IMAGE EVALUATION TEST TARGET (MT-3) // ^/ ^ .<^. ^sf ■^ ^ .<^ .« » 1.0 I.I 11.25 m m 12.2 10 12.0 Hiolugi'dpluu ^ScJHices Coroaration mm 4(ISt 23 WKT MAM STUnT WBnm.N.Y. 145M (71«)t73-4S01 •t ■ ■ 1.1. IfcJa.." . .>>"!, .J. *u (rnaanintf "CON- TINUED"!, or tha symbol ▼ (maaning "END"h whichavar applias. Maps, platas. chartt. ate. may ba fiimad at diffarant raduction ratios. Thosa too iarga to ba antlraiy includad in ona axposura ara fiimad baginning in' tha uppar laft hand comar. laft to right and top to bottom, as many framas as raquirad. Tha following diagrams lilustrata tha mathod: 1 .^31* 3 L'axamplaira film# fut raproduil grica i la g4n4rosit4 d*: / ^ SoclStS du Nu»*e -du SaafiMlr* de Quebec Las imagas sulvantas ont 4t4 raproduitas avac la plus grand soin. compta tanu da la condition at da la nattat« da I'axamplaira film*, at •n conformity avac las conditions du contrat da filmaga. Las axamplairas origlnaux dont la couvartura %r\ papiar ast imprimAa aont filmds on commandant par la pramiar plat at rnxx tarminant soit par la damiAra paga qui comporta una amprainta d'imprassion ou d'illustration. aoit par la sacond plat aalon la cas. Toua las autras axamplairas eriginaux aont filmda mt% comrhan^ant par la pramiAra paga qui comporta una amprainta dimprassion ou d'illustration at an tarminant par la ^miAra paga qui comporta una talla amprainta. Un das iymboif s suivants apparaTt^a sur la damlAra imaga da chaqua microficha. salon la cas: la symbols -^ signifia "A 8UIVRE". la symbols V signifia "FIN". Las cartas, planchas. tableaux, ate. pauvant Atra filmte A das taux da reduction diffdrants. Lorsqua la document est trop grand pour Atra taproduit an un aeul clichA. il est filmA A psrtir ds rengle supArieur gauche, de geuche A dreita. at de haut en bas. en prenant la nombra d'images nAcessaire. Les diograrhmes suivsnts iilustrant le mAthdde. ( 1 2 3 • * r * . 4 5 6 _ '7. ■ ■ .»■ .f :*. *'^^ m »*to INTRODUC *■'.- TO THE TREATIS: ON eOMMERCIAL ARIT OOMBININQ THX ^ ANALYTIC AND SYNTHETIC METHODS; DESIGNED AS A COMPLETE TEXT-BOOK ON THIS SCIENCE, # COMMOJf SCHOOLS AND ACADEMIE& BT THl CHRISTIAN BROTHERS. . nooMD iDmoii. ^^^^^^^^^^MM^MVMMM^ QtfEBBO : 0. DABYBAU, PBINTBR AND PUBUSBBB, 8, MOVHTAm BILL. mi 5 t-1 SCHOOL-BOOKS PUBLUHID BT TBI CHRISTIAN BROTHERS ' IN THK DOMINION OF OAN^A. '^'^'^'VS^VWWVW • bW^SoSK!* ^'^ 8A0««'> HI?Wmr, .ppr^d b, U, Or- th. Arch- " mo., of art nV«, Thi. -IS"'^^'''" ^'^ OOMMBROIAL ARITeMRTIO •traoainTft^fnTo^QLbUTsTL" "~""'"* "^ "»• Conadl^f Public iS •-f.f.tiSy£ir^l^^^ S*""?."""^ Aritbrnodclowblcbl. *»«• wo «r tMohon. '^^•** QaeitloM in MonUl Arithmotlo, for l-Wtal AifthmoUc, for tho w ofllLi'h.l ""*** ""* '"" '^'^ ,rfi4.ta tt. t«, UieOfflceoftheMiDirtwofAgriottlliM. . KS )A. hoola; rao« Um Irch- r bli Lonbhip 'OpofQaebee. UTHMBTIO, ofPnblioIn- ie. to whioh ii rithmetio, for Umi on Men- •nntion and ed Qnutions ROIALB, la iURATIOir Worktwen 1870; (TTitAITi d'KnidflM Ithmltiqaa Mukmaat. *0(MiUIalq( the Tear mov, in PREFACE.' Although this book is only an Introduction to a higher course of Commercial Arithmetic, we expect that it will be found a sufficiently complete practical treatise for Common Schools and even for Academies; in a word, for the great majority of learners^. Decimals following the same scale as whole num- bers, we have chosen to treat of them with the latter ; we have then introduced them with Numeration. We have essentially followed the decimal system, but without neglectmg tha old method. Our main object has been to supply our schools with a practical, and, at the same time, a cheap book, within the reach of the laboring classes. We think it contains mqre examples than other works of the same size. And amongst its particular features, it offers the pre- cious advantage of Mercantile Forms, followed by nu- .meroos examples Of appUcation. having for principal object to render the pupil familiar with figures. Some desire the answers placed immediately after the examples, and others desire them omitted. Both methodshave their advantages and their disadvantages m order, therefore, that pupils may receive the ad vant- ages of both methods, the answers to nearly one third of the examples in this book are omitted. They will be found, together with clear solutions of all the exam- pies, in a Key to this work, prepared for the use of teachers and private learners. r principEd ires. itely after -ed. Both Ivantages. le ad van t- one third rhey will the exam- le use of CONTENTS. SIMPLE NUMBERS AND DECIMALS. Paoi. D«flnitkfMi Slgpf 10 itomM NoUtkm 11 A»bio NoUtioD 13 irain»raUon Table... „ H Rate for NoUUon 14 Rale for Nnmsratloo IS DeoimaU , 1(J Applioation 6r the Prineiplea of Na- meration M Addltioo „ „ 23 PaOc. BubtraoUon 28 MaltiplioatioD > 36 Cootraotioiu in Multiplication 44 Diriiion , 60 Contractions la Divi«ion......„ 69 Decimal Carrenoy 03 Kedaotion of Decimal Cnrrenoy 05 Practical Problem! combining the Fundamental Kales 68 Bills and Aoconnts 60 Forms of Bills and Aoooonts 70 PROPERTIES OP NUMBERS. Exact DiTlson and Prima Num- bers „ 78 Table of Prime Nnmbedl 70 Factoring , „„ T» CanoellaU*..... ^WS^w. 81 Common Divisor w... ......... 82 Qreatest Common Divisor,.. 83 Least Common Multiple 84 FRACTIONS. Definitions, etc „ »..m..> 88 Bndnetion of Frac^ons. 88 Addition of Fractions 94 Subtraction of Fractions Oft MollipliMtfoo ofFraottoos w Diriaion of Fractions ; 09 Qraateat Com. Divisor of Fractions! 103 Least Com. Multiple of Fracdons.. 104 Practice by Aliquot Parts 105 MisoeUanaonsProblem«.......„ io» DENOMINATE NUMBERS. Deflnttioos, eto.....« 113 Old Canadian Money m English Money. 414 United States Money 114 Fieooh Mon^ , „, m Troy Weight ,15 Apothecaries' Weight jkj Avoirdupois Weight "" n j Linear or Long Measure ..'"', 117 durwyon' Long Measium us h^' .* A'^iSRjGUiVSi--^:;-^ ooNTiirts. Page eqnan M«Miin ^......, ng Burwyon' 8qa»re MtMui* ijo Oabfa or Solid Mo«ui»..^., no Liquid Mearan u, Drjr MoMura •••••••••^••••••••t««««**««» MeMnro of Tim* „.„. ijj Circular MoMor* ^...."...".' 134 MiMoIIaaeoni TablM ijj The Matrio Bytma of WaighU and MaMuraa „. ufl B«IooUon of Compound Nomben. 134 Raduotion of the Old Oanadian Oor< niMytoAheDeeimalOiirraiiQx. 141 Pioa. Badaetion of the Deoimal Ourrehey to the Old Oanadian Carreooy. 143 Addition of Compound Mumben... 143 Subtraction of Compound Numben. 144 Multiplication of Comp. Numberf.. 14a Mnltiplieation of Compound ^Num- ben bj Aliquot Parta^ 149 DiTliion of Compound Nnn^beri;... 153 . Longitude and Time ». 154 Dnodeelmali.....^ „ IM Multiplication of Duodeoimalf 157 DiTlaion of DDiodeeimals., „ 168 " MiMeUaneona Examploa 159 RATIO, PROPORTION, AND PERCENTAGE. / Ratio V ^ Proportion 1J4 <„8lmplo Proportion „ iw Compound Proportion................. I88 jiPeroentage ^ „.. 171 *^Miacellaneo!na Bxamplea in Per- oentage.......^ 175 Simple iDtareat . 177 Partial Paymenta „„.•... 134 Problenu in Intereit ., im Promiaonous Ezamplea in Simple Intereat , igg Compound Intereat 191 Promiaaory Notaa 193 Formi of Notes 195 Profit and LoM , J97 Commisiion and Brokerage 200 Plre and Marine Insurance 203 Aaiesimoiit of Taxes 205 Cnstom-IIoufle Business 207 Discount and Present Worth 209 B»nk Cisoount J13 Promiscuous Examples in Discount. 210 ^^^ •'•• .;,.. 218 Partnership „ 333 Exchange jjg Foreign Exchange,.... , 230 Equation of Payments „] 333 MISCELLANEOUS. AUigation Medial..-. w— ...... 3S« AUigatiw Alternate „ 337 Inrolution ^ ..; .. 341 Bfolation.... .^ ^H Square Boot....^. „.,.,. 343 Cube Boot 2ii Arithmetical Progression. 343 Geometrical Progression „ 2H Measurement of Lumber... 353 MisoeUimeona Ezamplea „..!!!! 854 .■^ c 4 inor 4 ber; G or a « 7 to ai T Isl eight 2di three thOM 3n fortj, strac H some yardi 1. 1 tud^ < arc ab Pioa. Oumiiey Inrreooj. 142 imberi... 143 fumben. 144 umberf.. 140 d^Num- 149 i|b«n;... IM . - 154 ,> 156 ii*l« 157 .'■^ \ INTRODUCTION TO Tl^E TREATISE COMMERCIAL iRITHMETIC i ~ i ~»~ i ~T*' i ~i~ M Ti>' i r ii \ i LW Jum. - 1»5 197 800 203 205 207 li 200 212 aeount 210 .....^. 218 223 228 230 233 245 248 - 251 253 ~ ...... 354 f DEFINITIONS. 1. Arithmetic ia the science of nambere. a. A Number ii a unit, or a collection of nnitfl. ' ». A unit ia one, or a single thing. }„««;. "'**^**™**' °' //•«*• «VaS* JJ*°'1ir^'*'^ *",ri T?'"P*';'*^ ''•* subdivisions, as/our. eight, &c. : thej are called oiMract integral number$. ^: "^ ' three un^^H/Z^!^^'^^'^^^'''''^'''''^^ '^^''"'•l ""bdiiisions as tnree untt$ fifteen hundredtha,~rix units tun hundred taenttufit^ thousandths : they are called abstraa deciny^ nSrs. ^-^ 3rd. And lastly, those which contain onlv dAnlmai -.,k,r • • forty l»*ndredthi,^eventy.fiv7^h^a^h!^'X^l iS^'Ill^*' ? $tract decimal fractions, or Bimply deciZh. ^ **"*^ '^^ 8. Concrete Numbers are numbers used with referent t^ Borne^ particular thing or quantity. Thus, J?!? TS^nJ^s T % ^^ ~ ' DlWNITIONt They aro alio robdiyidod hito three olMses : ». A Slmplft^iimber ia eithdr an abstract or a oonorete iiumW of batone denomination, as, ^.o, .en dollar^fz/CTat ^^r iS^J^PO^^ WnmlrtfJs a coHecUon of concrete units whose subdivisions are not deoimata, but represent wveral Zn! minations, taken collectively; as, n^ ^nni^ur shuSSL' „?«^ pence,i/;iwfeet>« inches, etc. ^ /our BHUhnga nin*^ 11. A Power is *he product arising from mtltiplyina a num. W or^quantity by itJlf, or repeating ft any numberytiiie?^ , 19. A Root is a factor repeated to produce a power. 18. A Demonstration is the process of reasonine by which a truth or principle is established. ««« mg wy wmon »•*?*• ^ Opwatlon is the process of finding, from riven anan- tities, others that are required. '^ ^ 15. A Problem is a question requiring^an operation. 16. A Rule is a direction for performing an operation. 17. Analysis, in Arithmetio, is the process of investigating " pnnc^les, and solving problems, independently of set rules. 18. The Principal ojr J-nndamental Operations of Arith- metio are Notation and Numeration, Addition) Subtraction. MultipUcation, and Division. . ""*«"""» ■ SIGNS. 19. A Sign is a symbol employed to indicate the relations of numbers, or quantities, or operations to be performed upon them. (.) is the decimal sign indicating that the number after it is a' decimal. $ means dollar. « { , 9. Whai M a BUnple number T- 10. What w a oompoand niunber T— 11 m^ u a power T- 12. Wkatu a root ?- 13. What u a ^emonaiSton C 14 K' w he equaWj of ntiofl. Thus, 6 : 9 : : 8 : 12, is read, 6iato9as8istol2 ■*i * * '-^ • J^^ NOTATION AND NUMERATION. or^JiJf^S?"'' ^ ^ ^"**" efaprt«% numbers by letten « JtlL?!??'***®'' " *^* P"**^ ^ ""^^ numbers when expressed by figures. •»-«., 29. Two methods of notatioa are in eommon ve^ the Somtm •nd the ulraMe. ^ ^ — 'i-iwimm ROMAN notation: iia2?;.iT^!i??^ WOUUOB, io ealled from its having origJ- J, iinaji^ hnndnd, thoonad. -*!!l.y^*'°»**"o° '- «• ir*al it nomentlon ?- M. Bbw mom mOmI. a/ ;iv' 12 y NOTATION. li Will bo seen from «he following Table, thtit all nnmbers may iTEl'" *'' "° °' ^'^^^ ^^""^' "^^' ^y repetition^ 1st. Everj repetition of & letter repeats its value- thus IT represents tzoo; III, represents three rXK, twent^eJ: ' ' ^h!f\7^T •** ^®'*®'" ''*' ?y ^*'"® " P'*'^^ «»««' Ohe of greater vaJue, It ad^ Its own value to the greater; but when %hoed Wore, Its value IS to be subtracted ; thus. VII represento £vm llntZtX: "'"^ '"^ '^^^^'^^^ "*■"'' ^' ^'^^ ^- '^ 3rd. A bar or dash ^-.) placed over a letter, increases its valu« a thousand-fold ;. thus V denotes Jive thousand; I?, /our thou- Band; X, ten thousand, oto. !;• is One. II- " Two. Ill " Three. I^. " Four. V........ « Five. VI " Six. VII « Seven. VIII.... « Eight. K " Nine, X. <' Ten. S " Eleven. XII « Twelve. XIII... « Thirteen. XIV.... « Pourteea XV « Fifteen. XVI.... « Sixteen. XVII. . " Seventeen. XVIII. « Eighteen. ^^.... " Nineteen. XX..... " Twenty. iJJJv " Twentj-two. XXIII. « Twenty-three. XX^. " Twen^.four. TABLB. XXVII is XXIX. " XXX... " XXXVI " XL « XLIX. " L " LX «« LXX... « LXXXI" XO « XCIV. «' C " 000... « Twenty^ieven. Twenty-nine. Thirty. Thirty-Bix. Forty. Forty-nino. Fifty. Sixty. Seventy. Eighty-one. Ninety. Ninety-four. One hundred. Three hundred Four hoadred. Five hundred. Six hundred. Nine hundred. One thousand. Eleven hundred. Fifteen hundrod. Two thousand. Three thousand. T^ thouismd. ^"^ One million. ..^*' NOTATION. si 13 EXERCISES IN ROMAN NOTATION. Exprefis the following numbers by letters: 1. Six. 2. Eight. 3. Ten. 4. Thirteen. 6. Fifteen. 4. Seyenteen. 7. Nineteen. 8. Twenty-fiye. 17. 18. 19. 20. 21. 22. 23. 24. Four ^ Ana. VI. 9. Thirty. 10. Forty-six. 11. Fifty.four. 12. Sixty. 13. Sixty-eight. 14. Eighty-four. 15. Ninety-nine. 16. One hundred and six. ,,,^ 'ed and nineteen. Eight hundred and eeventy-five. Nine hundred and sixty-five. Four hundred and forty-one. Four hundred and eighty-seven. Sjx hundred and ninety-fite. One thousand six hundred and fifty. One thousand eight hundred and forty. ARABIC NOTATION. 24. Arabic Notation employs ten oharaotere, or figares, to express numbers, via. : 123 4 56 7 8 9 One, two, three, four, five, six, seven, eight, nine, cipher. 85. The first nine characters are called significxmt Jiguret, because each has a value of its own. They are sometimes called dtgitt, from the Latin word digitxu, which signifies finger. The '^"'oSi " *'*^®^ naught, or tero, beoaUse it has-no value of its own. 26. In order to reduce the numeral ^JyurM to a small number,4,i> jre give each a second value according to the place it occupies. Thtu, ^e first represents Hie units ; the second, the ten$; the third, the hundretU; the fo»j^, unita of tJioutanda; and so on, each succeeding fi|^re to iiffleft belonging to a distinct order, a ' me unit of which is tenfold the value of a unit of the order to the right 87. Since the value jpf a number expressed by any figure d*- pends upon the place the latter occupies, it follows th«t figum have two Values ; the one a6«o/«<«5 or simple, that is, the value expressed by a fi^re standing alone, or, when in a coUeetioo. standing in tha nght-hand place : the other rtlnHnM nr IajuJ AmI ^r ^5 -■ -■ -^'^•*%" VXVmv A *r"^ "rill""! TmVmtmlfW vT WfOWTi WttMB ■ V- -f'- U. Bow man;/ ekaraeten are emplojfed in tlU Arabio NoUtion?— M. mat 27. How many valuta hatuflgwrtt t mw. AMvw many anaraetei art tk»fir»t %in» charaettrt eedlad f~ 28. How Van we'rmrMma'tM nouMt UnwutktktUnfigwrml—^'' " 1... . i — .^'^ - «» yww»w» 14 NUMIEATIOW. cipher becoiersignifoaTwS:^^^^^^ The by filhng a place which otherwise, ^Zii:t:T(^^^''^* hand i?S:'and'?i'C S^^trV.".' ^* ^«"- <^'' t^e' le/l figure of the fourth oX- • thf o- ^'** °f thousanda, becauae it is . and it8 local vXe 4 W i^t ""M^ »al«« of the 4iKS5J ii / fllie the vacant P Wthe^ndrJ?./*^^ ^•'"- ^ «^^»-J tui/pheJ NUMERATION TABLE. ** I -^^ •ft o III III III III 1 1| I § J I II BULB FOB NOTATION. ^!^ writ, in ilg„„, „, „„l„ ^y,;^ • • • « -*-ir(r« (rao soo «. JF»"ndre/and seven mVona.'^Si^h^acS^ and fiftj-four thousands, three hundred and twenty-six units. EXERCISES IN NUMERATION OF SIMPLE NUMBERS. 1. 2. 3. 4. 6. 6, BEAD' AND WWTB THE FOLLOWINO KUMBEBS. 400 7. 800800003 13. 6004 8. 87974015 14, 80067 9. 35000918 15 670006 10. 30150900 16 9006014 11. 708000549 17, 92100121 i 12. 4050300 18. 28754105 1000500 3008727 605054046 78592835 106406021 EXERCISES m NOTATION AND NUMERATION OF SIMPLE NUMBERS. ■XPBISB BY nOUBES AND READ THE FOLLOWWO NUMBIBS: 1. Twenty-seven, forty-eight, sixty-five. 2. Seventy-flve, ninety-three, eight hundred. 8. One hundred and ten, one hundred and twenty-four. 4. Three hundred and flfty-one, six hundred and two. 6. * our hundred and mnetyH)ne, nine hundred and nine. V One thousand and one, three hundred and three. ft S?!-!. J!!?** T ^"^^^^ »nd t'^el^e, thirty-six thousand. 8. Nme hundred and seventeen thousand five hundred and ^o. iJ' TJn°JZS!^-'"v "'1^^*° :i**°'^"'* threeiundred a^ ten. 10. Two milhons six hundred and twenty-five thousand. 1 1* S!I?S'"^!!2 "• S°'" ®}«^* l^VLX^AKd thousand and fifteen. 12. Four hundred mUlions three thousand four hundred. 1. 2. 8. 4. ■xpAbss the FOLLOwiira boman numbebs by fiocbu. IT IV X L J3 H 7. V 8. vn 9. IX 10. m lU . XV ni2.| KXIV \[ 13. 14. 16. 16. 18! XXXV XL XLIX LXV xeix cvi 19. 20. 21. 22. 38r 24. ». Wkatiaih* CD CMIV DCCXXX OMXLIX in. MM mm tn tH ont 6«i-ai'.'««» t; !»' 16 DBCIMALS. DECIMALS. '-T;i:Th'^Sone*:iL^'^^^ L"'^ -it •" called tained a hundred times i^'ttTf^xi *^*'»" ^^J are con- un^t; thetentbofthoui.Sr^XtL^^^^^ th^ fi%&hL'drStK'6?!2"orr^ii;^^^^^^^ *-!-•, and dedmal and eight thousandths. ' ^' *'»'i.decimal two hundred « I^UMERATION TABLE MB WHOM KU1CB.B8 AXD DMnCAM. ' DaoBrsnro Ptoousaiov. '" a^s u^.5 ^«^ ssg iisg ^Itllvi^ii?; * 2 0. 6 7?. 9 3. ten times ; the hundredth exn^^LT ♦t^ <«n w the uaitrepeated twamizednambwr PWB lo dwtnol jwrto f— Ji. r raoniALs. 17 JM.. From the foregoing illustrations, we deduce the following fJnt^1henALl^jtJ^/'T^\'''^''"^^ fJ^ decimal point , then from the Uft to the right, write $ucceuivelv the tentha the hundredth,, the thomandth*, Ic. «^»«'«i^ tt^e tenths, Thus, the number 3 unita 26 hundredths is written 3.25. v>UhJpK^ "^^ o/cfocfmai. he v,anting,fill the valant places * J'^V""' J^® ""ra'^' 12 units 5 hundredths is written 12.05, in nlacin-r In7»?f*?J*P"'?M^^ **"**»«' and 4 units 3 tenths 8 hiXdfe and 8 ten-thousandths is represented by 4.3808. "unareaiug /A ^^I-^ '*;?•««'« *"'»*a^o«'y, a cipher is put in the place of the units, and the decimals follow in their regular order 5 ^o^rdThs^foir'"""'^ '^ '•' ' ' **"^^^ ' ^-^'^^'^^^ o-s«> There is always a figure less in decimals than in a corresDondin.^ whole number, because the figure of the unit which i? iSSTn Ef whole number is not moluded in decimals. "«''uaea m me «??* j^°«i°«.«Pl»««». to decimals does not alter their- value ten a^^JL'/r°"^ ^'°* " ''"' ^H^"^^' ^^« P'^'t^ »re made ♦fm.- i!?i. *r^ .""^I® numerous, kt they are ten, a hundred times smaller: there is then compensation. Thus, 0.25 becomes 0.260 hf the addition of one cioher andO 2'>0n KXBBCISBS ON DECIMAL NOTATION AND NUMERATION. WBiTB nr noDBu m roi^Awnro wxkd kuxbgbs. 1 . Two hundred and eizteen, and three tenths. * . iSillf ?"■"«* and seyen. and twenty-five hundredths. A i^^V-**^^t •nd four hundredths. t* J^J»'»ndwd »nd twenty-one, and nine millionths. «. Forty-foor, and twenty-three hundredths. «. Thrte hundred, and ferty-two ten-thousandth . 7. Twenty, and forty-eight thousandths. 8. Fou r hundr ed andfuj and fly hiinrfrHfhg, ; ^i£3Jt:w,-S3i';?£ni^,ij*tKi7''- " * ""• n 18 THE PRINOIPLBa OP NUMlSaATION. 1. 3. 3. 4. I. 2. 3. 4. WUJIBEEa AHD SIHQLK DBCIMALS. Mixed numbera. 8.90 9.908 641.400 703.2004 0.004 0.000607 0.006 0.0007007 6. 354.0064 6. 352.06046 7. 76.26007 8. . 376.600506 Single duAmals. 6. 6. 7. 8. 9 10 n 12 0.4072 0.401950 0.9540626 0.076003 41.0040((4 452.010778 7667.008007 1898.04 . 0.69804445 0.736050210 0.000600019 0.00000501 APPIICAIION OP THE PWNCIPLES OF NUMERATION AS LAID DOWN IN N08. 27 & 31, :'«,<;»« midtor?-: .*,^l^pl -I-,.... «,' x.'\f 20 TH. PE0MRTM8 Of If OMlftAWOIT. •. B«id«rtheini»Hlnumbw42.I0««31 1«> 2<» 40 6» 10 100 1000 1 0000 100000 timea greater. 60 1000000^ a. Render the mixed aumber 4.20 I* 20 30 40 6» 101 100 1000 10000 100000 tiniee greater. 6* 1000000^ « 4. Render the deoim^ 0.O6 101 100 1000 10000 100000 10 2o 3" 40 60 69 1000000^ timee greater. «. Render t,he whole number 6706416 1« 101 2« 100 3« 1000 ,. 4** 10000 f "°*W greater. 60 100000 e^* 1000000 J 6. Benderthe mixed number 7610438.06 1* 101 20 100 3« 1000 4" 10000 f **"« ■mailer. 60 100000 6«» 1000000 J 7. Bender the mixed number 6.45 Ant. An*. Ant. Ant. An$. Ana. An$. Anr. Ant. Ant. Ant. Ant. Ant. Ant. Ant. Ant. Aat. Ant. 4210.64231 421064.231 42106423.1 42. 42000. 0.^ Ant. Ant. Ant. il»w.67064160000. Ant. Ant. Ant. Ant. Ant. Ant. Ant. Ant. 76104.3806 76.1043806 1* 101 a* 100 8« 1000 40 10000 6* tVOOOO 6** 1000000 times sinaller. Ant. Ant. Ant. 0.00645 Ant. Ant. 0.00000646 fefcit.;.f' -.. THB PEOPKBTlig Of NUMIBATIOM. 8. Rendar the decimal 0.06 21 10 ICi 2« 100 3* 1000 4° 10000 6° 100000 60 lOpOOOO timet imaller. ». Bender the mixed namber 209.007 10 101 2* 100 30 1000 4" 10000 fi* I 00000 60 1000000 times cmaller. 10. Render the mixed namber 1463.309. An$. Ant, Ant. Ant. Ant. Ant. Ant. Ant. Ant. Ant. Ant. Ant. 0.000006 10 20 30 40 60 6« 11. 13. 18. 14. 19. 16. 17. 18. 19. 80. 81. 83. 8S. 84. 8S. 80. 87. 88. 89. ^ lO'i 100 1000 10000 100000 1000000 times smaller. Render 88. 88. 84. 166. 10 3867. 100 2064.16 1000 640.4 100 *4. 1000 ^46. 10000 9.36 100 76874. 10000000 6.468 1000 0.46 1000 9.10 1000 0.06 1000 9.6786 10000 4.0000007 100 0.0007 100 14.666 10000 ■ 0.7 10 674.867 10000000 *«-. 40.6804 1000 60600867. loooo - Ant. Ant. Ant. Ant. Ant. Ant. times greater. " smaller. " greater. " smaller. " greater. " smaller. ;; greater. « u " smaller. M tl " greater. " smaller. greater. smaller. u greater. <. 0.00206007 146.2309 Ant. 1650. Ant. 38.^7 Ant, Ant. 64040. Ant. 0.074 Ant. 4'M. Amt. 460. 0.00006 *i«ff— rwoocwo 46.620 1000000 76840.3 Jjiannn 0.013 ^poOO « u u u " wna ll e x . ft $t « greater. " smaller. Asu. Ant. Ant, Ant. Ant, Ant, Ant. Ant, Ant. Ant, Ant. Ant, Ant, s 2r (I » ■ ADDITIOfN. ^ ADDITIOlf. OPERATIOK. 428 636 87 4 , 1937 Sum or Ajnonnt """"'*'"'• ""gl* nomber odlod th« defoLS!^ «• <*a. «»e ktod wLn they h.™ ae »n. £7xa«yfe 0/ a» Jrfrf^u>„ i^A „fiole number,. g^Whatisthesumofthe three following numbers: 428, 636, and S tenfl under the c^nmn nf tM^ „ ^^ «,» foUomng • / ADDRIOM. Sg . ADDITION OF DBOniALB. « fi^'^i^^'S^^** "i^l^. cfecmoZi, «| follow the tame prooe$t as /or Aeadditjmo/tchole number,; lut v>e cut of /rSm the right by a j^intae many decimal, a$ there are in the numler which eontatna the mo$t of all the numhere added, A^uC ^'Jqo *^?/o"o''»''« number* to be added: 35T9 anitB Mhim- 3 6 7 9.26 hnidn^. i tratli ud » h«iib^Stk^Wa 4 6 8 2.06 r"^** *5*..* >>"»<»w*?•» tkiw, land l»i» 8, .ad T«»lS 7 8 5 6.80 Md 8 «• 18 ««tlu - 1 unit ud's tenUifT wi TOto ttioS under th« wdamn oT tontlu udewrr "•*»«*totheorimnaor«nitij thu, 1 udft VB lOf 4kO« il«». 16 6 9 1.86 whioh ia read in the foUowiDr manner: J0091 nnitt85 han* dredtlu. PEOOP. «H?„*^;n^^®^!r^°^"*^*5™®*^«'^<'P«"tio'» " mother oper- ation to prove the ezaotness of the first ' ^ «^^V?'^!r"'^~** '** •?"**** *o^<*ddedinto tm parte • add each 0/ then tu>o parte and eet down their reepective\^' Then add theee two eume together; if their ama^i, ZZt^ the first answer, the work iepreetmed to be correct. " ^^ The numbers could be divided into • greater number of Darts than two. '^^ Exampie, PRoor. OPERATIOK. 123.24 349.00 66.26 . 149.34 967.32 Ane. 1645.16 Ifit Part. 123.24 349.00 472.24 2nd. Part 66.26 149.34 967.82 Addition of partial totala. 1172.91 472.24 764645 — ._. .w-Ew.xw 1172 91 which is read 1645 nnita 16 handndths. ^^."^iZ^^^ aemng price waSTu^' '^ Z' J^^""^ *\** '^^ ««>Z«ak« th^* u^"'f on^e\Td2d td'.lSj^r •* *r ^^-^ »»d twentj. twentythfee, + on JhundrJd t hn^. J^^- «el»ty-nine thousand and thousand a„k tiiZh^^ *^°'««°** «ix hundr^ and ,ten, + thre, _ 'd Beven tundredSSet^nr^-t.itf^L^^jS^.??'* ^«»" 5^todBT;eni^Sdand^«t. -^"^ +.*'«^* hunted indiei . thousaH + ^^n^XSatdr ^r.?ote^^ „:i^it.j / ASDiTioyr. 26 dred and thirty twor+ewtn^Jlillii^a^ T*^ ''*^*^"' + ^^^^ ^uo- and nine thouLnd „ite hSmU iin • *"'* ^r?^)'' + <>"« hundred 21. 40.05 + 104.8 +1003.026 + 7.38 + 2.16!\ 22. 0.4 + 0.^ + 0.0306C+ 0.01 + ol^S* iTofeT"^^^'- +-7.889. * *:^ + '•"> +/•'*„+ ■'■1t+%.1^+ 9.99 '26. 4.9S + %u + 8 69 4- 4 29 J. 51 ^"r'/nj-°5','.''°"*f5<'"»- 9Q IT f „ "*" '•^" + *-66 + 9.09 + 7.60 + 55 4S X 9 qS> dretfotSlteft^nt^^^^^^^^ fi-^nnlStfai'^ae-hi. twenty-fiveSou8in?h«!?«'^ ton® thousand three units an^ units LTfiftrhundU^^^^^ '^""^ hundiWlhs, 4- twJ 30. RoquifcHi he Tun, nfl.i^^ ^*- 1167.406. three hundred ten'houSindth^^A"^^!,^ S/ '^"^^^^^M +, Bandihf rS tVdrS tit A'^ + twet° a^-thou- sand^ih^ir^^efttnThT Ifc^e rS*--^/CAi« '^^^ ionths, + nineteen thouUdtiS ^"°^-*J^O"»ndths, + eight bfll- nine hutdtl tenSoTiSr^^ «ght hundred-thouaandtha, + eand hund^tU + ^SS^nli"*?"^ ^"^l^^ ^""^^ + one iJoJ eight hundS? t iw^ », ■i^°??''**"''^ + *^«°ty""l»onth8, + and nineSTSl/iotthe hundred-thouBandtha, + three thouSw* thousand Sllionths? '^'^ ten-miUionths, + 6ne hundred -vt S^5.s^rhuL*^Srfil^^ na^be M,e„,y.fl ^„ thooeandthg; one hn«H^ !il ^ ^^ f* hundred and fortv-threfl dredths; a^;en^i.^°;y,K^J^ g^rf."!?:?^"' .'^S^ .^'ghtynini K ^i«>twitndthflr- ' and -t h ree hun dre d and forty^igftt fi^nrreJ- 4n». 363.636487. ^ a -.1 ■"^ »v w 26 ADDITION. PKACTICAL PROBLEMS OR QUESTIONS IN ADDITION. i"*llS'?.^'** * '•''"'* f'>r«25840, Ipaid $1565 for right of DosseBflion and $238 lor repairs 5 what did it cost me ? possession, OPERATION. AKALT8is.-The wholi, cost of tho houso is oaual to th. $ 2 5 8 4 T'Sli 'iV"""" «P«°««""'. that Ib 268y + ffl + Si f2 7643/ln». n.u;tlSttmrgr$\*f2i^^^^^^ sumof$245.65; howmuch 2 6.20 246.66 + 25.20 = «J^0.86. aelU^g^rir ^^ ° ' *"*' "*' » 2 7 . 8 5i4n». ^ thfno^nMoilratdtSo^ZtT'h^-^''^ ^^-^^ -«^- spent during t;LX*eda,iT *'*" ^"^ ^""'*^= ^<^^^ '""«^^ ^^^ tho^''8'J;T4.7f4f*i U^-L^T"^'*^^ of Tuesday and Sunday. On Tuesday. 4.76 -jTe 90 + f flj-lf ii8*fl3 Jhn? ^'""'''^' r'*-""v +. 2.08 =/ $7.98. Then A A r '-vo — ¥18.63, whole expense for the three daya. '|69;lheSerTlJ8' at'r*'?' the butcher, $46 ; the ahoe-maker, family o^f'Hli?' ^'^d for house-rent, 145; how much does the 6 - Th?^^ }T '° J^,f • '•^ ^^^^t y«" ^i» ^' be 24 y^arfold ? Queb^ 64W ^S" ^^.^^'"t'^al " about 135000 aouls, that of Sv's 6300 So^.T f/.?!';'' u^^"?' ^*- Hyacinth, 4102, Point- pMtLo^r^^^^^^^^^ $J^nL7^'T^^ merchant sold during the year $9023 worth of cloth- Sino'^'^Yf *"" ' *V''i,r(.^"«^ "^ $n90ofSco;$856' 8 A .^; For how much did he sell during the whole year ? $240 ^oTe'2nd \h7«^2? 'T f ,«»<>»«7f h« Paid t/e Tat. time owes «92 Trn» ^Jy^/ ^^'^ ^'■'^' *^<>^-*0, alter which he yet owes $92. How much did he owe at first ? Ans. $818 16 mentinTP*"? of soldiers have fired 29682 cartridges in In engage- SX'^^^rX.:^'' 13403 remaining. HowU^. h^JS^ 86lS'ninTV?ra^°^^'^'"«««™"o"ght a new farm in a township; the iHt. year it yielded Jn«},^'?K''L?^'' the 2nd. year, 3697 busfi.'; the 3rl yearf-9982 manVh,i'i*wIT' •^^?^ ^"«^-' the5th.y^ar, 12760 buah.': how many bushels did it yield m the five years ? Arts. 37240 bush 15. How many years elapsed from the taking of Troy, which oo. curred 1184 years before Christ, till theyear 1869of theShriSner^ yei'did iJeTer"'^ "" '"''^ '"^ ''''' ''^ *"^« »«« o^j^' fS«f «* the3ri,l96.T; and the^i' i'^'^'whS''' ' ?f nd., ,8J5.40 ; them? ' ' ana lae 4U1., J798. What mun do I require topay tJ P J ■" "^ »f 'HOi'mnion of Canada i. comp^JS' *.?' Wtol- niL- Tj .flT" "''" ' 't? Pn>'i"oeof NovaScoUa, 19650»ilSS 21 A m«,„K- * • ,.• .^'^' •123,40. Whole sum $27'9.70. 29 ! . fu*^® '^"■^^'^ ' ^°^ ™"ch did he pay for it? ' UoSi^n • K*K?^'?^*'M^^1' *h« populatiorof Upper Canada was 3m2Jo v'^**'^"*"' ttiat of Lower fcanada, 1130800: NoJasUut fhZfji^"" ,^"'1'^'''^' 260000. How many inh'abitonts weS ten^ar ^""'''''^•'^ '°"r« ttf present DominSS 23. The battle of Marathon took placHgO S cSsS^^^H^w ^9! 1, ?*?®'V.**^'"*'^ horee-hides weigh 486 pounds; theylIaVelo.t 324 pounds in being tanned. Wh&t wis their mw weight f ^^* ^^' 59??: ^wS/tKi^SJV'''"™'''^ ^^ «*«^ ^^h^^SiS.'^' 27 ' Th« " f * P"''?^^* pound of prepartKl wool? Am.W0. thi^f V Pppwlauon of Europe consista of 278694707 inhabitant • that of North America, 43879348 ; that of South An erica. 22007ftM .' u»i, IB ueJMioi* pc^lataoD <3r the globe? .-^ '^r?-— T»?y__=:n. - / Am. 1020860878 iohAbitftata. 28 SUBTttAOnON. SUBTRACTION. 45. Subtraction is the process of findinff the differennfl betTOen two numbers of the same kind. ^ (nuerence ih.u-'^^] "°'^^\'' ^"^ '^* ^l^ich is to be diminished is called cesMr^erTnte!''"'*"''^^^ ^'^ rmainde^ ex- ■£^3?. From 647 take 324. OPBBATION. AsALTais.*- We Write the less ntaUliM> Jider tbe greater, .o that uDite of the same ordM^Fstlnd in the same oolumn; then, we begin at the right and 1 proceed as fallows : 4 units from 7 units leave 3 unUs. ZtfZl f"'"" ' u."?"" P'*""- T''" »«•" f«»D 4 tens leave two tens, which we write in tens' place . Three wo write in hundreds' nIacJ""Hll«"\,*i"?**'??' '"^ ' hundreds, which tens, and 3 units/or 2/3 ""^ •*•"* ^*" ^''^ remainder, 2 hundreds. 2 Minuend Subtrahend Eemainder Minuend Subtrahend Remainder From Take 11. 3692 — 12. 7634 — 13. 8742 — 14. 41763 — 15. 7839 — 16. .3724 — 17. 2945 — 18. 69524 — 19. 56247 — 20. 72365 — 467 325 132 (6.) 648 234 1212: 3132 ■ 6331. 11522. 5427 = 2502 ■ 832. 47321 . 16123 • 1243. EXAMPLES FOR (2.) 273 132 141 (7.) 376 164 PBACTIOB. (3.) 936 714 222 (8.) 857 622 (4.) 685 423 262 (9.) 498 175 (5.) • 974 631 343 (10.) 645 642 = Ans. 2480 = An8. 4602 ■ Ans.^ 3411 > Ana, 3ip241 ■■ Ans. 2412 ' Ans. 1222 Ans. 2113 Ans. 22203 Ans. 41124 Ans. .7-1122 Case II.- greater than 21. 1243 22. 48673 23. 34272 24. 79832 25. 16475 ■ 26. 15768 ■ 27. 982876- 28. 217951 ■ 29. 760142- 30. 391657- - 213'= Ans... • 16330 = Ans. . . ■ 13051 =.^n«... ■ 67411 "^ng... • 4G60 = Ans. . . ■ 4327 => Ans. . . •120341 = ^n,... • 6430 = Ans. . . 570031 «;!«,.. . 141322 = ^n»... -Jojuhtract when any figure in the tubtrahend is thejig u re ab o v i it in^^te^inumdr — • ret?; ^V»»'t'«"onT-Z).yJna minuend.- subtrahend.- 4«. ^<,«, f. ,A, the difference M. Sow M th« SUBTBAOTION. £.r. Find the difference between 863029 and 360476. 29 METHOD BY BOBROWINQ. OPERATION. •3 AitALTBis.— Having jdaced the smaller number under the greater, with units under units, Ao., as in Case I, we draw a line under- neath. Then, beginning at the right-hand *« say : 6 units from 9 units leave 4 units. Which is the differenoe of the units, and which IS written in the units' place below. We then proceed to take the 7 tens from the 2 tons above; but thii cannot be done, since the 7 Is greater than the 2. We cannot borrow from the next figure, as it is a cipher, we then borrow 1 from the 3 thousancb, which equals 10 hundreds, leaving 9 above the ci- pher, and add the 1 hundred equal to 10 tens, to the 2 tens, making 12 tens; 7 tens 4 from 9 leavM « «hi»i, — — «». • 'fo™ 12 leave 6 tens, which we write under; ? Jf"™ " w»7M 0, which we write in hundreds' place below. As we have taken 1 thousand from the 3 thousands. 2 thousands rSmainV nTuirhtfrrm 2Yeave8 2 so fromThl s'S.fn"/"'; ..^' "■5""* **''•' « ten-thousan'ds ?S /Z-t£ou.and8 ; fl ?i.^ , '»"">dred-thou8ands we Uke 1 hundred-thousand, which eauaU l6 ?"eni«^rfo^l?5„*^r *" *'/ f t^n-thousands, make i?Sn tholi.Sds" under HaS[»k^„ih'i'°°^'^u'^'"I^' ten-thousands, which we writ^ h,.n,i~j .if* J ° ^. ^nndred-lhousand from the 8 hundred-thousands. 7 leav2lh"uSrtt« Jf ' t '•"'^red-thousand, from 7 hundr;d-thous«ds Minuend ' Subtrahend Remainder 11 WhhWhp 863029 360476 4 92664 OPERATION. 8630 2 9 36 0476 4 9 2 6 64 METHOD BY ADDING 10. leave 6 hundreds, which i^wr"5 Slow N^' « hnndrwb fh,m 10 hnnd^d« or I thousand, to the mtonenrf w«^»n V ^°'' " ^' '*»^ "dded 10 hundreds unless we «dd 1 tho«K ttTonh^ftZLTl^h" ^^^T'""' *«<»'"«• thousand ; 1 thousand from 3 thouswd. I«~ ?1t^ the subtrahend, making 1 to take the 6 ten- fconssirfs from ihfkf.!!.r * "'?'"»l'd«. We then pnwed add l<>ten-thousMS. toffe 6 t?n5hon.^dJ' mllt-"^ ?h" **• " wec^^nTwe thousands from 15 tea-thoMMriea^ fi^fS%fc*'"°«.," ten-thousands; « on- ^,/T ^r^e illa.tn,Uo.8 we derive the fellowing «. Whatuth«ni9foriMbtraeUomf SUBTSAOnON. IT. Commendftg at the right-hand^ take each figure of the mb^ trahendfrom thejlgure above it, and uirite the reiutt underneath m, I/anyjwure in the subtrahend be greater than the corree- ponding figure above It, add 10 to that upper figure be/ore sub- t^& <^ oda one to the next left-hand figure of the mb- PROOF OP SUBTRACTION. \ 48. We make the Proof of Subtraction in adding the re- mainder to the subtrahend, their sum wUi be equal to Uie minuend. It the work is correct. Ex. From 35678 take 27899. Rem. Proof 35678 2 7 899 i 7 7 7 9 3 6 6 7 8 Amaltsib.— To nrove thia operation, we add the remainder 1179 to the aabtrahend 878M, and obtain 35878, whiehenmiseqaalto the minneod, or greater number. Hence we oonolnde that the operation is eorreot This method of proof depends on the principle, that the greater of any two numberi it equal to the leu added to the difference. UsB OP SUBTRACTION.— iS^Mifracfion serves to find the gain or loss on goods ; what we still owe on a sum of money of which we have already paid apart; in general to find the surplus of a number over another; the difference between two numbers, &c. We know that the solution of a problem requires a subtraction when we must find the difference between two numbers, or the excess of a number over another; and when it is required to find one of two numbers forming a total, that total or amount, and one of the numbers, being given. ■XAMPT.tt rOB PBAOnOK. (1.) Minuend 76618 Subtrahend 49359 Reminder 27159 (2.) 67813 38675 U9138 (3.) 13042 9176 3866 (4.) 260143 176158 73985 Proof 76518 Proof 57813 Proof 13042 Proof 2"60143 IS. 7o«) do jf the minneod, nolade that the It the greater difference. I the gain or y 0/ which surpltiao/a hers, &c. subtraction, or the excess >find one of and one of (4.) 260143 176168 73985 roof 260U3 6. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 26. 26. 27. 28. 29. 30. 31. 32. 33. 34. 36. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 60. 61. -S3r 63. 54. From « <( II u « « « « u « « « « « « « >( il tt (t It « <( (( (( (( « « « « « « « tt 3900 take 49469 « 79906 " 190540 U 478764 « 426542 " 740070 « 677406 " 406907 " 897462 « 8950076 " 14003325 '< 16989700 " 21530600 «' 97660054 •« 457662478 « 337008974" " 164400000 << 190054009 « 754674896 « 10007549 " I27321I55 " 418030450 « 945000090 " 809006409 " 490009076 " 84765.W54 " 850070462 " 546807675 " lOlOlOlOl " 663405995 " 677454864 " 764927074 " 960076074 " 466700760 " 876007054 " 753097607 " 400076546 « 487054554 " 432700769 « 954876754 " 453007527 '< 4D0700007 " 974600700 " 839457364 « 847654976 « 905207 246 « 4184546945 « 9457385700 " 9707000591 •< 16574 16134 30409 97125 179127 471097 198676 56596 608678 4137976 988827 164379 737898 14550045 49876679 40073049 91791994 4590489 64834795 9068073 1300475 27740761 1500734 3740055 6475904 74376676 97050664 277451794 9737350 476294474 495647562 676489672 475207464 45612496 798436495 194289778 93467897 98047776 71904267 677469519 276499619 203406604 93236945 746689835 39787496 14686547*-- ^^ # 178809709 17073969 19779883 31 ^na. 3649 33895 63772 160131 247416 268973 378829 388874 4812100 13014498 15836321 20792702 831100^ 2969.36925 > 62608006 185463520 689840100 939476 126020680 390289689 806266364 484533172 773277878 763019798 269366781 91272751 181807292 88437502 474868620 411088256 77571559 568807729 389006779 360796512 377406176 176507908 881264755 93767519 807867481 ^5836177* — r: 4006736238 r 9687220706 *^ StmTRAOTIOK. SUBTRACTION OP DECIMALS. Ex. From 86.7 take 69.354. . OPERATION. ANiiTBrs -Having placed the le«. Dumber under the 8 6.700 .i .S ' *•"*.* "«""" "^ '•>« «»""» decimal place stand 6 9 'I fi i T P» ■»™« oolumn, we write two ciphers at the rieht of ' . y^'t * I' "• Ofder that the minuend maj" hav» as many deoimS , 17.346 fi««>;M a« the subtrahend; then we subtract niinmhZ m1?„^"'^-"''.?'"'"y.P''«« **'» <'*"'°»l point in the re- mainder direcUy under that in the given number. ^3^l!f'^h '^"ff '^* ^' number under the greater, so that the decimal points shall stand directly under each other. iJL ^^l7f "'7? '^^J>knumlers, and place the decimal point tn the result directl^ und^ the points in the givm numbers. EXAMPLES FOR PRACTICE. Prom Take (1.) 12.067 9.71 (2.) 8.11" 6.7519 Ans. Pfouj « It ti n 20. ^8^.=^ 2.357^ 90.49 109.191 6409.055 764907.05 897450.07 465742.5 870079.04 400048.2I.S6 409004.9099 670076.9004 49.1019 610011.050 71079.0013 79073.07 126001.0001 191279.9709 40 1646.10051 700007.0236 411978.10.359 960945.00005 0.0707 0.0006 0.90019 1.3581 (3.) 36.105 7.L1892 28.58608 (4.) 1.0062 0.43 0.5762 ■t « « 0.0904 0.7009 0.09.91 take 39 " 49. " 4045 " 87929 , " 98776, " 76908, " 198789. " 9372 100 4053 35 31971, 7482, 7398, 98996. " 60056 '" 498 " 79797, " .36730, " 600979. " 0. « 0. " 0. tt rt « tt tt tt It tt tt 59 073 .997 ;795 095 07-5 958 .016 ,1?7 509 708 999ft 17.36 .1204 .9088 .0099 .6709 .0098 .09671 ,00007 000607 0000075 7300007 Ans. It 50.90 60.118 " 1 363'. 058 " 676977.255 " .388834 " 671289 " 390676, " 408904 " 566022. tt " 578039. " 6.3596. " 71674. " 27004. " 141223 " 401146 tt ,425 ,082 1976 7729 .3914 .0*501 .8277 .94^ .0913 9610 ,4296 tt tt tt 0070675 1.00289709 M 90007 .004500008 " 375248.00688 " 359965.99998 tt tt tt tt 0.070093 ♦ 0.0005926 0.1701893 - 0.0018325 0.08750291 0.610893 0.094599992 SUBTBAOTION. 32. 33. 34. 35. 36. From « 0.0779 take 0.900 " 0.19100 " 0.4500 " 0.09839 " O.OIOIIOOI 0.0019904 0.09900036 0.00550045 0.09500959 An8. « 33 0.06778999 0.89*80096 0.09199965 "0,44449956 0.00338041 PRACTICAL PROBLEMS IN SUBTRACTION. / What is the ^.1- A field which had coat $2360 was sold for $2628, OPERATION. $2628 $236 ?