,000.Howmuchdidhespendtobuytheextrafittingsmadeforthemotorcycle?
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SolutionLettheC.P.be`100. Profit =
10%,S.P.=`110IfS.P.is`110,thenC.P.is`100.WhenS.P.is`44,000 C.P.
=
11044000 100# =`40,000
\Amountspentonextrafittings=40,000–36,000=`4,000.
exeRcISe 1.2
1. FindtheCostprice/Sellingprice.
cost price Selling price Profit Loss
(i) `7,282 `208
(ii) ` 572 ` 72
(iii) `9,684 ` 684
(iv) `1,973 ` 273
(v) `6,76,000 `18,500
2. Filluptheappropriateboxesandleavetherest.
C.P. S.P. Profit & Profit % Loss & Loss%
(i) `320 ` 384
(ii) `2,500 `2,700
(iii) `380 ` 361
(iv) `40 ` 2 Loss
(v) `5,000 `500Profit.
3. FindtheS.P.ifaprofitof5%ismadeon
(i) abicycleof`700with`50asoverheadcharges.
(ii)
acomputertableboughtat`1,150with`50astransportationcharges.
(iii)
atable-topwetgrinderboughtfor`2,560andanexpenseof`140onrepaircharges.
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4.
Bysellingatablefor`1,320,atradergains10%.FindtheC.P.ofthetable. 5.
Thecostpriceof16notebooksisequaltothesellingpriceof12notebooks.Find
thegainpercent.
6.
Amansoldtwoarticlesat`375each.Onthefirstarticle,hegains25%andontheother,heloses25%.Howmuchdoeshegainorloseinthewholetransaction?Also,findthegainorlosspercentinthewholetransaction.
7. Anbarasanpurchasedahousefor̀ 17,75,000andspent̀
1,25,000onitsinteriordecoration.He sold thehouse
tomakeaprofitof20%.Find theS.P.of thehouse.
8.
AfterspendingRupeessixtythousandforremodellingahouse,Amlasoldahouseataprofitof20%.IfthesellingpricewasRupeesfortytwolakhs,howmuchdidshespendtobuythehouse?
9. Jaikumarboughtaplotof land in theoutskirtsof thecity
for`21,00,000.Hebuiltawallarounditforwhichhespent`1,45,000.Andthenhewantstosellit
at `25,00,000bymakinganadvertisementinthenewspaperwhichcostshim
`5,000.Now,findhisprofitpercent.
10.
Amansoldtwovarietiesofhisdogfor`3,605each.Ononehemadeagainof15%andontheotheralossof9%.Findhisoverallgainorloss.
[Hint: FindC.P.ofeach]
1.3.4 application of discounts
YesterdayPoojawent to a shopwith her parents
topurchaseadressforPongalfestival.Shesawmanybannersintheshop.Thecontentofwhichwasnotunderstandbyher.
With an unclear mind, she entered the shop
andpurchasedafrock.
Thepricelabelledonthefrockwas`550.ItiscalledasMarkedPrice(abbreviatedasM.P.)andshegavetheshopkeeper`
550.Buttheshopkeeperreturnedthebalanceamountandinformedherthattherewasadiscountof20%.
Here,20%discountmeans,20%discountontheMarkedPrice.
Discount=10020 ×550=`110.
discount is the reduction in value on themarked Price or List
Price of the article.
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AmountpaidbyPoojatotheshopkeeperis`440
= `550- `110
= MarkedPrice-DiscountHenceweconcludethefollowing:
Discount = MarkedPrice-SellingPrice SellingPrice =
MarkedPrice-Discount MarkedPrice = SellingPrice+Discount
Example 1.11A bicyclemarked at ` 1,500 is sold for ` 1,350.What
is the percentage of
discount?SolutionGiven :MarkedPrice=`1500,SellingPrice=`1350
Amountofdiscount = MarkedPrice–SellingPrice = 1500–1350 = `150
Discountfor`1500 = `150
Discountfor`100 =1500150 100#
Percentageofdiscount = 10%.Example 1.12
Thelistpriceofafrockis`220.Adiscountof20%onsalesisannounced.Whatistheamountofdiscountonitanditssellingprice?
Solution Given :
List(Marked)Priceofthefrock=`220,Rateofdiscount=20%
Amountofdiscount = 100
20 220#
= ` 44 \SellingPriceofthefrock = MarkedPrice–Discount = 220–44 =
` 176.
During festival seasonsand in the Tamil monthof ‘‘Aadi’’,
discounts orrebates of 10%, 20%,30%, etc., are offered toattract
customers byCo-optex, Khadi and
othershopsforvariousitemstopromotesales.
SinceDiscountisonMarkedPrice,wewillhavecalculatediscount on
M.P.
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Example 1.13Analmirahis sold at ` 5,225 after allowing a
discount of 5%. Find its
markedprice.SolutionKrishnausedthismethod:Thediscountisgiveninpercentage.Hence,theM.P.istakenas`100.
Rateofdiscount = 5% Amountofdiscount =
1005 ×100
= ` 5. SellingPrice = M.P.–Discount = 100–5=` 95
IfS.P.is`95,thenM.P.is`100.WhenS.P.is`5225,
M.P. =95100 × 5225
` M.P.ofthealmirah=`5,500.Example 1.14
Ashopkeeperallowsadiscountof10%tohiscustomersandstillgains20%.Findthemarkedpriceofanarticlewhichcosts`450totheshopkeeper.
SolutionVanithausedthismethod:LetM.P.be`100.Discount=10%ofM.P.
=
10010 ofM.P.= 100
10010
#
=`10S.P. =M.P.-Discount =100- 10=`90Gain =20%ofC.P. = 450
10020
# =` 90
S.P. =C.P.+Gain =450+90=`540.IfS.P.is`
90,thenM.P.is`100.WhenS.P.is`540,
M.P.=90
540 100# =`600
` TheM.P.ofanarticle=`600
Vigneshusedtheformulamethod: S.P. = Rs5225Discount = 5% M.P. =
?
M.P.=100 Discount%
100 S.P.#-
` j
= 5225100 5100
#-
` j
= 522595100
#
= `5,500.
Vimalusedtheformulamethod:Discount=10%,Gain=20%,C.P.=`450,M.P.=?
M.P. =100 Discount%100 Gain% C.P#-+ .
=100 10
100 20450#
-
+
^^
hh
=90120 450#
=`600
[OR]
[OR]
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Example 1.15
Adealerallowsadiscountof10%andstillgains10%.Whatisthecostprice
ofthebookwhichismarkedat`220?SolutionSugandanusedthismethod:M.P.
=`220.Discount =10%ofM.P.
= 22010010
# = ` 22
S.P. =M.P.–Discount =220–22=` 198LetC.P.be`100.Gain
=10%ofC.P.
= 10010010
# = ` 10
S.P. =C.P.+Gain =100+10 =`110.
