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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

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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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