~2T8 ■ ing tho cost Ss«fnP/*°•^'u'''"•'^°''''''l»«»>tr«ol- wf obtain the S ''^•*^" ^""^ "^^ '"L""* J™' »2828, ° .AIM, $268 gaioa .hi;.h''cZSf^^^^^^^^^ for $4825.75 goods OPERATION. $5174.10 $4825.75. $~3T87T5 buying price $5174.10, we obtainAe iS. ' .<1m. $348.3*6 lou. f «"<"''« d'fl'ifenoe between 70401 .„,i«oj9. 'J"*- »1227. 7 TVio „,/♦ i^/ ^ *'^' ^ "ow much do I owe vpt? ' . 1 he greater of two n u nibers in l in9 » ;; j ii • ? • Z what ia the smaller? "•"*'^" »« ^^''2, and their difference 981 ; ^ 8. A merchant sold in one dav *.'>k-\i An ii » , "'*'*'• ^21. 9. To what number must we arl,1 7fi ♦« • ^^' *I886.46. 10 Th#. /.itn^ r^fni « *"" '° *o increase it to 740 ? milM doe. the fonner exceed the latter !^' ""■ ' Y ^"^ "'•'■7 "qrare ,^ 12. A fiuher wa, 28 »eSe old tt th.w.,1, *•"■ ^WO Km. lo'^itT?" """"^ ■"°»' "» •d'l«"o 4 unite 5 h„ J;^^' Kv. «3206436.10,'aiSJ„3^KM2°,SnS ^S*" '''" « »«W^ ■o.I»rtatione«ceedth?^SSt,W '''■'^- ?<>' m"»h did ll,, . ^« -In*. 1769, . i 34 SUBTRAOTION. 18. An army consiisting of 41500 men, lost during a campaign ' 14704 men. How many remain ? Ans. 26796 men. 19. What number must be added to 3 thousandths, to have 12 hundredths? 4n«. 0.117 thousandths. 20. The population of Paris is 1953262 inhabitants and that of ■ London 2863141 ; 'how much does the population of London exceed /! that of Pariy? Ans. 909879 inhabitants. A-^ ,| 21. Alfred the Great died in 901 at the age of 52, after a reign of ^ 1 24 years : in what. year was he born ? Ans. 849. * 22. Charlemagne was born in 742 ; he was crowned king of France in 768, emperor of the West in 800, and died in 814. How old was he, 1st. at his coronation as king; 2nd. as emperor; 3rd. at what age did he die; and 4th., how many years elapsed from hig death until 1869? Ans. Iflt. 26. as king, 2nd. 58 as emperor, 3rd. at the age of 72, and 4th. 1065 years. 23. Murillo's picture of the Immaculate Conception, being auc- tioned, the first bidding was $30000, but it was finally knocked down at $117000 and adjudged to the French Government who placed it in the museum of the Louvr?. Required the diflference between the Ist. and the last bidding ? Ans. $87000. 24. The population of Montreal, in 1765, consisted of 7000 inhab- itants^; in 1851, it was 57715; in 1856, 75000; in 1860, 90000; and in 1868, about 135000. What was the increase of the population from 1851 to 1868 ? Ans. 77285inhabitants. 25. A farmer reaped 1689 bushels of wheat, and 965 bushels of » oats. He sold his neighbor John 890 bushels of wheat and 478 bushels oats, and the remainder to Joseph. How many bushels of each sort did he sell to Joseph ? Ans. 799 bush, wheat and 487 bush. oats. 26. Two merchants, in commencing business, invested a capital of $18500 ; the Ist. invested $6590.40 ; how much must he add to his investment to equal that of the second ? Ans. $5319.20. ^!j ?*^ ' $508.50 more, I could pay a debt of $1015.80, and would have $75 left; how much have I? Ans. $582.30. 28. A merchant sold $11630 worth of cloth, whioli was $876 more than cost price; how much did it cost him ? Ans. $10764. 29. A house which was sold for $14360, would have given a Srofitof$840 to its owner if he had paid it $300 less. How much it it cost ? Ana. $13820. 30. Gunpowder was invented in the year 1330 ; how long was this before the invention of printing, which was in 1441 ? Ans. Ill years. PRACTICAL PROBLEMS COMBININQ ADDITION AND /SUBTRACTION. - 1. A retail merchant places $45.25 in his drawer for change ; on Monday he sells/or $75.85 ; on Tuesday, for $68.40; on Wednes da y , for f8&; on Tturaffay, for $1^.60; on Prid^, for $54.85; and on Saturday, for $p2.16; after which he pays a Bill of $96.60, another of $43.26, and lakes $240.75 for his own expenses^ and then there remains to him in cash a sum of $150. Are his accounts right ? SUBTRACTION. 35 2. A market woman having 152 eggs, sold to one oerson 14 of 4. The waters of the St. Lawrence corer an area of 565000 eouwe m.Ies; two of ,ta nbutariea, the Saguenay and St. Maurice, Tv" the one an area o»-27000 square miles, and the other 21000 Zare miles. How much does the area of the St. Lawrence exceed tK5 Its two tnbutanes ? Ans. 617000 square S. 41. \.?'^ has bought four buildmg lots fortnesum of $16860. For £%/^-,^' paid $2070 30; for the 2nd., f 3674.60 ; for the Trd.' »41 75 : how much has he paid for the 4th. ? Ans. $6940 20 6. J deposited in a Savings Bank $8752.70 ; the first time I drew from It a sum o( $1286; the second, $1650.50; the third. $972.75. How much have I left in the bank ? Ans. $ 1843 45 7. Moses was bow about 1571 years before Christ, he left "E.»vnt with the Hebrews the year 1491 before Christ, and died on M^unt Nebo, in the year 1451 before Christ. What age was he, 1st. when he left Egypt ; 2nd. at his death; and 3rd. how long fh)m the period of his death to the year 1871 qf the Christian era? ^ Ans. Ist. 80 years; 2nd. 120 years; 3rd. 3222 years. «. A speculator gains $6570, and then loses $3762.40 ; at anothe r time he gains $4545.72, and loses again $5632.10. Tell how much his gams exceed hia losses? Ans. $1721.22. 41, o j"^?° • ^ i" 8^119 since 6 years ; the Ist. year he lost $356 : the 2nd., he gained $780.20 ; the 3rd., he gained $685..^0 ; the 4th he lost $2600 ; the 5th., he gained $4320.95 ; and the 6tA he lost again $3000. Did he gain or lose, and how much? iln«. $169.65 loss 10. A owes a sum of $690, plus $55.20 for interest. He reimbursed at different times $87.60, $210.00, $318.45; how much does he still 11 A r I Ans. $129.25. amcJunUfSq?:"^.^^"^'' ^^^•''^' ^^'' «g*'« effects to the Sw^lf^ft^Lrounu"^"' " paynient$704.65^whatis yet ^^M\pt^^^^^^^^ ^^^e p^ on account 13 Peter ha«, 360 sheep, Maurice 145 more fhan Peter and 36 ; 16. UULTIPLIOATIONk If I had aoM $20 more a piece of linen which cost me $360, 1 would have gained $30 ; how much did I sell it ? Ana. $360. 16. A speculator bought 217 corda of wOod for $1085. He gave in payment 1800 pounds of salmon valued at $144.00 ; 700 busliels of potatoes worth $210, and 1200 pound^ sugar equal to $72. How much does he owe yet ? . ■ Ana. $659. / 17.. I have three creditors ; I owe the Ist. $2500, the 2nd. $840, and the Srd. $764. On the other hand, I have 2 debtors, the one owes me $1800, and the other, $2644. Besides I have $37^8 in cash. Required what sum remains in hand after paying ray debts af. Ana. $4018. 18. How many pounds Of bread will 200 i^itands of flour give, knowing that it takes 114 pounds of water to if^ad them and that 44 pounds evaporate in baking ? > '\Ana. 19. Three boxes containing 1436 oranges hax,« cost $17.16, and $3 each box for draya^e ^ the first contains 240 orangee, the second 80 more; how inany does the third contain ? Ana. 875. ^>20. In adding $5.08, the price of an ox hide, to the sum expended by a tanner for 4 calf arid 6 horse hides we obtain a sum of $22.98. RequireiJ the price of the 6 horse hides, knowing that the calf hides have cpst $4.40 ? . Ana. $13.60. ,,. 21. A cloth merchant bought «0»yard8 more than he had at first and then sold 140 yards; after iJltftph, he has left half what he had in bis shop before his last purchase/' How many yards had he at first ? 22. A dyer bought at three different times 109 pounds of dye for the «uin or$3.84. The first time he bought 47 pounds and this quan- tify exceeded by 15 pounds his third purchase. How many pounds, did he buy in his second purchase ? Ma. 30 pounds. 23. A general starting for an expedition with I8OOO men, left 6O6 of then} to garrison a small town ; at the same time he received a reinfOTcement of 800 more, 450 of whom he was obliged to leave in hospitals. Having asked 3500 more, he received only 2730; of these he left 1750 at different posts. Required the number of men he had on reaching his destination? iln«. 18730 men. , MULTIPLICATION. 49. lli^tlpllcation is the prooesa of t^ing one number as "toany timeei asjhere are units in another. 50. The t&ma in Multiplioation are : 1st. The Hultiplicand, or number to be taken ; 2nd. The Bliiltiplier, or number by which we multiply, o» which shows how many times the multiplicand is to be taken ; 3rd.. The Product, or the result obtained. ^ S I. The multiplicand and multiplier are called Factors, because they jwwfecB^rarOTafelhepo^^ — -^ "^^'*" ' ' ' , ' , III . Ill 40. Wh ..I'ii,'! iitfit % ■ ' MDLTIPLIOATION. 37 me $350, I ' ns. $360. MULTIPUCATION TABLE. He gave in buslielfl of 1x1= 1 T $72. How 2x1= 2 3x1-^3 4x1—4 08. $659. 1 X 2 a 2 In ak 2x2=4 3 X 2 = 6 4x2=8 . $S40, and ne owes me 1 X 3 « 3 1x4=4 2x3=6 v2x 4 = 8 3x3= 9 3 X 4=12 4;* 3 -. 12 4x 4 = 16 1. Required 1 X 6 — 6 1x6= 6 2x 6 = 10 3x 5 = 15 4x 6-= 20 1. $4016. 2x 6 = 12 3 X 6=18 4^ 6 = 24 ^ flour give, ,^ ' 1x7= 7 2x 7 = 14 3X 7 = 21 4 X r — 28 1 and that 1 X 8 — 8 2x 8 =16 3 X 8 = 24 4 X 8 1 32 " 1x9= 9 2x 9 = 18 3 X 9 = W 4 X 9 — 36 15, and $3 . 1x10 — 10 2x 10 = 20 3 X 10 — 30 4x10= 40 second 80 1 xll — 11 2x 11 = 22 3x11 — 33 4x11 - 44 [ns: 875. ' -— - _1 X 12 = 12 2x 12 = 24 3x 12 = 36 4 X 12 = 48 , expended of $22.98. ; 5x1=6 6x1= 6 7x1=7 8x1= 8, r* c^f hides i^' $13.50. ^ 6 k 2 .= 10 6x 3 = 15 6x 2 = 12 6x 3 = 18 7x 2= 14 7 X 3 = 21 8x 2-16 8x3= 24 ■ dA at first 6 X 4 = 20 6 x 4 = 24 7x 4 = 28 8x4 =32 >^F^» m^^r 4A4 %J^r b he had in 6x 6 = 25 ■■'' 6 X 6 = 30 7 X 6 = 35 8x6= 40 ^ i ft le at first ? 6 X 6 = 30 6 X 6 = 36 7 X 6 = 42 8x6= 48 8x 7 - 56 *. of dye for this quan- ay pounds, pounds. ;■ 1, left 600 6 X 7 = 35 6 X 7 = 42 7 X 7 = 49 5 X 8 = 40 6 X 9 = 45 SxlO «= 50 6 X 8 = 48 6 X 9 = 54 6 X 10 — 60 7 X 8 = 56 7 X 9 = 63 7x10 =» 70 8 X 8 = 64 8x 9 = 72 8 X 10 = 80 X 6 X 11 = 66 6x 11 = 66 7x11 =77 8 X 11 = 88 ,<. ■ received a' » leave in J of these . 6x12 = 60 6x12 = 72 • 7x12= 84 8x12= 96 ■' : '• 9x1= 9 10 X 1 = 10 11 X 1 =11 12 X 1 = 12 en he had 9x2= 18 10 X 2 = 20 11x2= 22 12 X 2 = 24 30 men. , .a 9 X 3 = 27 10 X 3 = 30 11 X 3 = 33 12 X 3 = 36 > B 9x4= 36 10 X 4 =40 11 X 4 = 44 12 X 4 = 48 t" 9 X 5 = 45 10 X 6 = 50 11 X 6 = 5^5 12 X 6 = 60 9 X 6 = 64 10 X 6 » 60 llx 6 = 66 12 X 6 = 72 r u %^^ 9x 7 =63 10 X 7 = 70 11 X 7 = 77 12 X 7 = 84 9x 9= 72 10 X 8 = 80 llx 8 = 88 12 X 8 = 96 lumber as 9x 9= 81 10 X 9 = 90 llx 9 = 99 12 X 9 =108 ^, > 9 X 10 =» 90 10x10 =100 11x10 =110 12x10 =120 •/ ; .. ' 9x11 = 99 10x11 =110 11x11 =121 12x11 —132 ''-1 ^ 9x12 =108 10 X 12 =120 11x12 =132 12x12 -144 iltiply, 01 " taken : ■ — — 7-J ■ -r .^^'■•-To repeat the Table by naing the second colamna ia moltfpUuf. has, 1 time 2 & 2, 2 time. 2 are 4, 3 times 2 aw 0, 4 timei 2 are 8, etc *^ . i T Factors, '„, ■ , ■ J. ^ l»rodoot W • '^TB^ 38 MULTIPLICATION. «or4cl.^"T2?''' '•^*''- " '""''»>'»«»'^ »*«» t^ multipUer doe, i^^. Multiply 542 by 7. \ OPEBAtlON. Multiplicand 642 Multiplier 7 Product 371I4 AwAiTBi^Ip thfii exunple, it ii nqairad to take 7 tiniM, we\.hall t»ke the .nUra number 7 timei' of -the mult,pl„iknd. we proceed thai : 7 times 2 ^ next prodaot. Seven time, 4 Lns.™ &.*■"* 7ir'. *^' 1 ten to add to the are 2'J ton* = 2 hundrodl and ♦-!. '•»'"'. "^ 'he 1 ten in re.erre, added, / andro.oryetho2huSd«toa,W?n hi' ^"'^ '•»« » M»n. in the ten.' pU^ hundredaaroaVhS^r^nS the 2 h-^ "^ hondreda. Seren tile, i added, are 37 hundreChrh'*.?;,?^^^^^^^^^^^ EXAMPLES FOE PRAOTIOE. Multiplicand Multiplier Product (5.) 2893 3 (2.) 4276 6 (6.) 16812 5 21380 O.) 48739 (3.) 6793 3 17379 (8.) 68607 • 8 (4.) 8634 6 61804 (9.) 76598 10. II. 12. 13. 14. 15. 873 946 4781 5607 6924 8657 16. 27693 17. 51786 18. 45678 19. 36397 20. 634576 Oasb II. ceeds 12. 3 = Ans. 4 = Ans. 4 = Ans. 5 = Ans, 6 = Ans. 8 = Ans. X f7 = Ans. X 9 = Ans. X 11 =» Ans. X 9 = Ans. S: 12 = Ans. 2619 21. 3784 22. 16924 23. 28035 24. 41644 25. 69256 26. 193851 27. 466074 28. 602468 29. 327673 30. 7614912 31. 76394 97631 266532 636466 641378 367542 426985 676483 6932574* X X X X X X X X f X 397466 X 3746178 X 4 =. Ant. . 6 = Ans.. 1 = Ans.. 6 = Am.. 8 = Ans.. 9 = Ans.. 8 =si.Ans.. 11 ^Ans.. 9 = Ans. . 12 = Ana.. 11 ™ Ant.. ■To efect a multiplication wTim the multiplier ex- '£.r. Multiply 478 by 64. OPKRATION. Multiplicand 478 , Multiplier $4 I Partial ) 1912 ^jirodiwts. I 2868 - Entire product 30592 AuAtTSB.- We write the mnltipUoand ud mnl- tipher a. in Oaw I, and proceed tfiu^ Four S 8 amt. are S2 nnit. = 3 ten. arfd 2 unit. -we write t\L^^^ in the place of unit., and SddCstoM ^ ten. l'^*"*" "^ **«"• '^<"" timeir 7 tew are W w^ T A^t«MMe»ltenr^ S lraiidndrSdiWV= 4^,^^^"^^.*" '^«,e>^««t rfhundoHi;. FoMttoM 4 hundred, are l5 hundred., + 3 hundred, an /— M Uiplter 'does lairad to take of eaoh ordar nber 7 timM: th« unit Agon timM 2 nnitfl ito the 4 units to add to the Menre, added, / B tens' place, leren times 6 last prodnot ' Mlnot is 3794. (4.) 8634 6 61804 (9.) 76598 — > Ans, . . = Ans... = Ans... ■= Ans... = Ans... = Ans... =^Ans. . , ^Ans... = Ans, . ', = Ans... = Ans... tiplier ex- tnd ud mnl- Fonr times Its; we write Id the 3 tens tens are M Is and 1^ ten; and add the . Four times nndreds are MULTIPLICATION. 39 /^ 19 liandrads, whioh we write in its DtoMr n\»M Wa »h<>n >. ■•■.. •Hultiply by the « tens in the malyeTtakfriarTto *i"l thi^rsf H'^' * '^« 'J !^' !^'*?'"« ^^ P^"^"^ prodncts obtained by the tw "muitiiZIlS™; we flad the whole product of 478 bj 64 to be 30592. muiupilcaUons, NoTfcP-When there are ciphers between the sfgnifloant flgurea of the mnlH. pltor. pass ow them in the operation, ani multiply by thf signiaoant flwros 5*. Prom the foreg^ng illukrations we deduce the following" Rule.— I. Write the multiplier under the multiplicand, so that units 0/ the same order shall stand under one another, and draw a line underneath. _„„ _ i II. Multiply each Jlgure of the multiplicand by each figure of the multiplier successively, beginning with the unit figure, arid wnUthefirstfigureqf each partial product under the figure of the multiplier used, writing down and carrying as in addition. Ill: 1/ there are partial products, add themllkid their sum wUl be (he product required. "^ PROOF OP MULTrPLICATION. 58. The Proof of multiplication is generally made by another multiplication (1) in which one of the factors equals the half the third, or the fourth, etc., of one of the factors of the operation' and the other equals twice, three times, fourtimea^ etc., the other factor of the operation. Or, In multiplying the multiplicand by the multiplier diminished by 1, and to the product adding the multiplicand ; if the sum be the same as the product by the whole of the multiplier, the work u correct. • U8B OF MULTIPLICATION.— iftti/tpiica^ion serves to render anu number so many times greater; to take several parts of a number • to find th£ value of several units or parts of units, when one of them is known; to bring a number expressing units of a certain nature to another number expressing units which are subdivisibns of the first, Jcc. < Generally we know that the solution of aproUem requires a multiplication, when the value of the unity is mentioned and that the value of several is required, or that of some, parts of the unity. ' ' ■ »■ (')• Inmnltlplylng U»e multiplier by the multipUoftiid, ths Mm* piodaet nut . .A< 40 MULTIPLICATION. 4. 6. 6. 1. 8. 9. 10. 11. • 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. (1.) Multiply 8621 By 47 60347 34484 Atu. 405187 976 697 749 8386 763537 134679 824956 984765 6654 97248 689834 867894 807497875 84966 543956 96824 43208 90480 43 76496 7674 - 3696 69421 4321 756849 908708 4916 7654208 80097 900007 4300407 460004 960076 690800 7006924 786630746 416342505 8963(12456 EXAMPLES FOR PRAOTIOB. (2.) X X X X X X X X X X X X X X X X X X X X X X X X X X X X X , X X X X X X X X 37215 65. 186075 223290 Ana. 2418975 27 ' 34 46 67 68 79 .387 756 789 865 943 996 965 7649 9475 4696 4962 9007 89006 87969 12478 819162 21764 987654 74323 70469 69678 20963 74269 700608 700608 99804 90708 456007 540086 367894 987405 9 43 7 6 11 Ans. t( it (< K (( « « tt tt (( « « « (( (( (( (f « « « « « tt tt tt tt tt tt tt , tt it (3.) 167034 304 668136 6011020 Ana. 50778336 26352 23698 34454 478002 51240616 10639641 319257972 744482340 5250006 84119620 660613462 864422424 779235449372 649904934 5163983100 464686504 214398096 814953360 3827258 67292T6624 95756172 3027622752 1610184434 4267652934 56261288227 69109512052 342637048 160455162304 6948724093 630562104256 3012899647466 4691023^216 8708657^808 316009635600 3784341665464 281494634808924 411098671149626 84688888738 6840 463966449974016 681797676464532 614499429100310 ?27 43. 44. 496307429 X 767489007 x 879407864 x 936704 900076 698766 I* (4 tt .ii;.!./Xt MULTEPLIOAWOft. 41 (3.) 167034 304 668136 011020 0778336 26352 23698 34454 478002 51240516 10639641 319257972 744482340 5250006 84119520 650513462 864422424 '9235449372 649904934 5153983100 454685504 214398096 814953360 3827258 67292T6624 95756172 3027622752 1610184434 4267652934 6261288227 9109612062 342537048 9455162304 5948724093 )6621 04256 2899547466 >9I02Si)216 [08667^808 >009636600 t341666464 [634808924 1671149626 449974016 676464632 429100310 45. 46. 47. 48. 49. 50. 51. 62. 9154907089 457907842 8B6407809 6|74396856 1864321 2465783 , 7240036 908007004 MULTIPLICATION OF DECIMALS. X 600789 An$. 673697675093221 X 796807 « 364864173860494 X 305407 11 261552939723263 X ^85679 it 192661019425224 X 609649 i( 11.36581433329 X 3686407 H 9089879711681 X 4029008 it 29170162964288 X 600123 tc 454115186861492 Ex. 1. ftjnd the product of 4.35 by 8.26. OPERATION. 4..35 8.26 2610 .., 870 3480 AKALT3I8 — Wo multiply aa in whole nnmbere, and point off on the nght-hand of the product as mwiy figures for decimals "rrT*" "* decimal places in the multiplicand and multiplier. The reason for pointing off the decimals in the product is. that in multiplying 4.36 by 8.2«, or by 826 hundredths, which is the same thing, we take 826 times the hundredth part of 4.36 but we obtain the hundredth part in removing the point two fi- a-rHoTTi A S.""' towards die left (No. 37, 2nd.) which ^'H give 0.0435 ; d5.»310 ^JM.there remains then bat to repeat 826 times this hundredth part to obtain the product required. As the number repeated oon- «..!. ^#*i. a. . °' ten-thousandths, the product wiU be oompoaed of deci- mals of the sartie nature ; to separate the units it is then necessa^ to take its ^X^T'^J^^^St'-J^ *i?* i», cut off 4 figures by the insertion of a point at the foS.'L'ri^SiaSmuTpU^T"^'' •PP""'^""' '•"" ""'"'" *'^'«'' If the factors are decimals only, we multiply aa usualand cut off as many decimals in the product as there are in both factors : but if the product does not contain a sufficient number of figures, we fill up the vacant places by ciphers, placing one also for the units. Ex. 2. Multiply 0.064 by 0.05^. AKAiTsm.— Multiplying 64 by M, we obUin 3024: but as . i? V^*^^'"!*!* *° *•"• *»« faetons, we place two ciphers at the left side of the product and having put the decimal point, we place Mother cipher for the unite, and thus we find the number 0.003024^ whwh ii read 3 tiiousandths 24 mill- lontluu 0.003024 S4. Hence the following RuLB.— I. JfuWp/y a* in whole numben, and point of a$ Z^t^T'f^ ^cimaZ., in the^wduct, a, there a« dedmal, w the muUxpltcand and muUiplier. ,^Vt1^« ?re»otjump^J aure, in t he pwd nct^thtre ag e ^maTplaceMxnihe mulHptxoand and mulHplier, $upplu the de- ficiency by prefixing cipher$. ^ > rrsf '^»^ lfoni..^To muiaply dedaab by M, 100, IMO, ttc (Wa 8«). 42 JIULTIPLIOATION. Proof. — The proof is the same as in' multiplication of whole numbers. EXAMPLES FOB PBAOTIOE. 3. 16.27 X 9. Ana. 137.43 4. 6.36 X 98. 11 622.3 5. 7.41 X 676. ft 6001.75 6. 197.19 X 66. ' « 11042.64 7. 97.85 X 975. « 96403.76 8. 69.78 ' X 696. (( 41588.88 9. 947. X 4.65 « 4403.65 10. 869. X 6.96 « 6048.24 11. 345. X 3.95 « 1362.76 12. 67. X 9.475 « 540.076 ^ 13. 786. X 7.789 « 6122.164 14. 374. X 2.967 If 1109.668 16. 9.47 X 6.694 (( 63.39218 16. 39.47 X ) 28.9005 « 1140.702736 17. 676.49 X • 60.706 (( 41066.32646 18. 401.04 X 13001.4 (( 621913.456 19. 9617.09 X 4281.45 « 41176089.9806 20. 6789.06 X 1.3808.927 (( 93749640.72768 21. 3807.45 X 6321.807 « 20262510.2547 22. 489.04 X 37.00845 <( 180^612388 PRACTICAL PROBLEMS IN MULTIPLICATION. 1. If a workman earn $15 per week :' how much will he earn in 9 weeks ? Akiltsis.— In one week he earns $16 ; in 9 weeks he will aun nine timet more, beoaase he works niiie times longer; therefore in multipfying by 9 we ob- tain the sum required = 16 X = 136. Aiu. In 9 weeks he Mnu $136. • 2. How much will 125 yards of cloth cost at $3.25 a yard ? Ah^tsib.— Ifone7M|L«o8t$3.26, 126 yards will ooat 125 timet moi«; in multiplying $3.26 by 126, the required sum « 3.26 X 13<> » Ah*. $406.26. 3. When a yard of cloth is worth $2.40, how much will 76 hun- dredths of a yard cost ? Ahaltbib.— The yard being worth $2.40, Ihe 76 hundredths of a yard will be worth 76 times the hundredth part of $2.40: therefore, mnltiplyinc $2.40 by 0.76, we find the snm required*. 2.40 X 0-76 — $1.80. Aim. $1.80. 4. What will 1635 barrels of sugar cost, at |25 a piece? A. $40876. 6. What will 786 kegs of tobacco cost, at $36 a k^ ? A. $28260. 6. What will 5679 bushels of wheat cost, at 86 cents a bushel ? 7. How many pounds of flour are there in 387 torrelo, there being 198 pounds in each barrel ? Atu. 76626. 8. Ho w many lettera a re t h ere in a volume o f 719 p a go i^ , i w^h jHWft - containing 1639 letters ? Ans. 1106541 letters. iu A house has 295 windows and each window contains 24 panes of gW, how many panes in the whole edifice? Atu. 7080 panes. iiSj.:ji£i,. -H^ j^lVii^. ,,M n of whole 137.43 622.3 )001.75 .042.64 1403.76 688.88 . 403.66 048.24 362.76 540.076 " 122.154 109.658 63.39218 140.702736 066.32646 913.456 089.9805 640.72768 5U).2647 9^612388 3N. he earn in I nine timet by 9 we ob- inu %\Zi. . •d? M more; in . 1406.25. 1 76 ban- rud will be igfX.40 by M. f 1.80. S40876. 128260. uhel? lere being 76626. letters. 24 panes 'panes. UULTIPUOATION. 43 10. Required how many trees in a nursery composed of 95 rows if each row contains 178 trees ? T, iggiO S. - A V»e Circumference of the earth is divided into 360 deerees 12. Required how many hours in a year of 366 days ? Ata. 876o' 13. How many days in 1000 years? Ana 365000 do^s\tdTsi1rc;^e*yt^:5l2^t^^^^ therti^/7reams?^^ ""^^'^^ '' ^^«' ^- "MSo^" n„iL'51*i^'' f r°® °0°*aiM 213 quarts; required how many quarts in 136 casks ? • ' ^^J. 28968 quarts ^ 17. How many eggs are there in 37 dozen ? A^ 444 qfifi ^«5T ?i^*°^ ^'%^ ^*^ * P®"^" *S«<^ 8* yea's ^i^ed, reckoning 365 days to the year ? - ^^. go^gQ ^ "« or mS'S r*""^^" "^ "^^'^ •" '«^ ^-« each confining agross Dec 1869 inclusively? (Not counting leap years.) An». 682185 21. Europe produces yearly 3466 pounds of gold : whatis the value oo A *i-u . - .. • •"*'*. $5956321. 22. A library is composed of 75 shelves and each shelf contains 86 J0^^me8; how many pages are there in all the volumeHXpols each volume to contain on an average 420 pages ? Am. 27offiS ^ 23. A speculator has purchased 268 horsVand 274 times as manv sheep : how many sheep has he purchased ? " AmU^T^ QK ilf"t"®^^^*8^ofwheaton a truck, each bag containinir w^gKW^Tdr?*"^^""'^"^''""^"^^^^ v««/''°'''"°*''*"°''^*^^^= ^^ow much will he S'rn i'n" 7 »f •t'^?°'^ much will 240 pieces of cloth, each containing 44 yds. cost, at IMO per yard? 4n,. /57024 ** in th.viTfi™*"^ paijof shoes can be made in 265 days, in a factory 9« Tr P*"" ^° ^ "a*^® in 1 day ? tn««; ' * j°°® ^*^' * *P***» of ho«es can draw 2997 pounds: how many pounds can they draw in 327 loads ? i~ "» , now k^l\^ ^*K5 ^ acres of land yields 46 bushels oats per acre ; what is thevalue of the cro^ of the 7 acres at 10.40 a bush.? aZvT J "Ji??^"^ * ***e*P JP^e" 6 pounds of wool a year 5 how man^ peck. ; how many Aeave. coold 14 Uinm, Ibnih in 9 cSm md "hat wo<^ be the qnantitT oTgrsia obtaiad f '* AH. MtO Aeara and 18M) peoka grain. 44 inTLTIPLIOATIONi CONTRACTIONS IN MULTIPl^ICATION, OE MULTIPLICATION BY FACTORS. 55. In many instances, by the exercise of judgment, as it' will ««"' A ^P®*"**^"" ^'^J ^« very much abridged. o». Any number that may be produced by multiplying togoth- ? IK S™^""® """"^era. w called a Composite Number. Thus, « .U^°> "^ composite nombera; for 6 = 3 X 2 : 15'=^ 5 X 3; 18 = 3X3X2. ?7;.^*^® ^**^*^" *^^ * number are the several numbers which, Multiplied together, produce the given number. Thus, the factors of 24 are 12 and 2 (12 X 2 =^ 24) ; or, 4 and 6 (4 X 6 ^ 24^ • or2and3and4(2 X 3 X4 = 24). ^' tl,?^*~T.f""C'^K7n"'"*'"'* booonfonnded with thepart.of annmber. Thna. the/actor, of which 10 m composed, aro 6 and 2, (5 v 2 = 10) ; while the Jarto of which 10 ia composed are/and 4. (« + 4 =' 1^0). The/actori iTmSupS while tbejMrt* are added, fo prodaoe the nutnber. "•""j'lteo. Case I.— To effect multiplication when the multiplier is a composite number. „ Ex. 1. What will 45 acres of land cost, at $367 an acre ? AKALTSis.-The factors of 45 are 5 and ». Now, if wo mul- tiply the cost of 1 acre by 6, we obtain the cost of 5 acres l and, by multiplying the cost of S acres by the factor 9, we evi- dently (fbUin the cost of 9 times 6 acres, or 46 acres, the num- ber bought. Henoe the following f 16616 Ant. (58. Rule. — I. Separate the multiplier into two or more factors. 11. Multiply the midtiplicand by one of these factor »,, and that P^vct ly another; atid so on, till all the factors have been used. The kut product will be the one required. NoTi.— The product of any number of factors is the same in whatever order they are multiplied. Thua. 4 X * — 20 : and 6 X 4 « 20. """*''^"' °'***' XXAMPLKS FOB P&AOTIOE. 2. Multiply 2746 by 28 =« 4 x 7. Ann TfiSfin 3. Multiply 66742 {(y 36 = 6 x 7. ' , i' J!"I!Jp!^ IX bJ ^2 - 3 X ? X 8. An,. 6618592. 6. Multiply 36783 by 81. 4n».'2979423. 6. What will 66 horseB cost at $178 each ? Ana. $9968. 7. What will 436 bushels of potatoes cost, at 32 cents a bushel ? 9. In 1 mile there are 63360 inches; how many inches, let. in 46 mil^B ?— 2nd. la 64 miles? ^nt. Ist. 2861200 ; — ■ _ ■ ' W. lf»«l it a oompotit* Domber T— 67. What an th» faotois o/ my mmbn- 1 ;m k.l .'.ii'i-i"it.o»„<1i'^),^i !a. 4 \tk ^^J i 1^ i* , hJ, 'V ON, it, as it' will ring ito<»oth- ier. Thus, 15-=^ 5 X bers which, the factors 6^24); imber. Thns, hilethepart* re mtUliplied, plier is a e? r, if wo mul- of 5 acres ) or 9, we evi- M, the num- re factors. ,. and that heen used. ttorer order . 76860. ■ 618592. 979423. $9968. )U».hel ? Iflt. in 45 51200; Mj/mmbtrl MULTIPLICATION. .45 EXAMPLES FOE PaAOTIOB (p. 19). tMnght-Jiand of one or both of the factors. r "** Ex. 1. Multiply 1400 by 80. and 100, and the multiplier inta the factor* 8 and 10 ^.Z i. K Hoot'd 8 ' ^?;„T i4TrJ'??2"°in'd %'^:^^z^- moo; «d 1,200 X 10 4l}2«lhe"LrrJS S Tthl «0. From the preceding iUuatration we derive the foUowioff /;W^«'?;r^"/f'?* '^J^'M^fMre, of the multiplier und^ th,se of the multiphcand, and multiply them togetherf To S product annex as many dpher, as there are on the right of til multiplicand and multiplier. ^ •' ^ OPERATION. 1400 80 TIMb Multiply By Ans. • EXAMPLES FOB PftAOTIOB. (2.) 3764680 270000 2<>35206 752916 1016436600000 (3.) 1306950000 600800 4. Multiply 610430 by 700500. 6. Multiply 3070607 by 7007000. 6. Multiply 2020370 by 4030200p. Ana 8142.l9ST7innnnn 4 ,»=?"--- -isSiS- 1045560 784170 Vl5&-Zld6«000000d An8. 427606215000. Ant. 21515743249000. ilfw. 814249517400000. do. 4iM. 560114005776000. a »m "iJ!i^or:i!^^^i^ii,f^^<^<^f^'^^ rv»*.*aj^;^ < I : 1 '^ k 46 MITLTrPLMATION. 9. Multiply forty-nine milliona and forty-nine, by four^htindred and ninety thousand. .. ;^4n«. 24010024010000. 10. Multiply one billion and twenty thousand, ly one thousand and one hundred. >ln». 1100022000000. 11. Multiply ten billions ninety -six thousand and eight hundred, by thirty thousand and seven hundred. Ans. 309971760000. 12. Multiply thirty millions ninety-thousand and eight hundred, by six hundred thohsand and eighty. " Ans. 18056887264000. Case IV. — To effect multiplication when apart of the mul- tiplier is a factor of anothtr part. Ex. 1. Multiply 7439 by 328. * • " OPERATIOS. 7439 328 59512= Prod, by 8 linits. ^38048 = Prod, by 32 tens. 243999^2= Prod^ly 328. ANALTSrs. — Wo consider the mnlti- . plier as separated into two parts, 32 ten* and 8 unit*, or 320 -f 8 j of which the smaller part is OTidently h factor of the larger, since the 32 tens, or 320, is equal to 4 ten» X 8. We next multiply by the 8 units, obtaining the product for that, part of the multiplier. Now, as . this product js the same as that by the factor 8 of th» other part of the multiplier, tre multiply it by 4 ten», obtaining the product tit the multiplicand by 8 X 4 ten*, or 32 ten*. These products of the parts, added together, give the true product by 328 j and, 61. From this illustration we derjve the following IlULE. — Multiply fint hy the smaller part Sf the multiplier ; and then that partial product by a factor, orjkoters, of a larger part ; and so on,with all the parts. The sum l^th^ several par- tial products will be tJie product required. v'*\^\ EXAMPLSS FOS PRAOTIOE. . 2^ Multiply 6526 by 668. 3. Multiply 3785 by 721. ^ 4. Multiply 85065 by 2432. 6. Multiply 236428 by 64918. 6. Multiply 397821 by 23125. 7. Multiply 1146084 by 24816. 8. Multiply 6723606 by 4249784. Ans. 3706768. Ans, 2728985. Ans. 12984152904. Ans. 28441220544. Case V. — To effect the multiplication of decimals when the multiplier is 10, 100, 1000, etc. (No. 36, 2nd.) 62. BuLE. — Remove the decimal point as many places to the right as there are ciphers in the. i^ultiplier, annexing ciphers if required.^ . . Z,_ ■ ^ \ n. What i* the taUM muitiptying wKen apqrt of th« myttiplier i* a factor of anoiherpartf—ii2. What i* the tvlIo for effeetmg the tmMpHeatvm of deeimtO* wheti the midliplitr it 10, 100, 1000, ite, t ^ / fl W-4^^^^^^-^'-•'^'J^^^'^'■'^^^Jfc'^^^i''^"^' '■''''■•■'''%"•'•'■'■' ^^^ *• ■'''^!ii''''-^i-^'.-..-. . 'i^ i; ,. 4'k^:!3l^-iK'f*Lti'Ci n MULTIPUOATIOK. E3?AMPLE8 POH PMCTK3B (p. 20 and 21). 47 rlTeT ^^ tfeama?j>face, of the j^roduct: should ht Plaf^^-inthfpffict'-'''''^ '•'«'' "^^"'"^ -'^ three.decin.al operation/ 6.5628 687.5 32814 = 6.562 x'6 4594 = 6.66 x .7 +2 525 = 6.6 X .08 +5 39 = 6. X .006 + 3 37.972 Product. . Akaltbis — W« TOTene the order of toe flgures of the njuUipIier and write toem under the multipUoand ; and, since thoosandths is the lowest deoimal flgure to be reUined in the produot. we place the units' figure rf the multiplier under the thousandths' figure of the multipli- oand. Then, the unit of the product of any figure of the multiplicand by the fijj- nre of the muldplier that falU under it will be thousandths. When there are 63. From this illustration we deduce the following Rule.— I Writt^the multiplUr, with the order of its fiqures reversed andwxth the units' place under that figure of the^tt phcand which is the lowest decimal to be retailed inthepZdZ IL - /: 'h m m.. ■ « 48 HITLTIPLIOATION. liXAHPLKS FOB PRACTICE. ( 2. Multiply 472.35 by 64.3645, and 3.657389 by 0.0536423, re- taining, in the first, 2 decimal places, and, in the second, 6 decimal places. ^' OPERATION. 472.350 5463.46 2834100 188940 14170 2^34 189 23 3040.256 OPERATIOK. 3.657389 3246350.0 182869 10972 2194 146 7 1_ 1.96189 3. Multiply 751.20371)7 38.7136, retaining 3 decimal places in the product- , Ans. 29081.801. 4. Multiply 36.275 by 4.3678, retaining 1 decimal place in the product. 6. Multiply 843.7527 by 8634.175, retaining only the whole num- bers m the product. Ans. 7285109. 6. Multiply 4266.785 by 0.00564, retaining only 3 decimal places in the product. 7. Multiply 73.27593 by 0.075325, carrying out the product to the seventh decimal place. Ans. 6.519509K N 8. Multiply 1.7323152 by 3962.57302, retaining 8 decimal places in the product. PRACTICAL PROBLEMS COMBINING ADDITION, SUBTRlC- * TION, AND MULTIPLICATION." worth of bark, prepare it. 1. The hide of an ox costs $6.16; it requires 12 9 quarts of oil at $0.18 a quart, and $0.60 for labor to Required the gain if it be sold afterwards for $12.75? Akaltbis — ^The whole eoit of th« hide sb fd.K + $2+ rtO.18 v >s SI «7k / + 10.80 - «0.37 , $12.76 - f 10017 = ^ ^ *^^ ^ A^/jl^^^Z^ : 3r A muslin manu&eturer sold la one year, 640 pieces of it, viz. • ITOmeces to Montreal merchants; 86, to Quebec merchants; 130, toToimnto merchants; and the remainder to Ottawa merchants: "What is that remainder? iliw. 166 pieces. 3. A man bought 26 barrels of flour at $5.60 a barrel, and 40 bar- ,xek of apples at $3 a barrel ; what was the cost of all ? An$. $257.60. : 4.a paid for building my house »1889, for my farm 3 times as ^uch4fi88$892, and for my fanature $140 more ^aiii paid to*" building my house ; how much did I pay for all, and for each ? ; . .Am. $4776 : $2029 ; and $8693. 6. A young man receives $1000 salary, and pays $180 for board, i36423, re- 5 decimal aces in the 081.801. ice in the bole num- 285109. tnal places uct to the 195093! nal places JBTRlC- L of bark, )repare it. 9 »= f 1.83) ►f it, viz. : inta; 130, lerohanta : ) pieces, id 40 bar- 1257.60. times B8 • pKd *ar Bh? 118693. for board, A ' MULTlPtlOATlON. 49 $2l/'for clothing, $120 for books, and $166 for other exBenaes- ],«. much can he sav in 4 years? JS .loon 6. A merchant soH 75 yards of cloth at $2.47 oer vaH vIL Jl..- ^ the amount of his invoice? ' ^^ ** aL Jum^^^ "^ 8. Leo has $127 ^ Peter, 3 times as much minus $m • and John has as much as Le.> and Peter together: how mSoh hiiv« P«l: 5 John respectively, and how much lave they 111" f"^ *''** 9. A merchant boughi !f S'«?f "bt'cS' ""'l'^'' ^''.'' 37 yards, andl2pieceTof\laK<5th, 'ac'htonVniS^ sT^JS^ how many yards of cloth did he buy of^h\ two Wn dTilSgfthe^r?^ ' 10. If a cow cost $28, a horse 6 times as much ftnd*^«?Q V as much as the cow anj horse together minS 4112. hn» u*'™*' wi 1 the farm cost than 5 horses fndri?ow^a^\'h\^ame"r;^^^^^^^^ baJrd^:S^SSS?;£'a?$^^^^^^^ f ™°.° P'^-^' - at$9a barrel^ how'r^hdidt^ainor Jst? J^! SjnS'fToTJS'" 12. If an acre of landlproducel yearly 362 po^ds of flS a;^ 1 1 bushels of seed ; it is required to know Lw m^^i^nnds^C\l\ how many bush, of seed will 7 acres produce, and how m.Jfh^luS'* whole be worth, f the flax be sold at «0 ift «\^.,-j J^^" ^"^ ***® $2 50 per bush.'? Ans. ?J34 i^und's liif 77 C^s^.^t^ "' 13. In a dairy, there are 27 milch cows wSj^fl^'^^' ' average, 108 pounds of butter- what «nrn Jli IvT P- ***'^' °° »*» .ellinlliis but^r at $S.18 a ^^uSf """^ ^' *^' ^r^J^no"? t« ^^ 14. A farmer desiireS to manuroa field of 12 ni^lt^V 1 j • , manure worth $4 the hundred weightrand Jays $M?fof iw'* "^^ hundredweight; how much will Jt cost h^ to mLSr«T^ ^' suppos.ng he requ„^B 2 hundred weight per JJret S"$i^30 80 ' 16. A cabmet-makerearna daily $1.65: his wife Hi on. a!,TV- three sons, $0.66 each : how much ckn ke tay VeJe^J 4^ thf daaly expenses of the wWe femily being $2.68 ? ^ X, mV^® $3^2 i i«"^« B W9660, B lets A hav/bank Stock to tr^mout of $38®2, a farm 4 times atf much as the bank stoA -.Sii qqa ? j ^^ the remainder in ^h ; how much'STdSB^^^y 1?^! $209^' •nVo ^J«weUer bought a certain quantHy of ivorv at the r«I» \^ KiiSsS'f' H\«bo'^^t6jounds^ml7hfc^Joul?ha^^ ftsJq J, -f^IJ *"** Buperhitendence of a raSKad track c«t yearly 3!w »'Wj "sj 50 DIVISION. 99 yards Jong which is' 20 yards more than the second and 34 more than the third; what sum must be paid to the plumber for hia P'P^ ^ , ,, , . , , , Ana. $144.64. ZO. A handkerchief manufacturer bought 78 packages of thread of which 40 are warp, at $10.90 per package, and 38 weft, at $10 65* He pays $0.85 per dozen for weaving and $26.30 for selling expenses • what will be his gain, knowing that he has made 640 dozen of handkerchiefs, and sold them at the rate of $2.58 per dozen ? Arts. $244. IH y DIVISION. 64. Dlvifllon is the process of finding how many times one number is contained in another; or the process of finding one of the fiwtora, the product and the other factor being known. Thus, To divide^l2 by 3, is to' seek a number, which, being multiplied by 3, giyes 12 for product ; or, to find by what number 3 must be multi- plied, to obtain 12 in the product. The product is called Dividend, the known factor, DivlSOr, and the factor sought, Qaotient When the dividend does not contain the divisor an exact number of times, the part of the dividend left is called the Bemalnder. and must be less than the divisor. , ■«««'«» Case I. — To divide whm the divisor does not extked 12. «-?*f**T7''®° *?.? P'oo'*' of dividing i« oarried on in the ndnd, Ud the qaotient only ia set down, the operaUon is eaUed Short mvUion. "» f"** "* huohtob. Ex. 1. How many times is 7 coufaeuned in 994? OPERATION. Divisor 7 ) 994 Dividend. 142 Quotient. ANALTsn.— We write the divisor oil the left.of the dividMd with a line between them and another line beneath thedividend; then, beghudng at the left-hand, we say : 7 is ooii- ti^ed inV, 1 time, and 2 hundreds remain- ji_ij ^ ..... I . ^ . - -*°»» '^ ^^^ *•>• 1 direetiy under the 7, its dividend, tot the hundred^ figure ot the qaotient To 9, the next Ague of the dividend, whioh is tens. We unite the S handreds remaintag, whioh equal 29 ""jJP ^^^"^ ** ^°<^ ^^ divisor 7 to be eontahied 4 thnei^ and 1 ten remaioing • we write 4xe 4 for the taa** figare in the quotient, and the 1 ten remainhiV.l equals 10 units, which, united to 4, the last figure of the dividend, make 14 units ; in 14 units, 7 is opnt^ned 2 times ; writing tiie 2 ilmr the imtto' flcure of the quotient, we have 142 for the entire quotient. 