IfS.P.is`110,thenC.P.is`100.WhenS.P.is`198, C.P. =
110198 100#
=`180.Example 1.16
Atelevisionsetwassoldfor`14,400aftergivingsuccessivediscountsof10%and20%respectively.Whatwasthemarkedprice?
Solution SellingPrice = `14,400LettheM.P.be`100. Firstdiscount =
10%= 100
10010
# =`10S.P.afterthefirstdiscount= 100–10=`90 Seconddiscount =
20%= 90
10020
# =` 18SellingPriceaftertheseconddiscount =90 – 18=`
72IfS.P.is`72,thenM.P.is`100.WhenS.P.is`14,400, M.P. =
7214400 100# =`20,000
M.P. = `20,000.
Mukundanusedtheformulamethod:
Discount=10%
Gain =10%
M.P. =`220
C.P. = 100 Gain%100 Discount% M.P.#
+-
=100 10100 10 220#
+-
=11090 220# =`180.
[OR]
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Example
1.17Atraderbuysanarticlefor`1,200andmarksit30%abovetheC.P.Hethen
sellsitafterallowingadiscountof20%.FindtheS.P.andprofitpercent.Solution:LetC.P.ofthearticlebe`100M.P.=30%aboveC.P.=`130If
C.P. is `100,thenM.P.is`130.
WhenC.P.is`1200, M.P. =100
1200 130# =`1560
Discount=20%of1560 =10020 1560# =` 312
S.P. =M.P.–Discount
= 1560–312=` 1248
Profit = S.P.–C.P.
= 1248–1200=` 48.
\Profit% = 100C.P.Profit
#
=120048 100# =4%
Ashopgives20%discount.WhatwillbetheS.P.ofthefollowing? (i)
Adressmarkedat`120 (ii) Abagmarkedat`250 (iii)
Apairofshoesmarkedat`750.
1.3.5 application of tax
Very oftenwe find advertisements in newspapersandon
televisionrequestingpeople topay their taxes
intime.Whatisthistax?WhydoestheGovernmentcollectthetaxfromthecommonpeople?
Governmentneedsfundstoprovideinfrastructurefacilitieslikeroads,railways,hospitals,schoolsetc.,forthepeople.TheGovernmentcollectsthefundsrequiredbyimposingvarioustaxes.
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Taxesareoftwotypes:
1. direct tax
TaxwhichiscollectedintheformofIncomeTax,PropertyTax,ProfessionalTax,WaterTax,etc.,iscalledasDirectTax.Thesearepaiddirectlytothegovernmentbythepublic.
2. Indirect tax
SomeofthetaxeswhicharenotpaiddirectlytothegovernmentareIndirectTaxesandareexplainedbelow.
excise tax
Thistaxischargedonsomeitemswhicharemanufacturedinthecountry.ThisiscollectedbytheGovernmentofIndia.
Service tax
Tax which is charged in Hotels, Cinema theatres, for service of
CharteredAccountants,TelephoneBills,etc.,comeunderServiceTax.ThistaxiscollectedbytheserviceproviderfromtheuseranddepositedtotheGovernment.
Income taxThis is themost important source of revenue for
theGovernmentwhich
iscollectedfromeverycitizenwhoisearningmore
thanaminimumstipulatedincomeannually.Astruecitizensofourcountry,weshouldbeawareofourdutyandpaythetaxontime.
Sales tax / Value added tax
Sales taxSalesTaxisthetaxleviedonthesalesmade
byaselleratthetimeofsellingtheproduct.Whenthebuyerbuysthecommoditythesalestaxispaidbyhimtogetherwiththepriceofthecommodity.
ThissalestaxischargedbytheGovernmenton thesellingpriceofan
itemand isadded to thevalueofthebill.
These days, however, the prices include thetaxknownasValue added
tax (Vat).ThismeansthatthepricewepayforanitemisaddedwithVAT.
SalesTaxischargedbytheGovernmentontheSalesofanItem.
7777 7777
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CanyoufindtheprevailingrateofSalesTaxforsomecommoditiesintheyear2011.
1.Electricalinstruments_____%2.Petrol_____%
3.Diesel___%4.Domesticappliances_____%5.Chemicals_____%
calculation of Sales tax
AmountofSalestax = 100Rate of Sales tax Cost of the item#
RateofSalestax = Cost of the itemAmount of Sales tax 100#
Billamount = Costoftheitem+AmountofSalestax
Example 1.18
Vinodhpurchasedmusicalinstrumentsfor`12,000.Iftherateofsalestaxis
8%,findthesalestaxandthetotalamountpaidbyhim.Solution
Valueofthemusicalinstruments = `12,000 RateofSalesTax = 8%
AmountofSalesTax = 12000
1008
#
= `960 TotalamountpaidbyVinodhincludingSalesTax = 12,000+960
=`12,960
Example 1.19
Arefrigeratorispurchasedfor`14,355,includingsalestax.Iftheactualcost
priceoftherefrigeratoris`13,050,findtherateofsalestax.SolutionGiven:Fortherefrigerator,billamount=`14,355,Costprice=`13,050.
Salestax = Billamount–Costoftheitem = 14,355–13,050=`1,305
RateofSalesTax = Cost of the itemAmount of Sales Tax 100#
= 100130501305
# =10%
TheGovernmentgivesexemptionofSalesTaxforsomecommoditieslike
rice, sugar, milk, salt, pen,pencilsandbooks.
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Example 1.20
Priyaboughtasuitcasefor`2,730.TheVATforthisitemis5%.Whatwasthe
priceofthesuitcasebeforeVATwasadded?AlsostatehowmuchistheVAT.SolutionGiven
:VATis5%.IfthepricewithoutVATis`100,thenthepriceincludingVATis`105.Now,whenpriceincludingVATis`105,originalpriceis`100.WhenpriceincludingVATis`2,730,theoriginalpriceofthesuitcase
=
105100 2730# =`2,600
Theoriginalpriceofthesuitcase=` 2,600 \VAT =
2,730–2,600=`130
1.
Findthebuyingpriceofeachofthefollowingwhen5%SalesTaxisaddedonthepurchaseof:
(i)Apillowfor`60(ii)Twobarsofsoapat` 25 each. 2.
If8%VATisincludedintheprices,findtheoriginalpriceof: (i) An
electric water heater bought for ` 14,500 (ii) A crockery set
boughtfor`200.
exeRcISe 1.3 1. Choosethecorrectanswer: (i)
Thediscountisalwaysonthe_______. (A)MarkedPrice (B)CostPrice
(C)SellingPrice (D)Interest
(ii) IfM.P.=`140,S.P.=`105,thenDiscount=_______. (A) `245 (B)`
25 (C) `30 (D)` 35
(iii) ______=MarkedPrice–Discount. (A)CostPrice (B)SellingPrice
(C)ListPrice (D)Marketprice
(iv) Thetaxaddedtothevalueoftheproductiscalled______ Tax.
(A)SalesTax (B)VAT (C)ExciseTax (D)ServiceTax
(v)
IftheS.P.ofanarticleis`240andthediscountgivenonitis`28,thentheM.P.is_______.
(A) `212 (B)` 228 (C) `268 (D)` 258
2.
Thepricemarkedonabookis`450.Theshopkeepergives20%discountonitainbookexhibition.WhatistheSellingPrice?
3.
Atelevisionsetwassoldfor`5,760aftergivingsuccessivediscountsof10%and20%respectively.WhatwastheMarkedPrice?
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4.
Sekarboughtacomputerfor`38,000andaprinterfor`8,000.Iftherateofsalestaxis7%fortheseitems,findthepricehehastopaytobuythesetwoitems.
5.
ThesellingpricewithVAT,onacookingrangeis`19,610.IftheVATis6%,whatistheoriginalpriceofthecookingrange?
6. Richardgotadiscountof10%onthesuithebought.Themarkedpricewas
`5,000forthesuit.Ifhehadtopaysalestaxof10%onthepriceatwhichhebought,howmuchdidhepay?
7.
Thesalestaxonarefrigeratorattherateof9%is`1,170.Findtheactualsaleprice.
8.
Atradermarkshisgoods40%abovethecostprice.Hesellsthematadiscountof5%.Whatishislossorgainpercentage?
9.
AT.V.withmarkedprice`11,500issoldat10%discount.Duetofestivalseason,theshopkeeperallowsafurtherdiscountof5%.FindthenetsellingpriceoftheT.V.
10.
Apersonpays`2,800foracoolerlistedat`3,500.Findthediscountpercentoffered.
11.
Deepapurchased15shirtsattherateof`1,200eachandsoldthemataprofitof5%.Ifthecustomerhastopaysalestaxattherateof4%,howmuchwilloneshirtcosttothecustomer?
12.
Findthediscount,discountpercent,sellingpriceandthemarkedprice.
Sl. no Items m. P Rate of
discountamount of discount
S. P
(i) Saree `2,300 20%
(ii) Pen set `140 `105
(iii) Diningtable 20% `16,000
(iv)WashingMachine `14,500 `13,775
(v) Crockeryset `3,224 12½%
Whichisabetteroffer?Twosuccessivediscountsof20%and5%orasinglediscountof25%.Giveappropriatereasons.
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1.4. compound Interest
InclassVII,wehavelearntaboutSimpleInterestandtheformulaforcalculatingSimpleInterestandAmount.Inthischapter,weshalldiscusstheconceptofCompoundInterestand
themethodofcalculatingCompoundInterestandAmountattheendofacertainspecifiedperiod.
Vinay borrowed ` 50,000 from a bank for a
fixedtimeperiodof2years.attherateof4%perannum.
Vinayhastopayforthefirstyear,
Simpleinterest = n r100P # #
=100
50000 1 4# # =`2,000
Supposehefailstopaythesimpleinterest`2,000attheendoffirstyear,thenthe
interest
`2,000isaddedtotheoldPrincipal`50,000andnowthesum=P+I=`52,000becomesthenewPrincipalforthesecondyearforwhichtheinterestiscalculated.
Nowinthesecondyearhewillhavetopayaninterestof
S.I. = n r100P # #
=100
52000 1 4# # =`2,080
Therefore Vinay will have to pay moreinterest for the second
year.
Thisway of calculating interest is calledcompound Interest.
Generallyinbanks,insurancecompanies,postoffices and inother
companieswhich
lendmoneyandacceptdeposits,compoundinterestisfollowedtofindtheinterest.
Example
1.21Ramlaldeposited`8,000withafinancecompanyfor3yearsataninterestof
15%perannum.
WhatisthecompoundinterestthatRamlalgetsafter3years?SolutionStep 1:
Principalforthefirstyear = `8,000
Interestforthefirstyear = n r100P # #
=100
8000 1 15# # =`1,200 Amount at the end of first year =
P+I=8,000+1,200=`9,200
When the interest is paidon the Principal only, it
iscalledSimple
Interest.ButiftheinterestispaidonthePrincipalaswellason
theaccruedinterest,itiscalledcompound Interest.
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Step 2: TheamountattheendofthefirstyearbecomesthePrincipalforthe
second year.
Principalforthesecondyear = `9,200
Interestforthesecondyear = n r100P # #
=100
9200 1 15# # =`1,380
Amount at the end of second year = P+I =9,200+1,380=`10,580Step
3: TheamountattheendofthesecondyearbecomesthePrincipalforthe
third year.
Principalforthethirdyear =`10,580
Interestforthethirdyear = n r100P # #
=100
10580 1 15# # =`1,587
Amount at the end of third year = P+I
=
10,580+1,587=`12,167Hence,theCompoundInterestthatRamlalgetsafterthreeyearsis
A–P = 12,167–8,000=`4,167.
deduction of formula for compound Interest
The above method which we have used for the calculation of
CompoundInterestisquitelengthyandcumbersome,especiallywhentheperiodoftimeisverylarge.HenceweshallobtainaformulaforthecomputationofAmountandCompoundInterest.
IfthePrincipalisP,Rateofinterestperannumisr
%andtheperiodoftimeorthenumberofyearsisn,thenwededucethecompoundinterestformulaasfollows:
Step 1 : Principalforthefirstyear = P
Interestforthefirstyear = n r100P # #
= 100P 1 r# # =100
Pr
Amount at the end of first year = P+I
= P 100Pr+
= rP 1 100+` j
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Step 2 : Principalforthesecondyear = P 1 100r+` j
Interestforthesecondyear =1
1001r r
100
P # #+` j
(usingtheS.I.formula)
= r rP 1 100 100#+` j
Amount at the end of second year = P+I
= r r rP 1 100 P 1 100 100#+ + +` `j j
= r rP 1 100 1 100+ +` `j j
= rP 1 1002
+` j
Step 3 : Principalforthethirdyear = rP 1 1002
+` j
Interestforthethirdyear =1
1001P r r
100
2
# #+` j
(usingtheS.I.formula)
= r rP 1 100 1002#+` j
Amount at the end of third year = P+I
= r r rP 1 100 P 1 100 1002 2
#+ + +` `j j
= r rP 1 100 1 1002
+ +` `j j
= rP 1 1003
+` j
Similarly, Amount at the end of nth year is A = 1P 100r n+`
j
and C.I.attheendof‘n’yearsisgivenbyA–P
(i. e.) C. I. = 1100
P Pr n+ -` j
to compute compound Interest
case 1: compounded annually
WhentheinterestisaddedtothePrincipalattheendofeachyear,wesaythattheinterestiscompoundedannually.
Here 1A P 100r n= +` j andC.I.=A–P
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case 2: compounded Half - Yearly (Semi - annually)
WhentheinterestiscompoundedHalf-Yearly,therearetwoconversionperiodsinayeareachafter6months.Insuchsituations,theHalf-Yearlyratewillbehalfoftheannualrate,thatis
2r` j.