65. RuLK. — I. Write the divisor at the left-hand of the divi- ■ dmdfWith a line between than, and draw a horizontal linebeneath the dividend. f 64. ynat M ^vision T-— Wuu U (&• dividend 7— TJu divisor 7— The ajiti«i(t? he/ore. and divide a$ r a Meftit ''"yr'^^f^i'iend he hm than the d^vieor, vmte ^^^Pferin the quotient, andprefix thenumber to the figure of tU next bwtr order in the dividend, and divide a, be/orl^ ^ it afJ^fl^' ^ arenunnder after dividing the laet figure, place tt after the quotient, and write the divisor under it. fi,„^^^5^""r^?^*iP'y *^® '*^^^'' a°'^ quotient together, and to equal to tjie dividend, the work is correct. plterjin*"^ of proof follows from diyi,lon being the «ym of muWr O EXAMPLES FOB PaAOTIOB. 3. Divide 8154 bj 6. OPERATION. Divisor 6 ) 8154 Dividend. 1359 Quotient. (3-) (4.) 6 ) 714325 3 ) 893763 142865 ' 7. Divide 6376 bv 6. 8. Divide 5592 by 6. ^ 9. Divide 98776 by 8. 10. Divide 174321 by.9. 11. Divide 1643784 by 12. 12. Divide 46215796 by 11. 13. Divide 63412632 by 12. 14. Divide 2271582 by 7. 15. Divide 11357912 by 6. 16. Divide 4066360 by 9. (6.) 7 ) 9491 1»^ 1366871 ^i*. Divideda980400 6y 8. 18. Divide 42084795 by 6. 19. Divide 4507060 by 12. 20. Divide 15023620 by U. PBOOF. 1359 Quotient. 6 IHvisor. 8154 Dividend. (6.) 4 ) 662846 Qaotienti. 1275. 932. 12347. 19369. 136982. 4201436. ^ Bern. 6. a. s. 4.' 62 DIVISION. PRACTICAL PROBLEMS. 1. Nine yards of Bilk velvet cost $72: how much did it cost a yard? AjULTOB.— If the price of » yard were known, in multiplying it by 9. we would obtain $72 ; therefore, 72 la a product having for factors 9 and the price W a yard. Then, in dividing the product 72 by the factor 9, we obtain the price of • yard } 72 -*- 9 '-'Am. |8. Or agai*, aa 9 yards coat 1572, 1 yard wilf cost V Umesieu, because there are 9 times leu yards : then, In dividing 72 by 9. we obtain the price of a yard. o^ j > 2. If 6 ehillinga make a dollar; how many dollars in 8890 shilhngs? iliM. 1778 dollars. 3. A gentleman divided $89622 equally among hie 9 children : how much did each receive? " Ana. 19958. •/•tfto ^'"*"^''*"'®'^°'^^°"''; ***^* barrel, can be bought for « Li« . ,. >ln». 85 barrels. c tT, "'^'^^s make one foot; how many feet in 7501464 inches? 6. Eleven horses were sold for $2531 ; what was the average sum received fo«- eSfch ? , Ana. $231. 7. A boy spent in one month 260 cents for oranges, givin<» 4 cents for each; how many oranges did he buy ? Ana. 65. 8. A carpenter worked 11 months for $572 ; how much did he re- ceive a month? J ^n». $62. •i?eo^i'"*P'®^^°'^^*^*''°'^r>owmany cords will be had for in A -u . ^. ., ^ iln«. 192 cords. J A . F"o° wishes to distribute 168 apples equally among 4f> boys and 3 girls ; how many will each of them receive ? Aiu. 24. Cask II.— To divide when the divisor exceeds 12. ^ffdiSSm" ** '^°'° *''*~" "^ diviakm is written, the operation is termed Ex. Divide 4738 by 34. OPXRATIOK. Oivisw. Divd'd. Quotient 34)4738 (139J}. 2nd. partial dividend l33 ^ 102 3rd. partial dividend AitiLTSis — Taking 47 hun- dreds for the first partial divi- dend, we say : 36 is contained in 47* 1 time. The 1 we write in the quotient: 34 X 1 « 34, which we wnte under the 47 ; 47 — 34 = 13, to which bringing down the next figure of the di- vidend, which is 3, we form 133 ; 34 in 133, 3 times. The 3 we write in the quotient ; 34 X 3 =» 102, which we write under ..,...„ . the 133; 133 — 102 = 31. to [Oh Drinj^ng down the next figure of the dividend, we fbrm 318 : 34 in sig a Sis ^tli "'sSS"*" ?»*'• '^""^r ' " ^ » -306. which w^ wri^ 'JL; !?li 'ufl Zf."^ 12, a remainder, w a part of the dividend left undi- SJ^^n ^« *rf»« ^ tJ«» qnotiont with the divisor below it, thus oompleting 66. BuiiE.~I. Write th ejiv itor and dividend aa in thnrfitj. ^rOf^w acurmriine at the right-haneTof the dividend. 318 306 12 Remainder. M. Wua U tU rale to dwith when the divieor erneekk 12 ? ! i I irtVisrow. 63 • ir. Take for ihefir^t varlial dividend, the least number of III. Multiply the divisor ly this quotient figure, place the nmd uct under the partial dividend, subtract, a«V to 25 in MO la^iili3"^ »• •"•»«»«»•>• > the addlUon of a ci- W«^tHjJt?5a£M therefcre. :!-' !l?M£^Kf&>i, / 'v.- mprmos. 65 S «umi^ «»7' W«» ««»«*^ ie muUiplied to produce Ltd When a. certain given number of units or part* of units a4 Mown, as for insUince, the buying, the seUinf^ce if a lard l^^^ng their whole value at^ that of the unit, as for ii,^ the number of days that a lahorermmt work to ej% a^Z tohich expresses subdivisions of this unit, such as to findhZ wa^ hours there are in any 'given numbed ofminZs^Jl t7^^irnt'irK'^'*''\*'' ^^P<'^tsofunits,aregiven, r^t^ofmS. -^ ' ^"^ ^*"*^' ^ •'''* '** *"*^^ 'J/' ««»■'» or KXAMPHS FOR PRAOnc# 1. Find how man/ Umea u 72 coataineJ iu 23596. OFEalTION. © Divid'd. DiYisor. 72 ) 23596 ( 327 Quotient. 216 ~199 144 " "^ 656 — 604 62 Remainder. rROOr BY 1IUI,TIPU0AW0». 327 Quotient. 72 Divisor. 654 2289 . 23544 62 Bemainder. 23696 Dividend. 27939 38582 405683 743^1 954992 173469 497699 218579 611286 16 18 20 25 30 36 40 42 4t Quotient!. 1746 2143 20284 . SI833 12442 ~ I 3006 ^ 8662 14703 Bein. 3 8 & 16 2 21 19 11 4n2ff 4326a6 846002 867632 876701 49 60 63 69 60 4 15 6 29 ^ 6S 41 / 3i» 56 16. 17. 18. 19. 20. ^21. 22. 23. 24. 26. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. fiO^ 51. 52. 53. 54. '66. 56. 57. 59. 60. 4968 9iM25 446124 4728 3f9006 1679407 4306404 167008 7466029 6717890 i DtVlSIOIC. 64 68 70 75 79 80 86 87 90 98 QnoUentfl. 77 6368 493 60663 82844 To calculate with two decimals in the quotient. Aem. 40 61 64 3 69 47 49 66 69 88 , 67980 432101 470896 ' 680094 666648 767642 124674 964321 7246579 7890646 9120128 687621 3466604 4268901 2486930 4107129 I • 81267904 69267421 89064010 694736210 468904008 389006753 86742807 / 707070709 654380316 987664321 8606000041 61247680241 74238961401 9649646664 8674289646 4247698734 6312480086 46680108007 37894216118 4- 96 69 72 67 441 386i 126 216 612 367 637 4691 1279 1467 7614 7614 6174 7186 7908 9087 7064 8004 8906 4260 49060 49066 60041 74085 48647 42867 74551 94672 59856 300462 987684 Qnotiente. 708.12 6262.33 10160.65 1511.67 989.47 4464.44 162037.96 ■y^-i.- ,mL' ■I \ DIVMION. Bern. 40 61 64 3 69 47 49 66 69 88 Kem. 48 23 16 46 / 153 380 78 196 328 / 187 I 153 1422 240 435 4632 ""6126 3592 6794 184 7462 6768 2684 ■ 6914 4120 37440 39106 49042 13310 11893 32712 48424 aso6& £^. 1. Divide .3.456 hy 2.4. DIVISION OF DECIMALS. 67 2.4) OPERATION. 3.456 ( 1.44 2£ 105 96_ 96 96 . . AifitT3T8.-W6 dirido aa in wh^e nnm. ^n»- ^ihl ,' O'?** tl»e diviaor and quotient ffinfff ^''T' :P"?2~ **"• diYifend/w. point off two deoimal flgares in the quotient. to make the number in the two fMtdrs equal to the number in tlie product or dividend. ^'> Ea;. 2. Divide 0.526 by 7.5. ^M OPERATION. '. 7600 ) 525.00 ( 0.07. 625 00 AnALna.—AB the decimal plaoei in the div. idond exceed those in the d^ »"«£ them equal by annexing two ciphen to the JlToL*°9 • ^"IT"^ P^poeeded in the division aa in Oaa. 2, p. M, wo find the quotient to be a.Of, or U unit 7 hundredths. ^ "* 67. From the preceding illustrations we deduoe the foUowing Or, RuLK II --Jf the dividend and divisor have not the eamt number 0/ decimals, annex cipher, at the right-eide of tZ ^ v,h^h ha. the haeteo that if may have c^ma^dlrSj^, i. Non 1.-T0 divide decimals by 10, 100, 1000, etc. (No; 87). ^PBOOF.-The proof is the same as in division of whole nam- EXAMPLES FOB PBAOTIOE. Quotients. 31.64 4.174 0.066 «7i ynatUthtni»/ortM4divm ""' Z^^ " * «"" ">«lft.n 35T6T ;' A *^?S f^' TOlume, how muT yolunfee can te eot for Ml) » 134 lo how "^^ 4««iJ2. ■f^s>t^mrs^m« m^iSi, as one man in 806 days '. .,- - -_ ^ diiL'^3rh°s:r»°'!:5^,''^'-''^"'2ri££*' \ i«UL "Sst'" , >.• 34 " 74Q 18478 6984 296 1998 628 960 17178 34 240 4 yard cost? ! bj 4^ w» oton 46 and ill heeara 18 days. e o6at«]iMd iumliers is ». tm. ye 70344 7 Kndthe IredUu. thai per 676 T >rf697 s can be M. 216. M. 80. e bonght yartUf Mda #H1 Ox Work, ; homfta DIVISIOK. Ifi A HMlIn A. »U. /I J m ■, ^ .. 59 we obtained 16. A' train on the Grand'Triink Railwav nin« Rft «,:/*"** ^'J*' at the same rate, how long would k Seto ^Tr-fnnJ ?u^*' *°,>°"' ? distance being about 25000 n^les ? ^ 'Xi iiSf Tl'^'*' ^^'^ 17. The large wheels of a coach am ifi f^tt ' ^ t ^^^^^ ™.ii on« 6 ^ti how -n' s'^T.^r^r rT'&j^.j l«'P2 (eet 7 il««. Large, ^345 + t . fimall mpm T^^^ °' ?• f ^,S? \r"?^' whose p^uc't by ©.OOt^S t o ob026. 19. I bought a farm containinc 175 acres fn* •TqSk "•"'"'^'>' dollars did it cost per acre ? ^ ^^^"^ **^^ 5 j»o%many 20. A butcher gave $66 for sheep, at the rat* nf lui !»n I'^f* many sheep did he buy ? ^ ^ '*** °^ f3.30 each ; how 21.^How many pair of slippers must be made hv r It^ ^\^^\ earn »1.36 per dav, if he be'Sdd f 0J6 for^^f^^^? jj^^s? '° 22, The annual receipts on a railroad finn T^.iiTi **' f860«000. R<^i^dthSaver4T^^rt4iL^Ste ""'r?* *^ ceivedper«iile«.t.ually? ^rSja dSl?S(JipL^786"3:^t4« " ^ ' do pe»m|Ie$f2w; 23 water The aircontained in a puncheon weiehs & frS.*i?"*'':u it would contain woirid wSgh 7607.6^hm;. hn^ *^\'- *^' is the weiAt of the ^tergreatlrVaa iftof&e^rV i 7?^^^^ *""** 24. A chiartjoal maker places 127coid8 of w!^ ;« 'K*^"®^. cost him $680 ; he oonsqSies IS^M f^?Se «iii'" ''^^ theTalueofihecharcoaliObtained^SKSavK"***^ ""^ of 110.28 Mr ibttjAel. Biquired wSfbltif.?*^^ $^*^« »te produced*taf»cSl of 1^? ^ ^^ ^ 5« ^V^J^" 26. The^bpilation of Ihe globe i«ab(mt 13008«Sftn • t'Sf^''- supposed tfiafit is ttn^iideyeri arvlan^fi^!^/''^^'**''*^'* 00NTMO!b)NS IN DIVISION, ' OB DIVISION BT lAOTOBa. * .' £jr. 1. IMvide|169e^qMli^an»Oi^28peraon^ OPIBATIOV. A»jiT«is.— The fiMton oi'lfl u» ^ .. ^ • ™ tt»n by 4. Henoe the fidlowl^ •8. Wla»i»«»«nito/«.*rtaiiy6f «wmpoiit;nBBbJf- *j- it.*^** .•/ ■>-'W 60 DlVBIO*. required quotient^. '^^^'"^ The last quottmt^mli he the KAMPLM POE PHAOTIOl. 2. Divide 4636 by 14 = 2 « ♦ 3. Divide 9774 by 8«3J.6 J. Dmde l5637fi hy 76 = 3 x 6x5 ^IM. 324. .^fM. 643. Aru. 416. ilfM. 1686. ■4n«. 1652. Ans. 798. ^n«. 2938. Ans. 23703. 3) 10183 6) 3394 . 7 ) 678 96 ... 6 OPBBAtlOK. rein. I .4 X 3 = 12 6 X 3 =: 90 103 true mid. * * ^fAf"<* » "maSider of 1 nndlTidttl, which, beinsa p»rtof the gl»,n dividend, mnrt aim be » part of the teae remainder. The 3394 beinga qnoOent arising from >e whole An». 9876.64321. ' , DECIMAL CURRENCY. renoy j UmM oiOlcMl ^WmU u sm Uta ourrenay of ffieUnfted StaiM ^wTi. <*S^^^itJt.^ra^'i§{'^'*/ If»« ^d.. 64 WBOIMAL OUROBNOY. •77. The present Oolng of the Dominion of Canada are of silver and copper. The silver coins are the fifty-cent piece, the twenty-6ve-cent piece, the ten-cent piece, and the five-cent piece. -loD^tolJidSS?'*'**'"**''""* P'*""' '»»<"•«»> »Umn oiroaltUon. is no The copper coins are the two-cent piec(J and the cent. • 100 cents (cts.) make 1 dollar, marked $1 nickel* ^^* "•*»■ «f t^« United States are of gold, silver, and The oo?d coins are the double-eagle, eagle, half-eagle, quarter- earie, three-dollars, and dollar. o » ^ , an?h^"me°°""' "® ^* dQligr^lf-dol}ar,quarter-do.Uar, dime. The nickel coins are the 5-oent, 3-cent, 2-cent, and l^jentpieces. 71... mwce the meUl of ooina mora Mrrieeable, gold ooina oontain 9 Darb h* weightof gold and 1 part of m alloj oonsUOng o? .Uw knd wPMr^^S^ oouu oonuSn 9 parta of dlvor and 1 part of ooppor. ^^ TABLB OF THB UNITBD STATES OURRENOT. 10 mills (m.) make 1 cent, marked 1 rt. or c IQ cents " 1 dime, " 1 rf - 10 dimes " 1 dollar, " f 1. 10 dollars " 1 eagle,' "IE. T». The Dollar is the unit of currency in the Dominion rtf Canada and the United States. Accounts^ are kept rr^Lf cento, and mills, ' Dimes, cento, and mills, being fractions of a dollar, are separated from the dollar bjr the dwamal point ; thus, four dollars twrdimes three cento five nulls, or four dollvs two hundred thirty-five mUls are wntten $4,235. ' * 1 ^°,*'?Z!~ *x? °T*^' of cento less than 10, a ciphet must be placed betwewi the figure expressing that number and the decimal point; thus, 8 cento is wntten .08, or 0.08. EXAMPLES TOR PRACTICE. 1. Write fifteen dollars twenty-three cents. >!•>• si r os 2. Write seven dollars six cents. ALiHi 3. Write ten dollars ni ne cents.. ^' *^'"^' — 4r Write fony^two cento. p:r^n>. foi rT"" T?» '»;*?* «•« «*«ooin« of the Dominion ofC^ a^ 1 -79. mati,tl^xa>ito/cmrmegimtA,j?.a^U:sf '^. Vf»*»dStaU,t A •q ■nb A 6. 6. 7. 8. 9. 10. Write Write Write Write Write Write 11. Write 12. Write DIOIVAL OTTBBINOT. Are dollars eight milla. thirty centa. one hundred cents. one thousand mills. one cent fire mills. MTente^ollars four milk f 6 and ftents. 3 eagles 4 dollars 3 dimes 3 mills. 65 Ans. 15.008. DEDUCTION OP DECIMAL OURRENOY. 80. Redaction is the process of changinff a number of one SluS^^V*^^* *l =^ ^^^ °«°*« = 1000 mills; hence the iir fSJ^?^^" ^°^^^' ^ '"^'^i «»»«» i^ree ciphen. ILUlb change centa to miUt, annex one cipher. to!L^l^^;rJSC?SS''i"P"r'*>^""?«^ number, are oh«ged iMiBfcl !...-« :' "moTing the decimal point to the rlKht : and dollars and oeate. by amiezing one cipher and wmoyiDfUie decimal S't^crighl! Conversely, • ,• ^^.^-^-^ff change cents to dollars, divide hy 100; that ^^ff-^offttBO figures from the right. ¥ ^^ , ^mt W'jt^^^'^*^^^^^^^^'* point off three figures. •lU. ^0 change miUs to cents, point off one figure, .^-^-^ IXAMPLKS rOH PHAOTIOE. "" 1. In $7 how many mill :3? '^ - ^« TOGO mSr'' *^ *^'" "• '""° "'*''"* ^ »^ *'** «• *time. 1000 mUIa 2. In 366 Cents hoyr mi^ny dollars ? ' AirALTSn — In |1 there are 100 cents, therefore. »JL, of the nnmlM* «r ..»«. aqodi the number of dollan; ^of3M=.»3.S.^^ *^ ""*' 3. Change $464 to cents. *. Change 612 cents to dollars. # 6. Reduce $3.10 to mills. 6. Reduce 36 cents to mills. 7. Reduce 7046 mills to dollars. 8. Change 10426 cents to dollars. ' . 9. Reduce $4005 to mills. ^ 10. In 2064 mills how many cents? „' '^^^^^^^^^^^^^ ./in«. 46400 cts. Ana. $6.12. r >.v . \tim '"» eq PRAOTIOAL ^BLBM8. LA * ^ PBACTICAL PROBLEMS COMHININg' THE FUNDAMfiJIXAL ^ i infalSi'T''. '^"T ™V«h flour can be bou.hl ftuf cost? ^"*'** "'^ «t~wbeme8 cost $0.9376, wSt wiU J* qLtrt fo/ilto'^T'^'^ '{!T*^ number of chicke;us dJS,'S^^ at »0.376 a pound and th« fflaL^ ^ ^ P'l""'!? ^^- P^""*^" of coffee he receive? ' ' * reminder in. cash; tow much cash did J2. If » K.. cct W.26, how muoh «11 a„ d,^ of tmilS^u in the army! engageme^l; Jiow may mm an a»»,,!«t amount of the third? - ^ v^ f**, f ^fijf /7»5^.J ' <«iJf "c^tf ''^ cover «Hrta »l?.20Mi0r .a^^j, a ^^-i '5 , •»*W»* He paid out dunnff th« whflil »— . a^ ^"•' paid out dttrinff the whole year $m^- left A«innr>ain>r ^a kWJ «*i omV < T. A^^T 1 *^v-~>' ■'■V I tRAcnoAL twattJiB, a ^7 '^fiisJfrte'^s:^^ paid 6i ctfl. 23. Fran cOdf was born m 1867, m what year will he be 21 yeara ho^«J^yt.tS^i;?,-:i^« P«--^S tay; that there are3^™iu,Sf^ 1" **\^ ^^^ <»^ 365 davT nuppoeibg the same quX|P^ lfe«* J**' ♦l^^« J ^o^ °>ay /ardr Wng 180^Se!^E^ij^ ?"5f*^ to Montreal, the distikce day r reduSywlBH'*^T*"^ * ''•J^" ** *J»« ^te of 2-7 mimLr regains vet^t'oTMa^r^^^^^^^^ -e,ved e^^ and there coffeeatio 37 k nonWSl^i!?*?* •^'^^ * P°"°<^' *«S 108 pound, t^t volume. n^ESw him ill^ST S '""^"* H'* «»^ ^^ ♦O-IS »^ ^m per diy ? « dv» *o do the work : how much will he niiietl«ellit? "o M to jMn MOO: C>r ba«h did Imj:, gain br his t 3 cents a 5e7 the hat off ■ haMTMS^^TTk tL^T" "^^ "' ".•™a'^ overcoat f. ^ Jy«^«o« »fc«theoth<* J Kowmany or^n^ ^ib^J^^atiSt ?? ^«^ «^' £d fo?k M X^Jkf r ^i i ^^^ 6d 'V S'T^-^ ttJiOnalL PROBLIMS. 41. A person having an income of $3285, wishes to lay by f3 a dapr. Required how much that person can spend daily, the year being of 365 days . ' Ans. tG.OO. 42. A merchant sold 75 yards of cloth at $2.70 a yard, and has received in payment 132 yards of linen at 85 cents a yard and al note of $52.40 ; how much has he :^et to receive ? Ana. $37.90. 43. What is that number wmch, "being augmented by 85 and divided by 9, gives 25 for (quotient ? Ans. 140. 44.^ A millionnaire owes a sum of $6540 which he agrees to liqui- date in ten 4qual payments one every year for ten years. His annual income"i8.$5925 ; how much can he spend daily after paying the tenth agreed upon ? ilTM. $14.44. 45. what nttmber must be dividitd by 37 so that the quotient may be 13.25 and the remainder 0«35t iln«. 490.60. 46. At 39 cents a pound, how muoh must be paid*for 9 bales of wool, each bale containing 317 pounds? Ana. $1112.67. 47. If a pair of boots be 8ord for $3.16; how much must be paid for 20 boxes, each coniaininff 60 pairs ? Ans. $3792. 48. How much will 3650 Mths cost at 2% cents per hundred? 49. How many barrels of li||>ple8Contaiiun^'3 bushels each at 50 pents a bushel can I buy for $40.50? ^ns. 27 barrels. | 50. A litenurr wcurk consists of 6 volmnes; in each volum« there are 660 pages, in each page, 42 lines, and in eaoh liile, 40 letters. How rnany^ letters are tl^ere in the work, if it is divided into 60 chap- ters, and ifS blank lines are left between each chi^tet f 51. How many eords of wood at $3.25 a cord did I buy for , $136.60? Ans. 42. x 52. Sold 20 pounds butter f 3.26 ;; PJanpel, ® .67 2""»?g» 'a .12 ; Jme MusUn, ® .18 , " ^?>o#am '® -30 aoz. Shirt Bosoms, /a> 6.80 " WoolHoee, ^ 3.26 I 64 48 16 Beeeived Payment, B. P. HSALKT & Co. \ ^ per N. Btan. (Form 2.) 40 76 08 $168 65 ^Hr. A. Seymour, Montreal, Sept 17, 1870. Bought of T. MoGrrbvt & Co. May JuDe 10 ruly 21 « Aug. it 24 i< 3 12 Sept 2 4 16 3 4 7 15 10 160 bo»8 Oranges, &f 3.55 " Raisias, /» 2.90 cheats Black Tea, /9 26.00 " Green Tea, ......... /a 28.60 J,' ^'2P*'i**^«'^ O 45-10 bbls. Cofffee Sugar, /^ 27.20 sacks Coffee, . . ., /^ 18.60 bushels Corn Meali /a .86 Credited by Cash, fci Beedvtd PajftnaU, T. MoQruyt ft Co. ""TV a;- I $168 65 80 90 00 90 SOBMB or BILLS AND AOOOUMTS. (FOBH 3.) 71 Mb. D. Johni^ QuEBSo, Jlne 2, 1870. Bought of Btbnb, O'Brien & Co. No. 2 7 14 10 40 75 108 67 pair Gaiters, ® $2.30 " Rubbers, Pi..& .72 " Calf Boots, ® 3.80 " Thick " /a 2.65 CoppOTlge and Cartage, ' Iillurance, it , $ 92 64 410 177 4 1 00 00 40 55 37 30 1739 L. Jackson & Co., By ^|]lanadian Express Line." (Form 4.) Toronto, Oct. 5, 1870. 62 e To W. Prio* & Son. A-. 1870. ^uly 3 " 12 Aug. 9 1870. July 20 '.' 27 Aug. 4 Sept. 2 To 140 bbh. Floni*, . . . ir. ./9 9 7.60 " 95 « Fish, fQ 18.50 '' 36 chests j^reen Tea, ...& 81.80 * . # ■ ■ By 200 yards Broadcloth. ...M $5.10 " 75 " Black Cloth, . ^® 4.67 "280 " BedFUnnel,..T® .72 " 24 gross Silk Buttons, . . . /® .43 Balance doe V. P.* Son ... . $1064 1767 1144 $1«20 350 201 10 00 50 80 00 25 60 32 $3966 30 $1582 $2384 IT 13 W. Piioi ft Sow. I 72 FORMS OF BILLS AND AOOOTTNTS. o GO o 1 o a t d CO Mjoooefl Tjt Tt< «0 CO 00 ■* , —• oerytod 05 t- O 00 o> 03 r-l i-H M *o o a «» *-ooo» • \ eo Til «i o \ ^«^' • : ■» \ '■ - > '• * \ * V ■ii \ . 1 ^*4i \ M vji a a o \ S \ bus bbl lbs. tea \ . ' \ y H? -• w o tS \ \ >^- ~. V. V. \ m \ eo T(< rfl Qo oi S t-S Hj e -^ o 1 ■* % 00 CO 94 0> 00 '^ *» -* 1-4 <0 00 <0 C4 r^ >..< at >-" <-^ . l-H *o «» eq — « -.ji CO o »o o • ■— 1 • •-1 ts, • • PI • • I i r> .orHe»co o> t- M3 (O pH "^-^ o! P .1 ->. BILLS AND A000TTNT8. 73 XXAMFLXS TO BE UADK OUT, MBTSmOAJ^^ On Form 1. • 1. Sold in Montreal, Feb. 2, 1870, by John Hogan, to Mr. A. Larae, : . viz. : 7 lbs. chocolate, at 25 cts. ; 16 lbs. candles, at 22 ct8. : 12 lbs. sugar, at 16 cts.; 18 lbs. flour, at24ct8. Foptiogof the bill, |11.17. r . ' On Form h 2. Edmond O'Shea of Kingston sold to T. lJ^ Feb. 10. 1870, and L. Norris, his agejit, collected the amount of the bill : 16 lb8.^batter, at 17 ets. ; 26 lbs. cheese, at 20 cts. ; 760 lbs; maple sugar, at 9 cts. ; 278 lbs. coffee, at 36 cts. Footiag of the bUV 9176.13. On Form 2. . ' "^ 3. James Owen of Toronto, sold, Jan. 8, 1870, to W. C. Maher, 37 yds. sheeting, at 26 cts. ; 43 yds. laoe, at 82 cts. ; Feb. 3, 76 yds. Irish linen, at 45 cts. ; 209 yds. muslin, at 14 ota. ; 330 yds. dowlas, at 16 cts. Footing of the bill, $160.69. On Form 4. 4. Messrs. JB. Sharpies & Co., Ottawa, sold to D. Hall ; Feb. 12, 1870, 110 pur thick boots, at f3.75; 28 pur buskins, at 86 cts.; Feb. 20, 40 pair slippers, at 85 cte. ; March 2, 67 pair gaiters, at fl.l5; 120 pair boys' brogans, at$i.2d. On this are the following credits: Feb. 27; 'by cash, f 280: Murch ' 13.20. What balance was due B. S. ft account was settled T, 16, 110 bozes^ lemons, at March 23, when the An$. f 66<63. ^: ■m^ .t« On Form 6. (C 'ti. A. OoMfk Co., Ottawa^ sold to G. Morin & Bro., Jan. 2, 18IY0. ITydB. luroftdcloth, at $5.26; Jan, 16, 29 y^ cassimere^ ac 91.62 ; Feb. 3, 60 yde^^eached shirting at 17 ots, ; Feb. 7, 49 yds. . tickings at 27-ct8. ; Feb. L&.18 yds. blue cloth at $3.19; June IJf, 27 yds. gray cloth, at 92|p^ Aug. 3, 76 ' ~ Remitted by Gt. Morin ft Co.' in part payifii 1870„4i#L «83 ; June 26, U bbls. On M#|||g^ Note, at 60 data. Aug..4>t}w JmO. Tirli^IrM the amoant'<» t^e notef 'flannel, at 61 cffi. follows : Jan. 28, flora, at 97.20 ; , dne~^ A. C. ft ilfM. 9163.28 \7 Ofn Form t. '9.V>D ■oid'te^.TiKaurh>pt'Maroo~lji~**T^"18 Iwk OBtraau Bota'iO'TT''2aSQrrap. Bnrofrii xotvf -to ivm tobacco, at 32 ots. ; 26 lbs. snuff, at 40 cts. ; 72 lbs. tobacbb lea^ at 18 ots.; 64 lbs. sugar, at 12 cts.; 20 lbs. soap, at 14 cts.; ^pril 2, * 46 gijllons molAscM, at 37 ots. April 6, 0Mdite4 by cash, $18. What MaiiMWMdiifWivJ^aiKilftr ilfw. $36.66. onto: oings; byL.1llNola l8. MoDt^M apples, marked 4, at 92.95; ' 6, at $2.25 ; |6j^bl8. Harye2g,*in«^ed 6, at %\Mf m bblB. Ru«%, iMrked 8^^ >^5 ; ing, AM #13:49 forAiinatarttm^ aam le^ by th^* WesterPS^-^^ ^^^ • ^^>' .l^kerofQueW, 80ld:«S4St^i^dn^^ May 6, 1870i »«CQflfee, at 24 eta. : 60 lbs. W. L sugto, at 7 cts. : 76 lbs. at 13 cts.: 12 gallooB ^up^ at 66 eta. ; 90 IJba. butter j^Jf^at 8 cts. J 64 lbs. picm0 ; 90 spades, a^ 86 eta. j •n, at $12; JttVie 7. 14 12* Jane 7, credited by '6» What glance waa due 4i|»,|g9, 6 pair B, 60 cts. ipple^ at I, $7.^0. »uDt was 191.21. May 12, |: 86 cts.: le 7. 14 dited by wasdue 80,02. > ■ ,J:>^ . 00, Hay .■;.-i^ia BILLS AN^ ACCOUNTS. 75 illon wine,»toarked 8, at $99 ; 19 bbla. superfine flour, marked 10, at $7; 23 bbla. peae, marked 3, at $1.52; 42 chests black tea,' marked 5, at $17.50 ; 87 chests green tea, marked 1, at. $2Jrfs^ cooperage, $15; cartage, $6.80; Insurance, $32.50. Forwarded b/ the " Mame Express Gne." Amount of Invoice, $8193.01. On Form 4. 13. Messrs. Hall 4 Brothers, St. John, N. B., sold; June 1, 1870. toP. N. Walsh, 15260 lbs. pork, at 5i ctS. ; 7265 lbs. cheese, at 8i cts. ; July 3, 11521 bushels corn, at 50 cts.; July 10, 1560 bbla. flour, at $6.12i. On the above are the following credits i June 25, oLl ,u '^' *^*'°"' ** ^i *'**' June 30, by. cash, $750;vJaly 12, 8256 lbs. maple sugar, at 7 ots. ; 6460 gallons molasses, «t 37A cts. Whatis the amount of cash requisite to balance the account on •i^^yi^f, - , ilM. $12963.78. On Form 2. A^^^'^'iJ^n^i'iH' l^ought of A. Murph>ACo., publishers, Montreal: Aug. 4, 1870, 75 Juneau's Mentil Arithmetic, at 15 cts. ; 50 Smith's Practical Arithmetic, at 37 cts.; 2 doz. Miller's Reader, at $4.50: Aug. 12, 60 Henry's Grammar, at 7 cts.; 36 Kerney's Compendium of History, at 72 cts. ; S.^#l, 30 Walklngame's Primary Aliebra,' at 18 cts. ; Sept. 1, credited by 60 Commercial Arithmetic of the Chris, tian Brothers, at 40 cts. What bftlance was due A. M. A Co., Sept. 2 ? . ' Atif. $54. 27. — — On Fom 6. 16. S. N. Kelly bought of H. mm4tt Co., Quebec, Feb. 3, 1870. Js^i 'm W'lf.^^T"' *'*^^.^ ^*y *' 24 yds. merino, at 1 S^o ?* ^^^i'^ ? a^'^cts-J 126 doz. eggs, at 12 dts. : 160 bushe 8 oata, at 66 eta. Footing of the bill, $731.69. T !• w°^*^.'°. To'"<>'>to> -A^F'l 20. laiO, by Isaac Chambers, to Mrs. Julia Meredith, and the ImH paid: 3 do*, silver table forks, at $43.75 a doz. ; 2 doz. silver table spooni, at $36 a dot.; 2\ doz. silver tea- spoons, at $18.60 a doz.; lidos, ivorj handle knives, at $7.50 a Sa' t»^ ^""^ °^""' ** *^^^* Footing ofthe bill, $394.76. 20. P. Barry & Son, Kingston, soldlto H. Miller, March 6, 1870, as follows: 2 loaves white sugar, 62 lbs., at 16 cts. ; 4 bbls. extra flour, at $7.80 ; 9^ lbs. che^e, at 16 eta. ; 16 lbs. rairtins, at 16 cts. ; 7 lbs. black pepper, at 42 cts. ; 20 lbs. butter, at 23 cts : 3 bushels Deas, at 70 cts.; 6 busH. beans, at $1.10; 14^ lbs. bacon, at 16 cts. ; §, ' ™o'j?88e8, 60 cts. Footing of the bill, $60.83. « ,o.,« :«^*^ Nelson owes D. I. Hogan, Toronto, as follpws: June 6, ISJO,'© gross shirt-studa, at 86 eto.j June 17, 1$ doz. woolen Btockings, at$3.18i; 3 doz. shirt fronts, at $5.06 ; Aug. 2, 12A yds. ".^°^e** «« .°*^,- ; 30 pair Bilk glovea, at $1,374 5 * doz. line* towels, **l«^'^ ',2^^ y*^*- ticWne, at 46 eta. Footinijof the bill, $131. 37*. 22. G. Turner & Son, Quebec, sold to A. L Green, March 6, 1870, f \l P^!"" ^^^ »* ^3.00 ; March 18, 19 pair shoes, at $1.08 : April 9 \8a.pair hose, at $1.20 ; 23 pair glovea, at 76 cts. They received of" "-ra:. 1. Green, the following aa credits : ApriCifc 27 Second Readers, at I Third Beadera, at $3.90 ; May ff, 7 " " -^^ ' " ^ Brown'a Dictionaries, 20 cts. } 10 , , ^ „j . „,„„„ „ ^,u„uu«.oo *' ***I^ ' ,^3,C^olden ifonuals,"at'ti937 2o'chri*8tian Dutiesrat'slf eta. The balance due'^iB. T. k Son, which was paid, May 16. 1870. amounted to $44.0ftt , *« > / > » 23. ,Sold by Smikli ft Watters, Kingston, July 24, 1870, to 0. 8. Peters: 276 bbls. Patapaoo flopr, at $7.16 ; 160 bbls. Ontario flour, at $6.25; 170 bbls. Chicago flour, at $6.87^; 214 bushels corn, at 82 cts. j 326 bu8h. wheat, at $1,624; 300 bush, oats, at 91 cts.^ *®2.^"5^ ^^'J^ *^'®^- Footing of the bill, $6413.48. ^J,i' ^^^?}^ ^ ^imoa, bought of C. T. Adams, Montreal, April 20. if'^X'^r/" . • ^y^^l ^^^ °1°*? »* •^•50; 1 satin waistcoat, $5.50 ; Trimmings, $3.76 ; 3 yds. yellow linen, at 19 cts. ; 10 yds, jgray fringe, at 68 cts. ; 3 pieces of ribbon, at 31 cts.; 3 yds. black ' oassimere, at $2.26: 7iyda. alpaca, at 66 (»t^.; 16 yds. cambric, at lOi cts. ; 3 skein silk thread, at 6i cts. j 4 yda. wadding, at 6 ots. s iP 9 yds. white flannel, at 90 ota. ; i cnvttts, ^ %\.l2k ; 4 yda. green baize, at 68 eta.; 6 cotton shirte, at 66^ ot9. j 6^d8. men90,.at 80 cts. ; 10 yds. muslin, at 14 cts. Footing of the bill, JtO.Dl. «6. Sold feyP.lfcjynkOdftea.,H^ ... r<*B., at $8.46 a ydTi follows: June 8, 1870, 4 pieces muslin, each 37 yds., », ^.r,» . 8 ipiecea printed calioc^ eaoli 47 ^ds., at 82 «t$f. • jud j Jane 27. 1 (Mr BiLts AND Aooomrrs. 77 pieces Dutch hnen, feach 30 yda., at 70 ctu. ayard ; July 10, 11 pieces serge, each-19 yds^at 66 cts. ; Aug. 6, 1760 yds. Lowell cotton, at 20 eta. ; 974 yds. Manchester stuflfs, at 25 cts. July 30, E. O'Neil, paid in part $350. What balance was due P. M. ft Co., Aug. 2, when the account was made out? Ans $1284 46 26. Messrs. Fraaer, O'Donnell ft Co., wholesale dealers, Montreal, •Sl i^ ^yP' A Lane: Aug. 4, 1870, 18 fine dress coats. No. 62, at $27.50; 46 cashmere Tests, No. 20, at $4..30 each; Sept. 9, 3 doz. men s black wool hate. No. 22, at $12.60 par doz.; J doz. men's P^arl hate, No. 64, at $27 per doe. ; 6 umbrellas 28-in., at 11.75 : Uct. 12, 6 doz. men's white cotton hose, No. 7, at $2.60 per doz. - 3 black leather valises. No. 72, at $9.60. On this are the following ^J J^u ^uP,*- l^bycash, $400; Sept. 30, by cash, $150; Oct. 7, by 50 bushels coin, at 65 cts. What balance was due F. O'D. ft Co. i;^ ^°®° *^® account was settled ? Ans. $21 1 55, iQ?I* 5^"u^^' °^^- ^' Williams, Quebec, by H. S. Connolly: June 3, 1870, 75 lbs. maple sugar, at 61 cte^. ; 9 lbs. green tea, at 65 cts. 21 gals, maple syrup, at 70 ote. ; July 1, M lbs. pepper, at 25 cts. 10 lbs. spice, at 20 cte.; 12 lbs. ginger, at 18 cts.; 15 lbs. coffee, at 12ict8.; July 12, 20 lbs. dried apples, at 10 cts.; 18 1 ha, dried peaches, at 12i cte.; 2 bushels onions, at 80 cts.; Aug. 1,^ lbs. mackerel, at 8 cts. ; 9 lbs. smoked herrings, at 20 cte. ; Aug. 10, 25 lbs. rice, at 5 cte. ; 12 lbs. dried beef, at 12i cte. ; Sept. 4, 6 biV ' corn meal, at 80 cte. ; 6 sacks table salt, at 20 cts. f 17 lbs. so^ ?^*n w'-.?* ^ ^*1; ^?°"°* «>^*^« ^"^f *fi2.24, wiich was paid 1^ L. R. Williams, Sept. 7. on^^o,^n^^'!i'y . ^'"^*^' Montreal, to J. B. Poston, as follows : Oct. ZO, 1870, 48 pair tongs, at 37i cte. ; 26 doz. pewter-polished bite, at «5 cte. per doz. ; 96 doz. hingM, at 18 cte. per doz. ; Nov. 3, 32 doz. currycombs, at 45i cte. a piece; 20 packfte shoemakers' awls, at 68 cte. per packet: Nor. 12, 75 packete 3i in. screws, at 95 cts. per packet. L. Trudel received of J. B. Poston on account: Nov. 8, 2 casks l^doc wine, each 4i gal., at 80 cte. per gallon ; Dec. 5. cash, $60, What balance was due L. T., Dec. 6 7 Aim. $ iW. 7 1 ,oWn°':?^*^^y ^- Molson, Quebec, to V. R. Lewis, Ottawa, Feb. 1, 1870 : 2 c^es calf boots, No. 3, each 67 pairs, at $3.7^ thick iwots, No. 4, each'64 pairs, at $2.62; 2 cas^s gait^rslBWTT, each 75 pairs, at $1.12; peases buskins, ko. 10, each 27 pairs, at uu ' >T°*?^ ehppers, No. 14, each 36 pair^, at 70 cte. : 2 oies , 11'^"^- ^^' ^^^^ P*^ ** •^•0* 5 charged for packing cartage, q'n*^^p M . ^o «,.^ Footing ofthe Sill, $1439.76?' iQ?S* S ?• ^°^'* * ^* Hftlifex, flold to U. 8. Brown, Sept.^ 7, 1870, 50 yds. pnnt, at 12i cte. ; 16 yd«. cambric, at 9 cte. ; 6 yds "^T'f'.^^'n^-^?/ Sept. 25,33yd[8.8heeUng,atll^te.; 3 yards Jt, at $3.00; 6X yds. broadcloth, at $4.37i; Oct 29, 20 yards iQh pnnt, at 17 cts. ; 16 yds. merino> at 70 cte. Ori this bill are follQari^oredita^N. ^ »1. Ad Odd Number is a number of which 2 is not anexict divisor; aal, 3, 7,15. Every number must be either ^me or compotitle. k^ 9S8. A Prime Number is one which can not be resolved oli separated into two or more integral factors;* as 1, 3 6 7. Nons.— 1. All prime nnmben exoept 2 an odd namben. ^ ■ 8. Nnmbon are i|ime to' each other, vhen they havenooommon diyiiori thm. 7 and 13 are primeU each other, as are alao <(> H.dHLlS. '<^ 98. A Oomposite Number is one £at has other ^zact visors besides 1 and itself ; as 6, 9, 14. 94. The Prime Factors of a number are ita exacts divisors'^ thus, 1, 3, and 7, are factors of 21. ' ^5.T]|e Power of a number is the product obtained by taking thofttomber a certain number of times tt a factor • thus 16tsa^wiof4. ^ ' ' WoT»^Whenth«aumbefia taken oijee, it ia oalleii its first power; when ^tkt^o, as a faltor. the prodfiot is called its second power ; and so on. » ••• The El^Onent of a, power is a figure written at the 4 r^t of a. number, and a little above it, to show how mapy times ' J''^^^®'^*®*^?^'^' *^"'''^*^®®?JP'"'^*°°^*> the exponent ; * fcs?, and the whoP^s reiid:5 second power. «. Prom these pnJD^JIiif • , ^ ■. / hCAnjjnkum^mMmioitl easily dm^ one of two numbers wUldivid^/glJf product, i. 2nd. uifMpmte- which will exactly divide each of two num- ^ben will d^sfge thmr $um. 3rd. An^^umber which will exdcUy divide each of two num- ■" hen will divide their difference. . 88: Wftot MtMezaot divisor T— 89. WhatareaU numbertt—W. Wltat U on Tim n nmher T— ftl. Jt« nAU iiinnK»r T q9 a Ttrim. .mmiuL^* t pi 6#«priinolo«ae&o(A«•' "i tuoM, 2. When a faetor it eancelled, i is inppoaed to take itf plaoe. U^'^is'x^!'*** ^^^ ^*^"''* °" "* ^® "* ". " '^ ^^ the product of OPBBATIOV. T X 1«| X 1^ X 5 35 Dividend, Divisor, 1^ X 11^ X (» = 7 = 2J. Obs.— We have perform- ed thia diviafon without faotoring the diTidend and divisor, by rejeetiog the faeton that are"eommon to both dividend and divisor. , "Ml writin* the remaining / ^. natorlintbelrproperplaoes. Write the dividend above and the divitor line.. II. Cancel all the /acton common to loth dividend and divitor. S'n^Si ^^l^i^jL^ remaining factore of the di vidend lOSi hehio a '^"jV^rrZ^ ^♦wwmAiy/aelort of mrdivitor, andlKi result will be the quotienk , , »#• 101. W»rt*ioaaoeIUtlonf— 103/TFaal*tt«tul«/or«(m<»«a«i(p|f -— r 82 DIVISOBS OF NTTUBXBS. KSAHPLSS rOR PftAOTIOf. iii 38 = 3. 16 X 24 X 48 -^ 32 X 36 X 4. 12 X 7 X 5 -7^ 2 X 4 X 3. 5. 16 X 5 X 10. X 18 -=- 8 X 6 X 2 x 12. 6. 84 i< 12 X 18 -f- 21 X 24 X 9. i. 72 X 18 X 16 -f- 24 X, 16 X 9. 8. 22 X 9 X 12 X 6 4- 3 X 11 X 6 9. 10. 11. 12. 13. 14. 96 -r- 17 X 51 X 32 25 X 7 X 14 X 36 -f- 4; X 10 x 21 x 184 X 145 X 80 -f- 23 X 29 x 60. 28 X 27 X 21 x 15 X 18 -f- 7 X 54 X 76 xs^34 X 7 X 14 X 36 -f- 4: X 10 x 21 x 64. 12 X 5 X 183 X 18 X 70 4 & x U x 9 x 6 x 20 >i'e^44J^?9 213 X 84 X 190 X 264 -h. 30 x 56 x 36. ^ DIVISORS OP NUMBERS. 103. A CojnmonBl visor « or measure of two or more numbers is any number that will exactly divi^ each of them. > 104L. The Greatest Common Divilsor of two or more num- bers is the greatest exact divisor of each bf. them. 105. Gknbra^ PRINCIPLES. — I. One U a divisoT of uU integers. U. Every number is a ditisor of iiaelf. ^ ** m. Every prigM factor of a number ia a diviaofSf that numbt^ .IV. Every product of any ttm or more prime fact&te of.u num- ber ia a divisor of that number. VS) V. Every number eguala ike product of ita prime facl0r$4 VL A number has: no divisors earcept its prime factors a«i, product of every two or more of them. Hence, the product oj the pritnefaclora common lo two orjnore numbers ia their greatest cojnmon divisor. . . qOMMON DIVISOR. 100. To find a common diviior of two or rn^re nwmhe^a. ^Ex. Required a common divifor of 9, 16, and 21. j OPERATION. 9 = 3x3 1Sl= 3 X 6 '21 =3x7 ■ AvALTSis.-— We raaoire each of the ejiren BtpmlS^n iktQ twA faoton, one of whirfh is common to all of xhepi. 'la the operation Z'^ • iff the oommon factor, and ia therefote » eoihtaoi diviMr of.the nambeia. 107. Rule. — Resolve the mim numbers ,tnto tKtir prime Jfkciors, thtn if any factor be commm to all, it will be a common divisor, l03. What M a eommon diTiBor T— IH, <%« gmAts^ common dir|M>r ? '.■.■;> ft' an an II I •» \ ^ 1?^ .; ins. ns. na. 1 2 A. X'6.>^4^ I or more them. t lore num- integera. ' ' ict of 'greatest >exa. : the g!v«n r whi3h U Mrat(on \i aenfon » common. 0SBATK8T COMMON DIYISOB. XZAMP^XS FOB PBAOTIOS. Find the common divisors of the following mimbns : \ 1. 10, 15, and 2$. vlna. 6,, , 2. 15, 18, 24, and 36. 4n». ?. 3. 3, 9, 18, and '24. 4. 21, 77, 36, and 42. - 5. 28, 14, 42, and 35, ' , 1». lO, 35, 50, and 76. 7. 4, 12, 16, and 28: 8-82, 118^48, and 146. 83 Ana. 7. Ana. 6.- V GREATEST COMMON DIVISOR. 108. To find tU greaUst cdmmon dii^or of two or more numbers. "' ' '^ ' *" Ex. What is the greatest oommorf divisor of 168, 210, atid 252 ? VIB^TMKTHOD, " ' ' . tpPSBATIOK. 16r 210 252 84 106 126 ?8 36 '42 4 ■■ 6t' 6 AifALTSn.— First find the prime faeton oommon totheAamben; (99), vhuh we 2, 3, and 7. Therefore the greatest, eofamon divisor is 3 v 3 X7»41. (IQ6,'j¥D. 109. Rule. — Find theprime/actora common to cdlthe num-h hera (99), arid their product ynllhe the greateat common diviaor. ' SECOND METHOD. OPEBATIOK. TheprimefJf8 = Jx 2x2x3x7 factors of i 210 =1 2 X 3 X 6 X 7* lactorsot [ 252 = 2 x 2 xax«3 x 7 ««- v.-;V Ahaltsis.— The prime footon oommoo to the three nambers are 2, 3, and /.Therefore the great- est ^mmoo divisor is 2 x 3 X*«=«. (lOS.VI.) IIO. "RmMi.—liesolvs th0htmhera into tJieir prime /actora^ and find ibfi ptoduct 8/ the ^mmon prime factora,' ' Third method. HI. PBnf -^ ? m, A common diviaor ojim nwffbera ia a diviaor wT their aum, and cUao of their difference. h - - < IV. TJJe greateat common diviwr ofthe difference of two num- P^*and one of them, ia the greqj^eat'^nnmon diviaor^ of the two nifmtHira. ^ -^ ', , • ;>c ■ -K ^ ..•■;'"..■■"■ m t » ' i 84 LKAST COMMON MULTIPLE. Ex. Required the greatest common divisor of 117 and 1365. ] 117) OPERATION • 1365 (11 117 195 117 78) 117 78 (1 39) 78 \ 78 (2 Analtsis — Slnoo llT.is the greatest divi- sor of 117, if it be a divisor of 1366, it ijriil be their grf^teat common divisor. By trii, 117 ia foandi^ notto be » divisor of 1365, siuoe there is a remainder^ 78. if 78, the greatest divisor of itself, is a di- visor of 117, it is the greatest common divisor of '■S and 117, and also, of 117 rfhd 1365. (Ill, IVO Bj trial, 78 is found not to be a divisor -jItU, since there ia a remainder, 39. If 39, the greatest divisor of itself, is a di- Tisor of W, it i« the g^atest common divisor o^^3» and 78, alsio of 78 and 117, and of 117 and 1366. By trial, 39 is found to be a di- oommon divisor of 117 and 1365. ^""^ "^ ^^' ^^ "' *'"^°"'' ">* ««"'*««* OBa.-A knowledige of the Principle. (Ill), will render the above analysis &=Tx mT^rf "®' "^ *<*i™°'°f 1" = 39 + 78rMd 112. "^^Ji^^.— Divide ^he greater number ly the Im, and the dvvisorbythe'remainder.andsoon, till there is n^emainder. 1 he last divisor will he the greatest common divisor sought. k«^^'~^® greatest common divisor of three oi* more numbers can be found by floding the greatest common divisor of two of the numbers, then the^eatest ownmon divisor of this greatest common divisor and a third nimber. and so on The last common divisor wUl bo the greatest common divLor S the numbeiS: EXAMPLES FOR PRACTICE. Find the greatest common divisors of the following numbers : Aiu. 4. 72 and 168. 176 and 455. 169 and 866. 4. 84, 126, and 210. 6. 12, 18, 24, and 30. 6. 385, 462, and 154. 7. 12, 15, and 18. 8. 210, 350, and 770. 9. 70, 106, aod 246. V Ana. 24. Ans. 35. , Ans. \K Ans. 42. Ans. 6 10. 16, 20, and 24.* 11. 78, 234, and 468.- 12. 2041 and 8476. 13. 286, 429, and 716. 14. 1649 aud 6423. 92, 116, and 124.' 262, 630, 1134, and 1386. 49373 and 147731. 3013, 2231, and 2047. 15 16 17 18 LEAST COMMON MULTIPLE. ^ 118. A Hllltl{|>Iie is a mmtm exactly diwsible by a mven fe number, thus, 15 IS a multiple of 3. * '^^ 114. A Gommon Bl^tipto is a number exactly divisible bv t»o or more given numbers ; thus, 24 is a common muldSe of ^, S, 4, 6, 8, and 12. *^ IW. Wica M a multiple?— 11-^, a oonunon mnJUpleT V. of 3, Ifl m A pi boi 1 1'^ i , LRAffT^MMON MULTIPLE. 86 aoHv*diriSt^' Oommon Oultlple is the least nember ex. 116.. To/wd the leatt common multiple. FJtjElST METHOD. Ex. What is the least common multiple of 9, 12, 16, and 20 ? OPBBAflOK. 9 « 3 X 3 12 = 2 X 2 X 3 16 = 2 X 2 X 2 20 = 2 X 2 X 6 2x2x 2x2x3x3x6 = 720 iliM. X 2 V « OPBEAnoK. ANALV8ig.-Ee«»Wngthenum. MW into their prime factors, we find these to be 2, 3, md 6, The greatest namber of timed the 2 oeours u a factor in any of the given mumbera ia 4 times: the greatest flfeiber of times 3 oooura . i?.*"y "^ "»• 8>^««» numbers is offime8the6oeeur«in«nyoftheffivwifii.«ii»*^' "^ ^^ greatest number 3, and 6, must be all Se prime f^SS5Sj!ri"!^~- ?«"«»• 2. 2. 2, 2. 3, 16 apd 20. Therefor* rjTo, the Sf^daSS^""?. in oqp.ponng 9, X2. multiple required. "Ft^*.^* w ^ese faoton, u the least oommon /a "I" •^''''''•""^- ^^^''^ *^ k^ nmiber, into their prTt 11. Jafe rime to eMh other, their prodoot is their last comnjon multiple. ' 1. Whfljlg,M>7 of the givM numbers is a ftetor of mj ot the others it may be oanceleo. ' ^ -» ■ t» "^ BXAicpxjts roa pbaotios. Bequired the least comndon multiples of the following numbers : 1. 24, 36, and 20. Ans. 360. 2. 7, 14, 21, and 16. Ana. 210. 3. 14, 19, 38, and M. Ana. 798. 4. 8, 12, 16, and 20. $. 32, 34, and 36. 6. 20, 36, 48, and 60. 7. 9, 18, 27, and 64. . 8. 12, 16, 42, and 60. 9. 10, 46, 76, and 90. iln».460. 10. 12, 16, 18. and 36. Ana. 1260. 11. 26, 60, 100, and 126. ' ' 12. 22, 12, 44, tod 11. 13. 18, 27,36, and 40. 14. 270, 189, 297, and 243. 16. 64, 84, 96, and 216. 16. 84, 100, 224, and 300. FRACTIONS. 119. A FrACtion is one or more of the egual parts 6f atmit. laO. Two integers are requved to writ© a fraction, one to express the" number of parts into which the" whole number i0 dividedj^^Wi(J the other to express the number o| these parts taken. If an^^le be divided into 2 equal parts, one of the parts is called one half; if divided into 3 equal parts, one of the i^rts is Called one third, two of the psirtH two thirds ; if divided into 4 equal parts one of the parts is called one fourth, etc. ; if divided into 6 equal narta. one of the parts is called on^7f/?A, etc. . *~-^ The parts are expressed hj figures; thus, One half is written Ori« tUrd « Two thirds " doefoor^, « -t. ' ■ .^1 Three fourths Oiie ffih FotuimB .Vit^msmatlm »te /nMkt*^' .^ ~ S*:l>' -C ^,- /f >. ,ii »'.|': •^ 1 ,4 » ♦' .> -..^- , . ■^ '• ',,?' *^ VaAOTIONS. « ' 87 121. The two integers of a fraction are ita^erms } the one 6cfoio the line, the Denominator; and the one above, iho- Nur merator. 122. The Denominator names the parts, and shows how many of them are equal to a unit. IJ^*' The mumerator numbers the parts, and shows how many of them are taken or expressed by the fraction. ' ISfi. From the foregoing definitibns, it follows, I; That the value of a fraction in units, is the quotient of tho numerator divided by the denominator. II. That fraetiona indicate division, the hiumerator being a dividend and the denominator a divisw^ 125. To Analyze a fraction is to Bfcie the unit or quantity divided, the value of (me of its equal parts and the number of parts expressed. J5ar. Analyse I of a yard. ^ In I of 8 yard, the uUt of the fraction is 1 yard ; the part or fractional unit, 1. of a yard ; and-aie number of fractional units expressed or numbered 18 5. Sim a the denominator, and shows that the yard is oonsidereias « equal parts. Fin is the numerator, and shows that 6 of the^ equal parts are enumerated. 6U the dividend, and 6, the dirisor. Hence, | of a yard expresses 6 equal parts of such value that ff of theni equal 1 yard, the unit of the fraction. " . 126. Fractions are classified as Simple, Compound, and 12f . .The Simple fraction is distinguished us Proper and" Improper. % x- -e 128. A Simple Fraction is one whose terms are inteCTal • WSk9. a Proper Fraction is one whose numerator is less ^an its denominator ; as J, f , f^. ISO. An Improper Fraction is one whose numerator eauala or exceeds its dendtninator j as f , J, i- a ^*.»^ Oomppnnd Fraction is a fraofion of a fraction : as I of I of I, * X I X #.' » ' 182. AOomplez'Fraotlon'isone having a fraction w a mijwd number in either or both of its terms • as f 1- M H _ A Hixed Number is an integer and a fraction united m the same expression; as 5f.' 88 BIDUCTION OP FRA0TI0N8. 134. Since fractions are expreeaionB indicating jkfee division of" one number by another, it follows, ^; ' i 1st. That, if (he numerator be multiplied, or the denoini$mtor„ le divided, by any number, the fraction is multiplied by the Mome number. 2Dd. That, if the numerator be divided, or^the denominator multiplied, by any number, the fraction i* divided by the tame number. -3rd. That, if the numerator and denominator be both multi- pTi^d, or both divided^ by the tame number, ihefmction will not be changed in value. ^ • KEDUCTION OF FRACTIONS. 135. ^he Redaction oi «:'fraction is the process of ohangiog its terms, or its form, without altering its value. 130. UASi: I.^^ reduce a whole or mvted nun^kr to on equivalent impropex fraction. Ex. 1. Reduce 12 yards to fifths. ^, , OPEBATIOK. 5 X 12 = ^, Ans. AKALTsra.— In 1 yard there are 6 flftht, and in 12 yarda there are 12 timed 6 fifths = V. 187. 'RvLK.— Multiply the whole number by the given denom- inator; take the product for a numerator, under which write the given denominator. Ex. 2. To reduce 15| to fourths. OPK&lTIOir. 15| 4 Analysis.— In 1 there are 4 fourths; therefore, 4 times the number of whole ones equals the number of fourths; therefore, 16 « W, to which add i and- we h*iTel6i=r:y. 13S. RniiK. — Multiply the whole nuniber by the denominator of the fraction ; to the product add the numerator, and voider the sum write the denominator. ^ E^^AMPLES FOR PRAOTIOK. 1» Reduce 9 to thirds, /tns. y. 2. Reduce 12 to eighths. Ans.^. 3. Reduce 25 to fourths. 4. Reduce 3^ to fifths. 6. Reduce 40 to thirteenths. 6. Reduce 16 to ninths. Ana.H^. 7. Reduce 70 to tenths. 8. Reduce 62 to fifteenths. 9. Reduce 35 to sevenths. 10. Beduoe 81 t6 elevenths. 136. WAatM reduoUon n/a/raettonf— 137. What u tlu rul« for ndiMna a - tahoU ntimbtr to an eqmvaleiU improper fraction T— 138. For rmhciM amvMd nHm^srtoanejwvotoittaijproiw/nioltoiir * ~i^ T^- illDrCTIOW OF FBA0TI0N8. 89' Reduce the following mixed numbers to itiiproper fractions. 41. 374. 12. 45t 13. 92X U. 23g 16. 132* 16. 134 IT. 96 A Ant.^I.. Am. 1^ 19. 125X. 20. 172^.. 21. 260A. -22. 171*, 209|*. 331^. Ana. i||A. Ana. ^^. Ans. up. 44. 2 :M. 3 zr 1 189. Oa£W n.— To r«KfMc» an improper fraction to An eguiv aunt whole or mixed number. Ex. In Y of a y«d, how manj yards ? OPiaATIOV. AiriLTiis.— Since 8 eighth* mmke 1 yard Yss 37 -^ a s 41 . Ana *"""? ^y* ■" ™"y y»«*« *«» 37 eightha of i il}f!^ii^?^^77Pi^^ ^H "«»»«''«tor>j/ the denominator, and the quotient mU be the number required. \ 1XAMPLI8 FOB J?aAOTIOB. • RedHce the foUowing improper fractions to whole or i^ed number^ : Ane. 3. Ana. 6|. An$.24^, /An$. \l[ Alia. l2. 141. CabI m.— ^b redwafraclioM to their hweat terma. ^Sii^toSSX.'"'*''^'*^*^^ nwirtd^and denominiUor Ex. Bedoce || to its lowest terms. OFIRAnoK. S)H«Ar Or, 12)11 =» f ' r-rf: AxALTtu.— DiTidiog both tmni of the frac- tion by the lamo nambflr doM not alter the ▼alueof the fraction (134, 3ni.); hence, we divide b^th terma of || Jby a, both terma of the '•■nit, J Jf by 2, both terma of this leaolt by 3, y>d obtain ^ for the final reaalt. Aa 3 and 7 ait prime to each other, the lowest tenns of Instead nstaad of dividing by the faoton 2, 2, and MO^ WA«« « rt, role/or r«facmy «„ i„profer frai^ion to a v,hoU m^^^ ^- L. 90 RMDUCTION.OF PaAOTIONS. i^f'"^^"'^'*" .""^ ^^'"^^ ^y *•"« »'«»'•'" oommon diviior of th» givan tonna. and reduce the fraction to ita lowest turma in a single operation. Hence, tho 142. nuLK.— Cancel or reject all /actors common to both nu- fneraror and denominator: Or, 4* Divide both terms by their greatest common divisor. ' EXAMPLES FOB PEAOTIOK. Reduce the following fraAtiom|*o their lowest terms 1 ' - 9. \%. Ans. Ans. |. ( ^ns. f . ' Ans. L Ans. i. V A- • S: ■ I' Hi- 14a. Case IV.— Jo reduce a fraction to a decimal Ex. Jleduce 5 to its equivalent decimal. ♦ FIRST OPERATION. S = 18M = T^ = 0.875, Ans. Am. \. Ans. |. SICOND OPBRATION. 8) 7.00Q • ^0.875 Akalybib.— We first uinez the same ^ulnber of ciphers to both terms of the fraction; this does not •Iter its Value, (134, Srdl) ; i|re then divide both resulting terms by 8, the significant figare of the deaaui- inator, to obtain the decimal de- ' nominator, lOUO. Omitting the de- nominsitor, ai^d prefixing the jign, we have the equivalent decimal 0.875. In the second operation, we omit the intermediate steps, and obtain theresnit. praotioally, by annexing the three oipbera to the numerator, 7, uid dlTidkiff the result by the denominator, 8. / - | 144. Rule.— I. Annex ciphers to the mafi^ator, dfid dfet& by the denominator. , '~'" IL Point of as many decimal places in ihdresult as there are ciphers annexed. i..^°I^ —^l!''? division b not ekaot when a sufficient number of decimal flotew toj;; ir."tiu'r„r.ii5"er!'^' +^ "^^ •" •"""••*,'° "^^ -^^-^ toindicSs* EXAJ^ieS FOR PRAOTIOfl. Bedace the ft)llowiBg fractions to equivalent defeimals. J: I: 3.4. .4. J. 6. f An».0'lU+ Ans. O.fi. Ans. 0.76. Ans. 0.8. Ans. 0.04. Ans. 0.86. 11- i. Ans.0.Z3i + 12. U. 13. A. 16. - • }?'• 7**** <*« rale /or rtdueing fraetiont to thtir tovutt ttrm^^lU. Wkat u the. rule/or reducing a /ra«ti«»r to a (itwnalf ttrmT^HA.WM ■--^r,^ .'■•'"tf"jfff BIDITOTION OF rBAOTlONli 145. Cask it. --To reduce a, decimal to afmftim. Ex, 1. Reduce 0.876 to an equivalent fraction. OPERATION. «-876 = im = I- { 91 AH1LT8I8. — Writing the decimal fif^ures, .876, orer tho oommon denominator, 1000, wo •*"" iVlfe = *• H«°o«' tho *46. ^^3\x,—-0miiA}iA decimal pointy and supply the proper OPERATION. .51 = ^J = L I' = 1. " 10 - 30 15' 147. BtTLit. — Omitting the decimal point, write the denomi- nator under tkfi decimal, and reduce the fraction to its lowest terms (142), '' ' ' ,, .' KXAMPLBS FOR PBACTIOC Reduce the .following decimals to equivalent fractions : Ex. 2. Reduce O.SJ to a fraction. Ana Ant .1: 9. 0.000125. 10. 0.3i. 11. 4.00075. ,12. 0.66|. "3. 0.574. ,4. O.I6|. Ans. |. Ans. |. . Ans. f . 1. 0.06. 2. 0.76. 3. 0.12. 4. 0.126. 6, 0.024. 6. 0.666. 7. 0.0008* '■^ 8. 0.6j8. ' 148. Casb VI. — To reduce a compound /faction to a simple one. ■ Ex. 1. Reduce § of f to a simple fraction. ' \ OPEBATION.. AjuLnn.— By multiplying the denominator 3 X £ = 1ft Ana. ^^^7 S»-tl»e denominator of §, it is evident wo we dbt«»n i of ft = X» since the, parte into which the number is aivided are 3 times as many, and consequently only \ aa large as before; and, si4Be i ofi = J> , | of i will Iw twice ^ = J^ ••# ■ '" Ex. 2. Reduce lof f off of ^ ^i^of 3^ to a-eitnple fractiofl. OPSRATIOK. ^ .' ' ■ '■ ■ ■' " ■' 2 •" * ■• ■ ^ ? >< 6 'X M '< ¥ >« J = 7». itiw. 140, 147» Wh9ti»ik»nJ»fi. a iicimai-to T»/raetion t \ th ■% 92 aiDTTOTIOJf OS" VRAOnOMS. IT Multiply the remainina numerator, together for a new nu- merator, and the remaining denominator, /or a new denominator. b«%uMd*tlI Soi^.TA^.?'* "'!S/^M''*' '^" '^ eompouDd fraction, ma.t M reduced to improper fraoUon., before the required reduction is performed. ' 1XAMPLK8 FOB PEAOTIOE. 1. What is I of 4 off? 2. What is I of I off? . 3. Whati8|ofT«rof5of ^? 4. Required the value of j^ of i of A- of 21 5. Reduce 1 of J of J of | of X to a simpi; fraction. 6. What IS the value of | of | of 4 of A of fi? 7. Reduce VV of J of X to a simple fraction. 8. Reduce Uoi X of'fj of 9S to a whole number. ». What 18 the value and |. Ans. H, M Ana. Am, 4, 6. }', andX. , t, and i. ^ I and f f and ^ Ans,' and^. 4n»/ ,A»i, i, and f . If, 5'i, and i- i, t, f and 8. ■ " and I of T|. of6, ^nd JI). Ana. 154. Cask YIII.— 2b reduce fractxona to their leatt common denominator. , ■ ' 155. The Least Oommoii Denominator of two or more fractions ia the liast denominatoi^ to which they can all be re- duced, and it/fcuBt be the least common multiple of their defloin- inators. ' £x. Reduce f, |, and ^ to their least common denominator. Akaltsis.— Wo finil tbe leiiUom- mon denominator, by (117), |to'bo24. Wo then take «aoh a part off it as is >11 2)3^ 3)3, SPXBATIOir. 3 T 8' 12* 4, 2, 2 X 2 ZVyAna. •xprosied by each oftho frao^ona aop- »»tely for their revpeoti re new nn- meraton.Thu8, to get a new numer- ator for i m takA I of 24, tbe leaat obmmon oenraiinator, by dividing it by 8, and mnltiplying the quotient by 6. We prooeedyin like manner with caohofthe fira^tiona, and Write the nameraton tbag obtained over the least oommon denominator. Hence, the 156. RULB.-— I. Find the least common multiple of the given denominator8,/or the leatt common denominator. IT. Divide this common denominator hy each of the given dfe- nominators, and multiply each Numerator hy the correaponding quotient. The products will be the tieto num^ators. EXAMPLKS FOB PBAOTIOE. Reduce the following fractions to their least common denominator. !• i> if I, and A- Ans. H. M^ 2. f, ^, 6, and!. '^«*. i«r, JS, ;66. WhatU J " a' •-',4 - * • ' jj,a '. i ' \.- - ' ■ T«^ . • ' "l. » * ■•- . ' - s -■ ^^^^ ' t ■ ■ 1 ■ ■■ ■ ■ 1 1 ■ ^^H ^^ » ■ ■ ■ ^^H ■ ■ 1 1 ■ ^^^1 ^^^^ y ■ ■ ■ ^^^1 ■ ■ I ■ ■ ^^^^1 ^^^^r ■ ,«'•'• ■ ■ ■ {■HI ■ ■ 1 ■ HI ^ ^ . ♦ • *■ - ■d* « .'■*■.■' \ , • • * y > \' _'^^^' 9 ■ V «■ V n, • ' '' , / • > , 1 • > ■» ' - ■ 4 ( • ■ • 1 n \ t ■ / " i f ,<■•■'- » ' ' .^ • *: i \ « IMAQE EVALUATION TEST TARGET (MT-3) ^/ Jf :^ ^ .^ rj^i 1.0 I.I itt itt 22 Itt i*i tuoiogFapmc ^Sdsices CarporatiaQ ^. ^i- <.s 4^. 4^ 4!^, 7, 6, lyid 4" }*• H. A, 41, an/ j^. ADDITION or IBAOtlOm. > 4m. Hw. ^Wr, im, im - ^»«- At» ^, Av, m •V ^««. H. «, w, |« ^«- m, m m, /A ADDITION OF FRACTIONS. NoTM.— 1. FrMtioaif, to be added or inbtraoted, mnit be abitraet or ef like deoominatjon, ud miut bye a eommoa denomiafttor. to ^the"'^ '*"**' "^ "" "°" *'*"••*•»•*'»" «»«tioiua er Integral eu be added Ex. I. What ifl the sain o#{, |, and ^f OPKBATIOH. 9 + 20 + U l + l + A 24 M = Itt, An$. ^iLnn.-We flnt rednee tbe glTen fraetfom to a eommoD deaooliiator And a* the nenlting fraetioni. A, M, ud 14 bsfe the lame fraetlaial nnit! we add them by mMng their Sl^SUon ilS one ■am, miin7ir= 1 if* the aoawer. —., m^u.^ ^ _, ^^^ l^ar. 2. Add 7|, 8^ and I}., Abalths.— The ram itf the intacen, r, 8. and I, ia 1« { the earn of the frae- tioni, |, ^ aad |, ia 1|. Henee,the rap M+1|=»1T|. H-^eth.^"" OVUATIOir. 7 + 8 + 1 » 16 l + A + l-Jf ITfAlM. tt«/ra 12 ^A* £dr. 2. From 24| take Iff. riBST OFKBATIQV. 244 == 2411 AvA&Tnk— W« ndoea the giran (Vm- aoM to • «onuM^ (leaominator. and hav* A ^"* A *1P%«P«»M frtetiooal oniti oTUm fame 4^, Thm » (welfUu Urn 8 *"«MUtt eqasl 1 tMlfth . ^ thewnrer. eunotlakell IW« ||. w. «id 1 - wXJ ^hni»7||tethe«itlnnmaiiid«r. ^^ BtODTO OntAnoV. ABttTiii^lii ty« op„^0B wa ». .^ \- /. / * ■#«■, ■sir: 96 HXTLTIPUOATIOII 0» nUOTIOMS. 158. Rum. I. To robtraot fVaotioM.— lF»«n neemtmi, ri. duce the/ractiona to their least commoh denominator. StAtract the numerator of the tubtrahend/rom the numerator of tht min- vend, and place the diferenee of th» new numeratore (ner the oam- mon denonUnator, II. To subtract mixed nvimheTa.—Red.wse the fraeiumal parte to a common denominator, and then eubtract thefractionaland llZ^^r", •'^'■f ''•«'• Or,-i&tioB ta dirided remtia the lune. 1 '»«»»?<»«««»P«r«tlnb,an, integer mulHplie. theJractio^Z^nt%er. r»cSTnSLS:r.x»or.M^^^ i7ar. Multiply 24 by f . FIBBT OPWUTIOK. , 2f X I = Ifa a 20. BECONO OPBRATIOM. 24 X f «4 X 6:^20. THIRD OPiaA.TIO]r. >^ '^ on Y X ^ = 20. A]rAi,Tin._lB the flnt opem»M, we flnt iMWply the integer. i4. by the ntraerator of the fraction, then diTlde the product by the denominator, and obtain 20 for Oie aniwer. In the aeeond operation, we diTlde the In- *•««• 24, by the denominator of the fraction •nd obtain ^ of 24 = 4. which multiplied by », tte numentor of the fraction, giTerlof 24 ^ 3w. JHaiae, ' cJJclSS^^tS^ ^^^*^I-rtofthe n^tipl, fr«^sii^i5:rmrti'»o';s§'rt-^^^ - • 1«1. Cam III.-7b m^ltipfy a/twlum h, a/raetiom. fr«SS:""'^'""'"P'"*^»»^*'^<««-*» tod . fr«tion1 paK oCj V i7dr. Multiply ^ by |. riBM OPBRATIOV. SBOOITD OPIBATIOK. «. or the aolUpUpMid, JL. NWr, to* 6bla1n i A A. »• ••■ply multiiay the nameraton togeth- er for a new namerator, and the denomlna- wntogethMT fiir » new ienomlnntor (150). MVLTIPLIOATION OV rBAOTZOm. i it conyiound/raettoni to titnple one$. Vtcm the foregoing we deduce the following general 199. fiuLV.— L Beduoe ail integen and mixed nunU>pr$ to improper /ractioni. II. MidHpijf together the numeratore /br a new numerator, and the denominatonjbr a new denomitiator. \^ TSvtn.—l. Oaaed *U ftoton waumm to nomkraton aad datomlaston: 3. The word cf batwewi finotiou U eqairalent to the aiga of maltipUoeflon. 'iXAXSJja TOB PftAOTId. An$.b\. ilfW. 5f. 1. 1 X 7 s 2. f X 4. 3. i X 8 s 4. AxS- fi. A X 6 = 6. 12 X a. t. 13 X 8. 16 X 9. 19 X 10. 21 X 1| illM. 1^ illM. 7f' iliiff.6}f Ant, \. 16. 3| X I B If. Jt X 15. #«" ^. 1\ X 21. Ux 22. |.x 7] 23. 4) X 91. 14. 12| X lift 25. 4i X f . >«• I X |xt = 27. A X 4- X ft X I. Aw.A-|g.*fx'*lfxf 15. Jjx Jj 31. find the Tfllue of %l^ times ft of | < 32. Find the value of f of If ofVf of J^' 33. What is the prodaot of | of i of I by 11 7 An$. 2\. An$. 2\, Ana. 60^ iliM. 63}f. Am. U1\. An$.^. JM.f. M. - Xf a Ift'X ft X 2 X 6^. Ah$.2. -4~»i-j80,^x^k^ii^ -v 34. What is the prodnct of r2{| hj 5| tioiei 6|t PRA-CTICAL PEOPLBIMS. Non.— In b Mfa iM i tnasMttooi it to eutomuy t£edd 1 eentwlMlltefhiSttdk it eqnal to «r glMtar 1ha» a half of • eeaCi ea4 to oadt it vAvftll II lail iMft the halTofaeent. ThefreetlonisntidiwdiatliefbUowiaiawimi. *"^«^*^^ ifbUowiag V ' . Required the cost ot 1. 61 lbs. of ham, at 12^ ots. per lb. . 2. 7A yds. of tape, at 5| ots. per yard. 3. 9{ quarts of plums, ail 7] ets. p^ qt. 4* 69 lbs. of ohalk, at } of a cent per lb. 6. 7} yards of muslin, at 9| ots. per yard. 6. 7A Iba. of beef, at 5 otk per lb. 7. 6| Dush. of apides, at 74^ ots. per bqsh. 8. I2i bush. | eta. per yd. 7| lbs. of coffee, at # of a dollar per lb. 8f cords of wood, 'at |2f per cord. 12 cords of wood, at ^6.37^ per cord. 42 buah. of apples, at 63f eta. per buah. II cwt. of sugar, at $9| per cwt. 71 d(M. of «gM. at 12^ cts. per doc. . III We. ofaalmon, at $84 per bbl. at37i„ctt 99 baah. of potatoea, .cts. per bush. i ycU» Qf adicia, jit 87| eta; per yard. ; corda of nmjibB, at $61 ptfr cord. buah. of ry^ at f 1.76 per bush. • yds. of calic% at 16|,ots. per yd. aiP ^ P^'nuBiiM, at la eta. per lb. ^>d^oC4o4,at»3iDer.yd. 761 hmh' of whe^ 6t jPI per bush. 9^dw!. of adzes, at llojner doz. 6| buah. of turnipa, at 374 eta. per buah. 284 Qorda of wood, at f3| per cord. T6| lbs. bf kugto, at 7| cts. per lb. ai2} ib«. of bett^ at 7iota. p^ lb. 3| tooa pf hay-j at »12| per ton. i la bbla of vinegar, at »10| per bbl. 6| sal. of molaaaea, at 23| ota. per gal. 1^ tendkflNhwft, at I of ai dollar each. 13^ Iba. of flab, at 9| ota.j^lb. f i DIVISION OF FRACTIONS, iln«.«6.86i. ' Am. •1.62|f. 168. 0^81 I.— 2fe dMd9 ^/ra^ion h^ an integer, Ex. Dividajfl^e. "" / mu9 tfemnktion, SIOOHD OPSR4TtO|r. «-«=« = * /» V .lif?*??"""^ *• ■"* openiaon,w« " ffl^d* the namerfttov «f the frMtion br «. •ad writ* 4U ^wotlwit, Jl, ov« the d«Bom- uutw. .X . I"tb*»««»doperaaon,wemnltlplTtha . aeoominatoroftEfcflTMUonbythedi^lMr, / \ -i ;:: .4''l*''^J'^fe*'>"P'od"»otunderthenumor- -„ ator, U. Henoe, X>wt^«A«niimj^or«MiZ«^pJ^njr the denominator of a fiactum 2y my number dhndee <*«/nic 100 wvisioif o» TBAonoirg. 164. Oasi it.— ro divide an integer hy afrdction. Ex. How many timea will 24 contain f T FIRST OPKRATIOK. 24 -^ f = 168 -f- 6 s= 28. SKCOND OPBRATIOV. 24-f-f«=4x 7=5 28. AxALTsa.— Tho integer 24 will oon- Uio I aa vaxay times m there are $«v- «ntA« in 34, eqnal 168 MTentlu. Mow, ifMoontaini 1 aerenth 108 timoi, it will oontain f aa many times as 188 will oontab 8, or SSt .. , . ^ ^ la the aeeoDd operation, we diTlde Uio integer hj the nomerator oT the fraetlon, and ntultiplj the quotieat by the denominator, which prodaoes the same teaalt aa ia the first operation. Henoe, Dividing hy a fraction consisU in muitiplying by the denomi- nator, And dividing by tJie numerator of the divieor. 165. Cask III.^2b divide a/raetian by a/raetion. Ex. Divide { bj |. OPERATION. AirAi.Tni.-.We inrert the tanu of the diTiaur, and thea piooaed aa In aialtipUoation of fractioaa (18S) . The reason of this proeeas , r "^ffSlLU seen, if we eonddar that the divisor. I, is an expression denoting that Sis to be divided by 8. Now, regarding S aa an intepr, we diTide the fraotioa J l>y it, by moltiplying the denominator} ^^^' 8 X 3 ~ ^' ^"^ ^° divisor, 3, is 3 times as large aa it on^t to be, since it was to be divided by 3, as aeea lathe ori|^nal fraetion; therefon the quoUent. ^ is i as large as it should be, aad moat be aaltipllea by 3 ; thos, is =: fit the answer. By U^operaUoa we have mnltipUed the deaomi. nator of the dividend by the nameiffirof the divisor, and the naawrator of the dividend by the denominator of the divisor. From the forgoing we deriye the following general . 166. RuLR—I. Reduce integen and mixed numben toim- proper fractione. II. Invert-the temu of the divitor, amd plication of fractions (162). NoTH.— 1. The dividend aad diviaor may be redoeed aator.aadtheaameratorofthedi^deBd be divided by diviaor; this will give the same rasolt aa the rale. 3. Apply eaaeeUatiea when pnwtieable. XZA1CPL18 roi psAcrrjoi. An$. f I 4. 28 -h^. t. 6. I 4 a eommoa denomi- Bflae r atar of the 1. f -5- 3 - 2. i -T- 6. I* at in ninlti' 2 3. 4 ttn- Ant. 3|. 188. ]r»r"** Ana. ^t. 7. At $5 per yard of br^adiftloUi, what part of a f&A cka be boudit for f of a dollar? fitu. ^. Si If I pay 6} cents for riding 1 mile, how many mi\m can I ride for 1134 cents? » Atu.^Q. 9. How many pounds (^ tea, at$U per pound, Can be obtidned tottnil Am. Hi .10. If 9 men consume } of 9| pounds of meat in a day, how muolL , Jma each man oonsume ? • . ^^ ^ ^^^ ^^^ S*r'^ 11. A\inan hon^t 37} yards of calico for $6.61, how much did it bOitpiBryard?^^ ,.. Aim. to. 16. ; 12. How many tons of coal, at $6| per ton, can be bought for iC^t 13. A horse eats f of a bushel of oats in a day, in how many Aii triU he Mt 16} buehels? Ana- ^ 14. A merchant bought ^ ^eep for flOOff, bow motii^tHd' h«^ pve per head? , jiUil.fUM; 'An*. 26. 17. If $af is paid for 6^^ pounds of gvues, hew araoh is thai per pound? .,^A^j "...u.* . iig^,!*, $e;§^ ■ \ ^^■fes per QWMATUt OOVMOll DITIMB 0» tBACTIONS. 103 18. How many ton. of h.y en Be purchi^rf ror^$U9^, at $9} 20* a! 1 'ir'i***^ ,V^ '^•?J' da/«'inu8l a man work *?!; iafe, ? forVr * "'^ °^ • •^""" ^' «•"*'»' J>°* '"«ch beer caVS S^I get'for'/LWr "« '^^'^^ 3* --*•' wl-t part of an iljpiol^-lou wouWgfbuahelaauppI^yKeijSl? '*^'^°" »anvh i6. A young man, having tlO mvi* a ^f i,;- ~ ' -- ^••'" »3l p.r ™m ? Low Much Id hVC? *''''''' """'/.i^^ ■*■*' " CO J fKM. sf.MV ■». ,?i-rfe-s •» ch2;d to te^'"*°"' "»« •*8»' "»- »"y i-mi, oJ.VL. JJt- •Id'tsr.Sf r^TK^SJj I'.i'^H^"'^^ did he miin ? * "°'*'» ***** ^b'- i ^ov much . GBEATEST OOMMON DIVISOR OF FRACTIONS. ' 167. The Oraatest Oommon Divisor of twrt a- ««« * them, giving a whole nuttbor for •quotient . """ '*'° °* ^JJI^^Jtnd the grtaten amnum dimar of tuH> or w^ £?«. What is the greatest common divisor of 3J, If, and || ? ' . ,n,- OPBaATlOK. « ?J!^'^^'''*^'*~'°^*'"^^«*= * I ^"atestoommon Least «)mmim denominator oflhefracfci8»36fdiTieor requind titii --\},i Mf. mdft>iA«gmtwteoiHaioi)^lH^ ^^■-^'iF^i^ 104 , ■ LlAfiT OOKMON JIDLTIPUI OF nLLOTIONg. 169. RinJi.--->^ 6, 61, in and2i. Ifllft P^ACTICte, OR ANALYSIS BY ALIQUOT PABTS. 178. An Allqaot Part of any number or quanMty is saoh a f as will exactly divide that numbei: or quantity: thia 2 3 4, and 6 aro aliquot par tTof 12. ^ iJL * "> **' ALIQUOT PABTS OP ONB DOLLAft. ^ 60 cents = i of 1 dollar. 331 centa = f of I dollar. 25^centa = | of 1 dollar. 20 centes^iofl dollar. 16 J centa = | of I dollar. 12icent8=lQfl dollar.' 10 centsasXofl^iollar. 81 centa = A of 1 dollar.. «J centa = A of 1 dollar. 6 cents = A of 1 dollar. 174. To^nd the cott of any nvmhtr or qwxntitv, wKm th$ pnee 0/ a unit u an aliquot pm-t of one dollar. - ~^-^^''»% ^^* ^'^^ * ^"'^ ^^*' ''*" *^* ^"^ «rf^udin ooaif a|caiTioir. * '-■■ 8) 416 An$. 169 > AvALTSM^ir the priee wm* |1 a vwd, ttw eoil iliT * *• »"/ doUwi M there an Tarta: or. I « the price ispaxt of me dollar. ^ '*<'9*vemnmtoer XXAMPLBS FOB PBAOTIOI.' 4' . 6« . * \ • _ -' i '. , 1 V "' ■ '^■i'id-W ..^i.11-'..: .' i ' ■ - ' "^5^^ ^^r-.c l^t-1.* 1^i&£^.k ^ X'^-^jj 106 paaotioal quistionb by analysis. IV. ^ • 4j QUESTIONS ^ rtrVOLVINO THB BBLATION OF PRIOB, COST, AND QITANTITT. 176. Ca3B L — T^e prioe and the quantity being given, to find the cost. Analtbis.— Th* oMt of 5 units most be 5 times the prioe of 1 unit ; er of pound* expreued a$ tJunuandthe. SXAXPLIS rOB PBAOTIOl IN THS PBXOSDIHO 0A8IB. "^ 1. At $7.60 per barrel, how many barrels of flour ean be obtained for $217.60 ? ^Ane. 29 barrels. 2. If 1 yard of calico cost 23 cents, what will 31^ yards cost ? 8. WhatooBt 16 tubs of butter, each containing 70^ Ibs^, at $A a pound? 4. Whai u thefreight on 1244^ pounds iirom Montreal to Quebec, at$0.86perlQQiB.f Jiw. $10.^78 + . .6. If board fSBrfemily be $342,181 *» 1 yc»> ^oyr much is it per OVT iin*. $0.93|. 6. Howmaoydannof^gsxianbe bought for $9.24, at lO^cts. adoseti? iliM. 88. 7. What will 39Jil ftet of nin^ boards cost, at $17.25 per 1000? •&, JThai is the value of 210 kegs ofnails, eadi weighing 162^ lbs., at$l7|aton? / .9. At h| a boahd, how many bufihels of oats can be bought for $U«.0<|1' ; .■ 4IM. 76| bushels. 10. At cents a pound, hotr mainr barrels of oodflsh. each ooiAun* . ing 90 lbs., can be purchased for $94.fi0 ? Ane, ikAhU. " U. WlmtirSi bd the Qoet of 1620 Kppie trees at416i| per hundred ? . il At»87i«ts. a bi^hel, whatfril) I of 466 bushels of potatoes cost ? .M. How much vast be paid Mr 43ft £eet of boi^s, at $20.26 per IfWlP^i feet«(^tUpft at iMpUiFr 1«0) and 4378 feetoflaS, •^W»^ »W^7^ .^f^ ;, , A«. $66.317i. krjKnnds of Pane dhurter, at j^ ; t„v ilfM. |8.416|. [ ooit't97,60, what it the nrioe ^$6C>jS2tt Am. 6^1. ; wfitijtiaabatniiraiai? ■ ' *■* kt 4 *»v ■'iifc'? '■2S' '••af. L ■* »ail2fr|ier I0??j airfWftliWgl^^ •^^. W*i^ -?• *^ *• WW«>*» * !*• W P« gMlon J £o# A- < ^ 1 fl*MMlWaM66A cts: per bosfa.; 8 sacks malt, No. 6, each 2| bush., at f 1.37 A per btlsh. ; 16 iackf beans. No. 7, eaeh 2 A bosh., at 86 cts. per hu^. 87lWper ■ InmttMce and cartAge, $3.40. Amount of Inroioe^ W21.66. >ri8TO0lbt. ■ •T^T«»®^'"*^*^<''»'^^'°^*^'«™«'c^nt«>Halifex, sold to Lenoir ■ * 0*BhSif Montri^ as feUows: May 19, 1870, 85 pieces Norwich crapeo^ at 18.32 ; 102 pieces Liverpool cottons, at $7.63: June 6, 1761 7<|0. Antweip ihCetiW, at 24| cts. ; 6984 yds. Amiens relvet, at f 1.80 ; Atig. 8, 3701 yds. Yorkshire drab, at 65 cts. ; 8721 yds. Ab- beviHimemto^ atfl.i2}. On this are the following crediti: July 10, by 18 Wa. Canadian flour, at $7.60 : Aug. 12, by dn^ at 3 day's eighty ibr $800. Wha$ balance was due T. McC. & Co., Sept. 3, when the acoennt waa settled T 'An$. $3377.01. 8. 0^ N. StoadWtti^of Montreal^ sold to Mrs. F. Stephens, April 6, Vm, and Bd. Noraan, his clerk, collected the amount of^e ull : 3»{ |«l. oaniblei, ai Uiftts.; 47i yds. shalloon, at 32 cts. : 27|yd8. droM jit 46) et«. ; 19i yds. calico, at 11 A cts. : 41 A yds. ohtnti, at dOi^ r84| yds. ci^Mnoo, at S7|«ta. Amt. of the bUl, "~ Sod Qo^ Wdl/to M«Mrs70~ Cooper 'ft Co., bS^ E'^, 13|^ bUii»BedileioL.II.* Son, July l&f ^.$240^. 7. Binftet, bpoi^ of Tessier t Oray, Montreal, as Iblknia : *~ ""^ '«^il|iflh>, eapb&7i^., at $2.16: tlltieoes , um^ mct». i July 12, 4iWosB HoUaad^HoM, :i^57iots. ^-l^^ko^miii^^m vdfl^ at 48^ li0QtfM||^j||^««ft ; OWllds. ^1^^ fttiHed ttnmber, r Jj^ |P»J|» •»« I to eqwraltnt tea/e/^ ^v^ . -ir '>■/ ""I £'..$. Ail#4|af|i[^ <. aio mmmLLAimoxn nowmm. pi • f:.''^' B. WhatTnlll0)conl8ofwbodoo6tatioft94|Mreordr 9. How many pov^ . ^ . ., - . aecodd 680}, the thii 10. Andrew apent B, „ . „ how much h»d he at first T Asu, 138470$^ 11. A servant had ^ of his flaringB in oq9. bank, i in ADother^ and ue rmnainder, which yttm $77, in a third biank | how onojl inon«7 had he? J^mt.'$l4A. la. Leo had I Of§ of 74 times #7862, Mid paid id-jt^P f« » iBDi howmachhadf^afiii^mmmmn*»^^ S454 IK 1064Alb8., 963f Ibe., 901|faB^ M9|, how manv poundf T ■, bale of dodrfor $9(5.374 j >3 ., hedi^owaofitfor on.t.ja^j wwijjed the 14. Uen^ bonght a ^. ^^ „. f of the cost, and by so dplng^ loses Bumbftr of yards in thebaic. ^-j ; , < ifi. What is the value of 376U aoresof lm4 ^^ l£..Itiiit transportation ^-18| tbDajpfivaiToosts Uperrtoii? , . 'An»,f. 17. AoiBan inirohasedl orf^yaid of TdTfli at^emte of yard; #hat did it cost him %^ Jh^',"' 18. Charles has 634 sheegBlrtdoh is 94 mm Omit I - David's number; how manyhas David? - 4^"™. 19. A man travels 4 milesia | ofanlMlM W ^v fiU.)i« framl in 14:hours at the same ra|e 1 • . ^ J^Ww&i* _ao. A merchant owned.f of « shipb and sold 4 of lofhis aiR^lbr '•a400.j^ithat KUa, what was th« wbolawortht i«^D93Q0. i^. 21.. WtetirilL^cidfltoDsofiwalooBt, at AP^I^fte^^ .„, - tT; ■ ; :;■/; ' 3S. Iff of I ofllj ^tUed^l^wJbafr^ aq. B and Ck««B ~ mmmMaiifm^' l^^jjgm^j^l^l ^_ — .Mwil& affiOME'Utw Awi sitdidf «hs 8^eili|«estt«t , --~-,«wp^ ^^^i JiuMsobMnillram two flekb $44 boshds of oint; ;|ieklsdj as MjB^fp «b« sMond, i«i|i|ind the yield of each HMf |M« $4 miles per hoar, iM^^loii^ra^ .jMi i^ltr., 1 bp^t 16 loadMf tfOodL eMflkoontdaiac lit tM, w«n»«,«w bdtt «M1^ ■mmm i pi^sotfi MAat did eUalhlRMtit '«to>n|^ IMS 18« ^ «»s hrokwi into^tii^jiiMt : ooaij » htoM of eellLMMs ff^^ Um wMh «l|l »tabk c^ *f of a bureL aiid tiM Qtlwir f of ft litfna r ^ •V lOBOlLLAinWm • VBOBUMS. in « . \-i 83. A oerUin quaoOty of applea is to be divided among 6 boys ; William w to have 4, John ^ teter,V, Thoma.^ anTPAunhi '*?i"''A'!' '''^"•f^J'* 2*5 "'*»' " *J^« wlokj qnantitytoiwdiTair - **• yo" f r ***** °^''^ ^^^' o^^^'ico^ «* 12i otii. |itt^.^~ *°«"i# of muslin, at 181 cts. pgr yard ?. '-.. ^Im. $S.28f. __TJ',. ^R***"". T^?^* ship's caigo, valued at $493000; Daniel much as PhUip and much does ^M own T •J5aa°^*°^^*^*!?^*^*"**^" i of ™7 share to^JSren for 146000. What part of the steamboat have I left, and what is it worth S T^iS ^ . , ^fl». A left, worth $16000. "JV. ^,P°"'^*^ of maple sugar cost 34^ cts., how much must be paid iat 801 pounds? 38. Ajrocer bought 9 i tons of coal at $6J per ton, and paid for It IB ooB6e at I of a dollar a pound ; how many pounds were required to ray % the coal? ^,„ 133 lbs 89. Xhave $800 and wish to lay out $346f of it in sugar at 84 cts. a poBtid, aod thereroalpder in tea at 621 cts. a pound: how many pounds ofteado I buy f ^. 859Ut|'lbfc 40. Amerchant expended $840 for dry goods, and then had re- maining only « as much money as he had at first; how much money hadheatflnt? / ' 41. A fanner has three fleM«; the first contains 13 Jt acres, the ■ecood 88^ acres, the third 139|f acres. What is the largest-sized house-lotJf of the same extent into which the three fields can be divided, and also the number of lots ? ilna. Size of each lot, 7,^ a. ; 4 1 lots. 42. A man owning 135 J acres of land, sold \ of it, and gave | of it to hw son-j what was the value of the remainder,- at $87.80 per' ■®21' » ^ -. \. '. -f ^n*. $2288.61i. ^48. A merchant owns | of a fiwtory worth $48000. He sdli | of liiB share to A, and i the remainder to Bi How much does he »• od^ftvm A and B respectively, and what Mrt has he remaining? ^ ^^"^W> A, $26200 ; Frota B, $8400 ; has left, A. lit 267 theep, st$2.26 per head; he afterward J per head^ then sold* of the whole number ai !the remainder Ml $2.^2i; did he %ain orlow, > An*. Lost $35. 81 J, labask«t«fiiraiigMaaBiwht f of« yard of cotton for f f«ielect«hoet has 60 pupiht ; a4^cf4ha « i pay-^ $1.26 a month each, the | of theremwnder, $1.76, and the rest $2.60. How much does he receive from his pupils in 8 months ? An». $840.. '66. The diflference of time between two watches is 1 of an hour; • one of them gains 4} minutes per day, while the other loses 6J in the latueitotet m how many days will wey again mark the same time? * • ,, .,-.'. >, ..h DiNOMnrAni itombibs. 113 66. How many herrings were there in a barrel of which 243 were gelid at one time, then the |, and if there still remain }. Required also thkraloe of the whole barrel if the herrings werf) sola on an average ofSQ cents per hundred?' ^ Aru. 1080 herrings; ^.64. 67.\ A dealer in poroelaia bought a certain quantity of plates ; he sells ^ of them at 36 cents a doz., ^ at 38 cents a doz., and the re- mainder at 41 cents. How many dozen of plates did he buy, knowing that he paid 31 cents per doz n and gained $1.05 by his bargain ? 68. A man having nought 84 bushels potatoes, forgets how much he paid per bushel ; but remembers that there was a difference of $4 between the 'f find the f of the sum laid out. How much did he pay per bushel 7 ', 4n». $0.37^. 69. A dealer ih furs sold a certain number of astrakhan 8kin3.«t the rate of $1.70 a pieqe. Now, in adding to the proceeds of his sales the Aof the same proceeds less $9.60, he could buy 25 fox skins at 1 19.20. How many^ astrakhan skins di(Lhe sell ? 70. A farmer sold 4 sheep and exi)endla the ^ of the sum in pur- chasing 5 lambs ; the rtoiainder of his money is equal to ^ of the sum itself less $2.00.. B«quir^ the price of s sheep and of a lamb ? An$. $9, the pnoe of a eheep; $4, the price of a lamb. ♦DENOMINATE NUMBERS. 186. A Simpto Nambei' is eitiher an abstract or a denomi- nate number of bat one denomination; as 18,912, 40 rods, 15 oniHSM(9). 18*7. A Oomponnd Namber is a collection of concrete units of d^j^Btent denominations (10) ; as, 3 feet 4 inches, 5 pounds 6 ounces, 2 days 8 houM 24 minutes. Non.— Id limpla nnmban and deoimab the Mala is uniform, and the law of iatrttm Vi djaonMo is bj 10. In oompouid nomben, th« Male of iaoreas^ And deeraase is vaijiiit. 188. A Deaonifilite flfnmlier is any Oonorete number whidh exmreases some partioalar kind or quantity ; as 3 yards, 7 dollars. ■]^t|^ A BfllfflBllnftt9 FractiOli is ai^onoiete flraotioB whose integral unit is one of a denomination of somo compound nnmber. TJbni8,f of aboahelisa denominate fraction, the integral unit bdng one bushel j so are f of • day, -^ of a yard, eto., doaominate frtetiODS. lOO. ])enominato Nnmben ezpresB Ooireiioles, Welgltfl, j^ 189. Whtltiiayim^ mimlMrf—187. .A oompoond QotniMrt^ IflS. A da- nominate Bnmbtt t--* UD. 4 ^nwontai** flraotioaJ— 190. mat do dAnominato [i^^Ji^UL ■ <= 4 = 20 a Non:—fiv»i7 id. of the old ooinige U eqiial to 5 oents of the new.' III. English Monxt. TA^LS. 4 ftuihings (Jar. or qr.) make 1 penny 4.^ 12 pence " 1 shilling ^. 20 shillings " 1 poudtt or sovereign £ or «09. £ 1 Is 20 d. ^r. 1= 4. 12 a 48. 240 = 960. Ihini.— 1.' FwihiaM ue genortUjezprened u firMflont U a pehnyi thus, 1 fui, MmettmM oallaa one quarter, {qr.) >« id. ; 8 far. ■■ fd. S. The old /, the wiginal abbreviatioD for idiUUnci, wai fonnerly writtmi betwetn •UUuici and ponoe, ind d, the abbreTiatkm fiMr mms^ wai omitted. Thw S«. M. wat 'written 8/V. A etraigbt line it now ued m ptote of the/, awl •hflUnfi an mitten on the left of it, and penoe on the ri|^ Thai, Sfi, 7{3, elo. i. The present Talae (tf the eterUnc ponnd in tlw Domiidoii of Oauida Is HMM, and htooe the tslne of an B&{^ ahilUng ii Mi oenta." 4. ThaeoinsofSngtaodin general eireolation an t the soverrign ('m£l\f and tile half*eerMeicn (— lOcT), made xi foM; (be erawn (•■ 6*.), tSe h^- enmn (-« it. (M0> the florin (» 3«.), the shilling, the dz'peneob the fonr-penee, ai«i the thiae-penoe^ made vttihtrj the penqy, the half-peniqr, and the ftr> ttlag, made of fofffMr. t. The standard gold eotorfBni^wgi Is lljggttjwireaoMead 1 part ^J^, JMtmngoU An)*offtr, |4p«Me,ineoppwoo«^veii^aiio«adaT^^^>>#^ M.,Ku^' j„=a2 ?« 240 :» mo. ' Norn.— 1. Dleooiids, •te.rtn wcii^ hj carott, wad fraetioiu of « ouat. A tfant w«iglu i.yrcNiM Tr»y weight 3. Id ipaaldng of uie p«t%«rgQU^a«iSHrtiiiMHii 3^ put} M» ISearMafim, iBeaoiiig^pan|^iDd JL«Uoy. "' ~ ,»-. . j ' ungPMim ^&. 116 'VftSttBM* "^ / II APOTHMAIttlS' WkKIHT. , 194. ApotheoAriss' Weight is used by apotheotriee and physicians in mixing medicines ; but medioineB, in the (quantity,, are bought and sold by Avoirdupois weight.. TABLl. 20 gnuns (gr.) make 1 scruple, «c or 8. 8 scruples " 1 dram, dr. or s. 8 drama " 1 ounce, gz. or 8. 12 ounces " 1 pound, lb. or ft. §c. gr- dr. 1 » 20 oz. 1 = 3 S3 60 lb. 1 « 8 =: 24 s 480 1 =: 12 - 96' s 288 = 6760 III. Avomtoupois WmaHx. 