Inthiscase, 1 r21100A P
n2
= + ` j8 B andC.I.=A–P
case 3: compounded Quarterly
Whentheinterestiscompoundedquarterly,therearefourconversionperiodsinayearandthequarterlyratewillbeone-fourthoftheannualrate,thatis
r4` j.
Inthiscase, 1 r41100A P
n4
= + ` j8 B andC.I.=A–P
case 4: compounded when time being fraction of a year
When interest is compounded annually but time being a
fraction.Inthiscase,wheninterestiscompoundedannuallybuttimebeingafractionof
ayear,say5 41 years,thenamountAisgivenby
A = 1r r
41
100P 1 100
5
. .
+ +` `j j8 BandC.I.=A–P
for 5 years for ¼ of yearExample 1.22
Find the C.I. on
`15,625at8%p.a.for3yearscompoundedannually.SolutionWeknow,
Amountafter3years = rP 1 1003
+` j
= 15625 1 1008 3+` j
= 15625 1 252 3+` j
= 15625 2527 3` j
= 156252527
2527
2527
# # #
= `19,683Now,Compoundinterest = A–P=19,683–15,625 = `4,058
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to find the c.I. when the interest is compounded annually or
half-yearlyLetusseewhathappensto`100overaperiodofoneyearifaninterestis
compoundedannuallyorhalf-yearly.
S.No annually Half-yearly
1P=`100at10%perannumcompoundedannually
P=`100at10%perannumcompoundedhalf-yearly
2 Thetimeperiodtakenis1year Thetimeperiodis6monthsor½year.
3 I 100100 10 1# #= =`10 1 2I 100
100 10# #= =` 5
4 A=100+10=`110 A=100+5=`105Forthenext6months,P=`105
So,12I 100
105 10# #= =` 5.25
andA=105+5.25=`110.25
5 A=`110 A=`110.25
If interest is compounded half - yearly, the amount is more than
whencompoundedannually.Wecomputetheinteresttwotimesandrateistakenashalfoftheannualrate.
Example 1.23 Find the compound interest on` 1000 at the rate of
10%per annum for 18
monthswheninterestiscompoundedhalf-yearly.SolutionHere,P=`1000,r=10%perannumand
n=18months=
1218 years= 2
3 years=1 21 years
\Amountafter18months = r21
100P 1
2n+ ` j8 B
= 1000 121
10010 2 2
3
+#
` j8 B
= 1000 1 20010 3+` j
= 1000 2021 3` j
= 10002021
2021
2021
# # #
= ` 1157.625 = ` 1157.63 C.I. = A–P =1157.63–1000=` 157.63
A sum is taken foroneyear at 8%p. a. Ifinterest is
compoundedafter every three months, how manytimes will interest
bechargedinoneyear?
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Example 1.24
Findthecompoundintereston`20,000at15%perannumfor2
31 years.
Solution
Here,P=`20,000,r=15%p.a.andn =231 years.
Amount after 2 31 years=A = r r
100P 1 100 1 3
12+ +` `j j8 B
=1001520000 1 100
15 1 312+ +` `j j8 B
= 20000 1203 1
2012+ +` `j j
= 200002023
20212` `j j
= 200002023
2023
2021
# # #
= `27,772.50 C.I. = A–P = 27,772.50–20,000 = `7,772.50
Inverse Problems on compound
InterestWehavealreadylearnttheformula,A= rP 1 100 ,
n+` j
wherefourvariablesA,P,r and
nareinvolved.Outofthesefourvariables,ifanythreevariablesareknown,thenwecancalculatethefourthvariable.
Example 1.25
Atwhatrateperannumwill`640amountto`774.40in2years,wheninterest
isbeingcompoundedannually?Solution:Given:P=`640,A=`774.40,n=2years,r=?
Weknow, A = rP 1 100n
+` j
774.40 = r640 1 1002
+` j
.640774 40 = r1 100
2
+` j
6400077440 = r1 100
2
+` j
100121 = r1 100
2
+` j
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1011 2` j = r1 100
2+` j
1011 = r1 100+
r100 = 1011 1-
r100 = 1011 10-
r100 = 101
r =10100
Rate r = 10%perannum.
Example 1.26
Inhowmuchtimewillasumof`1600amountto`1852.20at5%perannum
compoundinterest.SolutionGiven:P=`1600,A=`1852.20,r=5%perannum,n=?
Weknow, A = rP 1 100n
+` j
1852.20 = 1600 11005 n+` j
.
16001852 20
= 100105 n` j
160000185220
= 2021 n` j
80009261 =
2021 n` j
2021 3` j = 20
21 n` j
\ n =3years
1.5 difference between Simple Interest and compound Interest
WhenPisthePrincipal,n=2yearsandristheRateofinterest,
DifferencebetweenC.I.andS.I.for2years= rP 1002
` j
Example 1.27
FindthedifferencebetweenSimpleInterestandCompoundInterestforasum
of `8,000lentat10%p.a.in2years.SolutionHere,P=`8000,
n=2years,r=10%p.a.
Findthetimeperiodand rate for each of the cases
givenbelow:1.Asumtakenfor2yearsat
8%p.a.compounded half-yearly.
2.Asumtakenfor1½yearsat4%p.a.compoundedhalf-yearly.
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DifferencebetweenCompoundInterestandSimpleInterestfor2years= rP
1002
` j
= 8000 10010 2` j
= 8000 101 2` j
= 8000101
101
# # =`80
exeRcISe 1.4
1. FindtheAmountandCompoundInterestinthefollowingcases:Sl.No.
PrincipalinRupees Rate%perannum Timeinyears
(i) 1000 5% 3(ii) 4000 10% 2
(iii) 18,000 10% 2 21
2.Sangeethaborrowed`8,000fromAlexfor2yearsat12½%perannum.WhatinterestdidSangeethapaytoAlexiftheinterestiscompoundedannually?
3. Maria invested
`80,000inabusiness.Shewouldbepaidinterestat5%perannumcompoundedannually.Find(i)theamountstandingtohercreditattheend
of second year and (ii) the interest for the third year.
4.
Findthecompoundintereston`24,000compoundedhalf-yearlyfor1½yearsattherateof10%perannum.
5.
FindtheamountthatDravidwouldreceiveifheinvests`8,192for18monthsat12½%perannum,theinterestbeingcompoundedhalf-yearly.
6.
Findthecompoundintereston`15,625for9months,at16%perannumcompoundedquarterly.
7.
FindthePrincipalthatwillyieldacompoundinterestof`1,632in2yearsat4%rateofinterestperannum.
8.Vickyborrowed`26,400fromabanktobuyascooterattherateof15%p.a.compoundedyearly.Whatamountwillhepayattheendof2yearsand4monthstocleartheloan?
9.