105. Avoirdupois Weight is used for all the ordinary pur- poses of weighing. ^ TABLB. r F 16 drams idr.) make 1 ounce, 1 pound. ox. 16 ounces (1 lb. 26 pounds - 11 1 quarter, . Vr. 4 Quarters 4 » « 1 hundred weight, cutf. 20 owt, or 2000 Ibe., u I ton, T. ,,iti^j#" ,. ox. dr. , fc''' ' " lb. 1 =3 16. li qr. 1 zm 16 =c 256. cut. 1 as 26 s 400 = 6400. 7*. 1 s= 4 ss 100 B 1600 = 25600. 1 = 20 B 80 =: 2000 s 32000 = 612000. K Not!.— The hng or groM ttg um. '• LONa»TON TABLI. • 28 lbs* make i quarter, marked gr. 4 qr. :*: 112 lbs. " 1 hundred weight, " cwL 20 owt = 2240 Ibe. " 1 ton. " T. OOMPABAVIVS tkSSM OF 'WSIOHTB. 1 poand 1 ounce 6760 aain& :*>: 6760 gnlmk >ii 7000 grains. 480 " =a 480 "v « 437.6 " 176 poaadfl, s 176 pwtltda^ s 144 poaads. * '^ MM>'i^&t3:iiA-^'s<'^!i^^iMfitiiA.'^ , '' i'-^ 11^ s? p j>;taS5 oSc"""' "• '^ *" "^"^ «*^ »H-ir ^oia ij MEASURES. ' ^ or !!^n^*""fl'? *^/* J^ ''^^ «=^«'»*. dimension, capacity IS ?, '^°"'*"°«^' /'f *«™'°«^i»g to some fixeS S^ n^F,. ^"^ be properly divided into tW'classes.-Mea.. * ures of ExtensioB, and Measores of Capacity. MXASUBC8 Of IXTUrSION. thiJSL.*'^*"'*"^ ^ ***"* dimension*- length, breadth and ^ JlllJ'hM only one dimension— length. A S?H?n? nJif t* ^? *^° ^^'"^"-ioM- length and breadth. ' thicW ^ dunensioM- length, breadth, and -• ■ , I. LiNiAB OR Long MiAsmti. I inch ({ft.)±s 12 inchee 3 feet 6i yd., orl6ift. 40 rods 8 Airlongs, or 320 roda 3 miles * 69i miles (nearly) /,. 360 dc^reei TABLf, 0.3363 French inch, make 1 foo^ Jy?»«di ; 1 rod, 1 forlong^ 1 mile^ 1 league 1 degree on the 1 great circle of « « « u ator, « earth. yd. rd. fur. mi. lea. deg. or*. ' *;■■ ,f ^ MB idUitnas. Nona^l. Kor «h» pnrpOM of tMararing olofii uA wbu gotOa mU 1^ lh« y«rd, the yard ia diyldad Into halrm, foviiha. elghtha. and lixteontJu.- Tba old Ubte of oloth mount* ii praotloalljr olMoUto. 3. In Marinon' Meainn, 12 UaM mako 1 inoh;4 iaoliM, 1 1 fathom ; 120 fathoma, 1 eable-length } 7 J eabU-tongtha, 1 m of th« oiraumfennoo of the«ait|i, 1 kno^ or f«>frapldo^^ oqnator, m.9 to fl9.0» miles in mlddlii' imndoi, •^'^^J^jj^ "^ ^^* '>~^ rMioni. The mean or ayerafo length ia aa 8tati«ipir Ubla. ^ ««JP^ « longitude U greatest at the eqnator, whei* tt la «».lf mllfca, ^|ul)t gradaally deoreaaoa toward the polea^ whera it la 0. atatote milea. 3. The length-of a degree of latitade varlea, beiqg TaBLB of THl OLD FbINOH LtNlAB MlAflUElS. 1 line . 12 lines, <<.) 12 inohes 6 feet 3 toises 10 perches 84 arpents 1000 French feet m 0.089 Engl. inch. make If II (( It nanbH'«QVJUi9 lOAauus. 1 square inch ($q. in.) • 0.007921 Bogl. fooL lil S!i"® *°°^*" ""?!" ^ «i^**» M •?. A 186 reet >- ' , " "* — ' ' ' - r I 9«>ise8 ■ 100 ppA>h«iP ^ ' 74)66 arpeota ; . «^ ^1 .A^oate' Jio •9. to. « ,r,t,iq|itoe league, «;. L. . by the MiMi,, ,^ «4 1^ tie n««i{ffio .,u«3JC ^ ^ »rf .1^ ^^U.^&. -^. %^ iil /. ."■ # . i ^ ! i. J- 120 MXASVBIg. «..^/j* *!""»*!?*** P»?nang of in(mldingi,eonioM,ete., the meunring-IiiiB is earned into all the mouldiDga and oornioes. •»>"»"ir"«> »» fJt' J,''*'*'^f'"8briok-lMin»by«ith«rtheBqu»royard or the iquro of 100 feet, the work u underatood toTw 13 inches or 1| brick thick. th J'wtuW.'"*^ '^'°**" "° ''*™»*^ **> ~^' 1 •»'»««. »>«ln« »*illllll^:*l y^'Wofntd dU toUd ocmimu o/aeuhe, mvitiplyiu Ungth, bf$adth, 0ttd thickneit togtthtr. ^ VXASUBI8. 121 TABLI. 1728 cubic ioChea tcu. in.) 27 cabic feet 40 cubic feet of round timber, or 60 " " "hewn " 16 cubic feet 8 cord feet, or ) 128 cubic feet ) 24| oubio feet make 1 cubic foot, •' 1 cubic yard, I: cu.ft. Ctt. yd. T. It u 1 ton or load, 1 cord foot, 1 cord of wood, ( or maHoniy. ) cd.Jt. Cd. TABLI or TBXNOH MBA8UB18. 1728 cubic inches 216 cubic feet' 1000 French cubic feet 1000 cubic toises make I cubic foot, cu.Ji. " 1 '< toise, cu. to. "• 1218. 186432 Engl. cub. feet. " 9745. 491456 cub. yd. »fc]'.*!!!^irL?''?"^ Mrf t«»n»port«aon«onipMrf«i eiUnuita Ught freight by the space it oooopies in oaliio feet; and heavy Seight, by weight* ^ .^" ^jH?A^*^ ® ?** ?*»«' * ^••* ^<*»» "d 4 feet high, oooUini one eord : »nd a cord foot none foot in length of «Bch a pile. ""v wru, ». A peroh of vtone or of nuMoniy ia Ui feet long, H feet wide, and I foot high. «♦.,** ^^*t l^^t^y'"' ««d masona, make an aUowanee for windows, doon, etc., of one half fte openings or vacant spaoei. Bricklayers and masons, in esO. maOng their work by oobio measure, mdce no allowanoe for the corners ot the Z.^f '^?";5?""^.»*°-'>* •«««n*»o a>«ir work by the «r<, that is, the' entire length of the wall on the oit(Md«. ' »•""•"»"»• 6. Iteginens, in making estimates for excavations and embankments, Uke the dlmendons with a Una or measnre divided into feet and deoimals of a foot. The oomnutattonsMe made in feet and deoimals, and th« nsnltsaniednoed toonbS yards. In oivil engineering, the onbio yard is the unit to whioh estimates for excavations and embankments are finally rednoed. «»»™»i«» lor 8. In sealing or measarin« timber for shipping or fteighting, 1 of the solid contents of round timber is ledaeted for waite in hewing or sawing. Thus, aloj that will make 3fl feet of hewn or sawed timber, aotnaUy contains 44 cubic feet r^sV«^im?art.jKrSr.J* "^ "'"""* '^ ^ "^^ "^•'•'^ «ldby:?at"i?XffirSS:;r**~*^ "BiKnr generally bonght and 8. A cable ibot of distilled watw at the maxhnom dendtv, at the level of the AtoWu^T'*"'***"*"^ *'•«»•''" welgMtoVJlMS! rJoOoS! MEASURES OP OAPAOIT^- 8P«wjty be i ng r e fetted to diffarant nnit a ^ - . - nines extent of spaoe. *•!• MeMarssofotpaoitjrmaybe properly snbdiTidad i»*^ two oIsAss, Meatuni qf LiqwuU and Measure of I>ruJ3uManem, 6 ■ ■ ' . ■^ '--{ 122 inABxntBs. I. Liquid Measubii. 208. Liquid Measure, also called Wine Measure, is now used for measaring all kiads of liquids. TABLE. 4 gills (ifi.) 2 pints 4 quarts Z\\ eallons 2 barrels 2 hogsheads 2 pipes, or 4 hogsheads make (( iWi. 1 > m. Mid. I = 1 =: 2 =: 1 = 2 = 4 = = 2 = 4 = ? = pi. gal- I 3U 63 126 252 1 pint, 1 quart, 1 gallon, 1 Barrel, 1 hogshead, " 1 pij»?> "jm, . ql,-*:i.f' f 1 = 1 %« 2 a 4 = 8 = 126 = 262 = 262 = 604 = 604 = 1008 = 1008 = 2016 = pt. gt. gal. bbl. hhd. pi. tun. gi- 4. 8. 32. : 1008. : 2016. : 4032. : 8064. -^ NoTO.— 1 . Tha Engliah Imperial gallon oontaina 377.374 enbio inoh«a or 10 lbs, AToirdupoii of pure dTatUled water, weighed at a temperature of 82^ Fahtsnheit, •od under a barometer presaure of 30 Inohea. 2. In the United Statea the wine gallon oontaina 231 oubio inohea, and the beer gallon 232 eubie inehea. the gallon of England ia therefore about equal to 1.2 gallona United Btatea Wine Measure. ' .«_»„„. 3. By an Act of the Imperial ParUament, 1828, the Imperial gaUoa of 277.274 oubio inohea, waa adopted aa the only gallon, and ia thorefbre the atandard for both liquid and drr meaanres. . ^ ^ . ..*.,« 4. Beer ia naually aold bjr the gallon : aometimila, however, in eaaka of 8, 10, 20 gala. ete. The beer barrel oonbdna 88 gallona, fnd the hogahead, 64 galkma. II. Dbt Miasub^ 209. Dry Measure is used in meMninng artioles not liquid, as grain, salt, fruit, roots, &o. TABUk 2pints(j»t.) 4 quarts ^ gallons 4peok8 36 ooshels mak« i< ft M ^ *1 bush. 1 86 * gat. 1 quart, 1 gallon, 1 peck, I Dushel, 1 chaldron, qt, gaL vk. hush, ch. pt. .A.==m. a. 1 4 144 a 8 « 288 « wm * 8 82 1169 16. 64. * v,- :' MBASTTRES. 12!^ MUurOi is now >)io inches or 10 ibs, of 82<>Faliraiihett. I inohei, and the Dre about equal to lgaUoaor27T.374 t&e ■tan. -i 32 " « a32nio. _ >f4 iA.- :.. I 126 VBOB WnBIO BTBTIM. THE METRIC SYSTEM OP WEIGHTS AND MEASURES. .* The meixto system of weights and measures — so called, because the metre ia the unit from which the other units of the system, whether of length, area, solidity, capacity, or weight, are deriyed — originatSed in France in IJ^O. It was determined and established as folfowB : a very accurate survey of that portion of the terrestrial meridian, or north and south circle, between Dunkirk in the north of France, and Barcelona in Spain, was made under the direction of Government, and from this measurement the exact lengthx^of a quadrant of the entire meridian, or the distance from the equator to the north pole, was computed. The ten millionth part of this arc was denominated a metre, and from this till the standard units of measure and weight are derived and determined. The metric system was finally made the only legal system throughput the whole of France in 1841. Since that timd, it has been adopted by Spain, Belgium, tind Portugal, to the exclusion of other weights and measures. ' In Holland, other weights are used^nly in compounding medicines. In 1864, the system was l^alUied in Great Britain ; and its use, either as a whole or in pome of its parts, has been authorized in Greece, Italy, Norway, Sweden, Mexico, Guatemala, Yenezuala, Ecuador, United States of Columbia, Brazil, Ohili, San Salvador, and Argentine Republic. In 1866, the use of the metric system of weights and measures, was authorized by Congress for the whole of Uie United Statev. TABLES AUTHO^ZED BY CONGRESS OF THE UNITED STATES. MXASUBIS or LIMTHS. Hetdo-Denominatioiu and VtlnM. MyduMtrt, KlloiB6tni>M... \ Hoetometre,... Deoametrai. ... ••••••.•.•• DeeioMtiw, ••..• CanUimtra>.... MUUoMtn.^... 10,000 uwUttt,.. 1,000 metres,.. }00 metrei,~ 10 metrea,., — — X ttete Cw^ iiAraO' of A metrat ... •f AaMtre.... tk nretrei BqalTalenU in Denominattona in oae. 0.3137 milei. 0.62137 miles, or 3280 feet, 10 inohea. 328 feet and 1 in^h. 303.7 inohea. ^ SOJf inehea.- — . ==;;==: »... I 0, 3.037 inohea. 0.3987 Ineb. 0394 iMb. '*^! ^^ HE UNITED minatiooa in oae. )0 feet, 10 inohes. THl MITBIO BtBTlM. MEASURES or SUHJAOtS. MT Metric Denominationa and Valaes. Heetare,., Am, Centiare,„ 10,000 sqnarenwtres, 100 square metrea, I square metre. Aioiralenti in Denominationi in OM. 2.471 acres. UO.O square yards. 1650 square inches. / MEASURES OP SOLIDS. Metric Dcnou^inations and Value,: Bquiralent, in Denomination^ Deoaater»,„ 8^«B«, Deoistere,... 10 cubic metres....... 1 cubic metre, 100 cubic decimetres,. I in Bse. 1S.070 cubic yards. 0.S7M«faoordorwood. S.63144 onUo feet. MEASURES OF OAPAOITT. Metric Denominations and Values. K^ames. v> KUoUtre,©* Store, Hectolitre,..; Deoalitre,.... LrrR% „. Decilitre, , CenUHtre, ..., ICUilitre, No.of litres 1000 100 10 I ttjW -Cubic Measure. EqniTidents in Denonrinations ' in use. I cubic metre, X of a cubic metre,. XOoubie decimetres,. 1 cubic decimetre,.... X of a cubic decimetre, 10 cubic centimetres,... 1 otfbic centimetre,. ...„ Dry Meaanre. 1.308 cnbKs yd. 2 bn. 3.35 pk... «.08 quarts, 0.008 quart,.... 0.10S3 cubic in. 0.0102 cubic in. 0.001 cubic In.. Liquid or wine measure. 284.17 gallons- 28.417 gallons., 2.8417 gallons., 1.0587 quarts., 0.845 giU„ U.3S8 fluid 01... 0.27 fluid dr.... WEIGHTS. Metric Denominations and Valoes. Names. Mitlier, or tonn^au,. Quintal Myriagramme, Kilogramme, or kilo, Heotogrammei Decigramm Centigramme,. Milligramme,... Number of Weightof whatqnantity of grammes, water at maximum density 1,000,000 100,000 10,000 1,000 100 1 i(roq esfsi Equiralesfs in De- nominations in use. Avoirdupois weight |,M.. ...#•«•••.. M. • 1 cubic metre, 1 (leotoUtre,..., lOUtras,. 1 Utre,. , 1 dMiUtre,. 1 Cttbie centimetre, , Jj, of a enbioeentimetie,., 10 cubic millimetre^ 1 cubic millimetre^ 2304.8 pounds. 220.40 pounds. 22.048 pounds. 2.2040 pounds. 3.5274 ounces. 0.3537 ounce. 15.432 gr. Tfc W. 1;6432 grains. 0.^543 of a grain. 0.0154 of a grain. 128 9Mi HITRta StSTEli. MVAStFEBS OF ANQLBS. Motiio Denominations and Valaei. EquiraleoU in Denomination! in nae. Ciroie Qaadrant, .. Orado, Minate Second, 400 grades, 1 oirde or 360o. y' 1 quadrant or 90Or &^ ■ 64 minutes. 7^ 33.4 seconds. 0.334 of a second. 100 gradei, 1 grade, GO H P I P4 I. MfiTBB, . NOMENCLATURE AND TABLES. There are eig)it kinds of quantities for which tables are usually oonstructed ; vii., Lengths, Surfaces, Volumes or Solids, Capacities, Weights, Values, Times, and Angles or Arcs. The table for Times is the same p the metric as in the ordinary system. The table for Angles is constructed upon a centesimal scale. The tables for the-other six kinds of quantities are constructed upon a deoim«l scale. In each of the tables for Lengths, Surfaces, Vol- umes, Capacities, and Weights, there are eight denominations of units, — one principal and seven derivative. The principal units are the mttrty which is ihe base of the system, and those derived directly from it. The two following tabular views present the fkcts regarding the principal and derivative units, which should be fixed in the memory. 1. Principal unit of lengths. 2. The badeofthemetricaystem, and nearly one tea-millionth pan of a quadrant of the earth's meridian. ^8. Equivalent, 39.3708 inches. . Principal unit of surfaces. A square whose side is ten metres. Equivalent, 119.6 square, yards. Principal unit of volumes or solids. A cube whose edge is one metre. Equivalent, 1.308 eubic yards. 1. Principal unit of capacities. 2. A vessel whose volume is equal to a cube whose edge is one-tenth of a metre. 3. Equivalent, .908 quart dry measure, or ' 1 .0567 quarts wine measure. /I. Principal unit of weights. * Ihfr w eigh t o f a on b e o f pure wBter !_ edge is .01 of a metre. 3. The water must be weighed in a vacuum 4" C., or 39.2" F. U. Equivalent, 16.432 grains. n. Abk, ni. StibKj lY. |i{tB%« ''••••Is! 1 V. Obammi, ' \ '•X iy iiu^i, *, - ^# / M. K vJ i30 «. TH« MBTBIO STBTIM. in their order, to the principal unite of the tables. The order of pragreaiion being ten, tl^e table of oapaoitiea will be written thus: — 10 Millilitres =^ 1 Centilitre. 10 Litres = 1 Dwalitre. 10 Centilitres = 1 Decilitre. 10 Decalitres = 1 Hectolitre. 10 Decilitres = 1 Litre. 10 Hectolitrea = 1 Kilolitre. 10 Kilolitres == 1 Myrialitre. All the tables peculiar to Ibtjietrio Svfitem are presented to- gether in a convenient form i^;^ two following tables :— TABLE OF SUBMULTIPI4S AND PRINCIPAL UNITS. Names of Ukiis. FRKFIX. BASE. , " W' ■' 4 lOMilll- Equal 1 Centi- 10 Centi- Equal 1 Deci- 10 Deci- Equal 1 Principal Unit. 10 Principal Units Eq ual Metre lAre ■I Stere I Litre \^ Gramme I Metre Are Stere Litre Gramme r Metre Are Stere ^ Litre ^Oramtne r Metre Are Stere Prohckciation. Mill'-e-mee'-ter Miir-e&re Miir-e-st&r Mill'-e-li'-ter '„ Mill'-e-gram .3ent'-e-mee'-ter Sent'-e-ftre Sent'-e-^t^r Sent'-e-li'-ter , '"1 Sent'-(S^|ram Des'-e-mee'-ter Des'-e-ftre Dea'-e-stfir Dea'-e-li'-ter Des'-e-grara Mee'ter Are. St£r Stmbols. ^^ .s .G aM ,A ,S .L aG ,M lA. i8 iL ,G M A S \ ID«foi^ I07-: UtM ^Grammie Li'-t«r Gram 17 G rho ordor of tten thus: — TBI lomia BTvriM. TABLE OP MULTIPLES. 181 Names o» Units. paariz. BASB. lODeoa- Equal 1 Hecto- lOHecto- Equal IKiV 10 Kilo- Equal 1 Mjria- i Myria- 'Metre Are Stere Litre Gramme r Metre Are Stere Litre ; Gramme r Metre Are . Stere Litre Gramme Metre Are 1 Stere Litre Gramme PaoHtnrouTiov. DeS'-a-mee-telr Dek'-dre* , Dek'-a^te Dfek'-arli'-ter Dek'-a-gram . Heo'-to-mee-ter Heo'-tAre Heo'-toHStIr Heo'to-li'-ter Hec'-to-graai Kiir-o-mee-t«r Eiir-dre Eiir-04t«r KiU'-o-Ii'-ter Kill'-o-gram Mir'-e-a-mee-ier Mir'-e-ftre Mjr'-e-a^tdr Mir'-e^li'-tep Mir'-e-a-gram M S O ^M v> 2 S "M S ■fi G 1 I ABBREVIATED NOMENCLATUBB. To secure the fullest advantage to business men by the nniyersal adoption of the new system of weights and mea8urM,it is neoe». sary that the names used should be short and easy to wiito and pronounce, that they should express dearly the relation of the qineren t ocnominations of the wanift tjMa t ^ a«/.it j^\.^ ^ j x i-,^. they should be identical in all langnagea. The last two of these requirements would be seoored by the universal use of tlje nomenoUtuf^ adopted hj the French. It^ ii C V 132 TBI Mimo VfOnM. ooAmopdlitan in its duunoter: it beloogi to their language no' more tha,n to any other. . The former, however, is not aeoured. It'^s evident to all, that, for bosiness parposes, the long ni^ines of the metric system are inconvenient, and that to shorten them would prove a great ^advantage. Efforts have been made to intro> duce short names; but these efforts have invariably sacrificed their nnive^rsa) and expressive character, which is pf more importance to the business world than their shortne The only true course which seems to be^pen, is to abbreviate the names already introduced, in such a ^ay as Ui retoin their peculiar characteristics. Toseoure this, the following plan of abbreviation is suggested: — First. Let the prefixes be abbreviated thus : Myr, kil, heot, dec, des, cent, mil. Seiond. Let the' initial letter of the names of the five principal nnit^ be used, instead of the names themselves, thus : For metre, use a capital M ; for ard, use a capital A ; for stere, a capital S ; for litre^ a capital L ; and, for gramme, a capital 0. Third. For the names of multiples and sub-multiples, attach to these initial capital letters the abbreviated prefixes, thus : KU M, pronounced kill-em' ; Kil S, pronounced kill-ess', Ac. By this method of abbreviation, the elements of the original twms are retained in such a form that each part is dearly indi- cated. The capital letter used after the prefix will always point to the base-word of which it is the iniUal, although the pronun- ciation is changed. * . , TABLES WITH ABBREVIATED N0MENCLATU||1. ■'■*•'■ t HftASnaiS OF LXNaiHS. Wtittan. IQMilM, 10 Cent M, 10 Des M, 10 M, 10 Deo M, ^ fleet ^, 10 Kil M, MyrM, ProDoaBMd. Hill-em' Cent-em , Des-em', Em, Dek-em', Kill-em', Mi^iem'. make <( (( It II . « . - II 1 Cent M», IDesMP IM. IDeoM. 1 Hect M. = 1 Kil M— \1 Myr Wt. t»-fc ' •<> THI mrAIO BT8TUC MIASUE^ ov suarAcxs. 133 Abbreviate retain their WritUn. 10 Mil A, 10 0«Dt A, 10 Dee A, 10 A, 10 Deo A, 10 Heot A, 10 K^, MjrA, PronoosoML Mill-a' / Geat-a, Des-a', A, Dek-a'. .mU ^^i^t-^^^^siifeiriw ^1 134 fitDtrOTION OT OOMPOUNO MUHBIRS. REDUCTION OP COMPOUND DENOMINATE NUMBERS. 214; Reduction is the process of ohangiog nnmben from one denomiqation to another, vrithout altering their valae. Reduction, is of two kinds, Detcending and Atcending. 219. ReAactlon Descending is ohaoging nnmhers to lower denominations without altering their yalue ; as pounds to shil- lings, yards to feet, etc. It is performed hj Multiplication. 210. Ilednotion AflCenduig is changing numbers to higher denominations without altering their value ; as farthings to pence, inches to feet, eto. ' It is performed by Dividon, aBDUOTIOM DlSOBNDINO. 217. CasB L — ^0 reduce a compound numhtr to lower da- nomination$, Ex. Beduoe £45 7«. Sd. to pence. OPXRATIOV. £45 7«. 8(f. _20 907t. 12 10892(1. AxALTSis.— Tb«i« u« 20*. in £1 ; therafor*, SO timea the nnmber of jC r=e the oamber of chillingi. SO timM 46 a 000*., to wlii«h w« mild 7*., and obtain 907*. There, an lid. in 1«. ; therefore, 13 timea the nnmber of ahillings eqnal the nnmber of penee. 12 timea 907 aa 10884(t,, to whioh we add 9d., aftd obtain 10892(1. Henoe the following 218. RuLS. — I. Mvltiply tlu highett denomination of the given number hy that number of the scale which toUl reduce it to the n«z< lower denominatianj and add to the product ^hnon^ tnofum repnNd, ,.. Ui SXAKPiaS VOB ntAOXlCII. An$. 8480. 1. In £36 6ff. 8(2., h6w many pence? 2. In £28 12«. 8\d., how manj fkrthings? 3. In 14/6. 10o«. I8pwt. 22gr, how many grains? iliM. 85894. 4. In liAT. ISoMt. 3qr. l9io. lioM., how many (j^iapeKf,; ( 6. In 23fl> 9l OK'29 13 »., hoif many grains? , it . , j o r 6. In 12r i. How jtiamy inohes in-8m<. 4jrwr. 82n>. lyrf. ? _..i p ; "" ^Tp BQBiy. Tji ir. ItO. 6ft., hm vamf feeTT . j s p;^^ g j^ I f'-^ ^ t ( * '•!« ''if '^t l>il«»S. . 4 1 EEDUOXrON OP CJlHrPOUND NUMBERS. i2^ Ana. 61630. in., to square !?• ?°7 *?*°^ ''"'^^ in '^"»t- 5/«»". 6cA. 30/. ? ??• n 1"**- '*^**- '2^-' tow many links? 12 T„I0>I iJ, OK V ,. j4n». 85937864 square inches. wJie^niw^*- ^^'^- ^^- ^^ '^' yi ^"i:f^' 136,?:A„, how many ^if nn»1!^! ,• , . . i*"*' 65296108 square inches. 1^. How many square Imks m 764. 4#o. c^.-18P. Tl8«7. 1. ? JK !*• **ow many poles in 3 townships of land ? S^Ia"**^ ®'i^*' ^!*V° ^^ <'°'^8 and 74 cubic feet of wood? lb SVoorda of wood, how many cubic inches ? In i\ggl. 4.26o«. 4.76gi-., how many gills ? iliw. 1901. In 57rim# SAAd. 60ga/. 3gf., how niaSy pints 7 * How man^pinta in 106». 3.5»ifc. Yfj^ I of. ? How many quarts in 676 tehaldrons, of 36 bushels each? T o il o J *??*•; ^^^ ^*"y seconds ? 4n». 362700. in dffffc. laa. lA. Imtn., how many minutes? Su^^ffl^a*!!^*^^"^.^*^^^ '**'•' 1870, to May 16th. 1871? -- n ***** ** °! *^ > *«»* "na^iy seconds ? Ana. 4820243". io. How many mmntes in l^C. IS. l^ 1'? , 27. Bedooe 38ft, 68 3» la, to grains. 28. How many days from August 30th 1771, to June let 1872 ? ai.iil.ifL"«rM« l"'*'P ?^i'"/V"''«' weighing 13/6. 9oz. What is MS Tttiutt at fl j38oA per ountie ? An* «22fi AiAi lOtCtauJk l5^(^.Avoi«l^pois weight to Troy^e^t ***' kV j?.T^*^ M- \^ ^^^^- of ^an**' at *l-25 per ^are foott ''^i**^fjff/?*^tMS^L . >ln/$187r7?87r* 32. Bought a hogshiMs^ ei^up at 40 ot». per gal., and sold it at 12 cts. perquartj what did I gain by the, harg£;?^ kw &8. W?/r^MilWi^RL»^ro redttet a denominaU^raetion to oite of " ■ I i .'.^l ,:::^i .^i.ha lower denomination. '^ ' "^ ^V ^««^^W iH.'?'* «aHon to the fraction of a gill. 11. 18. 19. 20. 21. ^2. 23. 24. 26. OPiBATlOV 4 *«,•»■* '^^ AirALTsn.— 7d ndnp* giUtoiu to Mi, m , ben in th« Male. And. •iaoe the cItmi nam- ' tor ii ft fhMtion, we indioato tiie proeeie. am '**' .f«Uhw^a^.^the|iu»«r. B^aoi^lSr '>>«. C^ *.• *?•; **«*--*^''t'^ the fraction of the higher demmina. ^' «>«8 ^ of an acre? , 13. Whatfraftlonofayardisf of Aofarod? 14. What part of a dram is r^j^ ofa hundred weight ? it. if^dr. 16. Reduce 0.03125 ofa mile to feet. 821. Casx'IIL'— ro reduce a denominate /ration to inf^trt of hwer dmominationt, Ex. What is the valae of f of a £7 OPBRATIOK. iMkVtm^l of £1 \» fke lame u '^ of £ 8. d.far. £»-8..M.i|Atr. H6DC«.th» ■ - 7 )3 OOP . ^ f 8 6 3f,i4n». 1^ "\- . . d22. Buu. — Contider the numerator o/ the frt^Uim . llox. ^dr. IthaA is the value of .2. 4 of a bushel? 8. I ofa shilling? 4. |of acwt? P ^6. i of a yard 7 . .;6. ^ of a lb. Avoirdupois? -a*t XofadayT . * ^« 8. f of 16 ctrt. ? ' U I of 2i pounds Apothecaries* weight? to. X of an acre? Ant. 2Jt firrf. ieq. yd. 5tq.Ji. I21-f^. iiL U.iof6Aton8? ti of » hh4. of wine ? ^ . . Ant. 6gaL 27*. Ipt. I^gi l^ofaicoflbofwoodT ^:^,\ .A\ lofasign? ^^ ^ ^^I2«»6l*j6»'f. , " to. l?fofl>"a piece oi Te^yw con«uaing e y a. o yr .— i ,BHt ■ yi" mgr x^ whatpart of the whole piece did I take? - . ;, Vf.—>To red/uct a dmonunate dvimedMt i^ttgm |UB>170fI0N Of COMPOUND NUMBlBS. Ex. Badace 0.628125 of a £ to Bhilliags and pence. 137 OPXBATIOV. £0.628125 20 12.662500*. 12 6.750000d. 4 3.000000/ar. £0 12». 6irf. Atu. S?^' y* u*?.?. '"■"'* M 12.. Md the deeimal jM» of a shilling. We then mulUplj thb d"d. •nd .75 of • rf. This last cfeoimal we mulUply by ^ to redaoe it to/or. or jr., and the reault is 3 ijd. ***'** ^•"'f' *•«> »n«ww M ^ 12». in^t^Lf^'T' ■^''^^F^^ '■^ 3iven decimal hy that number in the scale whiifh will reduce it to the next lower denominat^ and point of a. in multiplication a/decimaU '^'^''^''^^ 11. rroce^tnih the dtcimal part of the product in th^ ,n«u. XXAMPLES FOR PttACTIOB. What is the value of 1. 0.45iofa£? 2. 0.748 of a bushel? 3. 0.765 of a pound Troy ? 4. 0.7526 of a mile? 5. 0.659 of a week? 6. 0.2170? 7. 0.876w.Jj „., tbenamberofpeiuiyweightr ,— , SS' of the anmber of iwnnyweifhtf ■ aamber of onnoee. ^ of SflS'lm lM«fc, Md I8pw$. NDttlBteg; ifeji. — U».| tlMiefi«e,^cf tU 139 ElDTTOnON OV COlfrOTTSTD WtfllBSM. of oqnoei = the number of ponnda. ^ of 183 » 18»^ sad Tot. Mmalidag j I thenfbro, TSflM^r. » 1326. Ion. 18pwt.Wgr. Henoe, the I fflHS. RuLl.— r. Divide the given'numher hy that nuniher of \the oBcending tcale which will reduce iCto the neait higher denom- B-^iimtion, r [I. IHvideinUkei, manner the quotient thus ohiainedy and to ^ until it it brought tq the denomination required. The latt Ovfiti^ty^oiththe several remaindert annexed iti a reverted order, vfill be the antwer. .^ XXAICPLES FOB PBAOTIOB, - l.'Inl6452/&r.,'howmanyje? Ant. £U 2s. 9d. ■ 2. In 90720 pence, how, many £'f «. - i„ . 3. H6w many pounds in 4255? Ant. 4» 58 1». 4. In 78692gT., how many pounds Troy weight? ' 6. A physician who averages daily 5 prescrtptioDS of XO grains each, how many pounds of medicine will he use in one year, or 365 days? -A"*- 6* ^* ^'*- 6. How many pounds of standard silrer can be purchased for $1099.88, at the rate of $0,062 per pWt. ? 7. In 87320/6., how many tons? Ans. 43T. 13cu<. 20». 8. How much will 230to. bf hay cost, at $10 per ton ? 9. In 1265 pints, how many bushels ? Ans. 19bu. 5ph. Ivt. , 10. A* 6 eta. a pt., how much sirup can be bought for $3,847 11. How many francs in $176.70 ? Ans. 960. , 1?. In 2468 pence, how many half 6rown*? 13. In 90060 seconds, how many days ? AnsAd.lh. «*»»•. 14. What would be the cost of plasterittg a room Uft. long, lejI/Ti wide, and ^ft. high, at 22 cts. a sq. yd. ? , Aw. $22.44. 16, In a pond i^easuring 28/1. 6trt., how ?»anymtttoinBdeep W there? : 16. Hoir many bushels of oats in 270725^,?; Ans. 8466BtA. IT. How many days in 98960 seconds? * 18. The extent of a certain farm is ^und, by survey, to be 1877*9. cA. How many acres does it contain ? ^ Ans. 137 A. 211. 32|9er. 19. A load of wood is 12 ftet long and 3 ftet wide,;how hi(^ mttat it be to make a cord ? ■^. 3|/l.l»igh. 20. How many tons of round tiitiber in 622080 cu. tn.T 21. A cellarwall, 32>». by 24ft. is ^. high and 14/1. thicki How much did it cost at $1.26 a perch? - -Ant. $60.909 + . 22. Reduce 169.36 links to miles. Al*. 2Wt. »o^. 361. 23(. In 161384 inches, how many miles?. ^1. 24. How many beer aallonsis there in 16K. IgaL 2gt« winetoM*' 26. tn 6832000 square incfilB^ how many roods? 26, Reduce 20937 minutes to agns. Ans. IIS. 18* 67'. . 21. Change 16». 3o«. Ivwt. Igr. Troy wewht toi Avoirdaiwii iveighU • Ant.im.6ox.lpv4.1gr. .#(f -.'? I H(SC l-.^ ^J- fj. M^ V ' "^ ■ jtCDfrcmoN or oompourd mncBXBs. 139 29: A Bhipy during 3 days' atonn at sea, ebanged her laUtade 412 geographical miles ; how many degrees and mioutes did Bhe«change 7 2^. How many acres of land can be purchased in the city of Mont- real for $147600, at 65 ots. a square foot? Ans. 5 A. 33ver. 15«9. yd, 3«j.^. 119^%SM,. frarv lb: 4iq/a»., ud that £;i-fi!«M >W. One fkrthlBfl is ^ o( a £, ud 411^. « 410 li^«;i; IP |1A s. .lv«>-'n "UO BipuonoM or oompound numbers. — SS30. 'RjaiM.—Btduceihegivtnnumhertoitt Ipwett denomi- natum/or the imtneratOTi and a unit of th$ required denomination to the satM denominatiow far the den^iriator of tht required fraction. • ' * ^-TAVPT.M roB poAonbs. What part of - ' 1. a £ is 10». lOd. ? - '^ Am. ^. 2. a ton is icuA. 3gr. 12Z6. ? Ans. ^^. 3. an acre is 2/t ZOper. T ~ 4. a mile is 1/ur. 12r Or, 12«. W. 3/ar. » W6/ar. jBl I = 96q/ar. 1^ = 4eo.64af625, Ana. AiriLTin.— Sinoe there are 4 farthioga in Id., \ of the namber qC fartbiDge oquala the Dumber of penee. i of 3 s= ^.Ibd. whioh added to 9d. «■ VJid. There are \td. In 1«., therefore, ^ of the namber of penoe eqniUi the namber of ahtlUngs. JL cf 9.r0(<.»O.81»f. whioh added to i2t.L 13.81iS*. there tte 30*^ la £1, therefore, ^ of the Anmber of dilUlagi eqnaU the namber. of poaad^ ^ ^ 12312A « JE0.04083t. H«noe,the . 282. V»(JiA.*-^Diii%de the ioweat denomination avoen hy that nund>er in the acale which wUl reduce it to the next higher denom- ination, and annex the quotient aa a decimal to that higher. iVor ceedin tiu aame manner uhtU the whoje ia reduced to me dinom- itiation required. Or| Seduce the given number to a/raetion of the requireil denomi- / nation, and reduce thU fraction to a dednuU. KXAIOLU rOB PBAOTICS. Whatiiy»w>l iOf .. _„ ,. . „ ~"Tt»WrO.F376j'irt. i. a week is baa. 9a. 46mtn. 4&aec. ? 3. a mile is fi/ur. dbrd. 2yd. 2/2, 9in. T Ana. 0.V30U3219 -t- mi. 4. a bushel is Zpk. ^t. Ipt, f ^'-t^iiA,^'" x* C*? V .- W^-la • RumonoN or oubbinoxbs. 141 Ans. 0.886458^26. Ana. 0.8857257'. 6. a pound Troj ia 10o«. I2pu^. ISgr. f 6^ a fathom is 3%ft. ? 7. a ton is \6cu>t. 3qr. 16.45/6. ? . 8. 1 i bushels is 0.45 of a peck 7 9. Reduce 1 2 T. 3cirt. 2qr. 20/6. to hundred-weights and the decimal ofahuildred-weight. ^ Ans. ^43.1. 10. Reduce to the decimal of a pound, 19«. 111(/., 16«. 9ld., and I7«. S^d.,- and find their sum. ilfw. £2.710416 + • REDUCTrON OF THE OLD CANADIAN CURRENCY TO THE NEW OR DECIMAIi CURRENCY. - Ex. Reduce £72 13 9| to cents. QPERATIOK. £72 X 400 = 13«. X' 20 sa 9i = 39/ar. x 5 ;^ 12 = £72 13 9|=: or f 290.76i, Ant. and farthings by 5, and diride equal to ^ of a cant; Th«t each farthins; Is eqnal farthings ^r one «hilling) are and one farthiQg equals ^.of AKAtTiM.'— We mnltiplT £73 by 400, becaase each pound is equal to 4 dollars or 4flO eents ; next we mul- ttply lMhe.BlMnber of shil- Xxanyvf^, because eaoh -ritiflihg is equal to 3&^oenU | lastly, we multiply the nunv- ber of farthings in the penoe the remainder by 12, beoause eaeh fturthing is 28800 cents. 260 " 16^ " 29076^ " to Aofaeent.iaeTident from the fact that 48 equal to 30 eents { or 13 faribings equal ^ Mnti, • eent. Henoe, the following 233. Ruit*.— I. MuUiplif the pounds hy 400, the shillings by 30, and take_fiM-H$^/ths of the number mprttting Amu nuintf farthings there are in the givek pence and farthings. tl. A^ the three riiults together, and thnr sum vnll he the number cf cents required^- . III. Consider the last twofigikes as cents, and the result will be dollars. and cents. >. .v:.>r ; : fcXAXPLWf ioB PIUIOTI0JJL ■' ^ '- ' • . ■ ' ^ t: How many dollars and cents in 1. £ 4 3 U? ilfw. |16J62^. 2. 27 16 3^ , ^, , 3. ^7 16 IH? Att;$tU,S8|. 4. '69 16 6 ? 8> 14 84? Ann. $2.94^ «. 7T I? ^i i. n l6 5if i!iM. $71.29^. 8. 18 18 IoJt 9. 9 3 SX? ^fW.936.69|. 10. £16 6 2? ilfM. $65.23^. 11. 97 3 Hi? 12. 4« 17 7i? Ana. $181. 52^. 13.121 7? 14. la 9 U ? ^fW. 149,981 ^ 16. 173 13 4 ? ilfM.l694.66i, IT. 91 8 8 ? Id. ' 19 11 4| ? ifiM, |78.a7|f t.>.. i.'&W . i * 142 ■'4 AJDDrVWH Of OOMfOITHD KXnfBXRS. REDUCTION OF THE DECIMAL CURRENCY TO TgE V ' OLD CANADIAN CURRENCY. Ex. Beda(^ $246.88 to the old Canadian currericj. i OPERATION. 4) 246.88 £61.72 20 14.40ff. 12 4.80J. 4 3.20/ar. Ans. £61 14 4t + A = i/<»»'- AwAtTBis.— We diride 340.88 by i, tha nuniber of dollars in a pound, and the result is £(11 and 73 hnndredths of a pound. We multiply 73 by 30 (224), the number of shillings in a pound, and the result is 14*. and 40 hundredths of a shilling. Again* we maltiidy 40 by 12, the number of pepoe in a shillint;, and the result is 4a. and 80 hundredths of a penny. Lastfy, we multiply 80 by 4» the number of futhipgs in a pamy, and the result is S/art and 20 hnnuedths or 1 of a farthing. Henoe, the 284. RULK. — Difide the ^(ven number by 4> and the quotient will bepoundeand deeimaU qf c^ pound. Then jtroeeed as in No. 224. fXAMPLKS rOft FBAOTIOI. BedoM to the old Canadian currenoy :— 1. $162.30 = 2. 716.12 ' 391.37 = 637.37i 82.19 *. 207.16 569.09* = 17.36$ 3. 4. 6. 6. 7. 8. 9. An$. £40 11 6 Ant. 97 16 lOf AiH§. 20 10 11| Ana. 142 5 6^ An$^ 231 4| 10.$319.13ia Ant. £79 15 8^ 11. 933.04^ 12. 601.63 = 13. 298.17 Aj^ 160 7 7^ 14. 39.06i» An$, 9.16 3^ 15. 436.99 1«. l5a.l8As 17. 846.071 Anii is 11\\, « 18. X19.Ua; Atu. 179 16 6} ADDITIQN Of COMPpUND NUMBERS. 389. Addition, Subtraction, Moltiplioatioa, and Diviaiop, of Denominate Nombera are pwformed by tbe same general methoijbi as are employed fwlike operations in Abstract Numbers. TIm only difforenoe arises from varying^ instead of wiiform seake. Ex. 1. What is the«am nm^ w M o h to^i S s.^— 4» ^«ir " 8 15 6 HaT^(w4timmlAi.w4>f^««!itf AitHags, w^ .4 13 .9 MR7t]i4£3(ottk»oolanuiofpoaiida,aiid find tiw antin ^itDITION or OOMYOUND MtTlIBlKS. Ex. 2. Add^ofa£tof ofadbilling. 148 OFBBA.TtON. ^ of a 8. = Oa. 8rf."2;^/ar. iitw. 'io»ro3r2f/Srr |iJ£ = 10». Qd.^far. AvALTsn.— W« lint find the rala* of each fraction in intonn of less denom- inationi (221), and then odd theresnlt- inc or equiralent oompoond namben. Or, we may lednoe the giren fractions to fractions of the same denomination (219^. then add them, and find the ral- ne of their sum in lover deneminationf . Henoe, the following 2S6. Bull — I. Ifanyofth« numlers are d^nominaU^frao- ttoTu^ori/ant/ o/thedenominaiiotu art mixed number$, reduce the /racf ions to integers of lowet detiominatioiu. II. Write the numbers to thft pnitt of the tame dettominatioru will stand in the tame column, III. Beginning ieith the hwett denomination, add at in simple numbers, carrying to each tucceeding denomination one for at many units as il takes of the denomination added, to make one of the next higher denomination. Ji^AHFLBS FOB PBAOTIOB. (1.) T. cut. qr. lb. oz.. dr. 71 19 3 27 l/L 13 (2.) 14 13 2 15 16 15 14 13 1 11 M 12 11 17 3 16 15 11 13 18 2 13 U 13 127 3 2 , 11 8 yr. da. h. mtn. 'tec. 12 10 13 42 27 16 102 18 24 36 19 8 21 64 67 23 13 19 49 48 29 18 23 68 66 (8.) deg. mi- fur. rd. ft. in. IH 19 7 61 47 6 78 32 6 It 59 1 "^- ■ - • ut* 2(l& ■ iiiiilje I il I II 15 11 1 39 10 11 14 9 9 36 16 10 30 16 1 17 I4i 8 (4.) A. R. per. tq. yd, tq.fi. 140 3 17 27 6 320 1 30 14 a 111 t 8 214 2 16 22 7 100 3 6 I 26 1 36 8 104 2 9 14 26^ d 1 IV 16 2 6. Whati8theBumof20/&. 9o«. 19|>trt;. 23gT., 10Z6. loz. IBjnat. l3^r.,Uoz.8^:,Anillb.QQ;^.Up»t.2igr.rAn»,3itb.lq^A5v^^ - I Si^S 'M t.i^I^^J, v" *V.' -■* 144 SUBTBAOnON 01* COJOOVm mTlIBKBS. 6. Find the sum 6r81ft> 111 6s Is Ugr.. Utb lOl 1% 2» ISfr.^ 14Ib 9s 7ft 19 Ugr., 37Ib 8B U la llg^r., 61D> III 3S 2s Sgf. Ana. 2720) 41 8» l8gT. 7. Add 197»o. yd. itq.Jt. \U\$q. in., I22sq. yd. 2aq.fi. 27|«9.tn., baq. yd. 9aq.ft. 2\$q.in., and 237«9. yd. 1 sq.ft. 128|«9. in. Ana. 663m. yd. Aaq.ft. 118.826*0. tn. ' 8. What ii; the snm of 17mt. 6/ur. ich. 5rd. 24/., 16mt. 3fur. tch. ltd. 2U., 41mi. Ifur. 9ch. 3rd. 19/., 19TOt. 6fur. 6ch. Ird. 16/., 31mf. Ifitr. Ich. 20/. ? . AtU. 133w». IJur- *<*• 9. Add 3S. 22" 50', 24^ 36' 25.7", 17' 18.2", 18. 3« 12' 16.6", 12° 36' 17.8", and 57.3' Ana. 68. S® 33' U.6". 10. Find the sum of ^ of a mile, | of a mile, A of a fttrlong, and ^ of a yard. Ana. 6fur. 29rd. iyd. 1ft. min. 11. Add 4 of a ton to ^V o^^f^ c^^ 12. Add ^ of a week to ^ of a day. Ana. Ida. 9h. ISmtn. 13. What 13 the sum of i of an acre and f of a rood 7 Ana. 3R. lOaq. rd. 8aq.yd. baq. ft. 113^«9. tn. 14. Find the sum o^4 of a cwt., 8flb., and 3Aot. bylongtoatable.^ 1 6. A farmer received 60ct8. a bushel for 4 loads of corn ; the first con- ^ tained 42.4bu. ; the second, 28661b. ; the third, 36}bu. ; and the fourth, 39bu. 29Ib. How much did he receive for the whole? iln«. $100.83 + . 16. Add I of a yard, f of a yard, and ^ of a quarter. SUBTRACTION OF COMPOUND NUMBERS. Ex. 1. From £36 6«. 10(/. Ifar. take £14 16«. 8tf. 3/ar. AxALTiia.— WriUng the mbtrahend un- der the miouaiid^plaeuis uoits of the Mine denomination nndisr saeh other, we begio at the Tight>hand ; tinee we oaattot twe ~ S/or. from Ifar., we add Id. or 4far. to OPKBITIOK. £ a. d. far. From 36 6 10 1 Take 14 16 " 8 3 &em. 20 U 1 2 yar., vuMag S/or.: and taUiTg 8/ar. froqi V<^M we write Uie I , . ) remainder, il , underneath the oolvmn of farthing!. Hav« ing added Id; or Afitr. to the minuend, we noi^ add Id. to the 8 in the lubtrahend, making 9d.; and 9d. from lOd. leaves Id., Which we write in the rem^der. Next, as we oaonot take 16«. from 8«.,.we add £1 or 20«. toCt., middng M*., i^id: taking IS*, from 28«., we write the remidnder, 11*., under the denMuinaUon of Bhitlinn. AdiUng £1 to £14, we snbtraot £16 ftom £35, ai| in rimple nomben^ maA wnte tt^eltibmidnder, £20, under the eolumn of £. E»* 2,. Pro^'l of a mile subtract ^ of a Airlong. OPBRAtlOir. ' \mi. rsiAftir. Xird'. Ayd. 0, f «t. =4/ar. I7rrf. Ayd. Oft. lOtn. |/iw. = 22 4 i 1» Ana. 3 34 44 I 8| AirALTSii.^W« perform the ■•me rednetioa •• in edditioB of denominate fnutioni, (SS4), and then anbtraet tlwJmi valve firem the greater. ^^^^ Or, ^rtU. X 8 - ^fftar, M/ur. Urd, A^jd^ftTsiin. \- BVBlipAaTlOS qjt OOMPOUMO MUHBEBS. 146 337. IluLi. — I. Wrji^ the aubtrahend under the minuend^ to that units of the same denomination shall stand under each other. • II- Beginning at the right-hand, subtract Sch denomination ieparateljf, as in simple numbers. / III. ^any term ef the minuend is less than the corresponding term of the subtrahend, add to that term as many units as are required of that denomination to make one of the next higher, and from the sum take the term of the siAtrahi^, and aM 1 to the next term of the subtrahend before subtracting. IV. Proceed in like manner with each denomination. KXAMPLES FOB FBAOTIOI. do T. ctet. qr. lb. oz. dr. n 18 1 13 1 13 19 19 2 16 8 5 61 18 2 .21 9 8 (3.) (2.) lb > s B gr, 16 7 3 I 14 11 9 7 2-^9 3 9 3 1 1& (4.) deg. mt. _ fur. rd. yd.fi. in. A. r: P- fi- in. 95 3 7 31 I 1 3 96 I 13 100 113 18 17 66* *= 1 6 =1 39 31 13 1 2 HI 1 2 7 8 6 89 3 17 200 117 7 b 6 6 1 1 35 35 172 140 : 36 76 65 7 6 1 1 ~i 32 6. Prom £23 18». 3J(f. take £13 I4». 14)W. Am. £10 3s. 5ld. 6. From 71/6. 3oz. Upwt. I5gr. take 16l5. lOoz. llinet. 2Qer. 7. Subtract 3ft 88 25 29 Igfi-r. from lOft 7» 4S Id \bgr. ^ 8. From 1717, 3AM. Bgal. Iqt. Ipt. Igi. take 9971 Ihhd. \9eat. ^' ^i^-^^.\. .*. ,« „^5f- "^- '^Md.blgal. lot. ipt. 2gl. Prom 66il. lA. 19^. 119/1. llOtn. t^ke 17.1. 3R. I3p. Ulft. 3qt. 9. 113in. " ' Ana. 384. 2R. 6p~264ftV33in! 10. From l«nn. IJur. IQrd. 3ft. Un. t«ke 9mi. If or. ISHL isft. ,, ™ .,,_.,, 4«». 6j»<.7/ur. 38rrf. 2/^llin. 11. Prom I of a buehel take ^ of a peck. Am. Ipk. iM. Ipt. 12. Prom f of a week take J of a day. Ana. 4jo. a" 13. Subtract * of 9c«*. IVom |J of 5 tone. 14; From 6i6W. take f of a hogahod. Ana. 4&6<. llgal. 3h. '-m "^6. Sabtrset0.6&9^week'^roiD:s weeks 3f daye.' ^ 16. Prom a hogahead of sirup contaiaiag 100 galloi ,.* — ja-**u .-J -.,. _.^^j quantity ■Ana. UgaL 0^, I||pt. out, and} of the remainder wa« iQidt what quantity eini remained of it leaked 146 V- MTOTDPLIOATIOH OF COMPOUND NUMBERS. PRACTICAL PROBLEMS IN COMPOUND ADDITION AND SUBTRACTION. remowd, how much still remains to be taken out ? J0L3 2«r fiSi ■? "^""g^fw^ghing ^ctct. 3yr. 21/6.; sold w w lis. u ' *o Bernard 2cw/. 3qr. 24/6.: and to Tl^omaa ^ r^ K ^ howmuchremainsunsoldt ^w. U/. Sgr.lTlb _ 4. Joseph and Henry start from two places 120 miles Imrt. and other time, lOrrf. 1» .How much still repi'ains to be-^buUt ? 6. A merchant sold goods to the amount of £397 18». 6kd • and 7. A hogshead of wme, ost by leakage, on an average, for 6 v^ars. .eluding two leap years,! one gill of -;;- d^ l^'w^m^^^^^^ 8. Supiwse a person was bora February '29, 1792: how manv anniversaries of his birthday will he have had orfVeb. 29 844? ^ ^, ftb, lloz. Uptfft. 23^.; W much unwrought sW 14gr.5 a vase, remains? KfiP "ifr™ » P«le of wood containing 423 cords, I sold JTone Ume. nsbllZ'-^' ^"^^^'"tim^^a'C. 113«.. h.j at anorrSm?, , « ■ 2 ^' "^^'^ ™*''y '><*' " remain unsold ? 12. Suppose a note given Sept. 10, 1856, to be paid March 6 1868 How long was the note on inlerert, if we Ciiunt 30 days tothemontii ? How longy if the time is ooinputc d by days ? ^ v^ ^"•- 1"*" Myr. 6ino. 26Ai. 5 2nd. 4135 dajf§, MULTIPLICATION OF COMPOUND NUMBBBa * t Ex. h Multiply £8 9t. M. by 6. OPXBATTOir. 8 9 6 ■■iSS) imte tta« M. under tha Mnoe, and add the U. with the prod^t ^yy.^ tl^^>., Ji'^tiS pooBda. fame, £8 «. iE«8. aad X» « S«o whioh eA-£M£.*«r^ ^''•*^? tt«»£8 tfc £S6 16«. 6«r. i. 'ION AND house-Iota, ; how much 8 to be made 1 have been 21/6.; eold to Thomas 3qr. nib. apart, and >ry f, of the 8//. 7f m. tee ; at one ' and at an- t? B^d.; and 8 due ? fordy^ara, V much re- how many 1844? and made no. Uda. h made 36 Ooz. I3jnct. >ught silver Iff*. 20^. > one tim^ lother time, chS, 1868. themoatii? ZSdayt. BfiBa moltipUoation of compound nombbbs. 147 ^ 238. RULE.—I. Wriu the mul&plier under'the bwe$t dawtn- inatwn of the multiplicand. ««*cw»»- ir. Multlnly a» in timph nuniber$, and carry at in addition of comjiound Humbert. , «•«*«»••«/» Tisable to multiply by the component faotorg. ^^»»rtu m w. .ni-«j^^*" ">"'«?!•« i« large, and is not a eompo*ite number, it mar ba m •dved Into any convenient p.rta, and multipllcationCde by th!Se .eTOjl^ru" Ej:. 2. What will 45 yards of cloth coet, at £2 3t. 6d. per yard t AvALTBiB — We find the Qam> 'ber 45 equal to the prodaot of ft and 9; wo therefore multiply' the price of 1 yard by 5, and then that product by 9; and the last product U the aonrer. Hence the .OPERATION. t. d. 3 6 = price of 1 yard. 6 = price of 5 yards. 17 1 It. ed. = price of 45 yds. »»». Rule.— When the mnltipUer is a composite nmnW. multiply by xttfactort in tuccetsion. ««u»«r, Em. 9: What cost 643 barrels of-flour, at £2 6«. Id. per bbl. T 0PlaATI05. ^ £ s. d. £ t. Ibbl. = 2 6 7, X 3 « 6 16 10 d. ■ 9 a value of 8 bbL 10 bbl. B 22 15 10, X 4 s 91 a 10 4 m value of ' 40 bbl. 100 bbJ. a 227 18 4, X 6 « 1867 10 Am. 1466 10 4» * value of 600 bbl. 1 m value of 643 bbU A»iMBig.,-Sinoe«4»tonota eompodte namber, we «amiot raiolTB it into ^if»:tr«k ' ^ "''J'^ i" ^.+ '• ^o *^^ operation, w« «rtt mnltifly by ft, and obtain theva^ue of 10 bamla, and thb prodoet w« mnUlply by 10, ani obtiUn Mt prodoot by fl J and to find t|M. value of 40 banela. we mulSJ £.^S. «f SJJSr^^Hen^ thS* "^'"* P^"°*"' '* *»'*^ *l4?5 loT lA fcr the a4: Bulb.— When the multiplier is not a oomporite nm^- i^,rist0lv^^Hnt».myimmiimi pcfrtt,at vfiaiiarim, W^ "s. 148 MULtlPLIOATION OF COMPOUND N0MBEB8. f; 1.1 ' 1XAUPLS8 FOB FRAOTIOli. (1.) (2.) (3.) cwL qr. lb. oi. 18 3 17 10 6 Jb. ox. piet. gr. 82 8 17 12 8 lb 8 5 8 33 10 6 2 gr. U 11 113 2 6 U 261 11 13 3 427 10 2 14 (4.) (5.) (6.) mi. fur. rd. ft. U 6 36 14 9 A, R. p. aq.yd. 1 1 33 21 6 deg. mi. fur. 18 12 6 rd. 18 8 T. How mach cloth will it take for 8 suits of clothes, if^ach suit require Byd. \qr. 3fia. ? Ana. 61yd. 2qr. 8. A man gives each^of his 9 sons 234. 3R. l^p., what do they all receive? Arts. 2144. 3il. 12j?. 9. How long will it take a roan to saw eleven cords of wood, if it take him Sh. 45mtfi. 60aec., to saw 1 cord 7 10. If 1 share in a certain stock be valued at £13 89. 9^d., what is the value of 96 shares? Ana. £1290 4«. Od. 11. If afomily consume 12gal. 3qt. Ipt. of molasses in one week, what quantitj mil they consume in 1 year? 12. If a man be 2da. 6h. 17iltt». 19^ ''h^^^f1^'^, ^'k i^ /■>■> ^ H^LtlPLIOATIok BT ALIQUOT PABIS.^ ^^ 149 MULTIPLICATION OF COMPOUND NUMBERS "^ SOLVED BT A;.IQU0T PABTSJ TABLE OB ALIQUOT PABT8 (173). Parts of jEI. 68. 8(1.= i 68. =i 48. =i 3s.4d.=:X 28.6(1.= I 28. =^ l8.8d.=^ l8. 4cl.=iV Is. 3d.=:^^ Parts of a cwt. (1) of 1121b. 10s. = i 56 lb.= i ' "' "28 lb.= l 16 lb.= l 14 lb.= I 8 ib.=:^ 4 lb.=X 3ilb.=Vj 2 lb.=X Parts of lib. Avoirdupois. Parts of loz. Troy. 4oz. 2ox. loz. Parts of Is. Parts of a quarter of 281b. 6 d. 4 d. 3 d. 2 d. lid. 1 d. = i I — I = i =A 14 lb.= I 7 lb.= 1 4 lb.= 4 3ilb.= r lilb.=A Parts of lib. Troy. 6oz. =: A 4oz. = I 3oz. = [ 2oz. = I loz.lOpwt= I loz. =^ SpwtOgr.sii 4 •' 0"fc 3 " 8 2 "12 2" 0"=^ I /' 16 "=S Par^ofa year. 6 months 4 3 2 Parts of 1 acre. H 1 = i Parts of loz. Troy. lOpwt. Ogr. = i 6 '' 16" =i 2R. » I IR. =4 20per. s I I6per. =^ Parts of 1 rood. lOper. 8per. Parts of a month. 16 days = I 10 H 6 6. 3 2 1 i< (( It u it « = i =1 (1) The aliquot part! of tho short ton or new owt. of 1001b.-in tbis •nmple, toe pnoo bting farihing§, we maltipiy number by a penny; but, as id. is not an even part of a penny, we < it i||> id. and id. ; id.ia the half of a penny, and {d., the fourth of a the war of id. We then take the i of je3 18 8 for id., giriog forresall 2 d. = io{6d.; Ans. £58 3 = 4t tl 8Jrf. Araltsr — The price being pme* n,ni fnrtking§, we multiply the given num- ber by a shilling, mw, as 8^ d. is not an aliquot part of a shilling, we deoom- poie ft into 6d., 2d., and ic^., and then proceed as in the foregoing example. Ex. 3. Find the price of J^52 yards ef meriao, at 3». 9^d. per yd. OPEEATIOJT. " 262 yards at £1 = £252 3«. 4 d. = I of £1 |£42 = price at 3«, 08. 5 d.= i of 3». id. \ 6 6 0= " " 0».Oi^«, we multiply the given ..1.1 1 v_^ -- ij ! . decompose a penny, or . , . , •— , o B —^rosultjEl 19 4; then id., or i of id., that is, one half of £1 19 4 = 10«. ScT, which #e add to £1 19 4; the sum then gives £2 19 Q, for the answer. Ex. 2. What cost 16381b. of sugar, at 8id. per lb. ? * OPERATION. 16381b. at 1». = 1638*. = £81 18 6 d.=:iofl».; iof£81 18 = £40 19 = price of 16381b. at 6 d. iof£40 19 = £I3 13 0= " " " " 2 rf. iof£13 13 = £3 8 3= " " " " Jrf. tt "Jf!?" -,^,-3« iply the given we deoompose of a penny, or aBaU£ll9 4; iohWe add to MUfctlPLIOATIOK ■Bt ALIQUOT tARTS. a cwt. )0 L3 )3. )0^ " I6i Antwen. £ 9. d. r 1 M 51 6 H i:\ 8i 9, 10. II. 12. I.*;. 14. 15. 16. Jr. /8. 19. 20. il. 22. 23. 24. 25. 26. 27. 28.- 29. 30. 31. 32. 6728 6430 2436 2147 7028 2708 6491 4936 4967 2522 2897 7509 1870 2244 392 676 465 425 1349 7045 2426 1454 3632 X 6741 X X k X X X X X X X X X X X X X X X X X X X X ». .0 1 1 3 4 6 7 6 9 2 Anneirt, £ 8. d. 173 67 17 8 6 d. n 3 6i 3i= 3U 6 8i= 241 11 61 7i= 171 11 lOA 8|= 179 19 2 10^ 11 = 116 11 10|= 129 15 Hi 75 19 9 10 21 «. d. 4 10i 8 2^= 11 i= 6 7 248 10 868 14 9" =: lli= 103 8 9i = 7i = 111 8 = .382 7 61 84 6 4i 9 n 4 4 4i= 6i 7 = 6i= 892 1 2i 1740 6 863 13 8 91 £ 33. 1893 X 34. 604x0 36.2916x0 36. 6348 X 37.3720x0 10 38. 1509 X 14 39. 878x0 11 40.4571x0 13 41. 64 x 1 2 42. 62x17 43. 17x4 3 11 = 44. 24 x 3 13 5^= 45. 472 X 6 10 3^ 46. 1958 X 1 18 8 = 3785' 9 47.2471x6 14 9^=14179 18 48. 972x3 16 10 49.1077x7 12 3 = 8198 13 60.3714x2 13 115=10023 18 61.1415x4 11 lol ' 52.2150x9 16 Ii-=2108{J 8 53.2175x6 17 I0i=:1281& 1& 64.7251x8 7 1% :. 55. 6494 x 6 19 51=45288 17 66.7122x913 4|=68860 16 An»ieeri, £ s. d. 6i--^ 1960 15 6 = 1094 6 ^ 7i= 31I3 19 10J 9 = 61 8 6 4i 71 88 4 81 3 H 9 H H 9 242; Case IE.— When there is h fraction ia the iiven qaaa- tity. tJo!. Required the price of 1581 ywds of cloth, at JEI 2 II per yd. 2«. 6d. = £1 6d. = J of 2a. 6d. iofjei 2 11 |of lU. 5^d. OPERATIOir, 1681 yards, at £1 2 11 ^ - 19 15 = price of 168 yd.1a;t 2». 6rf. 3 6 10 = " " « " •* 0». ad. 11 6J= " " A " 6 81== If f X X X X X X £». 1 17 6 3 1 3 3 4 2 IS 16 7 14 4 6 d. '8 = 6 i, 10 2 s 2 9 n 8 a 9i = , 18 lOA 9 4a 3 18 9 a 2 7 6 3 15 10 s 11 8 s 7 6 3| 1 10 10 =, 4 19 11|=, 2 16 91 9 11 6 = 8 19 lOi = Ans. Ans. Ana. Ans. . Alia. Ans. Ans. Ans. Ans. At^. £ 353 106 r. 2 16 3650 6 917 11 125 1630 14 13 7 12 1272 2^9 ^54 16 103 2 Ans. 369 12 Ans. 1877 7 d. 6 104 2 9 H 9 6 1 ' H Ans. HI6 16 lOi Ans. 6736 6 9^ ^£r^Wh«taih.co.t ofMcrt. 2,r. 15(6. oflobacoo, at « 12 6 OPEBATIOW. IO».Od. ^.£4. 2».6rf. = JofI0«. lJ/6. «iof2or. fiifc-iofioA. x^6 « £470 = cost of 94cwL at £5 47 11 2 » 15 « 16 3 » 11 3 » 6 71 = 2qr. 10/6. 6/6. . ^il#....£632 8 I J= cost required >KOTHgR MBTHOD. iUlo; «^ _^fiji«=008t0flct£*. fL VU^ = ^^28 16 - cost of 94cutf. '« 10«. " 2». 6( 11 II II U 2 18 at £6 12 6 perowt. 2 9 ! 7 91 I 12 9 t 16 6 1 2 1 per ( a « u <( I (f II ( u-"^ It It u U ll' , «• DIVISION OF COMPODND NUMBBR8, 1S8 1. 2. 3. 4. 129ctt'/ 6. 144cwt. 6. 168cw^ 7. 2S5cwt: _^., 8. 346ctt»/. ]yr, 9. iSlcw/. 3qr EXAMPLES FOR PnAOTIOE. 65cwt. 2qr. lib. at £0 17 78ctt>/. 2qr. 2ilb. at £4 14 I9cu)t. Bqr. Ulh. at £4 11 Ulb. at £2 I'i H/6. at £4 6 17/6. at £2 15 7/6. at£l 18 10 4/6. at £1 J2 7 53/6. at £2 1 3 4 \qr. ^qr. \qr. 3qr. per cwt. << « « II « ^IM. £ 74 3 4»«. £ 372 11 Ans. Ans. Ans. Aha. £339 14 £ 620 6 £ 554 19 £ 564 3 10. 175/on«. iHcw/. l<7r. at £38 13 per ton Ana. £RT9 - V 3 7 XI tient wiaiog bjr the otkw. the -..fa; nT^i^ ... ' *i'»' i ^ mJ^ .... V^ '?>i^ -Li^' /'■ 154^ 'BIVIfllPM Of OOUPOnMB NUMBIES, iKr. 3. Divide £360 8 4 by 173. (£i OPXBATIOH. £ 0. d. 173)360 8 4 14 _20 ' - ' 173) 288 (1#. ' 173 116* 173 ) 1384 ( 8rf>Vf 1384 A»iLT8M.^We divide the ponnda hw »7». •nd obtain £t for the qaotieot, and £14 remain- ing, which we redooa to (hillings, and add the B». ; and again, divide by 173, and obtain l». for the qaotieot The remainder, 116«., we reduce to m^, and add the Ad., and again divide by 173, and obtain 8d. for the quotient. Thus, the method i» the same as by general rule (244). By unittog the several quotienU, we obtwi »2 1 8, for the answer. Ex. 4. Divide £24 3 9 by £3 fij. OPBBATIOV. £24 3g. 8rf. 0/ar. 23216/ ar. « 3 o«. 6d. 2/ar. "** 2902 /or. ~ ^* AiTiLTsn — ^Reducing both dividend and divisor t« the lowest denomination mention- ed in'eitber, and then divid- ing as in simple anmben, we have 8 for the quotient (1.) T. cut. Ml 7) 4g ih 26 J 10 76 ■ZAMPLI8 FOR PaAOTIOB. (2.) lb. oz. dr. 9) 143 6 6 16 14 13 (3.) hhd.gal.qt.pt. 12 ) 9 28 2 49 2 1 1 iL*„f "" ^ ™*°*** *™''®^ ^**«»'- ^A*"- how fer does he go in I' if ?fi r*?* f ''*'*?' "^i ^}^J 3' 'h** " the price of.l yaHT 7.Z15J280 61' 27.766" by 2.764. ^~- ^^^ ^^2^42?^ J. Divide 12764. 2Al6jwrr22yrf. Bft. 32t«. equaV LfongM „i; '* i!l u f?"^ of a certain etock are valued at £1290 4». erf.. whftt would bo the cost of 1 sh ve ? ' ,■ 10. If a towft 4 rtile^ square be divided equally into 124 farms, how how much will each farm contain? Aria S2A^R\^^^ a. Divide 675r. l9c«tf.42/6 140Z. by 12^* ^" *^^^''*^' 12. If a man walk round the earth id 2yr. 68uWi ittaks hin^tewidfc t ti^ree, »iJowtm 13. Divide »16to». 3/i»f. SOrrf. 10/?. Sin. by 47. 14.. How many timea are £6 10 10 oontaiiicd in £637 10 .10 ? I9h.5i min., h ow 366J~day8 to ft~ 17mi'n. Idaet. t -Mi^^^r V'-i'f*'- <- ^ "Afl-.l^ *H^l - *.St«> iC -^ ,-K 1 « IiONCtlTUDX AND TDO. 151^ 16. Divide 12fllb 3S 2S Is 4gr. by 13|f i«I. V^ merchant sold to each of a certain number of farmers 6bu. ipk. iqt. of grass seed, and to them all he sold 716». bat. How many lanwera were there ? ^ ^^ 1 1 "^ 18. Divide 3794CU. j«i. 2001./?. tOHcii. tn. by 33i. * ' LONGITUDE AND TIME. A ' ' 846. Menrftan* of Longitude are direct liqes on the globe, from the north pole to the south pole, crossing the equator at right angles. a47. Longitude is distance on the globe, east or west of a deter- mined mendian. In the British Isles and 6n this continent, also gen- erally on the ocean, the meridian of Greenwich Obeer?atory^EngJand 18 the determined mendian. All parts of the earth ou thi Mne are considered to have no longitude. «,TflnJ"^''!^* longitude any place on the earth oaDhareis 180«eaBt. or 180° west from the determined meridian. ^f^; T^^ Equator and parallels of latitude beioR elides, are divided into 360°, called degrees of ion^iftKfe. N0TM.-I. The earth revolves on id uii flrom wut to Mit onea in 24 hours. w^Ch ooMtituto a solar day. The middle of this day is IS noon: When the ioq te^ireotly over the meridiao of a plaoe. it is noon at that pUoe. and at plaoes wtet of this meridiaa the time is before noon j at those east, the Umb to aOv noon. Ji^The whole oirole of the earth - SW> wUeh pass «ad«r the en M Mhonn. llmmutepassesj^ofl^t^iso^ia.iy. Onemtairte - M ,«H^d., h|noe, in.l seconS pwse. ^ on»' =. g' - j- , w. Henee, theMowii« * COMPARISON OV LOMOITUDl AND TIMI. 16** of longitude 16' of longitude 16" of longitude 1 hour o^ time. 1 minute of time. } seoMtdoftime. a49. Bulb.— I. The difference o/lotuitude between two^Iaeee, ^prewid %n degree; niinutee, and ueonJe, divided bu 16 wOigitfi their difference m t^nu expreeeed in hm$, minuUe, and eeooiub, II. The difference o/Hme in two plaeee, expre$$ed in houre, ?*!"!?f"l*'"'^'^"'"^> «y^P««^ fty IS Witt give their d^eren ct ii iongitud0expreMed^indegree»,in^ium-mamn^ J-rflhZilf'n^JtV''?.!* ^^if^ •"* *• •*« «■ w^ loBgltad^ the differ. K£'ffil£,ffil&^ »^' «*ictb. sumSISJ^th-.iao'S: » i!VA t:j.hl. JA «■ >vf^ 106 DUOOIOPIALS. «.?1.5^ "?'"' appMn to more from eut to wort, when U is oxaolly 12 oolook at one place, it wUl be <»•( IJ o'elook at all placei eaet, and Mitr, 13 at r P'"?«" »•'*• Bono®' «f 1»« differenoe of time between two plaeei, bo lubtraettd !romtlioHmeBUha«».«eWyplaoe,therMultwiU be the time at the westerly £??t' ^\^^ *' differenoe U added to JUm time at the wetteHg piaoo, the nuuU wiU be the tune at the easterly plaoe. / .. XXAJIPLK8 l^B PRAOTIOI. I. Quebec is in longitude 11^ 16' west, and Toronto, 19" 21' When lids 12 o'clock at Toronto, what is the time at Quebec ? west. 15) 0?IRATI0V« 79" 21' 71" 16' 8° 6' Oh. 3Zmi. 20«ec 12 i2A. 32mt. 20 6*. DiTiding by 15 and changing to time erea 33m«. 20we. for the difference of time itween tbe two places ; and, as Quebec is east of Toronto, the time is later, and we add the difference of time, which gires 12A. S2mi. 20tM. the time at Qaebeo. 2 J The longitude of Haliikz is 63° 36' 30" west, and that of Ottawa 18 7A*> 41' west; when it is 10 o'clock 12mtn. A. M. in Halifax, whs " \) time IS it at Ottawa? 3. The lonritude of Valparaiso is 71« 37' west, and the longitude of fiome IS 20^ 30' east; when it is 11 o'clock 15«tn. A. M. at Va YO- iO' west: what is tl iOacc^t Philadelphia? Atu. 1h. 20mtn. 62»ec. 6. When it is noon at St. Paul's, Minnesota, longitude 93o o* west, jtiiratBan^rlA. 37JIIMI. 12see. P. M.j what is the longilude of B»?g™; Maine? ^^ , iln». 68" 47^ west. 6. The longitude of Jerusalem n SS* 32' east, and the longitude of Montrei^ 73o 26' wedtj when it is 10 0*01. A. M. at Jerusalem, what time 18 It at Montreal ? Ah$. 2A. Umin. Usee. A. M. 7. The longitude of Boston is 710 4' 9" west, and when it is 10 o clock A. M. in Boston, it is 8 o'clock 63mtn. 57|#«;. in Chicago • what is ihe loagitode of Ohicago? Ans. 87° 34' 46^ 8. The longitude off Constantinople is 28«» 48' east, and of Kingston. Canada, 76o 41' west: when it is 3 o'cl. P. M. at the latter place, what time is it at the former? An». 9A. 67TOin. 56sec P. M. 9. A captain at sea finds by his chronometer that it is $k. 40mtn. ap$ec. P. M., at Greenwich, when it is Ih. 10mm. 46»ec. by solar time on board hia^veawl) in wbat longitude is the vessel? .4iM. 370 26' 15" west. DUODEQIMALS. 9S4^. Dnodeoiaialt «re deoominate numbers, the denomina* iiotu of wMdi ineveaM aoeordii^ to die $edle of 12; or denom- ■Aj.JS».J y v/^tir^U -,if«r' S'»^\'7 is oxaolly 12 id be/ore 13 at I, bo Mibfraeted the westerly Imo, tho rasuit 9" 21' west, bee? >flongitadeia inging to tima renco of tim« as Qaebeo ia ir, and we add h girea 12ft. 9. kt of Ottawa^ ilifox, wh^ 9 longitude L at Valpa- t 5 P. M. liladelpbia, cl. 20mtft. n. 52ffec. 3° 5^ west, >nei4ude of 17^ west, ongitude of ilem, what c. A. M. en it is 10 Chicago: 34' 46^ 'Kingston, Iter place, c.?. M. i. 40mt». by solar >" west. lenomma* r denom- MVLTIPUOJlTIOM OT DUODIODfALS. ■% ' • 157 inftte fraetipDB, whose deDominatora are 1, 12, 144, 1728, etc. la praotifle, ^QOctcoimBls ve applied to the measuremeat of extension, the foot bong taken as the unit. TABLX. 12 fonrths, barked (""), make 1 third, marked 1'" 12 thirds *' 1 second, « 1" 12 seconds " 1 prime, or inch, " 1' 12 primes, or inches, " 1 foot, « ft. ^-oriBATiov. 9ft. r 1ft. y 7/1. 2' 3" The marks ', ", '", •'", are Galled indices. SUIfl. JDnodeoimals are added and subtracted in the s^e manner aa compound numbers. MULTIPLIGATION OF DUODECIMALS. Ex. How many square feet in a floor 9/Jf.* V long and Ift. 9' wide? > AVAITUS.— Beginning atthe riefat, 7' X 8' « \ 68" — 6' S" : writing the Z» one pUoe to tlie right,, V we reeerre the 6' to be added to the next produoC Then, »/(. X y + 6' - M' =. Ifi. %', wLh we write in the plaees of feet and primes: Dext, muN i ' wjftfa* the I' in the plaee of primae, we resell the 4/1. to be added to the next prcduot Then, 9ft. ^ 2^+.^-,-= •V*'" ^^^^ w« writ* in theptooe of feet. Adding the partial piodaots. we'^ave TV*. 8' »- fte the ptodnetTaqoired. Henee. the ^ T 35SI. lUru.— I. Write the teveriU temu o/themultiplier\ der the eotretponding ternu of the muhiplicand. IL MuUipljf each term t/the nwitipUcand hy each term of t^^ multiplier teparatefy, h^fimring with the hwett denomination in the multiplicand, and the highest inthe multiplier, and torite the first figuri of each partial product one pkue to the r^ht of that of the preceding procbict, under its corresponding denominittion, car- tying I for every 12. III. Finalfy, add ih§ several partial products; iheir sum voUl he tA« re^piired answer. 61Ji. V UJt, 3' 3" iTAin»T . iB roa paaohoi. 1. How many square feet in a ineoe of marble 12/t. T 1 ft. S'^wideT _^ Ant -2. wiiat isThe aam entHdof,^ lEeTen^ of which -^2. wiiat isThe tarn entHdof,^ lEeTenaii of whiiK iS M and width 3/?. r t iliw. 34/?. IQ' 11" 6"' and W 11% , J*-?**y."*"7"9°*^*«*»° 1® boMd^ elujh lift. W Idtog and ,^y?.8'wid»T Ans. SUft.wl*: < ^i0:m£k,^ii '■ * \, .,;f&.'«:*M'iv* ).. iMk",-* .w . ' r ^a*^a'\■,> I, ti 168 I! DIYKIOW OF DtrODtOIMAM. ein. hich> *" «"/'• «"•• will", wilh a oloM ffcnce r/(. 6. What will thep?Mterin, of. mom oott. .1 is'^^i"'-''- *'• ^/ * 4n«. $38,667 DIVISION OP DUODECIMALS. OPKEATIMT. 11' 3" 11' 3" SA^aUmM. MulUplytog the whole prodnot which wesabtraAtfrom the MiTMpoadingdanominstioiu ««^ 1. Divide 184ft. 3' by 40/». 11' 4". 2. Divide 41/1. 8' 7" 6"'^by iff. 4'. 3. A table whose leneth is 6fi. 9» 7". 3' ■XAMPLKS lOB PAAOTIOi. "^ ^ Am. 4/^.6'. rAne.5/1.7* 6'' ll"2"';-wbatleTte"wilAr''-'*' ' ' ' ^ " *"' i' '%-1. J w^tt /2A vW ^'"^ ""^^ ^^^ •- ^r^«i f • r ', 7, A8tickoftimberi8 3/>. 2'wide 2« 11' twt* IVi * 1 bu.ld.ng. The height wa. 11/i. 9in. ; what waa the lengUi^fSe '"^*^ ^--- ilii#.%7 feet. ^t8aMIM.JnOVB BXAHPLIS. mSCBLLANEOUS EXAMPLES. 159 ioh ^S? " "^"^ ^ ^"«''' ^' ^^° 2 ^^^i* ^»»*t " the cost f What ia ths/value of 16c«*. Stt. 14/6. of tea, atl^SoliAwi. T DO tne exnense of makinor A fii«innit<^ (>>rJLv u^..^ « wi: ; ^ , . "' ^"cw** OTT. i4to. or tea, at f950 mr cwt i IW^isT^^ the expense of making a tu7njn^78iSu. Vfur. 9 f^r ^r--^'*;-."*- °f ^*^ *° *h« decimTof a-C^gS:" - »^- 2**^^* ^J^ AvoWupoiB weight, 16/6. opium,'S S . a dram Jt Iwi *^' '^"' by Apothecariet' weight, in^ dos'es onOir?. S at K ote. per doee ; how mnch did I gain ? An» tefri 40 ' U J;!l!|*P*rt of 44-0/. 3,/. is 2qt. lft.2gi.1 A^u'. DletL iS^LToTSf-^^^'S:!"' ?'^'- ?A""-^2rrf. of road j afterVom- pietiqg 4 of It, i of the number of men left. What distance did each man oonstnict before and after i of the men left ? °""°^* *"** •^^ i,«t;l«- v^ ^«^ ZOmtn. f# a man to cut 1 coid of wood howmany daya of 8 hour»each wffl be required to cut 746 ^S .iv leetr Ans 311^ lik in«ii* wii*.L,nJl!!^»r*'"*' ?fu *"^x^H°i!r°P«'*^ *^ the valueofje9 12*91, tJL£!'*Hti?^.?*'<''*°**™*«^ •»«h containing 2i6i». at $8 7W&. and Bag. mote than lWwa«!. How much did <^htS?I?^ » Amkn l,f!SA^l*^i^$*^' ^ ^?3J'*' *nutfd, S^ riven for >l?'-r?5'-<^JP*»^«««*!L»*37icent8agXn? ' * ^ ^ **s.-rhe wall of a cellar is 20 feet musM on tii« .'n.i.ii> q •»?»-OTnwasBe8, at 37i cents a gallon? m aad irfclir?l.i!*t.^'l^ ^** *** "^"T °" *^« ••»J<»«> 8 feet hid,, ana h Mt in (hrokneas; how many perohea of maaoniyare thertV ^S > ^ ^A^T> - '^A-^Jit^lU^, "•r^-il J.. ,^ 150 MISCBCLANEOUS EXAMPLBH. 24. Sod \5ctvt. 22/6. of rice at $3.76 a cwt dnd rS ii r s^'ite .1 r J X. - =°'' ■"-■' -"'^ "S 1^- Ko''A''Mrf? l»5"«;'»df b.l.«n London antst tti^ 36. Express an acrea and the decimal of an acre the m^'^J'^q square Ota, each measuring 5rd. 6ft. tin. oul side! *^ °^ ^® on\'n,^e;^e*^i^r;nife^ ?r :-r^,^^ buildingsoccupying remain^^occ^ied ? '^^ ^' \^\f^^Z% fo*« .»«'«V 38. Bed„ce*5iol4y/6. totheftm^ofk^hS^^^^^ in 'SJ^.^?''", '^**^ in Montreal until he was 18yr. 8mo. 24e\H»6-cent loaf weighs f.n« o*. both jlz ifou"^^^,. ?'"* -■" ■' z ^av»» inVuToYtir^-''^^^*^^^^^ • c««e. How many lotewnitWeS? ^^ ''^ '^* ^'^^l"' P^j^le 68. Paid 3 debt* successively; each of which jjjk' ??S '^**"- r added to it thJt I mS sell it at 6^ olJJ^' how much water must be the sale of it? ^ ^" °*°*' P*' 8»^'o°' and gain «16 iu Ant.sogal. 4 .Ki- 5 r'«f i62 SIIBPIIXANEOUS EXAMPLES. 60. Sold 125 equal loada of wood, measuring 115Crf. ^cd.ft. 1fitt./t. for $492.50. What is the quantity pet load, and price per cord ? Ana. USlcu.Jl. each load, $4.26j per cord. " ^1. How many francs must a merchant in Paris send to Montreal in payment fpi a de'bt of $15989.862 T 62. If a mati fill ^ of a cask with brandy. \ with wine, and I with -water, and if it lack 211 gallons of being full, how many gallons will tliatcask contain? An$. lOOgal. 63. If by selling cloth at 10«. 6(2., \ of the price is gain, what part of the cost would be gained by selling it at 13s. ? 64v A ship's chrQiK>meter, set at Greenwich, points to 6/i. 45min. 24wc. P. M., when the sun is On the meridian. What is the ship's longitude ? Ana. 86° 21' E. 65. A grocer boqgjit 15 barrels of salt, of 4 bushels each, at f 1| a barrel, and retailed it at '| of a cent a pint How much was his whole gain ? Ant. $4.60. 66. James owns A of a field, and Leo the remsunder; | of the difference between their 'shares is 5 A. 3R. \6{per. What is Leo's share? Ans. 20 A. 3 R..9lper. 67. A gentleman desirous of giving Is. 6d. apiece to some needy boys, found that he had not money enough in'ihis pocket by 6d. ; he therefore gave them U. 4d., and had 9d. left. Required the number of boys. ^nf. 7. ' / 68. A liquor agent has 50 gallons of wine of superior qnali^, worth $7.60 a gallon ; he wishes to reduce its quality by the addition of ^ater, so that he may sell it at $6.25 a gallon. How much water must be add 7 An$. 2l^gcU. 69. A clothier has 960 soldiers' coats to make, each coat contain- ing 2^^. of cloth llyd. wide, and lined with drilling }yd. wide. How many yards of lining will be required 7 70. A ship captain, sailing from London to Portland, found, on taking^an observation, that the sun at noon was Sh, 25mtn. AO$ec. earlier than the London time, t» shown by his chronometer. How many de^jrees west had he sailed ? 71. My father's garden is 10} rods long, and 8) rods wide, and 8u^ roundeil by a fence 7j( feet high ; he haslaid out a walk around it, within the fence, 7^ feet wide on the two sides, and 5} feet wide on the ends. How nauch remains for cultivation? Atu. 21296*9. yt. 72. A boy having been sent to a store with 5 ^ do2. of^gs, was directed to purchase with them equal quantities of sugar, coflTee, butter and tea; he disposed of hie ^s at the rate of 2 for 6 cents, and piUd for the articles purchased n, 28, 371 and 137} centa per pound, xespeclively. What amount of each did he purchase T cord? per cord. o Montreal ind \ with alloDS will lOOgal. what part ah. 46nitn. the ship's » 21' B. h, atfl^a 8 his whole 8. $4.60. ; I of the it is Leo's ame needy by 6d. ; lie le number Anf. 7. ility, worth idditibn of uch water 21fga/, It contain- vide. How found, on tn. 40«ec. ter. How e, and 8u^ around it| ride on the Waq.Jl. tgga, was ur, coffee, r cents, oeota per • T BATIO. RATIO. 163 254. Ratio is that relation between two numbers or quan' tiities, which is expressed by the quot^nt arising from the di^sion of the one by the other. Thus, the rmo ofl2to4isl2H-4 = 3. 255. The Temift of a ratio are the two numbers compared. ftSB. A Couplet is the two terms of a ratio taken togethop. 257. The Antecedent is the first term, or dividend. 25S. The Consequent is the second term, or divisor. 259. A ratio may be expressed either by two dots (:) between the terms ; or in the form of a fraction, by making the antecedent the numerator and the consequent the denominator. Thus the ratio of 8 to 4, may be expressed as 8 : 4, or as |. 260. A ratio b either direct or inverse. 261. A Direct Ratio is the quotient of tho antecedent by tiie consequent Thus, 8 to 4 is | or 2. 262. An Inverse, or Reciprocal Ratio, is the quotient of the consequent by the antecedent. Thus, 8 to 4 is | or ^. 268. A Simple Ratio is that having but one antecedent and one consequent; it may be either direct or inverse. Thus, 6 : 3 or i : J. ' 264. A Compound Ratio is the product of two or more ratips. Thus, the ratio compounded of 6 : 3 and 8 : 4 is ft v * = H = 4,or6x8:Sx4 = 4. ^* 265. From the foregoing we deduce the following principles of ratio. 1st. Multiplying the eotueguent divides the ratio; dividing the eoneequent multiplies the ratio. 2nd. Multiplying the antecedent multiplies the ratio; dividiM the antecedent divides the ratio. ^ * 3rd. Multiplying or dividing both antecedent aud consequent by the same number does not cuter the ratio. IXAMPLKS FOa PRAOTIOl.' What is the direct ratio of 1. 2. 3. 4. 6. 54tO 6T 108 to 18 7 7 to 21 1 lTto68» 60 to 12^7 1 t w ',Jl- jt,i*A. . ' 1. Ans. 9. An$. ^. 6. 13 to S27 7. 63 to 212 7 8. 12yd.io9yd.1 9. 6Qmi. to4yiir.7 10. 3qt. to 20gal. 7 'ViVin' An$. {. 4M.19Q. 164 PROPORTION. Required the inverse ratio of 11.27 to 81. Ans.3. 12. 72 to 8. 13. 16 to 48. 14. 42 to 6. 16. .02 to 2.503. 16. 256 to 32. Afu. |. lo* wu'^^ '^*^® greater, the ratio of 86 to 240, or of 45 to* 72 ? to 19 ? *^- '^^^° compounded of 35 to 40, 60 to 75, and 21 19. If the consequent be 32 and the ratio 4* what is the ante- Ans. 7 20. If the antecedent be 7i and the ratio f, what is the conse- *1"^°*^ Atm^U, PROPORTION. 266. Proportion is the equality of ratios. It is indicated thus, 6 : 3 : : 8 : 4; or thus, 6 :'3 = 8 : 4, and is read 6 is to 3 as 8 18 to 4 ; or the rdtio of fr to 3 = the ratio of 8 to 4, Hence every proportion has two couplets and four terms. o2^" '^^^ Extremes are the first and fourth terms. ZSr' '^•^^ Means are the 8eoQn<} and third terms. 2»». Since in a proportion, the ratio of the first to the second . term is equal to the ratwijf the third Xo the fourth term, the pro- portion 6 : 3 : : 8 : 4, becomes | = |, multiplying each member by 3 and 4, wo have 6X4 = 8x3. Hence, In every proportion/ the product of the means is equal to tho product of the extremes. 270. From the foregoing principles and illustrations, it fol- lows that, any three terras of a proportion being given, the fourth may readily bo found by the following ari. Rdlb.— I. Divide the product of the esitremes btf one of the meant, and the quotient will be the other mean. Or, II. Divide the prod^ct of the meant hy one of the extremes, and the quotient will be the otKer extreme. m NoTi.— We will denote the reqniied term of a proportion by the letter x. KXAUPLS8 FOB PRACTIOB. 1. Find the value of a? in the proportion, 9 : 16 :: 36 : x: x=: 16 X 36 9 = 64, Ant. '■\ i\ 1. 24 : 96 2. 7 : 42 3. 16 : a: 4.'42:70 What is the v al ue of a? in each o f the following proporiiona ; "^s 14:ar? : jr:96? 10 :40? : 9:xl a- Ant. 66. Ant. 16. Ant. 64. 6. 974110 :: 36&U. : xbu.l 7. 2yd. : 8yd. ::$^:x^l 8. 7.60: la :: xox.: I^ox.1 ^\ '^'scm and , SIMPLE PBOPORTION, I85 'simple proportion. ?lLl%ufS"°^ " '"^ eciHalityof twoBimplo ratios, SnittteSr ?:!?fe^ from thaoiroau.- ' '•'°" being gJTOn to find a fourth fLl\^ie'? '"'^' **' '''''' «««' *30, what will 42 yards cost at yd. 12 OPKBATIOir. 42 :: ^30 : ^2 the^Ir Jr~^'*5"'*«'' *''« K''^"" numbers in n?!v?«n 5u* P'"?P«''t'<>n, or ttate the question, we kind M the required /ottrt* term: and, aa Mm 2 thaiX^fv^^ ?"'"'"°° *'"' »atter'mu8t*rgre.U «th«, 1 ' *'"r* **'u"'' *« ""»''« 'he greater of th« -?H"j V ' J?^ ^"°' *^® prodnotof the meua di- SS?m^ th, given tx£Z. giva. the le^ SAMB BXAMPLB BT ANALT81S. If 12yd. cost $30, ryard will cost JL of S30 — 112 fin . fk«» •<• 1 *j o«.2.6o, 42,d. ,m 5»t 42 «™ fe/.'L" ?.Jvl t:;!S,"ii Ex. 2. If 49 soldiers coil day^ hew loog will it take ' 12) 1260 3! = tlii5,Ai Soliien. 70 OPBRATIOir. Soldiers. : 49 di/g. 1^8 7 14 *6 19f, Ant. *r lime a certain qoanWty of floor in 28 l^Boldiers to consume it ? \ . 3"*''''^™'— S'no* th* required <»^- Sir»I!"Jl*^!' ^^ ?■''<' the given Of^" ^n **•''* ""•"• Then, w the >•?: *?" n?t >Mt 70 .oldiers '«> long M>? w.1149 aojdiere. we make 4»«ol' tor>,4heimallerof tbv two termf. Oie •fc^lifljterm, and 70 Mlitea the firm term j"»Bd proceed aa in the firat •*'?P?' «<»P*^that we ahortan the work by eancellatlvn. THB sImb BXAMPLB BT ANALYSW. If 49 BoldiefB consume the floujt in 28 dare, it willtftk* 1 «aM!«^o 1372 days, 70 soldiers will consume iHn ^.of 1372 days = 19|diytl a7». RULI.— I. WriU the tm given numbers, which are of tTtK^''^^^"^*^ ^^rtd fourth te^, oVZuI 'the vthtrtwo opor ^„^. ^^nmiien, write the larger for tHie MgMmt third tenn; but write the UuJ^ the second UrZ andl^kira^ for thej^t, ^ the anewer should be Uu than ^ Ivrf S C *. ■'#t !. ^ik-V^^^^; A«,., t . -A- ' /iS' ' ' h tm SIMPLE PROPOB^ON. III. Multiply' tU second and third term together, and d !„{dg their product by the first,; or divide the third <*' «»* V 6«- per oo T iln» jBI 18 3 J^Eji^'ii"S'"°'?£? »'''*"* ^ •"■lUiediolai.Joutoi bow .Tilte^' """"^ '"•-*«' •""nS?"- > '/t -'"•' Jrrt?*^^ knowing that t^e aeoondhiJTIjrd. ionger than the q^ W«-wJji. -nsx/L . iliw. iBb e7yd. ; 2nd. 78y(J. «fbi52;j!tei'i.S:s?,'^'* b«* of w«i, i,^ 4.,. ||V»ko«*U»M«i*hri».Mr«llt(on Ouli' I Aoald'wJJirM,. r i TiMT B H i f rtlTIn i l i <^« o f ««»« M l i^a^ i iiM t l 11 It I - — #*=l^ I- ' /' 5* ► « ,f/ 168 everj^lOtfJ hoWPuch dllpsETSr^ ?^f ^ lo« f 4.60 on ♦ ' f ^^li^^'l "^^nwnoSK^te tl.lO ; . forlfiwir mncb should I re- M »**#lft at tlw^tef »0 00 every'flQOO? ,, , An$. fl.lSA. ^toetallio WBjwe 6i cte. a^doien, b^ touch will 101 i35!»f^^f^*"''^'L!S8^''^'*>t»>» how many ^& ?"' f ^^l*^5,i%'<™^«ei%^ the smaller ;^ 48. Twopi«5eBofcr6th«irere8pe«tively41aDd36 yards; thelflrst piece costs $46 more than the second j wquired' the price o/eaol^y 4^t^henwheat f^^^.^^'^^i^^^^ wMrfis 9 ounces; v^af ftould be the weight if whekt is but 68. the bushel? ^ '^- j^^ lllo» . .^ 4tt. To draw success on % business, I propose •toirfre' tfi to tha 63. A &ther earns 63. $d. per di^, biasba, 3ft tuBBI^E^A i. ^. jnUtheyha^eeoonomi^dXl 108, ifthVei^JTw J4^^fiowmugiBriurtI^^^ i^-^igp^ ^ , W444A, wide^ rf 1426flq. ft. cost |34i ? >I^*? ™^ - 6«., -^ro ghngs coAfpoeea of ^Oaiii«o rae^^^ yank^ofk oertaih WtoVt in 26 (toys | how. rnqphwoSwrtbeyWdone had thei^ number been augmented by 16 r^^ Af». ISfioJawlu V ^-i^- O^e I'U'xired degrees of 0«tttigr»ie i|» aq^valeaifesd degrees ,^^**?55" 5 to hdw m*ny degrees of. Bd^iJ*«i|] 231 4|«eSTf .•^C5entigNdiiei|Ml*v,:^. ' V-*|^$^;2 ^' ■. ..:iiili Jj8||?^0fi|^^ ■fe «.,v/ COMPOJp) PROPORTlb^: " IW I I \ ftiZi\'^i\ •~r7 L Jl^:' \v t; li74« OoHnpoiiBd ■:D6tw«tti a'eoihpottnd ' niuMi Thus, "^ »le ratifl^ or betHMa two •..'■■■ y» i^ ' .. COMPOUND PBOJ^BTION. , 169 -£iT''~^*""P'"'^ l^roportion embraces tbat class of qnestions whose solutiim cTedffi[:irr.>rr''''^^*'"'' '° ^r^ proAS itt "moS^. how. many dollara caa 9 men ear^n 6 days, by working 12 hours a No«.~To aid in remembering the question and in forming tlie ratios, the ;^pil should write the conditions upon hiq slate, or blackboard. As in the margin. STATEMENT. Hen. $. Da. 6 72 10 9 X 6 Hr. 8 12 METHOD BY PBOPOBTIO«r. '* OPBRATIOir I $: 9 : 9 ) 9 :!«> ■•' oPBiuMojf n. '■ ' % 9 , ... ^ ■■ METHOD BT"XWALT8I8. -^ If 6 men in 10 days of 8 hours each earn $76, 1 maq in the same timt wrll earn \ of $7^^ f 12 ; and 9 men will earn 9 x $12 = $108. If m 10 days of 8 h. Bulb.— I. Mahe that number toUch i$ ofthiaamt kind 08 tht antwer required, the third term of a pr&fiortioiu '^ll. Then taTce tlie other nimhers in pairs, or tioo of a kind ana arrange them at in airnple proportion. * *Ji-^'*j5'>%'^'^'^.*^^*'"^ o/t^'^condterm, by the third, and dtvtV&ty^f^bg the product of the first term. The quotient wiU be^'/fhttikidttiif or anavoer. f*«v IZAMPLES FOft PBAOTIOI. • \' VWA^^ oan ploagh U acres of land in 6 days: hownmay M irill ft require to plough 33 aOjres in 18 days ? Ana. 10. r?iL^^''^"*w'' '•* » wontbfl, gOiat siim will $450 produce ft ^ffff^mj K LJor^ 15 JJ^^RJrork of X horses, each drawii^ ««^JS*f^i^f?^P'*P?'^'»'^' bowy&ucbTshould bT'l^S 4. % iwJliBg fo otter ski profit of $24 } how mii^^ would F V iiginia sUyeced fox nkfos, efteh drawing with a powei^ of " ^n«. $27. -cost me $3.60 each, I made » lave guned in proportion on 40 ►^TTTOtoch? iii#.^30.80. ims " '' . ., *«**.. 170 OOHPOTTND PHOPORTION. „J-V}^ ™«°» in S 23. A piece of cloth 3Qyd, long, | of a yard wide, was 26 lb. of thread j what will be the* length of a piece | oft •Biiiriliidi^K^re8 32^Ibr;i0f^feidf Ana,"' was woven with fa yard wide^. 2^ btf th signii etc, 2-: comp' ft"* thebi = $1 a: of the . ^h Th Th ihe P Th iheD 2j< or ac .„M. ■9- * 200ft, long, ach, will 30 ;k? how many Ans. 40. can make ours a day Ana. 9. .ny can be Ans. 8. f potatoes, 44 days. nen during e increased »6050 lb. ployed at a ds will 36 .238yd. vill be nee- 5k? days, after rork? ere obliged >do4l|yd, 12 men. Dg 8ft. ia id for the ig 2 doors $14.40. tain debt, bt 3 times lushels. its 124.68, f 64 miles .135 + . ers for 6 lily nation iMBOlNTAOB. 171 PEBCBNTAGB. 276. Per Cent, or Rate pet Gent, also written ^, signifies ht/ the hundred. Thus, 6% means 6 of every hundred, and m%y signifj^ 6 cents of every 100 -cents, 6 dollars of every 100 dollars, etc. 2TT. The Base, is the number on whioh the percentage is computed. 27S. Percentage is the reqnired number of hundredths of the base. Thus, the percentage of $200, at 5^ is t^ of $200 = $10i 279. The Amount or Difference is the sum or differenee of the bai^e and percentage. , Hence, ^ ^ The Amount = the Base -^ the Percentage. ' • The Difference = the Base — the Percentage. The Base = the Amouni — the Percentage, or the Difference -\- the Percentage. * The Percentage = the Amount — the Base, or the Base — tJ^e Difference. . ^^ 280. The rate per cen|* may be expressed eithi^y a decimal or a comrnon fraction, as shown in the following M Symbols. 1% of a number is ^% 6% TABLB. Deoimab. Common frMtloni. II lOOfo 125^ <( « <( (( ti « (< « « tt n of it (( « ,11 tt tt It .01 .02 .04 .05 .06 .08 .10 .18 .76 UOO 1.25 .005 .0076 " « .075 " " « II II It II 11 (( 11 ti (< (( « It It II u 2S1. Cask I. — Oiven, the htue and rate, tojmdthejpereentage. £Jar. What is 6^ of 512 yards of cloth? AiTALTsis,— 6% i= .06. Thewftwe, £« tiM2y 4.^ Mot &ir m , 3 0. 72yd; - ~ OPSRATIOV. — 512 -J^^ , ^ Or,6«=JL. Th«refoM.6«of6Uyd. 30.7 2yd. Ans. U ^ of 613 jSOb =» S0.72ycl. Or, ^x^»^i=30.U7d. Ans. mSH/t - !jf-.(t h'M'^ >£;■ ■>- ■^=.?4i» /■ '•'?i--?%^i^-i;i-ar Or, 100« = ^fil^d. ,. Or. If 100* ^"^ -■ 6.l2yi. '612yd. = 6.ig "TT =. 30.72yd. 4fi». =k72yd. Hence the foFowbg Kn*rf thatj,art ^f the hose which the rate f. is of 100 ■ BXAMPLES »0R 1>«A0TI(!«. " ** , ^^1. Wh« h 5^ -of ,,462? 4^ i ,560^ 8:8 of,W.m25M^ of ^,3. w..t.335 „f,„o.eo, 4w or «-'^|^^,n,f of 6. What is 15% of IM^ of «fln ? 9 l^"tQl^,°'T*; ' ^-25 J etc. . 6. i merchant havfng 1^6 hilViSi^^^^^^^ ^1^ of ^20 15 8? draw 1856 ; how much will rem JP' '°"*^ ^anl^^jphes to with- much did he spend for each ? ^ *' '^^''^ 'W* ^o» «, i!»M. For P. $692.05; T. $986 75- C 9.\ttR im.. itc what it cost. How niuohlid he ^ v ' ■ '"i^I'ii^s";;;;*?' '■*•) ■'4* ^# iifW. * TQiie 5©, 'hat per cent, of f460 is 27? '" ■4. *;iWff ®f ♦*5'>5 therefore. $27 is J 7 of 100 «J W* of 1005(5 = 656 of $460. ' JA^.