Ariftookaloanof`80,000fromabank.Iftherateofinterestis10%p.a.,findthedifferenceinamountshewouldbepayingafter1½yearsiftheinterestis
(i)compoundedannuallyand(ii)compoundedhalf-yearly.
10.
Findthedifferencebetweensimpleinterestandcompoundintereston`2,400at
2yearsat5%perannumcompoundedannually.
11.
Findthedifferencebetweensimpleinterestandcompoundintereston`6,400
for2yearsat6¼%p.a.compoundedannually.
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12. The difference betweenC. I. and S. I. for 2 years on a sum
ofmoney lent at 5%p.a.is`5.Findthesumofmoneylent.
13. Sujathaborrows`12,500at12%perannumfor3yearsat simple
interestandRadhika borrows the same amount for the same period at
10% per
annumcompoundedannually.Whopaysmoreinterestandbyhowmuch?
14.
Whatsumisinvestedfor1½yearsattherateof4%p.a.compoundedhalf-yearlywhichamountsto`1,32,651?
15.
Gayathriinvestedasumof`12,000at5%p.a.atcompoundinterest.Shereceivedan
amount of `13,230after‘n’years.Findthevalueof‘n’.
16.
Atwhatratepercentcompoundinterestperannumwill`640amountto
`774.40in2years?
17.
Findtheratepercentperannum,if`2,000amountto`2,315.25inanyearandahalf,interestbeingcompoundedhalf-yearly.
1.5.1 appreciation and depreciation
a) appreciation
In situations like growth of population, growth
ofbacteria,increaseinthevalueofanasset,increaseinpriceofcertainvaluablearticles,etc.,thefollowingformulaisused.
A= rP 1 100n
+` j
b)
depreciationIncertaincaseswherethecostofmachines,vehicles,
valueofsomearticles,buildings,etc.,decreases,thefollowingformulacanbeused.
A= rP 1 100n
-` j
Example
1.28Thepopulationofavillageincreasesattherateof7%everyyear.Ifthepresent
populationis90,000,whatwillbethepopulationafter2years?SolutionPresentpopulationP=90,000,Rateofincreaser=7%,Numberofyearsn=2.
Thepopulationafter‘n’years = rP 1 100
n+` j
\Thepopulationaftertwoyears = 90000 11007 2+` j
World Population Year Population 1700 600,000,000 1800
900,000,000 1900 1,500,000,000 2000
6,000,000,000In3centuries,populationhasmultiplied10fold.
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= 90000100107 2` j
= 90000100107
100107
# #
= 103041
Thepopulationaftertwoyears = 1,03,041Example 1.29
Thevalueofamachinedepreciatesby5%eachyear.Amanpays`30,000forthemachine.Finditsvalueafterthreeyears.
SolutionPresentvalueofthemachineP=`30,000,Rateofdepreciationr
=5%, Numberofyearsn = 3
Thevalueofthemachineafter‘n’years = rP 1 100n
-` j
\Thevalueofthemachineafterthreeyears = 30000 11005 3-` j
= 3000010095 3` j
= 3000010095
10095
10095
# # #
= 25721.25 Thevalueofthemachineafterthreeyears = `25,721.25
Example
1.30Thepopulationofavillagehasaconstantgrowthof5%everyyear.Ifitspresent
populationis1,04,832,whatwasthepopulationtwoyearsago?SolutionLetPbethepopulationtwoyearsago.
` P 1 1005 2+` j = 104832
P 100105 2` j = 104832
P100105
100105
# # = 104832
P =105 105
104832 100 100#
# #
= 95085.71 =
95,086(roundingofftothenearestwholenumber)\Twoyearsagothepopulationwas95,086.
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exeRcISe 1.5
1.
Thenumberofstudentsenrolledinaschoolis2000.Iftheenrollmentincreasesby5%everyyear,howmanystudentswillbethereaftertwoyears?
2.
Acarwhichcosts`3,50,000depreciatesby10%everyyear.Whatwillbetheworthofthecarafterthreeyears?
3.
Amotorcyclewasboughtat`50,000.Thevaluedepreciatedattherateof8%perannum.Findthevalueafteroneyear.
4.
InaLaboratory,thecountofbacteriainacertainexperimentwasincreasingattherateof2.5%perhour.Findthebacteriaattheendof2hoursifthecountwasinitially5,06,000.
5.
Fromavillagepeoplestartedmigratingtonearbycitiesduetounemploymentproblem.Thepopulationofthevillagetwoyearsagowas6,000.Themigrationistakingplaceattherateof5%perannum.Findthepresentpopulation.
6.
Thepresentvalueofanoilengineis`14,580.Whatwastheworthoftheengine3yearsbeforeifthevaluedepreciatesattherateof10%everyyear?
7.
Thepopulationofavillageincreasesby9%everyyearwhichisduetothejobopportunitiesavailableinthatvillage.Ifthepresentpopulationofthevillageis11,881,whatwasthepopulationtwoyearsago?
1.6 fixed deposits and Recurring deposits
Banks,postofficesandmanyotherfinancialinstitutionsacceptdepositsfrompublicatvaryingratesofinterest.Peoplesaveintheseinstitutionstogetregularperiodicalincome.
Differentsavingschemesareofferedbythesefinancialinstitutions.Fewofthoseschemesare
(i)FixedDepositand(ii)RecurringDeposit
(i) fixed
depositInthistypeofdeposit,peopleinvestaquantumofmoneyforspecificperiods.
SuchadepositiscalledFixedDeposit(inshortform,F.D.)note:Depositscaneitherbeforashorttermorlongterm.Dependingonthe
periodofdeposits,theyofferahigherrateofinterest.
(ii) Recurring
depositRecurringDeposit(inshortform,R.D.)isentirelydifferentfromFixedDeposit.Inthisscheme,thedepositorhasthefreedomtochooseanamountaccording
tohissavingcapacity,tobedepositedregularlyeverymonthoveraperiodofyearsinthebankorinthepostoffice.
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Thebankorpostofficerepaysthetotalamountdepositedtogetherwiththeinterestattheendoftheperiod.ThistypeofDepositisknownasRecurringDeposit.note:
TheinterestonRecurringDepositiscalculated
usingsimpleinterestmethod.
to find the formula for calculating interest and the maturity
amount for R.d :
Let r
%betherateofinterestpaidand‘P’bethemonthlyinstalmentpaidfor‘n’months.
Interest= 100PN ,r where
2
1n nN 121 += ^ h; E years
TotalAmountdueatmaturityisA=P 100PNn r+
Example
1.31TharunmakesadepositofRupeestwolakhsinabankfor5years.Iftherateof
interestis8%perannum,findthematurityvalue.SolutionPrincipaldepositedP
=`2,00,000,n=5years,r=8%p.a. Interest = 100
Pnr =200000 51008
# #
= `80,000 \Maturityvalueafter5years =
2,00,000+80,000=`2,80,000.