* |^^~^"''*>'i' 100^ Jy th0 percentage anddhddehsf « OPBKATIOW. r \^oo ~ ^' 460)27(pm j^ • 27. « 656, Atu. 1^ &a«e. F^Jiid that part of im per cent, which the percentage i$ of the Mr pBRcniirFAai. y^ 173 ^KXAUPLKS FOB PBAOTIOI. 1. At what rate per cent, must we place $20 to have $27- $5 to have $0.25? $1440 to have $21,60 7 £160 6 to have £12 16 4| ? $4 to have $0.30? Ant. 10^5 ; S^g; etc. 2. What per cent, of 40 is 1,67 of 480 perches ia 24per. ? of 3|| is X? of i is i? Of 92gal. is llgal. 2qt.7 Ana. 37 i%; h%\ etc. 3. What per cent, of 148 iS 24|7 of 301b. Avoirdupois is 11 lb. 4oz.? of 7201b. is 601b.? of 620yd. is 46iyd. ? of 1401b. is 771b. 7 Ant. 162%; 3'^i9^; ^^'^' 4. Wliatner cetit of $678 is $26.01? of $260 is $80? of | is ^7 e#tfe3 15 »3s. 9d. ? . Ant.A\(f,\ aiti. 5. What per.ctht. of $300 will give 2556 of $727 Ant. 6%. fr. Bought e horse for $840, and sold him for $66,0; how much e $37,150, andthe rate 4S||%;'what ia the kmt » iin*. $1600. ^ 9. Afkrmersavedannoally $146.60, which was 33|% of hia annual inoome ; required his income 7 4Si^C|w!^ ...* " •mr^j 174 aST. Cask iy^(Wt,«,, the rate per cent, and amwnt or difference, to find the base. Ex. What number increwed by 6515 of itaelf is equal to 477 ? OPCRATIOV. 1 + .06 = 1.06 477-M.08=r460,ilM. Or, U = 477 5^= 9 |g = 460, Ana. AxAiTSis.— A number increassd by ioL of lUelf, a(]u>ls IWtjf,, or 1.06 of itsolf, which, by the ootadition of the question, is 477; hence, once the number equats 477 ^j- i OS = 460. ^ Or, 65(5 of a numbet-is ^"^ = g"^ of tbo number, which being increated by JQ. the _ 477 rrii-i .r iL. . A-nO: nnmber .nnttU «« „r*t. .. -""nor, wmon oeing increaied by IQ, the the naai^ilM77Tri^^^ "^i"' "" f"""'- "= *"' tv o^ Ueitoa thV " ^^' '"""''•'■» «1"*'» *0 times 9 - 450. rf«?^w' ^"^?— ^»»^ '^« «»noMn< % 1 plus the rate %, expretBed ^?^i& *"" ""* <»'»'»»?»/'-«cow flMoy bu«hels were ia the pile I ^ ^L Sf^ ' Jt^JrS!'''^ Increased my eapftel by 1 6 ijg *"«'» <*J«llfe« horse cost him. and wh^^ I ii < !*?«!* * . fV -; piBotofTAoi. 175 MISCELLANEOUS EXAMPLES IN PERCENTAGE. ^ 1. Find^ofTOcwt Iqr. 12Ib. itfW. 14.074 lb. 2. *l§. is 1% of what number ? Arts. $180. 3. Find a number which, diminished by 10% of itself, gives £48. 4. A merchant owes $4500; hi3 property is valued at 1225)5 ; what , rate per cent, can he pay? Ans. bl%. .^' J^ superior officer, having 1600 men under bis command, lost 9% of them in a^ battle, and AQ% of the remainder by Biekness; how many remain? i«»«. 819 men. 6. I sold clolh at £1 10 3 a yard, which is but &b% of the cost : how much ditl it cost a yard ? Arts. £2 6 61 -f. 7. A man expends $18, which ia 331^6 more than his weekly wa«r9 : what are his wagee ? A na. $1 3;S0. 8. After paying 42^% of my debt, I find that $2650 will settle the balance ; how much did I owe ? Ana. $4608.69 + . 9. What per cent of £40 will giVe 20$6 of £7 15? Ana. 31%. 10. A little boy laid out 40% of his money in play thing*., .So* in sugarplums, and has 12 cents remaining; what did his purse con- tain ? Ans. 48 cts. 11. What per cent, of a number gives 22^% of the i of this num- ,J A An*. 181%. 1 i. A cargo of barley having been damaged, the owner was obliged to ee 1 the whole for $1999.20 which was at a loss of 32%: how mnoh did the cargo cost him ? An$. $2940. ^ el^' / ™*fchant having $2160 deposited in » bank, desires to draw - 15% of It ; how much will remain ? Ans. $1827.60. 14. There remains 25^yd. of a piece of linen, after having sold 16% of It ; what was the length of the piece ? Ans. 30 yards. in^A" ^^« n"«nt>er of deaths in a certain town, during the year, was 1950, which is 31% of the population : what is the number of its in- habitants? ^n,. 60000. 16. A fish-monmr Jwd 720bbl. offish, and sold 288bb]. i what per cent, remained nniold 7 ,^ 60% 17. I81b. 16oB. is 12i% of hoW mtmr lb. ? .liw. IMJb. Sob! 18. Gave to a Benevolent Society 29bu8h. of wheat, which was 14i% of my entire crop; how many bushels had I remaining? Ans. 171. 19. What per cent, of ^ of | of f gives i 7 Ans. 26%. 20. J6»Bph having received a legacy, deposited 76% of it in a bank. Ashortf^^after, hedrewf&rth30%ofhi8 deposit, and there still «W«<>;;^80J7 6; what was the legacy 7 Jim. £2439 16 24. 1 - ii!!T^'^"" ^^^ ^^^^ "^ ^^ P*'^* copper and 4 parts nickel ; what p»»(fe^}t. is the copper and nickel ? A. copper 84%, nickel 1 6%. .22. A gentleman has an annual income of $2700; if he expends 20% for nourishment, 8% for clothing, 3^% iu alms, 6% in books, and 14% in casual expenses, what are his annual expenses 7 il. $1363.60. ~4t.Hi Iir«n engi^raent, 6% of the army won? killed on tfarfl6>dlir~~~~ battle, and 6% of ifce remainder died of their wounds in the hospitals. The difference between the number of the dead and the number of the. wounded was 164 ; how nianv men composed the army ? Ans. 22000. " m 176 PteOBNTAQE. • 25 An Rr«/? • ' \°"' ^^^ ^™'" ^'^^ ^'^^ <^o<^« t'^^ 'a«er Jive ? to iqiin ^' 'u*'""^ been twice decimated in battle, is reduced v«arlv. llalf?!' ""^cantile establishment amount to $131000 fhfmetfflierct\^'""^^^^^^^ ''''^^' ^^^ --»^^'' r ^LFJl^P'^P^^'^'of^ieflof a factory, sella 2456 of his share to nionV^slfoiok.l^'f'^^'^Pl"';'' "2?"^^' ^"d if 6556 orSul'a ^ qn nil ^ ^r^° ^' ^"'^ '""^^ ^'a» Paul more thaii Leo? voo, *^Ji'';opofpo*atoe8thisyeari8 956 greater than that of last •42fiLn •''S'i?''"''"'°"'<''' ">« Do-aJnion of Canada, "„ 1869 «■., r^nfii^f?^ °^'^'^'"'/°"« *« «»«h article, i lllHiiii l enUrr^&£f5S0 ?n °? ^^ on ^e coffee j^he received fmm his chandiS? ' how much did he pay for each sbrt of mer- than mS^ aSl^l'^** l^"' respectively 656 and.456 more f£ey „:fu\^^'9!^^^J^^'^jj>o&metice»hiimness on the let." (JjfSSarv *' Tr t foK? Pef 1i "^^ *''\.^"^ ^^'' "'«"^'^«' I read inTs SJ '4f > s; 8IMPLB .INMBBST. SIMPLE INTERESTi 177 2H9. Interest is the compensation mado hj tii?^ borrower ta the lender for the use of mop^, 2»0. The Principal if'the sum iSit - . ' 2»1. The Rate per.ceiit. is the interest paid for the loan of SI 00, c£lOO, etc., during any time whatever, whioh is ordina- rily a year. NoTt.— Tho rato per oont. is cbmmonly. ezpratted deoinially tm haDdredthf. 2X2" Ji^® Amount is the sum of the principal and interest.' ^»a!. Simple Interest is the sum paid for the use of the principal only, during the tin^e Of the loan. 294. Legal interest is the rate per, centi established bylaw * It varies in different countries. iroTi!.— When no rate is montionod, th« rate eitablish^ by the UwB of the coantry in which the transaoUon takes place, is aiwayi nodentood to be the on« intended hjV the parties. - ' 29S^Sury is a higher rate % than is allowed by law. ' Now.— The law prohibits usury and makes it iubjeot to » penalty. 296. To Jhid the interest on any. *uf», at any rate %, for any * nnmher Of years gndnionths. Ex. What is^he interest of $780, for 6 years and 3 months ^fil years), at 1% ? What is the amount ? "» \«t Akaltbib.— The UitereM of |l fbr 1 jw, 1% is $0.07, and ef |780 it is 780 tlittM $« = $54.60. IftheinterMtof|780 for 1 year, ^i loL is 64.80, for hi jt^vu it is h\ tlmei M4.0o' = |li8«.66. ** t. 1 OPKRATIOA'. 1780 Prin. .07 Rate. $54.60 Int. lyr. 5i $273.00 " ""Syr. 13.65 " 3mo. % 286.66, " *iyr. . 7&0.00 Prih. added.* 0'' xiff °^^''' prinoipal «. the Interest for 1 JL year at 7^. The amount is found by adding tho prinoipal and interest together. $1066.65, Amount., 297. KuLE. — L Multiply the principal by the rate % ex- pressed decimally, qnd the product will give the interest for one year. ^ II. Multiply this product by the number of years, and tl^ months' as a fraction of a year, for tht interest required. » ,^j , The amount i^ found byhdding the principal and interest '{»• gether. ^ NoTB.— When part of thc( time &r ii^rest !• glren In mon^ or daytL on* month is oonaidored as ^ of a yew, ana one ^y as JL of a mobtii. ^ 8« ^v »■ .?•. ;il €^. -•*- .4* 178 SIMPLl IMTBBEST. IXAMPLM FOB PRAOTICB. 1. 2. *3. ■'4. 6. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. What is the interest of $450 for 3 years, at 4% 1 *b56 for 2 years, at 756 ? «??? ?// ^J*'" 10 »%onlh8, at 75ft ?^ «?« «S ?' o ^**'' ^ "^"'hs, at ef ? «7fi fiS ?■■ ? ^"^^ ^ ""Wths, at 856 ? Hi; ?/°/ V**'* 2 month? at b%1 $444.44for6year8 at6|5g? II^a'. ?/**/ ^ /**' 2 nionihs, at 75^ ? »I671.32 for 14 months, at 6$^? What is the amount of «5?ft*?«^/ V**^ 6 ttonths, at 5%l\ J81.81 for 8 ye^ra 4 months, at 65^? Sfi.5fi ii?,"" *"'* ? ™°"t^«' at b% 1 WP^Se.fat 4i months, at 75g ? • •894 for 20 months, at 6% 1 .•^Sp.ljrejrears 7 montl^ at 6J5^? Ana. $54. iljw. $8.96. Ans. 1155.52. Ans. $435. Ans. $166.86|. / Ans. $45,27. ^n«. $8,258. ^n«. $12,892. Ans. $8.28 + . Ans. $78,414. - ilnff. $656.5125. Ans. $116.99. Ans. $60.39. iiiM. $1110.234. Ans. $3797.26 + . Ans. $122,716. af^. foJindOuiaUrest on any «,m, for any time, at any raU%. " SI* ntB CENT, METHOD. ^To^findthk interest of$lftr«,y time, at 65«; also, at any other T!'«'^?9^^"":r^* ^^ P" a^^""* *^6 interest on$l. For 12 months is $06 " 2 months (A =si of 12 mo.) " .01 , ^^" (tof6da.«Aof30da.) " .'oOOi. Hence, ^^J l'fT^-^l^*H^^ ^^ ™0'»th, or $.01 for avery 2 2 ^2* JS5f'*~^' «*!/'^ ?«/?** -.-Call tvfTyy tar $.06,e9mr ^"-^^ SIM^Li IK TM HM T. 179 II. To find tKeinttrest: — Multiply the principal by the rate. Ex. 1. What is the interest of |660, at (o%, for 3 years 7 moBths 27 days ? • . OPERATIOK. Int. of II for 3yr. Amltsis.— The int«reit of UiiMOtimM for 3 yean 7 ^. ,g -thesivanprinoiMliiMOtimoa — »U. 1 B the intereit of f 1 for 3 ri = 0.035 months 27 dayi. As the int. = 0.0044 of»lforjTr.is$.Ofl, for Syr. it will be $.18 ; ana since the Interest for 2 months is $.01, for 7 months it will be aa many times $.01, as 2 is contained in 7,or31ct8.^ftgain, since the interest for days is $.001, fojr 27 days, it will be as mtdur times $.001, as is eont^ned in 27, or 4i mills. Adding these three results together, we hare $0.2I9| which equals the interest of $1 atOa^ for the Multiplying $660 by $0,210^, we obUin $144.87, the int. requiivd: " 27 days. " ."^yr. 7mo. 27da. = |0.21i4,M#kte.v.i.'.j. i^J't^^f^^~^ if fi^'t'^i-4 "■* *f-7'*- fv*. «• 180 siKRUi unriHisT. P«- cent, will gi„4e i«o i^."' or fSJ S m ** ""'"P"""^ "^^ '^^ «''«» Thw, if th« JMe bo 85g. the iuMrj«tT^21^*""''"'°"5^ per anflum: mxtr^" 'l^T; ^f"''" ^^° use the following . METHOD BT ALIQUOT PARTS. if..Wl.tiathe.oJe«stof,42^ opw^Tioir, PruiLcuiiaL BftteSU, Interest for 1 year^ ►* liai.60 .09 137.9350 / i'3 Int. for 3 years, "|l 13.8050' Int. for Jnip. = i of lyr.'" 18.9676 Int. for ISda. = \ of2m o. 1.580 61 Int./or 3yr. Smo. IS^a. IliOfseJ, ilfM. AiTALTSrs^Havine found tte totewat for l/r. a^ then for 3yr., the int for 8ino. « obtained by first taking 4 of 1 year's int., »r Jmo.,' and then * of thi»Ia«tlnt. for2mo. And ""?" i*^"y»aro 1 of Ima, or i of 2mo., we take' 4 of 2mo.'« int. for 15 day*. The Wt. as found for the weralp»rt8«fthe whole time, added together, gives the interest required. -^^Tk^'^'i lif t«<.mf /or the monk and day, hy aliauofparU ' ^ ,, Thcum o/thepamal int^,u will be the inlejlr^uireT " METHOD BT IIOWTHS. 6.06 18,5 ±t}AltR NotM.-l. 4,r. rmo. IMa. -. «6.Smn. * •^'*? ^ ^'**' ^^^ ^"^" a. The above is the prodiwt^r the iwin- Pipal. rate per*oent. dJSmSy eSwCl lao«i by 13.=> 3 X i. ~^' "t^ of v' • », . ♦^■t-^ i-i^.;*'' 55.5 . "6363 '.. % ■ , tint. tor 6.^6 mi^ .^T^ • 20da., at 6«7 .15. #r6 11 3 for i>p. 4mo., ft 7*? 1$. $336 for 5roo. 15da., at 651^ 7 . 17. $1265.60 for 5yr. 2mo. 9da., at 1%i^ 18. $72.1 2| f6r 6yr. and 6mQ,i at ^^f , ,19. $4?7.36for lyr. 6nK>. iaa.,at5^? • 20. £19t 6 4 ,for 2yr. flrao., at IjU? ^ / 21. $76tl.09 rof Syr. 8rtio. 5da., at 8*f >25|. $4a»80 fer2yr. arid Mmo., at 75lSf " M. $35(^.80 feilSiDO. HDif 84»., ii^lO^"? Ans. $56.SyL 4«*. $1347.85 W Atu. $14.84. ilfw. $111,177 + . Ans. $31.46. Ans. $4,254. 4«». £8 16 2X. Ans. $6.01 + . J4n». $16.85 + . ilni. $36,396 + . Aw. $i 36.^48 + .. ilfw. $20.04}. M0. $6,468. Atm $75.2081. Ant. £1 1 bl. Afu. $1.14. ilw. $469.a4 + . 4i«». $18.61+. • Vvin*; $37.37 +v 4ft«. $21,039 + . .dlf«.$a2$8.70 + . .^if^- ." 1 y-.'A H:i* 182 SIMPLl INTSBBST. It' It^^^t'Jr '}}'"''■ 29da., at 7 5ft? 25. £24 18 8 for l«nio. and Ant. $i56.m. 26. |5l.l7 for lOmo. and 29da., at'4 ^'? 29" iSfinV^'^ ^T- ^•2'^*-' '^^'^^ft? ««• *^*'»«50 for 6jr. 6mo. 4da., at 6 ^ ? 31. 1680 for 4yr. Imo. 15da , at 6 J ? 34. Jl 90.016 for 3m6. 24da., at 4^ J ? „A « ^^ ^ ^ ^or lyr. 5iao., at 6|<5^? * 36 |708.20 for 2yr. 2mo. 12Wit 4^? 37. |65t0.70 for 8mo. and idJZ^ tl S o 20da., at ,t4g? Atu. £1 11 Oi + . ^n*: 12^3.72 + . -4na. 1296.19 + . Ans. 1168.30. ^»w. $164,888 + . ^n*. £34 16 4 + . 8n,o. and 26daV,,at 6^ % ? ^„,, ?S' il-^'^^ ^?' ^^"°- ^"'^ 28da:, at 6i «? 39. $9a0 for 4yr. 7mo. 9da., at U%r if- f81 10 for'2|i.. and 5mo.;at4|*? 41. $160.80 for 7mo\ and 20da., at 7i /? 43. $601.20 for 4yr. 2mo. 3da., at 8* <*;? 45. .^319 10 9 for lyr. lOmo., at 41^? lJo?n^^/°/ '*^'- »'P0- 19dav at 6| ^ ? .«I*f/*'„^y'' *'"*>• 27da., at 7i^? J160.76 for 2yr. llmo. 4da., kt S?^? >184 18 8 for lyr. 9mo. 6da'., at 3%? Whatia the amount of $26.037 -f'. ^«». $361,178 + . -4n». $6.98 + . » ^fw. $i2l3,36 + . ^n«. $102,618 + . ^»M. $360.30 + . ^n». $724.24 + . h\' ^•1*5 for 9yr. 9mo. and 9da., at 6 «; ? 62. $1061.60 for 2yr. lOmo., at 7 W ? * S' l\fAl 5^' ^y'- '^"o- 3da., at 6 * ? 64. $100.2fi for 2mo. and 29da., at 4 * ? 66.-$1.011 Ibr lOyr. lOmo. lOda.. at 6 *? Jf • il^l^^J^""' 3°«>- 29da., at 6i 5ft ? Ro K8^k*$ ^r ^^* ^'"°- *°«* iOda-> at 6i 5ft ? 68. $2ffD0 for Imo. 6da., at 61 5ft ? ' "* » * 69. $0.06 for 20yr. lOmo. 16da.Tat 8 5ft? «?• SJH^i^' 2yr. 9mo. 12da. at 6^^? 61. $498.96 for 6yr. 6mo. 6da., at 61 5ft? m. $2660.76 for 4yr. Smo. 26da., ^t 6i|? -w ^W.^mat«th^mte,^tof$1660 from April 9, to Novemb« 10, ary V2, i§6t;t%T"°*'^^^^-''' ^"^ ^""^ ^9, W60, toFebru- iln». $0.23 + . AilB. $1260.046 + . Ana. $253. H 9. Ana. $101,241. Ant. $1183.18. Ant. $2013.121. ^n». $384.09 + . «► SIMPLE INTljIXSf^- 1^3 68. "Whati8the1ntere8to1jM3 2 § from March 17, to December 7, at 7^56? / jins. £2 5 1^ + . 69. What is the interest of $1630.60 from February lO, 18G8, to January 25, 1869, at i 56 ? Ans. $7.33 + . ' 70. What ia.the amount of $168.30 from February 17, 1868, to December 30, 1871, at 7^ ^? 71. What is the interest of $1728.19, from May 7, 1868, to July 17, 1869, at i 56? ' ^n».$5.l6 + . 72. What is thp interest/of 432 8 9 from September 25, 1867, to July 9, 1869, at 1 5(&? C 73. Whae is the amount of $89.96 ffom June 19, 1870, to Decem- ber 9, 1871, at 8i ^ ? Ans. $100,886. 74. What is the interest of $990.75 from October 5, 1868^ to Jan- uary.I6, 1869, at 1| ^ ? * . 76. What is the interest of $1030.10 from November 8. 1867, to March 3, 1869, at 85 56 ? Ans. $120,625 ■♦- . 76. What is the interest of £45 10 4 from December 10, 186G, to May6, 1869, at|%?- EXAOT MBTHOO OF OOtHPUTINQ INTEBBST. • 805. In the preceding methods of compatiDg interest, which are in general use, we have reckoned 39 day^6 the month, and 12 months to the year, which allows to eadn year 360 instead of 365- days. Hence, the results obtained in these oalculations are n9t^triotly correct. Tl^e following exact method is used by bunnes^? mien in coin* puting interest when the time is short. , . i ■ ^ Non.— The ex«et time, when it is leu than a year, ii fonnd by the table on pagelJ4. 800. Rule. — Multiply the interest of the principal /or 1 yeorr bjf the exact ntm&>er of days it has been on interest, and divide the product by 365, the ^(uotient will be the interest required. 1. Whajk is the interest of $345.60, flrom Fbbruary 6, 1869, to Aug. 20, 1871, at 7 ^? Ans. $61.374 + . 2. What is the interest, tA5l%, of $426.50, from January 8th., until November 20th. ? Ana. $20.26 + . 3. What is the interest, at 6^ %, of $140.40, from Aug. 29th., 18-70, to No*, 29th., 1871 ? Ans. $11,426 1 . 4. What is the iaterMt^ If-^^, of $4560, from May 18th., 1«68, Uy Sept 26th., 1871 7 1 5. What is the interest, at 1^%, of $3790.45, from July 20th., 1869, to Sept. 12th., 1871 ? 6. What is the interest, at 4^^^ of £48 16 3. from Sept. 12th., 1868.40 Aug. 28th., 1871 ? f . . * » N., 184 -, PABTIAL PATMBNTS. . .^ vartial paym:e3nts. The payments aJo af nfS^d h! '* ^'^T^'lSr'' creditor on the back ofthT^lt.^ fe ""^^'P*^ ^#^° V the Indorsements. °' obfigatioD, -^hicTare called ^J'^j'^rL-^^l^^^^^^ be paid by days --MuUiply Payment L ZZ SubTJuhJn^^^^^^^^^ remainder bu the nut^ZZfi ^^'"fi'^^P'^S^^nt, and multiply the "nd second ^aZr^fJ^',"'^''^ f^'^d between tUfirU ^is remainder by te «tS^5 T^.'^r^'^^*'' ^^'^ »««4^^ second and third Z^^B €t^ , IT^'^ -^T'^ ^''""^ ^^e ^^^^^^:ei^^J:):r^^ ]^XZeTj!itimr^ "^'^«* ^*^^ ^ *« t'^^^ on the fol- / *i-» Quebec, Sept. 8, 1868. $ 420. Not. 20, 1869. « ' *i2.28. "•^ »»».; . ;;;;;;::;:::4f;»?: From Sept. 8,1868, toQet.^1 ]s 23464.76 for 1 day. > ^terest = that of *^ifeW993.23 fori day. Balance Whole 4^ — ^ r- VAXSUS. PATMKNTS. 18:> Infereflt nn $391993.23 atl^ for lyr.\^ ^ $27439.5261. Uence, the mt. for 1 day = $27439.5261 -r- 365 = $75.1767 + . Then interest due =$75.1767 4-. Balance on note = 99.8600; $ 450. Principal and interest due $176,026? + . '*> - Montreal, January 13, 1869. 2. Nine months after date, I promise to pay Louia Merrill, or order, • '"Jl""^ and fifty dollars, with interest, at 6 56. for value re- ceived. 1, A. N. Moreau. Indorsed M follows: Oct. 7, 1869, $126.10: Ang. 25, 1870, $225.35. 'How maoh remained due Sept. 19,1871? Aw. IU2.88034-. 1325^. Kingston, July 26, 1866. 3. Four years after date, we promise to pay Lawrence Boyce, or order, three hundred twQpty-fivo and ^^ dollars, with interest, at 7 %. Val- ^e^ecel\?ed. L, r. Whelan & Co. o«%T,?,?^C*!?'*"'Jl-" Jan-.^O. 1867,1121.185 March 14, ISnS, |72.46; July M, isotf, $ijj.65. How much remaioed due Sept. 8, 1870 T Am. ^1.01-f.:. $1737^5. Toronto, March. 6, 186§. 4. On demand, we promise to pay Fisher & Howe, or order, one thousand seven hundred^jjuly-eevpn and ^ dollars, for value re- ceived, with interest, &iJ^^ T. Johnson & Bro. indorsed as follow?: /Jui^l, IB68, |623.80j Sept. 10, 1868, *700. How much was due Jan, ^1,1869? / * An*. $466,763+. $1240. Ottawa, Aug. 18, 1869. .6. For value received, I promise to\pay R. N. Kelly, or order, twelve hundred and. forty dollars, on derfiand, with interest, at o %. Joseph Rogerrt. ,Jj^£1*'*„"^'°""'"'' ^opl^l 26, 1889, $95; Oct. 28, 1869, $217.86; Deci 12, 1869. $432.36 ; April 6, 1870, $120'.20 ; July 3, 1870, $366.60. How muclf re- mained due Sept 10, 1870 ? < Am. $43.768- ^^ > ' ,', I . - " >- ■ . £ 304 6 6. Hal'iftkz, j;une2, 18681 6. For value received, I promise to pay N.,' J. Webster, or order, on . demand, three hundred and font poCftids six shillings and six pence, with interest, at 6 %. , ' A.. G. Murphy. ^ „^2?^ilf *>"'^» ' J^'y "» ^®** ^1 ^« A Oot «. 1868, £52 8 ; Deo. - 1MM8, £80 4 «,i mjeka y. 1869, £104 9 U." How much w^ due Oct. 7. 18Wt %alBfe , . • ' A«.,X21 13 0^^^. . ■u%' 186 « 14696.60. St. John, June 17, 1806. ^»». $3596.31 + . PHOBLEMS IN INTEREST To fl.d the ra.e«; V. Toli •«,"«„/ "''"'"'P™"''*'; I^. , The Case I. a„d II. have already been Wred (296 298) "" "^' "--r^^rr^' ---'««. -i^?.^ S-. W,,.. principal in 3 „.„, „ e^, „;„ ^,„ „,.„.„^^,,, OPERATION. ' * _ OPERATION .06 int. of $1 for lyr. 3 ^ ^)m.70 (^265, 4«,, 18 cent.. It irifi require a prinoipfl of " r°^ "^o''*" to gain W.f0 M ♦0.18 18 pontinned times in'|47.70- dividing, we obtain »2««, the na^ By propoition. $100 :aj :: 16 X 3 : $47.70. t«.7/ 6. Iheprmcipul.-^ ^ '"* ''"^ °"^*'«'^' and thiquotient , BXAMPLES FOE PBAOTIOB. , What principal will in . ' 1. 6yr. 3ino., at 6 %, give $66.25? ♦ , ' ^i. .,^„ * 2. 1 rr. fiino., at 6 %, give $1.2924 int. ? - ^m^II^sS" •■-'■X": vaxmiMMB m nrriBiBT. 187 1866. IV. 3. 4mo. 18r 183 days; at 7i %, is sutficicnt to produce $619.16? ^^ ^ » «w > ' 819. Oasb IV. — The jtrincijfMl, time and intereat being given, (ofind the Rate % Ex. The interest of $750 for 4 years is $180, what is the rate %r OFBIUTION. $760 .04 $30l0O ) $180.00 ( 6^, Ana. 18000 By proportion. .^ $750 :^ J? X 4 : t\£0. ' °AHAi,Y8t8,— We find thelntewit OB the principal fgr 4 years at i ^ . 6ino« tbiB kiterest of $1 at Ife for 4 years ia 4 cts., the interest of $760 will be 7&0 times as muelh, or $30. Now, if $30 is i%, $180 will be as many % as $30 is eontained times in f 180 ; dividing, we obtain 6, the required rate ^ . Hence the 13. Rule. — Divide the given intereat hy the interest of the ncipalfor the given time, at 1 g&, and the quotient will be the rqte % required. JIPLES TOR PBAOTIOB. fX^ Required the rate per cent, if thi interest of J 1. $500 for lyr. St^o. is $66.25. Ans. 9 %. 2. $40 for 2yr. 9mo. 12da. is $13.36. ^ Ana. 12 %. 3. $540 for lyr. 2mo. 6da. i8'$38.34.. Ans. 6 %. 4; £37 16 for lyr. 4mo. is.£3 10 6f. Ans. 7 %. 6. $126 for 3yr. 6mo. is $32.37J. Ana. 7| 56. 6. $1500 for 3yr. 3moi 29da. is $274.77. 7. $124 for 4yr. 3nio. lOda. is $29.17f Ana. 6i %. 8. $36 for Syr. 8nao. 19da. is $8,034. 9. At what rate % must $1, or any other sam^ be on interest^ to . donble itself in 14^-! years ? A na. 7 96. 10. A man invested $4500 in the Montreal BankJ and received a semirannual dividend of $167.60; what 9^ tiras the dividend? ■ - V 814. OAsb Y. — The principal, intereat, and rate % being given, to find the TutXi. ov IMAQE EVALUATION TEST TARGET (MT-3) -r 1.0 1.1 11.25 Iiil2.8 ■50 ,^^" |2£ Itt lU 122 U.ttg Ih « 1^ U Si-- Ih u& 14 1 11.6 '^ .Sciences CarporatiQQ 33 VnST MAM STRHT WIISTIi,N.Y. MSM t^U) ■72-4903 '^ ^^y- j^N <»"•-. ^-t .• *^i^Hiii ""-y **-, J;'iiW^ ,■«» j",*' w^-i^iiiifel.^-f*.. ^j; >.*.-.;iv:-s/'!'.\--^^iv'& -. : dl^v.- .it' ■■. ,■.;:;■;-, .,»: - ..sf^^-'*' ■. .G'j.l'fe'. 188 r& PRQBLI1C8 IN INTIBIgT. 1 Ex. In What time will |460 gain |64 interest, at 6 ?g 7 OPERATION. 1450 ■ 06 •27.00 ) 154.00 ( 27r. Atu. : 54 00 By proportion. , ~ $100 : 1450 :: 6 X :r : f54. AwAtTSia.— We and the intereit on the giren prinoipid for 1 year. Sinee the inter^t of |l for 1 year ir 8 centf, the iotereet of $460 will be 4i0 timei m much, or |37. Now, if . it require 1 year for the giren prin- cipal to gain ^1, it will require as many yean to gain |64 ua $27 is contained timet in $54; dividing, we obtain 3 years, thtf required Ume. Hence the 815. Rcrj!.— Z)»VtW« the giien interett hy the interest on /he pnnapal/or 1 year, and the quotient will be the time required in years and decimais. ft EXAMPLIS FOE PRAOtlOI. In what time will 1. $26, at 6 %, give $1.95 interest ? 2. $280, at 6 56, give*84 interest? 3. $45.25, at 6 %, give $1.81 interest? 4. $98, at 8 %. gain $25.48 ? 6. $240, at tf %. amt. to $280 ? 6. $70.50, at 9 'if,, give $31.72J interest? 7. $408, at 7 5i$. amt. to $434.18? 8. £120, a.ii\%, amt. to £140 8 0? 9. $1, or any other sum, double iteelf, at 5 9$ int. ? An$. 20 vr. 10. $2365.24 double itself, at 7 ^6 ? ' ^ aontbi * Ana. lyr. 3mo. Ana. 5 years. Ana. 8mo. Ana. 2yr. 9 mo. lOda. Ana. 1 1 mo. PROMISCUOUS EXAMPLES IN SIMPLE INTEREST. What principal will in 1. Syr. 4mo., at 4 %, give $2048 int ? 2. fimo. 6da., at 6 %, give £186 3 6 int.? 3. lyr. 8mo., at 6i ^, give $97.60 int. ? 4. 9mo. 21da., at 6 %, give £15 16 int.? 6. 3yr. 5mo. 18da., at 5i ^6, give $288 int.? Ana. $9600. Ana. £5237 10. Ana. $900. ilfM. £38^ 13 9 + _. _^.. „...v,. »■— ., o. i.j;^,.,5iT0»«00 Ull. f ilfM. $1682.42. 6. llnio. 9da., at b)i%, give £466 2 6 int.? Atu. £9000. 7. 4yr. 6mo. 14da., at 5 %, give $150.37i >«»• ? Ana. $675. 8. 3yr. 6mo. I7da., at 51 56, give $1451.62 int.? Ana. $72«|^71. > In what time will f In what time will 9. $626, at 69$, give $262.S0 int ? !«. £67 10 0, at 4 ^ give £24 6 int. ? 11. $1779, at 6 %, give $296.60 int ? iffw.7yr. Ana. 9yr. Ana. 3jT. 4nio. .M^ Dmt the interest ipw for 1 year, ll for 1 year it of $460 will be or 937. Now, if the giren prin- will require ai 1 164 aa |27 is 64 ; dividing, we < required Ume. nterest on fhe < required in need to montbi . lyr. 3mo. i«. 5 years. Ana. 8rao. )ino. lOda. 1 1 mo. An$. 2QyT. SREST. na. 19600. £5237 10. Ina. 1900. 89 13 9 + I1S82.42. w. £9000. Ina. $675. •72^71. Ana.ljT. Ana. 9jT. 3jT. 4ino. ^OBLKMfl IN INTKBI8T. 189 Ana. 4yr. 9mo. 12da. Ana. 2yr. 2oda. Ana. 7yr. Ana. dyr. 3mo. Ana. 8yr. Required the rate % if the interest of $978,20 for lyr. is $48.91. ^£110 12 6 for 50da. is £1 16 $1290 for 124da. is $19.99^. $4340 for 3yr. is $586.90. $675 for '{4mo. ie$l42.31i. $7500 for 48da. is $60. $11004.75 for Ivr. is $550.23}. £12Q for Cnio. ISda. is £32 10 10^. 0. Ana. 5 %. Ana. n%. Ana.^%. Ana. i\%. Ana. b\%. Ana. 6 %. . Ana. 5 %. Ana. IH}%. 17. 18. 19. 20. 21. 22. 23. 24. 25. The annual sales of a starch manufacturer amount to £2737 10, supposing that hia profits are 5 j6 per year, in how many years will they reach £323 18 9 ? Ana. 2yr. 4mo. 12da. 26. An individual disposed of the | of his (unda at 4% and ^ at 5 % ; every year he draws as much aa will pay the harnessing of a horse which harness is worth $117.60; what is the amount of his funds? iln». $2800. 27. What is the interest of $17.18, from July 29th., 1864, to Sept. Ist., 1863, at 6%? iliw. $4.214 + . 28. What will 1« the amountof £19 15 9, at 7i^, from Feb. 17th., 1864, to Dec. 30th., 1867 ? Ana. £26 10 7 + . 29. If $1756.76 is plaqed on interest, June 29th., 1866, what will it amount to Feb. 12ih., 1869, at 7 96? Ana. $2078.869 + . 30. What principal, at 5 %, during lyr. Smo. 12da. will amount to JE231 12 Uf? iln«. £213 10 0. 31. On Aug. 15th., 1860, I lent $5269, at 6%; what amount will be due me on May 1st., 1868? ' Ana. $7692.164 32. An individual buys 65|^ acres of land at the rate of $609J3^^^ per 100 acres; if he pays only at the end of 3yr. Imo. 16da., theist. . will equal to i of the principal ; what is the rate 7 Ana. 4 %ii 33. A pertH)n placed a certain sum on interest at 4 %, which pror duced £427 10, in 3 years ; what is the sum ? Ana. £3662 10. 34. What is the interest on a bill of $257.81, dated March Ist., 1865. and payable July 16thk, 1867, at 7 ^ ? Ana. $42.86 + . Find the amount ofTl7041.20, at 4| ^, for lyr. 7mo. 2dda. What sum is that which will give an mterest of $900, in lOyr., k-» 35. 36. at4i^? 'J ilfw. $2000. 37. A principal of £112 10 was put on interest, and at the end of Syr. amount^ to £144 ; at what rate was the principal placed? 38. A boy has accumulated a sum of mone^ by his savings^ and wishes to obtain an anni^l revenue of $140 ; if the rate is 5 ^ what principal must he have 7 Ana. $2800. 39. A merchant borrows the sam of £938 12 3, which is owned bjr a minor aged 16yr. Smo. 20da. He keeps it until the owner is 21 yean old ; what sum will be then due, at 6 ^ simple interest T 190 PBOBIiKMS IN INTXRX8T. 40. WW will be the interest of 1326, from June 6th., 1866/ to July 4th., l«68, at7i%? iliM. I49.02 + , 41. Amerchantsays that hie gain, during the nine years he ca^ -ned on bumt.ess, equalis tiie price of 3659 yards of cloth at |2.08 a yard ; what was I.ih annual revenue, supposing he placed his capital on.nterestatSJ? *'»~°« W $380. 636. *^. 1 roiu I8o7 to 1867, the population of Syracuse augmented 24Js6; knowing the last year'snmnbcr of inhabitants to be 102296. tell as 7*Q Ti ^^'^ P<^P"'"''«^n in 1857 ? Ans. 82000 inhab. #fi4' ,„T "",""","„"' ^^ '''''^^f"' ^°*«"«'*» ^^'^> to amount to £627 18 6 in 2yr. lOrno ICda.? '^ Atu. £563 2 Ik. RlAVff'Q«TJ^*'''V.''?V'''^'''*^ '*"''«?'»«'«'' «n '"'e'eot a sum equiv- enn« n?«S^n i"" "'."''' ''^ ^'H^ * y"''' '^^ ^iU secure an annual rer- enue of «163.1>li > what must be the rate ? An$. 6 56 45. i'rom an investment of $35680 in commercial concerns, I withdraw a gain of3223 per month 5 what is the annual rate of the Tr J ^ ,. Atu. 11%. «o k' t^I^^'^y was sold for £2830; the conditions were £800 in cash, £875 in 6 months, £C25 in 10 months, and the remainder in lyr. dmo., with interest at 7 ^ ; what was the amount paid ? ' aini^^lnSrrS^n^l"^ "'r*^,' '^»"9gt^e « y**" ofiis business,- » capital 01 J29G5.10, desires to know in what time he will receive f8H».6d as interest at 5 5i5? J^f^g g„j, 48. An individtial borrowed £375Q,at 7 %, and then lent it 'at 6 5^ ; li^ J'jin 1 '*"* '"1^^ ^^n' l^ ^^' y^*'"' f*^' ^^^ fi"t transaction, con- sists of 360 days, and that of the second, 365 days ? ^JS' ^T%1}^'^*- *'"'® '""*' * <'^''t*'n sum b« oninterest at 4* ^ to produce font? iln-Hk 9mo. lOdJ! 50. In selling merchandise at 128. the yard. J ^^E profifcof 6i • -what 18 the price per yard? « •^■T 1 iTsi + *.* ' {. fi' "® t of a sum of money is lent at i%f andthef, at5 S^; what Ko A "' ^"°^"« tfa»t the annual interest iS $28.82 f ^in*. $666. ^ 52. An apparatus for astronomical purposes cost £49 ; but, as this earn could not be paid before 3yr. 9mo., tEeprice was ftugmented A of ite primiUVB yafue ; what waa th» rate ? An$ 4 %^ ^^' '^'^*° Placed on interest, at 4 J6, a certain sum of money which produced in 6 years the funds requisile for the purchase of 368 lbs of jpreserved tamarinds, at 46i eta. a lb. ; what was the sum ? 04. A merchant has invested in businesa a capital of |21840 which produces him 12 ijj annually; but, for sanitary reasons, he retires ttom mercantile affairs, and loans hia money at 71 5*: how much will -♦ V;^ -If '.' *^*)oP.«'".'''P*^ *« * o^ ^^'ch at 6516, and the remainder at 7^ will give $4340 interest ? ""'Ant. $70000.00. -„i * ^f'l^i*!?' '^*"'^'' *° P""5i»M« » tract of land, containing 450 acTM, at £6 17 6 per acre, and, for this purpose, borrows money at f * fa f * *''• expiraUon of 4yr. 1 Imo. 20da., he eells the | of the fand 5if\ 1 *° •°'«' »nd tl»« remainder, at £8 2 9 the acre f how much . does he loae by the transaotion ? ' \ ■V- Ih., 1866/ to 49.02 + . ears he car^ b at 12.08 a id liis capital $380,636. iente4l24i^: !295, tell as 000 inbab. > amount to 563 2 IJ. sum equiv- D annual rev- ' Atu. 6 %. concerns, I 1 rate of the liw. 1i%. :r« £800 in eraainder in lid ? lis business,- will receive Ans. 6yr. It it at 6 9g ; taction, cou- it at 4^ % to no. lOda. STsi + d. t5^; what tu. 1655. but, as this ;mented ^ \n$. 4 %. onejr which >f 368 lbs. m? 1840 which he retires much will 636.86}. remainder oqoo.oo. Aining450 money at 3f the fand liow muoh « ooxrouHD nrriBiBT. COMPPUND INTEREST. 191 816. Oomponnd Interest is interest on both principal and interest, when the latter is not paid when due. Non.— The limple iateraat maj be added to the prinoipal annually, semi-an- noally, quarterly, or monthly, aooording to agreement A creditor may receive compound interett without being liable to the charge of oaory, but cannot UgaUy demand it. Ex. What is the compound interest of $390 for 3 years, at 6 ^ 7 OPKBATIOir. 1390.00 X .06 1390.00 19.60 $409.60 $429,975 $409.60 .06 = 20.476 $429,976 .06 — 21.49876 Interest for 3rd Principal for Ist. year. Interest for 1st. year. Principal for 2nd. year. Interest fpr 2nd. yeai". Principal for '3rd. year, year. $461.47376 Amount for 3 years. $390.00000 Given principal. $ 61.47376 Compound interest. 817. RuLI. — I. Find ih0 amount of thepiven principal at the given rate for one year, and make it the principal for the sec- ond year. II. Find the f;^ /\rt p ;?'^Ty ^ """l* payable to a particular persT as in fZ^No 2 ' n, fn ,1^ person who is the bearVr or hoIdSr of it,' A^o 3; ^ ^JV^! ^ *? tJ^e o'^" of apersonnamed ini^^ No. 4. ^ '^ ""^* P*y*^'« »' a particular place, m in fZZ rnI^)^::^lI:[Z?tnr ''''''''''''''' payee demands its pa(. Bou«.-If .H, Um. fa fl«d. In a not«, for payment, it I. payable on dam«.i. another perLnTh?^?i'1??^ A w-u''*'' '"tf^^'^^ ''*? ""^^ note in bLk or oth?^." ^'^' ^- ^*'^'" °*'"'*^' ^y ''«!'>"ing the The Note, Form No. 4, is payable only at the Bank named in it. 826. The race of a note is the sum named ia€^ u^^T'i.^*^ Of Orace are the three days usually allowed bv understood, «hI If • note p»S faL,«l?lth„n^?^ '?f'°'*" '»°"**'« «• the legal inUetof th^eoiSTwhfarJir^^L '^i""* *"" "*" %,»»«>•« cii&*'„'iarss'Sd • ^■** ""• ••«- '^ ;« Of in««S2 ^^^«»t. pwtirt- eo Bwrfay 0, a h„l koUday. tt mart b. paid on th. dv >te which (M0 No a, / IS tiated with|mt ■am for whioh Form No. I in FortM ular perBob, holder of it, aaiaFor'm 18 in Fott ds its pa| I on dema t the ezt^- I it, to the te in blank, >ereon la#- payable to I cap make oreing the Died in it. llowed by the time ' the days tholntneit If tlMtima nontha ai« i , it bean oh does Aot of interatt « Biamaday * PV it if onthad^ rOfMB or N0T88. 19S 339. A Basinets Note is a note giren for a valuable con- sideration. It renders the maker liable for the amount to the payee, or to any subsequent bona fide holder. , 830- An Accommodation Note is^ a note ^vcn for no val* uabic consideration. It does not render the maker liable to the payee, but makes him liable to any bona fide holder after it has been npgoliated for value. Non.— Acootnmodation notes are ncuallj giTon to en^Ua the payee to borrow monojr on the credit of the raakelff of the notes. 331. A Joint Note is a note signed by two or more persons, who arc jointly liable for its payment. A Joint and Several Note is a note signed by*"two or more persons, who may be held, either jointly or singly, for its payment. 333. A Produce Note is a written promise to deliver goods to a specified amount. 333. A Due Bill is an acknowledgment of a debt due in money, or its equivalent in goods. . — ^ . 334. A Bond is a written obligation, authenticated by a seal, to secure the payment of a sum of money of the perf^^rmanoe or non-performance of certain acts. 335. A Mortgage or Mortgage Deed is a conveyance of property given to secure the payment of a bond or debt, on con- dition that when payment is made, the conveyance is^void and the mortgage is discharged. FORMS (IF NOTES. Form No. 1. — Dbmand I^oti./- (yn c£)maru/, Q^j^iomt^/o Aay .^! Q4^ S^iitl^, (STAMP.) ^^u0 ^~&aineau. IW rOBMS Of NOTSS, Form No. 2.— -Ndrlt Payable to Bearer (Nbootiable.) (STAMP.) Form No. 3. — Note J>ATABbE to Ohder (Neqotiablb). C/ne yea* a^M ^ci^a, Q^ ^iom^a fe-^au fy ^ oU/d o^ Q/, Srf. (^ntdon, ninelu.-mne ant/^ (^ ^, UH^^ mUia^, ae /^. Wa4ie iececi^. Form No. 4. — Note Payable at a Bank (Negotiable.) ♦ Sim. o/oioneo, q4/iu//2, /<9/0. <^oUu e^oMS a^H e/cUe, of^ ^iomcio {o /toy ^ ^an4, e^4^-6evan am/ ^s e/o^i6. ^a/tie tecoM^. /^ PBOTIt AM> torn. m ^ Form of Produce Note. Z S^. <^^ife/, on (/effianc/, /*'/^/-'*9^^ ^^^ ^^ do^lS m acoc^, at oui ^oia. (STAMP.) e$f Mi>^u/ ^ ^o. - Form of Due Bill. 1103. (9ltau^a, ^^e p, /<$7^. " ^/ue S(/ioai(/ ^. .^kildcn, ^ vadte Ueeivect, one nunc/iei/ MHi/ miee c^/mU, wU^ t'nMaSH. (STAMP.) ' ^^^/i Q'^/^yn. PROFIT AND LOSS. . 83G. Profit and L088 Wpbmmeroial terms, used to express the gain or loss 'in business traiisaotions./ 337. There are four terms or quantities to be considered in Profit and Loss, viz. : — 1st. The Goity or original number, which is the BaSA. 2nd. The Rate % of gain or lo$», which is the Rate %. .- . 3rd. The Gain, or L6t$, which is the PercentagO. 4th. The Selling Price, which is the Amoiint, or Difference. The questions follow the same rules as in Percentage. Setting Price = Coat + Gain, or Cott — Lo9$. Coat = Selling Price — Gain, or Sdling Price + Iioaa. Gain = Selling Price — Coat, Loaa = Coat: — Selling Price. 198 In f PftOriT AND lomI ttAJfPLBS rOB PRAOTIOC k 1. 1 bought cloth, at $2.60 ppr yarJ, anJ sold it so a»to gain 26 % i for l»ow much did I sell it a yd. ?^ , win*. |3.12i. To iolT* thU Kzample, m* Cms I., 282, Kcli. 2. A farm was bouijht for $4600, and sold to as to;gaia |9ob: how much was the gain %'l Ana. 20 %. To solra Ibii BzMnpU^ m« Cm* II., 284, Ron. - •<• >"> - X t 8. By selling a building lot, a man gained $175, which was 12* Ofthe cost; what was the cotit? ^ ^. $l458.3;{i. To Kin this £zampl«, m* Cue IH„ 286, KolsN » '4. A gentleman sold a hor8e for $180, and theteby giiincd 20 95; what was the co&l of the horse ? .-Wis. 1 1 50. , To iolTo thU Kxamplo, .i. Umo IVr., 288, RoLi. . 6. A naerchani lost 15 ^iJ on hia oltl stock -CfgooilH ; how fmich did he lose on those that cost 12i ctsl, fCj, ■• , fCj, ;jyi ctH., 3^ ctH.,and $18J? Ann. 15 ct.". ; $1 ; 5 J cts. ; etc. 6. Bought sugar, at 12 ctfiapouud, and Hold it so as to gain U cts. a pound ; required the gain %. 7. Sold butter at I of a dollar a pound, which was at a gain of 26 % ; required the cost per pound. Ans.Gdi cts. 8. A market woman sold orangeg so aa to gain | of a cent isn each orange, which was at a gain of 3^%; what was the cotit of an °T^ u , . ^"^ 2 cents. ». Bo^ a horse at 33^ % gain, and with the hioney boiiglit another horse, which I sold for $120, and lost 'ii)%. Did I gain or lose by my trading? and how much? 10. Iflmakfeaprofltof 15/r56 by^selling paper' for $0.85 above the cost per ream, how mdch must I advance on the price to realize » profit of 32i^? ^ln«. 93|ct8. 11. What should I sell a barrel of flour for, that cost me £1. 2 6 to gain 161^? "^ ' Ans. £\ 6 3. 12< A neighbor offers his house, which cost him JCilDO, for 20 5^ less than cost ; what is his price ? Ana. $5520. 13. A merchant sells cloth for $5 a yard, which cost him $3.7.5 a yard 5 what is his gain per cent. ? Ana. 331 %' 14. I bought 640 yards calico at 15 cts. per yd., and sold it at a wdoiied pnce of 2* 5J ; what did I lose ? Ana. $2.40. ^ 16. A grocer sella coflfee at 7§d. a lb. which cost him 9d. ; what 18 his loss per cent. ? Ana. 111%. 