Example
1.32Vaideeshdeposits`500atthebeginningofeverymonthfor5yearsinapost
office.Iftherateofinterestis7.5%,findtheamounthewillreceiveattheendof5years.
Solution Amountdepositedeverymonth,P =`500 Numberofmonths,n
=5×12=60months Rateofinterest,r = 7 % %
21
215=
Totaldepositmade =Pn=500×60 =`30,000 Periodforrecurringdeposit,N
=
2
1n n
121 +^ h; E years
=241 60 61# # = 2
305 years
Themonthly instalmentscan be paid on any
daywithinthemonthforR.D.
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Interest,I = 100PNr
=5002
3052 100
15# #
#
=`5,718.75
Totalamountdue = nP 100PNr+
=30,000+5,718.75 =`35,718.75
Example
1.33Vishaldeposited`200permonthfor5yearsinarecurringdepositaccountin
apostoffice.Ifhereceived`13,830findtherateofinterest.SolutionMaturityAmount,A=`13,830,P=`200,n=5×12=60months
Period,N =2
1n n
121 +^ h; E years
= 121 60
261
2305
# # = years
AmountDeposited=Pn =200×60=`12,000
MaturityAmount = Pn100PNr+
13830 =12000 200 r2
305100
# #+
13830–12000 =305×r
1830 =305×r
\ r =3051830 =6%
1.6.1 Hire Purchase and Instalments
Banksandfinancialinstitutionshaveintroducedaschemecalledhirepurchaseandinstalmenttosatisfytheneedsoftoday’sconsumers.
Hire purchase:
Underthisscheme,thearticlewillnotbeownedbythebuyerforacertainperiodoftime.Onlywhenthebuyerhaspaidthecompletepriceofthearticlepurchased,he/shewillbecomeitsowner.
Instalment:
Thecostofthearticlealongwithinterestandcertainotherchargesisdividedbythenumberofmonthsoftheloanperiod.Theamountthusgotisknownastheinstalment.
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equated monthly Instalment ( e.m.I. )
EquatedMonthlyInstalmentisalsoasequivalentastheinstalmentschemebutwithadimnishingconcept.Wehavetorepaythecostofthingswiththeinterestalongwithcertaincharges.Thetotalamountshouldbedividedbytheperiodofmonths.TheamountthusarrivedisknownasEquatedMonthlyInstalment.
E.M.I Number of monthsPrincipal Interest
=+
different schemes of Hire purchase and Instalment scheme
1. 0% interest scheme:
Companiestakeprocessingchargeand4or5monthsinstalmentsinadvance.
2. 100%
finance:Companiesaddinterestandtheprocessingchargestothecostprice.
3. discount
Sale:Topromotesales,discountisgivenintheinstalmentschemes.4.
Initial
Payment:Acertainpartofthepriceofthearticleispaidtowardsthe
purchaseinadvance.ItisalsoknownasCashdownpayment.Example
1.34
Thecostpriceofawashingmachineis̀
18,940.Thetablegivenbelowillustratesvariousschemestopurchasethewashingmachinethroughinstalments.Choosethebestschemetopurchase.
Sl. no
different schemes
S. P. in `
Initial payments
Rate of interest
Processing fee Period
(i) 75%Finance 18,940 25% 12% 1%24
months
(ii) 100%Finance 18,940 Nil 16% 2%24
months
(iii) 0%Finance 18,9404 E. M. I.
in advance
Nil 2% 24 months
CalculatetheE.M.I.andthetotalamountfortheaboveschemes.Solution(i)
75% finance P =
`18,940,Initialpayment=25%,Rate=12%,Processingfee=1% Processingfee
= 1%of`18,940 = 18940
1001
# =`189.40- ` 189 Initialpayment = 25%of`18,940 = 18940
10025
# =`4,735
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Loanamount = 18,940–4,735=`14,205
Interest =100
14205 12 2# #
= `3,409.20- `3,409
E.M.I. = Number of monthsLoan amount Interest+
=24
14205 3409+ =24
17614
= ` 733.92 - ` 734\Totalamounttobepaid= 4,735+14,205+3,409+189 =
`22,538(ii) 100% finance Processingfee = 2%of`18,940 = 18940
1002
# =`378.80- ` 379 RateofInterest = 16% Interest = 18940 100
16 2# #
= `6060.80- `6,061
E.M.I. = Number of monthsLoan amount Interest+
=24
18940 6061+ =24
25001
= `1,041.708- `1,041.71 = `1,042 Totalamounttobepaid =
6,061+18,940+379=`25,380(iii) 0% interest scheme Processingfee =
2%of`18,940
= 189401002
# =`378.80- ` 379
E.M.I. = Number of monthsLoan amount Interest+
=24
18940 0+ =24
18940
= ` 789.166 - ` 789
Totalamounttobepaid = 18,940+3,156+379=`22,475
AdvanceE.M.I.paid =
`789×4=`3,156Hence,0%interestschemeisthebestscheme.
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Example
1.35Thecostofacomputeris`20,000.Thecompanyoffersitin36months,but
charges10%interest.Findthemonthlyinstalmentthepurchaserhastopay.SolutionCostofcomputer=`20,000,Interest=10%p.a.,Period=36months(3years)
TotalInterest = 2000010010 3# #
= `6,000
\Totalamounttobepaid = 20,000+6,000
= `26,000
MonthlyInstalment = Number of monthsTotal amount
=36
26000
= ` 722. 22
- ` 722
exeRcISe 1.6
1.
Ponmanimakesafixeddepositof`25,000inabankfor2years.Iftherateofinterestis4%perannum,findthematurityvalue.
2.
Devamakesafixeddepositof`75,000inabankfor3years.Iftherateofinterestis5%perannum,findthematurityvalue.
3.
Imrandeposits`400permonthinapostofficeasR.D.for2years.Iftherateofinterestis12%,findtheamounthewillreceiveattheendof2years.
4.
Thecostofamicrowaveovenis`6,000.Pooraniwantstobuyitin5instalments.Ifthecompanyoffersitattherateof10%p.a.SimpleInterest,findtheE.M.I.andthetotalamountpaidbyher.
5.
Thecostpriceofarefrigeratoris`16,800.Ranjithwantstobuytherefrigeratorat0%financeschemepaying3E.M.I.inadvance.Aprocessingfeeof3%isalsocollectedfromRanjith.FindtheE.M.I.andthetotalamountpaidbyhimforaperiodof24months.
6.
Thecostofadiningtableis`8,400.Venkatwantstobuyitin10instalments.IfthecompanyoffersitforaS.I.of5%p.a.,findtheE.M.I.andthetotalamountpaidbyhim.
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1.7 compound Variation
IntheearlierclasseswehavealreadylearntaboutDirectandInverseVariation.Letusrecallthem.
direct Variation
If two quantities are such that an increase or decrease in one
leads to acorresponding increase or decrease in the other, we say
they vary directly or thevariationisDirect.
examples for direct Variation:
1.Distance andTime are inDirectVariation, becausemore the
distancetravelled,thetimetakenwillbemore(ifspeedremainsthesame).
2.PrincipalandInterestareinDirectVariation,becauseifthePrincipalismoretheinterestearnedwillalsobemore.
3.PurchaseofArticlesandtheamountspentareinDirectVariation,becausepurchaseofmorearticleswillcostmoremoney.
Indirect Variation or Inverse Variation:
If two quantities are such that an increase or decrease in one
leads to acorrespondingdecreaseor increase in theother,wesay
theyvary indirectlyor thevariation is inverse.
examples for Indirect Variation:
1.
WorkandtimeareinInverseVariation,becausemorethenumberoftheworkers,lesserwillbethetimerequiredtocompleteajob.
2. SpeedandtimeareinInverseVariation,becausehigherthespeed,
theloweristhetimetakentocoveradistance.
3.
PopulationandquantityoffoodareinInverseVariation,becauseifthepopulationincreasesthefoodavailabilitydecreases.
compound Variation
Certainproblemsinvolveachainoftwoormorevariations,whichiscalledasCompoundVariation.
Thedifferentpossibilitiesofvariationsinvolvingtwovariationsareshowninthefollowingtable:
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Variation I Variation IIDirect DirectInverse InverseDirect
InverseInverse Direct
Letusworkoutsomeproblemstoillustratecompoundvariation.Example
1.36
If20mencanbuildawall112meterslongin6days,whatlengthofasimilarwallcanbebuiltby25menin3days?
Solution:method 1:
Theprobleminvolvessetof3variables,namely-Numberofmen,
Numberofdaysandlengthofthewall.number of men number of days
Length of the wall in metres
20 6 11225 3 x
Step 1 :
Considerthenumberofmenandthelengthofthewall.Asthenumberofmenincreasesfrom20to25,thelengthofthewallalsoincreases.SoitisinDirectVariation.
Therefore,theproportionis20:25::112:x ...... (1)
Step 2:
Considerthenumberofdaysandthelengthofthewall.Asthenumberofdaysdecreasesfrom6to3,thelengthofthewallalsodecreases.So,itisinDirectVariation.
Therefore,theproportionis6:3::112:x ..... (2)
Combining(1)and(2),wecanwrite
::
: x20 25
6 3112S1
Weknow,Product of extremes = Product of means.
extremes means extremes
20 : 25 ::112 : x 6 : 3
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So, x20 6# # = 25 3 112# # x =
20 625 3 112
## # =70meters.
method 2number of men number of days Length of the wall in
metres
20 6 11225 3 x
Step 1:Consider thenumberofmenand lengthof thewall.As
thenumberofmenincreasesfrom20to25,thelengthofthewallalsoincreases.Itisindirect
variation.
Themultiplyingfactor2025=
Step 2:
Considerthenumberofdaysandthelengthofthewall.Asthenumberofdaysdecreases
from6 to3, the lengthof thewallalsodecreases. It is indirect
variation.
Themultiplyingfactor 63= .
\ x = 1122025
63
# # =70meters
Example
1.37Sixmenworking10hoursadaycandoapieceofworkin24days.Inhowmany
dayswill9menworkingfor8hoursadaydothesamework?Solutionmethod 1:
Theprobleminvolves3setsofvariables,namely-Numberofmen,
WorkinghoursperdayandNumberofdays.number of men number of hours
per day number of days
6 10 249 8 x
Step 1:
Considerthenumberofmenandthenumberofdays.Asthenumberofmenincreasesfrom6to9,thenumberofdaysdecreases.SoitisinInverseVariation.
Thereforetheproportionis9:6::24: x ..... (1)Step 2:
Considerthenumberofhoursworkedperdayandthenumberofdays.
Asthenumberofhoursworkingperdaydecreasesfrom10to8,thenumberofdaysincreases.Soitisinverse
variation.
Thereforetheproportionis8:10::24:x .....
(2)Combining(1)and(2),wecanwriteas
:
:: : : x
9 6
8 10241
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Weknow,Product of extremes = Product of means. extremes means
extremes 9 : 6 : : 24 : x 8 :10So, 9×8×x = 6×10×24 x =
9 86 10 24
## # =20days
note: 1. DenotetheDirectvariationas. (Downwardarrow) 2.
DenotetheIndirectvariationas- (Upwardarrow) 3. Multiplying Factors
can bewritten based on the arrows. Take the
numberontheheadofthearrowinthenumeratorandthenumberonthetailofthearrowinthedenominator.
Formethodtwo,usetheinstructionsgiveninthenoteabove.method 2 :
(using arrow marks)
number of men number of hours per day number of days6 10 249 8
x
Step 1 :
Considermenanddays.Asthenumberofmenincreasesfrom6to9,thenumberofdaysdecreases.Itisininverse
variation.
Themultiplyingfactor=96
Step 2 :
Considerthenumberofhoursperdayandthenumberofdays.Asthenumberofhoursperdaydecreasesfrom10to8,thenumberofdaysincreases.Itisalsoininverse
variation.
Themultiplyingfactor=810
\ x = 24 2096
810
# # = days.
exeRcISe 1.7 1.
Twelvecarpentersworking10hoursadaycompleteafurnitureworkin18days.
Howlongwouldittakefor15carpentersworkingfor6hoursperdaytocompletethesamepieceofwork?
2.
Eightymachinescanproduce4,800identicalmobilesin6hours.Howmanymobilescanonemachineproduceinonehour?Howmanymobileswould25machinesproducein5hours?
3.
If14compositorscancompose70pagesofabookin5hours,howmanycompositorswillcompose100pagesofthisbookin10hours?
4.
If2,400sq.m.oflandcanbetilledby12workersin10days,howmanyworkersareneededtotill5,400sq.m.oflandin18days?
5.
Working4hoursdaily,Swaticanembroid5sareesin18days.Howmanydayswillittakeforhertoembroid10sareesworking6hoursdaily?
6. A sum of ` 2,500depositedinabankgivesaninterestof`
100in6months.Whatwillbetheintereston`3,200for9monthsatthesamerateofinterest?
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1.8 time and Work
When we have to compare the work of several persons, it is
necessary
toascertaintheamountofworkeachpersoncancompleteinoneday.Astimeandworkareofinversevariationandifmorepeoplearejoinedtodoawork,theworkwillbecompletedwithinashortertime.
Insolvingproblemshere,thefollowingpointsshouldberemembered:
1. Ifamanfinishestotalworkin‘n’days,theninonedayhedoes‘n1
’of
thetotalwork.Forexample,ifamanfinishesaworkin4days,theninone day
he does
41 ofthework.
2. If the quantity ofwork doneby aman in oneday is given, then
thetotalnumberofdaystakentofinishthework=1/(oneday’swork).Forexample,ifamandoes
101 oftheworkin1day,thenthenumberofdays
takentofinishthework
101
11110
#= =` j
=10days.
Example
1.38Acandoapieceofworkin20daysandBcandoitin30days.Howlongwill
theytaketodotheworktogether?SolutionWorkdonebyAin1day= 20
1 ,WorkdonebyBin1day= 301
WorkdonebyAandBin1day =201
301+
= 603 2+ =
605
121= ofthework
TotalnumberofdaysrequiredtofinishtheworkbyAandB= 12112
1 = days.Example 1.39
AandBtogethercandoapieceofworkin8days,butAalonecandoit12days.HowmanydayswouldBalonetaketodothesamework?
Solution WorkdonebyAandBtogetherin1day = 8
1 ofthework
WorkdonebyAin1day = 121 ofthework
WorkdonebyBin1day =81
121- =
243 2
241- =
NumberofdaystakenbyBalonetodothesamework= 24124
1 = days.
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Example 1.40
TwopersonsAandBareengagedinawork.Acandoapieceofworkin12daysandBcandothesameworkin20days.Theyworktogetherfor3daysandthenAgoesaway.InhowmanydayswillBfinishthework?
Solution
WorkdonebyAin1day =121
WorkdonebyBin1day =201
WorkdonebyAandBtogetherin1day = 121
201+
=60
5 3608
152+ = =
WorkdonebyAandBtogetherin3days =152 3
52
# =
RemainingWork = 152
53- =
NumberofdaystakenbyBtofinishtheremainingwork=20153
53
120
#=
= 12days.
Example
1.41AandBcandoapieceofworkin12days,BandCin15days,CandAin20
days.Inhowmanydayswilltheyfinishittogetherandseparately?Solution
WorkdonebyAandBin1day =
121
WorkdonebyBandCin1day =151
WorkdonebyCandAin1day = 201
Workdoneby(A+B)+(B+C)+(C+A)in1day = 121
151
201+ +
Workdoneby(2A+2B+2C)in1day = 605 4 3+ +
Workdoneby2(A+B+C)in1day = 6012
WorkdonebyA,BandCtogetherin1day =21
6012
# =101
WhileA,BandCworkingindividually can complete
ajobin20,5,4daysrespectively.Ifall jointogetherandwork,find in
howmany days theywillfinishthejob?
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` A,BandCwillfinishtheworkin10days.
WorkdonebyAin1day
(i.e.) [(A+B+C)’swork–(B+C)’swork] =101
151
303 2
301- = - =
` Awillfinishtheworkin30days.
WorkdonebyBin1day
(i.e.) [(A+B+C)’swork–(C+A)’swork] =101
201
202 1
201- = - =
` Bwillfinishtheworkin20days.
WorkdonebyCin1day
(i.e.) [(A+B+C)’swork–(A+B)’swork] =101
121
606 5
601- = - =
` Cwillfinishtheworkin60days.
Example
1.42Acandoapieceofworkin10daysandBcandoitin15days.Howmuchdoes
eachofthemgetiftheyfinishtheworkandearn` 1500?Solution
WorkdonebyAin1day = 101
WorkdonebyBin1day = 151
Ratiooftheirwork = :101
151 =3:2
TotalShare = `1500 A’sshare = 1500
53# =`900
B’sshare = 15052 0# =`600
Example
1.43Twotapscanfillatankin30minutesand40minutes.Anothertapcanempty
itin24minutes.Ifthetankisemptyandallthethreetapsarekeptopen,inhowmuchtimethetankwillbefilled?
SolutionQuantityofwaterfilledbythefirsttapinoneminute=
301
Quantityofwaterfilledbythesecondtapinoneminute=401
Quantityofwateremptiedbythethirdtapinoneminute=241
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Quantityofwaterfilledinoneminute,whenallthe3tapsareopened=30
1401
241+ -
=120
4 3 5+ - =1207 5-
=1202 =
601
Timetakentofillthetank =160
1 =60minutes
=1hour
exeRcISe 1.8
1.
Amancancompleteaworkin4days,whereasawomancancompleteitinonly12days.Iftheyworktogether,inhowmanydays,cantheworkbecompleted?
2.
Twoboyscanfinishaworkin10dayswhentheyworktogether.Thefirstboycandoitalonein15days.Findinhowmanydayswillthesecondboydoit
allbyhimself?
3.
ThreemenA,BandCcancompleteajobin8,12and16daysrespectively.
AandBworktogetherfor3days;thenBleavesandCjoins.Inhowmanydays,canAandCfinishthework?
4.
AtapAcanfilladrumin10minutes.AsecondtapBcanfillin20minutes.
AthirdtapCcanemptyin15minutes.Ifinitiallythedrumisempty,findwhenitwillbefullifalltapsareopenedtogether?
5.
Acanfinishajobin20daysandBcancompleteitin30days.Theyworktogetherandfinishthejob.If`600ispaidaswages,findtheshareofeach.
6.
A,BandCcandoaworkin12,24and8daysrespectively.Theyallworkforoneday.ThenCleavesthegroup.InhowmanydayswillAandBcompletetherestofthework?
7.
Atapcanfillatankin15minutes.Anothertapcanemptyitin20minutes.Initiallythetankisempty.Ifboththetapsstartfunctioning,whenwillthetankbecomefull?
abbreviation:
C.P.=CostPrice,S.P.=SellingPrice,M.P.=MarkedPrice,
P=Principal,r=Rateofinterest,n =timeperiod,
A=Amount,C.I.=CompoundInterest.
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Chapter 1
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Percent means per hundred. A fraction with its denominator 100
is called a percent.
Incaseofprofit,wehave
Profit=S.P.–C.P.;Profitpercent=C.PProfit 100#
S.P.=100
100 Profit% C.P.#+` j ;C.P.= 100 Profit%100 S.P.#
+` j
IncaseofLoss,wehave
Loss=C.P.–S.P.;Losspercent= 100C.P.Loss
#
S.P.=100
100 Loss% C.P.#-` j ;C.P.= 100 Loss%100 S.P.#-
` j
DiscountisthereductiongivenontheMarkedPrice.
Selling Price is the price payable after reducing theDiscount
from theMarkedPrice.
Discount=M.P.–S.P.
M.P.=100 Discount%
100 S.P.#-
; S.P.=100
100 Discount% M.P.#-
C.P.=100 Profit%100 Discount% M.P.#
+- ; M.P.=
100 Discount%100 Profit% C.P#
-+ .
DiscountPercent= 100.M.P.
Discount#
When the interest is
(i)compoundedannually,A= rP 1100
n+` j
(ii)compoundedhalf-yearly,A= 1 r21
100P
2n+ ` j8 B
(iii)compoundedquarterly,A=100r1
41P
n4+ ` j8 B
Appreciation,A= rP 1100
n+` j ; Depreciation,A= rP 1 100
n-` j
ThedifferencebetweenC.I.andS.I.for2years= r100
P 2` j
Oneday’sworkofA= 1Number of days taken by A
Workcompletedin‘x’days=Oneday’sworkx x