16. A merchant buys at auction $9562.50 worth of goods ; it he sell them at an advance of 20^15 on the cost, what will be his net profits, deducting $600 for expenses? Ana. $1312,60. 17* How much should I sell different qualities of sugar whiTic«i is 10 % below the buying^ price ? An$. $28.2176, IsrBoiigl.la-hofBelorliaO, paid |r> for his nourishment' during 6 weekH, and then sold him for 1120; what was my lobs per cent ? iO. liou^'ht codfish at f4.25'the cwt.,^nd ^Id it at |4.l>3; what was my gam per fcent. 7 ^n,, jg i 21. A grocer sold tea whic^i cost .3d. IJd. for 3s. i>d. per lb. ;.sukr wluch r $189 ; a iMrt havinz been lost by leakage, be sold the remainder ar$3.99 per gallon, and found that hia loss was 5% on tlie cost; how many gallons did he lose by leakage^ ■ . -- in*. 18 gallons. 26. Sold a cargo of corn for £4000,1 j^t 26 ^-profit: what did the <»!:«ocoft? '*' • ::i„i. £3200. 2t>. In Ht'lhug tea at 90 ctfi. a lb., I gained 20515; -lioV much would Ana, :]?.i '■/(,. I tlie cost ? ich cost I have gained had I sold it at $1 a pound ? ^.,„ ., 27. I| V selling cloth at $4 the yard, I los^20 % ; what was t) 28. What Willi gain per cent by selling 'silks at $5 whi **'^^^ Ana 17i4^ 29. By selling lard at £1 168. per cWt. I gain 76 %^ how nluch* % would I cam or lose by selling it for 18s. ? Ans. lose 10 * 30. Sbld wheat at $1.25 the bushel thereby losing 15 * ; how much per cent, would I hymyniaM had I sold it at $l.647,V the bushel? 31. Lost 15 56r^ selling a lot of paper for $480; for hew much 32. Sold a field contaimng 106A. 3B. 30id., at $120 an acre, thereby maTimg a profit of 18 5^ on the cost j what'did the field cost ? 33. Tea, sold at 25 % loss, is $1.26 a lb. ; what would be the gain or loss per cent, in selling it at $1 .60 a lb. ? Ana.i% loss. «4. A lumber merchant sold 36840 feet of wood at £5 5 74 per M and gained 28 % j liow much wouW he have gained or lost br selline the wooii at £4 6 per M. ? •> - * 35. The retail prices of my 'goods are 40 56 above the cost. I supply my customers wholesale at a reduction of 12 % on the retail price • what is my profit on the goods sold by wholesale ? Ana. 2315^.^] '/ 36. 4in engineer sold an engine for $8812.60 and lost 6 * on ther' cost ; what should it have been sold for to gain 1 2 A 5|5 ? 37. I sold a horse at an adVance of 30 $6, and with- this money bought another which I sold for £46 10, losing 1 2i Ji^ : what did each horse co^t me ? An$. 1 st. horse £407 2nd. ho»^ £52. - 38. A speculator sold the goods of a store at a reduction 'pf 71 %, and realised a profit of 6 56 ; at what rate of redaction were tlie i^q • bought? ^ -^'^ ^W. W ■" '%J,V>.\>> 200 OOUMKBIOM AND fillOKEBAaE. arfLt Jf ^^!IP"t^ ^?^F^^ *^^<**^ " *^-^5 per yd., by which I make aprofit of 33y6. I sell 100yd. by wholesale at 30 ^ reduction on the Hi! Jo f* J7, " "^Bain or loss per cent., and how much do I receive a yard? • ^n«. 6i % loss ; $3..32i a yd. « ^w.flr' ?lf'5''*°l*'®^« '»"«" 2i eta. more than the cost and realizes a profit of 8 56; what 18 the cost of a yard? Ana. 31^ cts. .\1 f,S^°^^' demanded tot a certain quantity of prunes a price 22 % aWe the cost; but bfipg a little musty, he sold them at 10% less than hiaflret demand, and thu)8 gained $98 by the sale; what was his first demand? ,, ,^ "' ^„'g 11220 43. At what price BhoaW I sell 6odflsh which costs lea! 54d. per cwt. to realise a BTofit of 12i % on the cost, after deducting 1 h % of ™TifT°1^4 ^*?"*"*i'y of cheese at 12 cts. apound. Supposing the 7.r}^^ i^ !*** ^^^^ *^*^ calculated, and 10 % of the sales to be ZrT rA ^' ^P'^ ™"<'^ """^^ '^ be sold a pound to make a net profitofl4 5lSonthecdi % profit 5 what was their total profit ? Ans. $1482.00. COMMISSION AND BROKERAGE. das. Oommisslon and Brokerage are the percentages paid an a^nt, 9iiee. *^ t.^ Ji ~5r«^ *^'*" buBlnejB olBoe i> remote from a consignor is sometimes MrSe^Sr^"*' '""'"^ "" " *«•"* ''^ ^^^ ^"^ ^^° consigns to .rrti:""^Iir.*I'n-5f'i?!2.??''^°«u'" ""• °**""» °f *•"• "»'w «"»«! contracts they ^hS^L Tl»»".»|««-4i-o*«rtoonewho negotiates the disooant on bills of ex- onMge. etc ; a Rtai-Mtaf hroktr is one who nojcotiates the sale of houses and UaA, In,wnmc,.broker, SMp-broker. Slock-brolZr. Pawnbroker, etc. or !.*.«„ i!!'fl!Sy ^*^\u '"»'''•" of settling accounts between individuals, rf„.L?V *^.?°?T!f Of *»>e iowmment,as a d>Ue«or oftlu Pok,i,h« — **^* .^^ "•* Pnioiito MTfieliiHoaht reccTveOfom a sale or oolleotioo, less the oommission and other charges. QuestionB in OommisBion ant^. Brokerage follow the same rules as tiipee u» ?£«0QaUge. r licli t make ction on the much do I 32 i a yd. ind realizes .31^ Ct8. a price 22 % it 10% less ; what was 9. $1220. 3. 5^d. per g in% of ;i 1 1\\. pposing the ! pales to be iiake a net a pound, of $6840 ; jmainder at 51482.00. itages paid id is esti- 5 sale, pur- or Gom- 381^ for aa- id to traiuaot :h, can tran- «ndenpe are or; theper- 8 Sometimes } oonstgns to mtraots the/ bills of ex- houses «nd ). individuals, /whose ba- 'rom a sale lamo rules OOMMiaSION AI^D BBOKBRAQl. EXAMPLES FOa FBAOTIOE. 201 1. A broker sold $15800 worth of stock for Cj required his broker- age at J '^ ? Jiw, $39.60. To solve this Example, •<« Case I., 283, Rutii. 2. An agent received $1600 for selling a house and lot for $25600 ; what waa his rate of commission? Aim. 6^ ^. To solve this Example, tes Case II., 284^ Rau. 3. A commission merchant receives $84 for selling wood, at 6^ ^ ) what is the amount sold ? Ant. %\b'V!>. To solve this Example, •«« Case III., 288, Bo|^^ 4. An agent receives $3105' tp be invested in dry goods; ^fter re- taining his commission, 3^ ^jg; how much was invested 7 .^nlB. $3000. To sotve this Example, •<• Case lY., 288, RuLi. 6. What is the commission on $874, at 1\%'i on $71.50, at Z\%1 on $1580.70, at 4|%? on $309.10, at 5^%; on $4706.20, at ti ^ ? Ans. $19.66); $2.50^ ; etc. 6. What is the commission on £15 9 10, at 3%? on £170 10 6, at 4) 96? on £630 9 0, at6i56? on £96 12 3, at SJ 9^7 on £918 7 0, at6i96? , ^M. £0 9 3i + ; £7 13 5i + ; etc. 7. How much will I pay for the brokerage of $750, at \^\ of $1540.40, at i96? of $3610.80, at 1) )6 7 of $823.50, at |9g7 of $1560.70, at li96? ' An». $1.87A; $7,702; etc. 8. Sold merchandise as follows : Ist. for £942 16 0, at4)9$ com- mission ; 2nd. for £15 11 6, at 5 96 ; 3rd. for £310 6 7, at 6 9S ; 4th. for £530 5, at 3 i 96 ; what is the total com. 7 Am. £80 7 5^ -t- . 9. What amount of brokerage must I pav for exchanging green- backs, as follows: $590, at 26%; $746.30, at 28%; $1616.72, at 30 % ; $4532.09, at 32 % ; $87.30, at 29 % 7 An». $2322.385 + . 10. A farmer paid a broker \ % toinTest'$Il730, in Ontario bonds ; what is the brokerage ? Ant, $102,637 -f- . 1 1. A broker received $465 for buying stocks, 9,\. \% brokerage ; how much stock did he buy 7 Am. $74400. 12. A flour merchant remits to his agent in Toronto $4740 for the ptirchase of grain, after deducting the commission at 2 % ; how much will the agent expend for his employer, and what will be his commission? Ans, $4647,06 — , for grain ; $92.94 + for commis. 13. An agent sold real estate on 4% commission, and remitted $10095.36 to the owner as the net proceeds | for what price d^d he sell the property, and what was his commission ? .=^14:iAa. agent receives $4920 to expend 4n purchasing coWs at $32~ a head ; after reserving his commission, 2) %, how many cows did he purchase? An». 160. 15. A merchant having on hand 4700 barrels of sugar, save an agent 3^ % for selling it ; what are the net proceeds, if sold at $16 a bbl, T 'II .-.■i4i*;S&-.'j:.Tii.i!:s' l^S^ii 20^ COMMISSION AND BBOKBBAQB. 16. I purchased 6000 bushels of wheat in Buffalo, at $1.37^, and shipped the same to my agent in Kingston, who sold it at^$ii62i. How much dill I make, after paying $543 for expenses and acomrais- eionof24%? " ^n«. $723. 17. A broker charged me Ss. Sd. % for the exchange of £681 4 10, in greenbacks; what was his brokerage ? Ana. £.ib 15 3^. ] 8. A commission merchant sold a consignment of oats for $12686. He charged $66 for storage, ande^^ commission; what were the net proceeds? -dfW. $11827.124. 19. An architect charges | ^.for his plan and survey of a building which cost $24000, and li ^6 lor superintending the work ; how much did he receive? Am. $450. 20. I sent to ray correspondent in Bordeaux £2097 10, with advice to invest in the purchase of wines, &W'r deducting his commission of 3 J % ; what was the sum invested and what was his commission ? An*. £2026 11 4 J, wines; £70 18 7^, commission. • 21. An agent having a debt of $1670 to collect, compromises for 909(5; wliat was his commission at 5^1%? ^n«.$77.714. 22. Paid Folger Brothers $5.46 lor exchanging $364 in United States' money ; what was the rale of brokerage? Ana li%. 23. A consignee in Glasgow informs his constituent of the purchase of Dry Goods to the amount of £396 16 6; what is his commission at2i5i5? Ana. £8 18 1 + . 24. Bought at Halifax a cargo of wheat, 9500 bufhels, at $1,20 per bushel, and sent it to my ag-ent in Portland who sold it at $1.50 per bushel ; what did I realize on the whole after paying $320 for expenses, and commission at ■ii%? Ana. $2031.25. 25. My correspondent at Bordeaux charges $74,20 for purchasing 264 cwt. of honey at $10.60 per cwt. ; what was the rate of commis- «on/? Ana. 2U%. 26. A broker receives £2086 7 6 comprising the sum to be invei^ted in Railroad stock at £20 16 a share, and his brokerage at j^; Kqw many shares can he bay, and what is his brokerage ? 27. A certain piece of land was told for $3925, but the owner re- ceived $3866.1 2^ as the net proceeds; what was the rate of com- mission? Ana. li■ .37^, and acomrais- 8. $723. £681 4 10, 15 3^. or $12686. were the J27.r2i. a building how much 8. $450. -ith advice mission of. :88ion ? iiiission. • )mi8e8 for 577.71i. in United i purchase )mmi98ion 18 1 + . at $1.20 at $1.50 $320 for 1031.25. urchasing [' commie- € invented at i%; owner re- ! of com- »• H%. in Bank 'estment? nmission, 18. 2 %. a broker ; 1 9000. purchase !c>ires me >r at i% mmiesioa commiB- £140. png it to FIRE AND MABINK INSimANOB. 203 gold ; and, after deducting the commission at 2 ^, employed the bal-. ance in the purchase of fruit; what was paid for the fruit: and what was the commission? An8^ $661.99, fruit; $13.51 commission. 34. Remitted to my correspondent at Rouen £255, for the purchase of oalico at 9d. per yard, after deducting hiscomnjisflion at 2 ^; how many yards will I receive ? Arts. 66663yd. .i5. A speculator receives $4113.50 as the net proceeds of a sale, allowing 5 % commission ; what was the value of the property? 36. A coTii mission merchant who charges 5 % commission on sales and investments, receives 260 cwt. of cheese, at 6d. per lb., and £748 10 6, m ca8h, with advice to purcha.se a cargo of cotton for the whole anroupt ; what will be his total con mission ? Ana. £97 10 1 Ifi .37. A Halifax agent buys 34 boxes of chocolate; he pays $7.50 for freight and cartage, and his commission is 1^% on the amount of the purchase. He sends me a bill of f740.83| for the whole ; what was his commission; and, allowing 2501b. per box, how much did I pay per lb. for the chocolate ? Ana. $10.83| com. ; $0.08^ per lb. 38. A commission merchant receives 126 barrels of flour from A, 150 bbl. from B, 225 bbl. fr6m C; he finds on inspection that A's is 10% better than B''8, and C's is 5^% better than A's} he sells the whole lot at $7 per barrel, and charges 4% commission. How much must be remit to each ? Ans. A, $842.30 j B, $918.87 ; C, $1598.83. INSURANCE. * 841. Insurance is a oontraot of indemnity, by whioh one pMty engapes, for a stipulated sum, to insure another against a risk or loss to whioh he is exposed. 34S. It is of two kinds : insurance on property, and insurance on life (1). 843. The Insurer or Underwriter is the party taking the risk ; and the Insured or Assured, the party protected. 844. The Policy is the written obligation or oontraot. 845. Premium is the sum paid for insurance. It is alwaya reckoned at a certain per cent, on the yalue of the property in- sured, varying according to the degree or nature of the li^ aa- sumed. ; FIRE AND MARINE INSURANCE. 84«. Insurance on property is of two kinds: Fire Iim&emefif and Marine Insurance. f>rE ',-»■' ft4T. nrilhsnraBCSli an Indenmificaiion of dnmage and loss caused hjjire or lightning. (1) LUb ioaoranoo will be tnat«d of Ii is44i.'iM'.Ali.imMlA,-k''. M,lMi!.'.~t ■lSAii:'^-:^MaiXL , 204 FIRE AND MABIint IITOURANCE. 'n.^t?' ^"inolMWWCeisan indemnification of damage iill T'^^ ^y *^« P^"'« l^^liar to navigation, principles "*''"'^°''®' *^^ calculations are baseij on the following I. Premium ia percentage. (278) TTT mt® ^""^ ^"""■^'^ " *^« ^« of premium. 111. A he sum covered by insurance is di/«rfnc«. EXAMPLIS FOR PRAOTIOB. w»4ouw, atJi%? An».m2.50. To solTo this Example, tMCaie I., 282, RnL».„ ' wtiat was the rate of insurance? ^^, » ^y ' To soiro this Example, .li^CM© II., 284, Run. w^^p'atT^r Fr"""" '"''5 '?«""'ig a tannery for | of its value, at 12 <^ was f I46.t,0 ; required the value of the tannery. Ans.$m4*r! To Bolre this Example, «ee Cue III., 288, RuLt. 4. What muet be paid for an insurance of $5728 at 1 J ^ ? 6. What premium must be paid for the insurance of a vessel and cargo value! at £364J| 8, at .sf^? Ans. £1 18 12 11 + W ilf Tf' jn^n^^* for»6000, at 2^ 5)5, was con.pletely wrecked • how much of the loss was covered by the insurance ? A. $4887 50 ' 7. A ho.tel valued at £3760 is insured for i of its value, a 8« The policy and survey oflhe premises are charged 78. 6d. what is the insurance? . ^ AnsMlbO •su?;Af9'"^?'^'*''*?\"*''**''^^'^"'>5 ^*^at sum 'must be 'in- ^9 ' wd?' *ft;*^''" ^*^ property and premium? 4w. $6500. in T *' " *"® premium of insuring £696 U 8, at £6 13 9<*;? 3iof5h^I*"T^**urT"~"^'°'"'J^"*>™'7'*nJ tl^« sum is n WK 't™*'"ru ^"'^ ''*''.*'^ ^ *™ ^°«"«d 5 ^hat Is the amount ? £Q ifl b!^o '* ^* PWmium for aa insurance of £1486 1^ 9. at 19 A?. ^ . * iliw.£66 19 9i + . «„i ; r business man, having $12000 worth of goods, gets them in- So o?tL°lfif^ "*K"f' *S*.^' in in a conflalSio'nAe saveS but * , S" S^ *^^ ^^<^^' ^^a* re*^ low will he sustain ? Ana. $472 13. Por what sum must a house, valued at $8274, be insured, at 1 A 95, to cover the entire loss, in case it ia destroyed by fire? A. $8fto 14. My goods are worth £1663 12. For what sura must I insure them to cover, in eaae of Iom, both premium and propSy, li il J^ «he pre mium of a Bchoo^houBe,inaared at U^ ^faliS^ V ==whHt renrwas iTinsOTetfT — —- ^^ v-Af^M »w^^ior offr«I!ium ? ' ^*' the i at 3 56 ; what is the balance ppr ja^ik'ai:fe|Sl^.it?.ai;Sii^iM.iiiiafe«J J,*, ^88B88MKNT OF TAXES. 205 of damar;o be following the {amount ^12.50. his house; 'e, at n^ '.$11648. ; 6? vessel and 12 li + . y wrecked : 14887.50. 'P. at 1 51$. I- ; what ia £9 16 0. lust be in- . $6500. 13 9%? jis sum ia lOUOt ? 5 1^ 9, at 9 H + . h them in- I saves but ». f4T3. ured«,atU ■ IS^OO. t I insure tj, at £2 £1600. 17. For what sum must goods worth £1938 12 6 be insured to cover both premium and goods in case of loss, the rate being SJ %? 18. A bng estimated at $40000 is insured for | of its value at 1 i %, and its cargo, worth $36000, at f % ; what is the insurance ? 19. A merchant paid $1450 for premium of insurance on a cargo of cotton coming from Havana, the rate of insurance -being 2^%; what was the value of the cargo ?* Ana. $68000. 20. I paid $18 for an insurance of $1200; what is the rate of the premium? _ Ans. l^%. 21. To £579 16 10, add 7J5(5 commission, and find the insurance of the sum, at ^%? . Ans. £27 5 4| + . 22. A merchant, having a cargo of 500bbl. flour, has it insured for 80^6 of its value at SJ %, and paid $107.25 for premium ; what was the price per bbl. ? Ans. $8.25.'' 23. A ehip-owner has two of his vessels insured for $30000 in the Royal Insurance Co., at J 56, and for $45000 in the Colonial Insurance Co., at ^5^ ; what is the rate of premium for the whole insurance? 24. A house estimated at £300 was insured for (i of its value, dur- ing 3 years, at 1 96 per annum. Towards the end of the third year, it was destroyed by fire ; what is the actual loss of the proprietor without any allowance of interest? i4n« £106. 25. My house was insured for $45000 during 5 years. The first year I paid $1.50 for the policy and plans, aqd | % premium ; every succeeding year, I paid i 56 premium. The house having been de- Rtrojred tlie fifth year, what was the loss of the insurance, no interest having l)€on allowed? iln«. $43817.25. 2(5. I piii'i $46.75 for insuring a store for the ^ of its value, at 1 g % ; what is the store worth ? Ans $6800. 27. I took a policy of £3011 5 for the the value of both property and preriMum; what is the worth of the insured property, the rate being 1 56? , - Ans. £3000. 28. A shipment of wheat was insured at 2f %, to cover | of its val- ue ; the premium paid was $44.07 ; the wheat being worth 80 eta. per bushel, how many bushels were shipped ? Ans. 2825 bu. ASSESSMENT OF TAXES. s S^O. A Tax is a sum of money assessed on .the penon or property of an individual, for public p9rpose8. SSI. When a tax is assessed on property, it is apportioned at a oertain per cent, on the estimated value. When assessed on the person, it is apportioned eqxmUy among the male citixens lid)le to asse^iment, and is called a poll tax. odd. Property is of ttro kinds, Tnf^rrecr^ tsttUf, vnAjMrsoimt' ~' property. f 853. Roal EstGLtO is fixed or immovable property, snoh as lands, houses, etc. t«^ $4000. tofbaild-. le balance r. $360, i*u,S «.. • ..*«.-. 206 A88X88H1NT 0^ TAZtS. 854. Personal JProperty ia nuyvahh property, such as money Btooks, furniture, cattle, etc. 855. An Inventory is a written list of articles of propertv with their value., ^ •" 356.' A Schednle is a list of taxable property with its owners' names and 4t8 value as estimated by assessors. ^857. Assessors are officers appointed to make out a schedule of taxable property, and apportion taxes thereon. Ex. A tax of $840.76 is to be raised in a town containing 65 polls • the taxable property of the town amounta to $48000, and eacH poll tax 18 75 Qt8. : what will be the tax on a dollar, and how much will be C 8 tax, whose property is valued at $5600, and who pays for 2 polls? OfERATIOK. . f 0.75 X 65 = $48.75, amount asseseed on the polls. |840.76 — $48.75 = $792, amt. to be assessed on the propertv. $792 .-r $48000 = $0.Q166, tax on $1. prupenj. $5600 X 10.0165 h= $92.40, C's tax on property. $0.76 X 2 =» $1.50, C's tax on 2 polls. " *" ' $92.40 + $1.50 ^ $93.90, amount of C's tax. Hence the ' 858. RoLB.— I. Find the amount of poll tax, if any, and aubtract it from tJie whole tax to be raited; the femainder will be the property tax, II. Divide the property tax by the whole amount of taxable property; the quotient will be the per cent., or the tax on Zl. III. Multiply each man's taxable property by the tax on $1 and to the product add his poll tax, if any ; the result will be the whole amount of hi* tax. BXAMPLIS rOR PRAOTIoa. 1. The tax assessed on a certain town is $1485: its property both personal and real, is valued at $42000, and it contains 300 'poUs. which are assessed 76ct8. a piece. What per cent, is the tax: thai 18, how much is the tax on a db\l&r ; and how much is A's tax who pays for 3 polls, and whose property is valued at $2250 ? o wv . • .V . , Ana. 3 cts. on $1 ; $69.75, A's tax. Z. what iS the tax of a non-resident, having property in the same town, worth $7900 ? 4ns. $ 3. How much will B's tax be, in the same town, who ijays for 3 polls, and whose real estate is valued at $32000, and his personal property, at $18880 ? Aim. $ 1628.66. 4. What sum must be assessed in order to raise a net amount of $11123, and pay the com mi ssion for col l ecting a t 2 \%7 6. The' expense for repairs of a publio building was $2521.06^, which was defrayed by a tax upon the property of the town. The rate of taxa- tion was 3^^ mills on one dollar, and the collector's comtaission was 3^ %"f what was the valuation of the property? Aim. $803843.69 + OUSTOM-fiOUSl BUSIN188. CUSTOM-HOUSE JUSTNESS. 207 359. Duties, or Oustoms, are tazps levied on imported goods, for the support of government and. the protection of home industry. 800. All goods coming into the Dominion of Canada from Foreign countries are required by law to bo landed at t^rtain places or pOrts called Ports of Entry, Every Port of Entry has a Cuttom-jHome. 861. A Cnstom-Hoase is an office establiehed by govern- ment for the transaction of business relating to duties. The t>ffi- cers attached to it are called Custom-House Officers. Their businoes is to inispect the cargoes of all vessels entering at any of these ports ; to inspect the invoice of goods, collect the duties, etc. NoTCa.— 1. — Beridea the dntiea on merchandise, all veueb engngod in xom- merce are required to pay certain charges for the piivilege of entering the port, etc. ; these charges are called harbor dues. 2. 3^0 carry on foreign commerce secretly, fritbont paying the duties imposed by lav, is unuggling. 862. Duties are of two kinds — Ad Valorem and Specific. 868. Ad Valorem Duty is a cef tain per cent, on the cost of goods, as stated in the invoice. 864. Specific Duty is a tax computed on the weight or njieasure of the goods, without regard to their cost ; hence, allow- ances are made before computing the duty, 865. An.InVOice is a statement of goods, from the seller to the buyer, or importer, showing the quantity and prices of the articles. 866. In the United States Custom-Hooses, certain legal al- lowances are made for draft, tare, leakage, etb., before Hpecifio duties ate imposed.' In Canada, however, these are not known, the tare being found by actually weighing one or more of the boxes, etc., containing the goods, and the leakage by gauging the cask. Non.— At present, the varions kindi of spirits are the only articles npon which speeifio datiei are charged by the Canadian Tariff. 367. — To compute ad valorem duties. Ex. What is the ad valorem duty, at 18 %, on an invoice of merino which cost 1256.50? OPIBATIOH. $266.60 X. .18 = $46.17, Atu. ""^ AXuniV. — Aeoat^g to Caae X,' (282^, we multiply the inroiCe, $256.(0, whicn is the bate of the duty, by ttm Siren rate, and obtain tbe duty, $46.17. encetbe "I 'yA: i L, .■ .;*:- )H':...,:^.m 2oa r! OUSTOM-HOUBt BUBUfttS. *i**^* ^^^^'-^^f^ *f^ P^oentaae on the invoiced value of t^egoodl at the given rate of tariff, and the remit will be the ad valorem duty. 869. To compute specijic duties. l^S"^*!.^^**"^^*.!^"*^ *l" * hogaheada of eugar, eaoh weighing 1^80 lb., gro88 weight, at 2| cts. a pound ; tare 14 56 ? OPERATIOK. 61201b., groea t^eight 1280 X ,6120 X .14 = 716.8'lb:, tare. 6120 - 716.8 = 4403.21b., net value. X .02| = $121,088, dutj. 4403.2 Analtsis.— We flrat ilndths whole weight of the invoice whioh is 61201b. From this amount we deduct the allowance for t«re, 710.8l^b., and oompute the duty on tlie remainder; iie&oo the following >i:*??*'"^^J'''T-^*'^"<^°''''""*"*^*'» if necestaru, and compute the duty, at the given rate, on 'the net value. KXAMPLES^^FOB PBACTIOE. • ^*- Y}^. i «*^.® *^ valorem dotj, At 19 %, on 15780 lb. of cordage. mvoicedatlficts. perlb.? - iln«. $449 73 2. At 7 ct8. a pound, what ia the np^ific duty on Sie-'kees of "to- o*^/*^n^ weighing'130 lb., allowing 6* lb. per keg for tare? d. At 30 cte. per gallon, what is the epecific duty on 40 hhd. of wme, each gauging 58 i gallons? ^ •i«a^«A*,' " **** ^"*^ ** ^^ ^' °° * ^^e of Holland linens which cost Kw2.t*uj. .«« iltw. $525.85*. 6. What is the duty, at 20 %, on an invoice of broadcloth which ®^' >n i^iverpool £657 1 0, the pound sterling being valued at $4,863 ? 6. What 18 the specific duty, at 10 eta. per lb., on 25 chests of tea. eachweighmgl201b*} tarelOjft? ^ 7. What was the rate <)(, of duty on whose invoice value was $2250 andfor which $337.60 duty was paid? Ant 15% ' 8. A merchant imported 64 casks of Hrine, each containing 42 gal. net, the duty at 30 % amounting to $1036.80 ; at what price per hi. was the wme mvoiced ? *^ ^ . •*, t "^'^^if "' '° Montreal makes an importation of goods invoiced at $16448. On goods mvoiced at $2400, tlie duties were at the rate or4%; on goods invoiced at $3360, the duties were at the rate of 15%; goods invoiced at $4800, were free of duty: and on the 're- mainder,, the duties were at the rate, of 30%; what was the whole amount of the duties ? 41,^^ 1 2366.40. 10. What is the duty at 18 % on 60 kegs of prunes, each weiLhine J. < 5 wt^ invoiced 4tt7|^^t8.Ber^Jfe. - tare at 31 % ? -^ ~ — ^ 11. A. TTamol A: ntv> i%/n..„U«L : i A'^ »r^ .y-,4- Hamel ft Bro., of Quebec, import from Manchepter46 pieces OAA §'*",®'^"'& *•* y^- e*ch, purchased at 5b. per yd,, duty 24%; 300 yd. Of merino, at 48. per yd., duty 19 % ; 160 yd.lriah linen,lt I valxie of be the ad 1 weighing flrat And the nroice whioh J amount we ice for t*re, ite the datjr ile&oo the i compute •f cordage, $449.73. tegs of to- ire? 10 hhd. of wliich coat 525.85i. oth which at|4.86|? ists of tea, vas $2250, ». 16^. iig 42 gal. e per gal. Is invoiced t the rate le rate^ of »n the 're- :he whole !366.40. weighing is pieces ut7245l5; linen, at DISCOUNT Ain) pRisnrr Worth. 209 28. 6d., doty 16 9$ 5 and leather to the coe^ of £90, doty 4»do at different times without Interest, find the present worth of e.ioh paymeut separately. Their saoi will be the present worth ««™ Jr^M r*'""^u''' *"'' '•"'« •""' subtracted from the sum of the sereral payments wiU leave the total discount. "»"»« EXAMPLtS FOR PRACTICE. : What, is the present worth of the following notes ; (1) 1. Dated Feb. 3rd., amounting to $104.60, on 6 months' credit, dis- counted June 6th at 5 % ? .Ifti. $104.20 + . 2. Dated March 4th,^ amounting to £58 10 6, on 7 months' credit. d.ecountedAug.lOth., at4 56? Ana. £bH 3 6+. A. Data! Apnl 2;id., amounting to $206.16, on 4 months' credit, discounted May 30th., at 4^ ^ ? Ana. 1204.564 + . 4. Dated May 16th., amounting to £135 9 0, ou 8 months' credit, discounted Nov. 16th., at 6 56 ? ' Ana. £134 2 2 + . 6.. Dated Aug. 7th., amounting to $8000.00, on 6 months' credit. diwsountedDeo. 6th., at6 56? /Ifw. $7931.699 + . 6. Datedyan. 3rd. amounting to £90 3 6, on 9 anonths' credit, discounted Sept. 20th., at 7 96 ? ^n*. £89 18 111. 7. Dated June 14th., amounting to $1660.90, on f months' credit. disoountedAug. 2nd., at6 56? ^iw. $1660.049 + . 8. DatedSept. 8th., amounting to f795.10, ou 10 months' credit, discounted Feb. I2th., at b% ? Ana. $779,297 + . 9. Dated Nov. 25th., amounting to £875 6 8, on 7 months' credit, discounted May 1 1th., at 6 9^? Ana. £868 19 21 + . 10. Dated Dec. 6th., amounting to $630.50, on 11 months' credit, discounted Sejof. 18th., &ib%'i Ana. $626,324 + . 11. Dated Oct. 9th., amounUng to £95 16 0, on 9 months' credit, discounted June 7th., at CJ % ? Ana. £95 4 6 + . 12. Dated July 16th., amounting to $208.95, on 5 months' credit discounted Oct. 12th., at 4| % ? Ana. $207.20 + 13. Dated March|2nd., ^mounting to £140 16 4, oroS moa.' credit, discounted Sept. 28tir, at 6^ 56 ? Ana. £ls39 19 1|. 14. Dated. Jan. 7th., amounting to $780.50, on 11 months' credit, discountedNov. 3rd., at7i^? iln». $775.19 + . 15. Dated April 10th., amounting to £780 5 .3, on 10 mos.' credit, discounted Dec. 4th., at 4J^? Ana. £773 10 61.^ 16. .Dated May I7th.,_amountiogio f436.76. on 3 aioatha' credit. -=x discounted June 22nd., at 6i 56 ? Ana. $433,110 + . 17. Dated March 14th., amounting to $600.00, on 7 months' credit. discounted Sep t. 7th., at 7 % ? Ana. $595,714. XO We leokoil only 30 days to the month for all the notes in true dlsooont. DIS^COtlWT '» Aip PBSnNT WORTH. 211 to the rate ie ducount' iqaivalont to d amount are Dreat bjr lub- irest, find the nrcsont worth r tho foreral credit, dis- 04.20 + . iths' credit, 3 6+. the' credit, 4.564 + . the' credit, t 2 2 + . tha' credit, 1.699 + . is' credit, 18 Hi. tha' credit, 9.049 + . ha' credit, J.297 + . the' credit, 9 2i + . ha' credit, 5.324 + . ^ tha' credit, 4 6 + . hs' credit, 17.20 + . 09.' credit, I 19 1|. ha' credit, '5.19 + . 08.' credit, : 10 6i. bs' erediiF 1.110 + . tha' credit, >95.714. Ibooont. T.^.- 18. Dated Feb. 9th^ amounting to JE850 18 0, on 5 montha' credit, discounted April 13th., at 7i * ? Ans. £835 IH 6i + . 19. Dated Nov. 11th., anioiAiting to $176.30, on 7 months' credit, discounted Mav 4th., at 6 % ? ^ Ans. $174,225 + . 20. Dated March 6th., amounting to £701 9 6, on 4 mos.' credit, discounted June 9th., at 7i ^ ? Ana. £697 11 Oi + . * 21. Whatis the preseBt worth of $117.60, payable in 1 year, at 12^1 Ana. $105. 22. What ia the present worth of a debt of £96 6 64, due 5mo. 15da. hence, at 6^? .Injr. £93 15 0. 23. What should be the discount on $373.75, paid 11 mo. before the term of maturity, at 6i ^ ? Ans. $21.01 + . 24. What ia the discount on £200 12 6, at 74 %, payable in lyr. ? 25. A note of $139.94 ia payable in 9 montha ; what ia the present ' worth, diacouiU being 5%? Ans. $134,881 + . 26. Discounted a note of £76, payable in 4 years, at ^i%; what sum shall I receive 7 , Ans. £(il 9 6 J. 27. Whatistheactualdiscountofanoteof $429.98 J, due in lyr. 6mo. Ida., at5i56? Jna. $32.82 + . 28. The sum of $195.10 ia payable in 13 months; what will be the discount, at 4 %, by immediate payment ? Ans. $8.10 + . 29. What is the present worth of £169 13 9, payable in 3yr. and 7mo., at 7i % discount? 30. Bought clpth, on 21 months' credit, for £140 7JJ; how much ready money will acquit me of the debt,^ if 1 5(5 discount per month, ia allowed 7 Ana. £129 3 74: • 31. 1 sold a house, which cost me $2964.12 ready money, for , $3665.20 payable in lyr. 6mo. ; what will be my gain, in ready money, by discounting at 8 56? -4"*- $->08.38. 32. I bought silks for $43713.60, on 15 months' credit; but, by paying before the time due, I will obtain 5 % discount; at what epoch should I pay the debt, so aa to disburse but $41632 ? Ans. In 3mo. 33. A 4our-mill was offered for $2500P cash, or i^r $12000 payable in 6mo., and $15000 payable inl5mo. Accepting the latter condition, I would like to know whether I gained or lost, and how much, money being worth 10 ^ ? " > Ana. Lost $238.09 + . 34. Louis bought goods to the amount of £8 2 63% pn 20 mos.' credit; at what time did he pay^ knowiijg that he obtained | % dis- count per month, and-that he disbursed but £75 19 ? Ana. 8mo. 36. A merchant gave out two notes : the firsts of $243.36, payable May 6th., 186T; the seooud, of $178.64, payable SepU 25lh. 1867; what sum is' required to pay the two notes Oct. 11th., 1866, discount at 7^7 36. What quantity of produce must be bought at 69. per lb., on 22 months' credit, in order to pay but £50 19 lOJ, after deducting the discount at 7*? , , ^ ', , , 37. Ok 9 months' credit, I baight 120 bales of cotton, each bale ~ ^ ' — ~nw^ wefghing 4881b., at 5Ad. the lb. ^SHnginninjedijftely forjtttirH oasb, I paid mj own debt, and refieived 856 discount ; how much did I rainT Ana, £390 8.. 1 m •A. 212 . r t. -si-'' :• BAHK DIBOOUNT. 38. IpaMf320lh» a sum I owed; what waa thi« BOm, knowing that 6J % discount vt* allowed T Ant. 933Gi80. 39. Paid £23 16 for 60yd. of cloth} having received 6 56 discount, how much djd it cost me per yard ? Ana. 9b. 1 l^ftd. 40. IsitTOore advantageous to purchase flour at $6.25 per bbl. on 6 months' credit, or at $6.60 on 9 months' credit, discount being An$. Flour at $6.26 is the more advantageous. !^ BANK DISOOtNT. « 374. A Bank is a corporation, legally established for purpose of receiving and loaning money, and of furnishing a paper circulation. 375. Bank Notes, or Bank Bills, are the notes made and issued by banks to circulate as money. They are payable in specie at the banks. Obs.— A bank wliioh ino^i notei to oiienUte u money, ii oalled a bank of iMue; one whioh lends money, a ba»k o/dUootml; and one which takei charge of money belonging to other partiee, a banh EXAMPLES VOB PBAOTIOB. Jind the 1. What is the discount and what are the process of a note of $1000, due in 60 days, at 6 51J ? Ana. Dis. tlO.60 ; pro. $989.60." 2. What is the present worth of a note of £2000, payable in 60 days and discounted at the Quebec Bank 7 Ans. £1979. .3. Desiring to loan £260 of a Montreal Bank which dispounts at 8 5l5, I gave my note for £243 16 payable in 60 days; how mpch must I add to complete the amount I require 7 Ans. £9 13 3. ^4. A man sold his fkrm, containing 195A. 2R. 25p,^at $27.60 an 'iKre^ and received a note payable in 4 mo. 15 da., at 7 9^ interest. Being in irai;nediate need of money, he discounts the note at a bank; how much did he receive ? Ans. $6236.168 + . 6., Find the day of maturity, the time of discount, and present value' of the following n(^ieb : — £40 2. Quebec, Deo. 3rd., 1868. Six mojitha after date, for value received, I promise to pay Daniel heS'S^ou, or order, forty pounds and two shillings currency, at the Btolrof^rtbefc v ; - A. T. B.mitiStt^=^ .Discounted, April 3rd., 1869, at 6^. Am. Doe June 3 | 6 1869 1 term of dieo. 64da. ; pro., £39 13 64 + . ■^l ■ • ^i 4 t/^^r^ ' I, BANK DISCOUNT. ,$1066,T^ff. Montret Ninety days after date, we promise to pay C. Simson, one tboasand sixty-six and ^^^ dollars, at the Union Bank, for value receired. ^ Rappb, Wcbbeb, k Co. Disconnted May 8lh., at 7 5)5. Am. Due July 18 | 21 ; term of disc., ?4da.; proceoda, f 1061.40 + cotttT^isij/t'^smoraTT^^r"*''''"^'''^""* ^'^'^ ^*"^ *"^ ^.mt^ofi'i'^fo'T^^'*'''?*'^**'"*'^^^^^^ a°d the bank aisBcmnt ol AJOOO 9, for 6 months, at 3 5^ ? 38J^. Cass l\,—Tht ^oceed* of a note being gi^en, to find \. the /ace. cofT«lYt^fbanW*T^°**'^\^±P*y*'''«'°«« days, which dis- counted at a bank, at 6 %, gives 1989.60 for the proceeds ? OPBBATION. AHAliWM.^SlB««tO.»8»5fathei*rO. $1.00dB «M«of $1, the notoof which $889.61} U Int. offl for 63 days .0105 SfJ£S!?^'??'*l**"™"f «'»"»"■• ■D J ..*■ W.98M M contained in $»8»io. HeDea Proceeds of $ I fo.9896 the «»~«« « ♦w».og. ueooe 989.50 -v- 0.9895 .■= $1000, Ant. ./•???' ^,^^^;-^*>»*^ the proceed* of the note, hy the proceeds 0/ 91, for the time and at the rate mentioned; the quotient will l>e the, face of the note. By proportion. ^ 100 - (6 X ^) : 989.60 :: lOO : * =5 theflwe, KXAMPLKS FOE PBAOTIOB. 2. A merchant desires to draw $5000 from a bank anr] «./♦».;- 4. Bought goods at Toronto for the sum of $1486.90, and nv« in gx A m erohant^wk hca to bor r o w f 760 «k b>nki -tH«t%hrtrifet 'i^ the face of his note, pa;AWei?I^'2wS^^ I li ^ «*!?,"y r^ *^ ®'^.**'V» ** •debt of I168 leYtf SJSSS^il X i {It monthly, what was the face of th« note ? * " » " awoount w Amt 96.20 ^£l.^^S!l.d^h* "o** »iog.ao' I i:ii - "«*"» laoe oi me note »10g.ao. Cj O Wfrh*rrtfaeiirtlirfWi8raien<^ TSK MOff.SO. thm t.,,.^^ MOA T"! Amt '• " 9106.20 16.20 + 0.278775 = 22i^5«,aii#, fl0«.20, the Intemt, W.20,Td the time, 93 dan. to And the rat» per cent., which ia dctMaoocnUnc toUMpsNodiagoMe. Bmm* tlw • IsjEKd t JSM.i£iJ^\ Uii>^*i. > 216 PROMISCUOUS EXAMPLES IN DISCOUNT. 887. Rule. — I. Find the interest and the amount of $1 or $100 /or tJte time the note has to run. II. Divide the interest by the interest of the anwunt at 1 %/or the same tima. ^ By proportion. ^ 100 + (24 X T^^) : 100 :: 24 : j? « 22^56, Ana. ' EXAMPLES FOB PBAOTIOE. 1. At what rate of bank discount must a note, payable in 60 days, be discounted to obtain 6 % interest? Ans. 5JJ|^%. ■^,^- 2. At what rate must a note, due in 30 days, be discounted to ob-f^P tain 6% interest? Ans. ^%^i'j6^ 3. At what rate must a note, payable in 120 days, be discouDti^ <$. ' 4. What rates of bank discount, of notes payable in 30 days, cor- respond to 5, 6, 7, 10 ^ interest? Ans. 4^^^%, 6iMt56, etc. 6. What will be the rate of bank discount, on a note payable Syr. and 4mo. hence, without grace, corresponding to 6 ^ interest ? 6. At what rates must notes, payable at 60 da^s, be discounted, to pay a broker 1, 1^, 2, 2^% per month? An$. U^fSf^lf), etc. PROMISCUOUS EXAMPLES IN DISCOUNT. What was the present worth, at true discount, of the following notes, when discounted: — 1. Dated Feb. 3rd., discounted June 6th., amounting to 1313.80, payable in 5 months, at 5 5l$ ? Ans. $312.62 + . 2. Dated March 4th., discounted Aug. 10th., amt'g to £175 113, payable in 7 mo., at 4 9^ ? Ans. £174 10 3 + . 3. Dated April 2nd., discounted May 30th., amounting to $618.45, pavable in 4 mo., at 4J^? Ans. $613.55 + . 4. Dated May 15th., discounted Nov. 15th., amt'g to £406 7 0, payable in 8 mo., at 6 ^ ? Ana. £402 6 6^ 4- . 5. Dated Aug. 7th., discounted Dec. fith., amounting to $8000.00, payable in 6 mo., at 5 56? Ana. $7931.69 -t- . 6. Dated Jan. 3rd., discounted Sept 20th., amt'g to £270 10 6, payable in 9 mo., at 7 ^? Ana. £269 16 10^ + . 7. Dated June 14th., discounted Aug. 2Qd., amounting to$4682.70, payable in 3 mo., at 6 %? Ana. $4650.14 + . 8. Dated Sept 8th., discounted FebVlSth,, amounting to $2385.30, pa yable in lOmo., at 5q^? Ana. $2337.89 + . trDat^lTovrfSth., discounted May TIffi.7imt*g lo"^«2Baf «"3, payable in 7 mo., at 6 9^ ? Ana. £2607 2 10^ + . . 10. Dated Deo. 6th., discounted Sept 18th., amounting to$1891.60, |»yable in 1 1 mo., at 6 % ? Ana. $1878.97 + . mt of $1 or ntatl %/or Aru, e in 60 days, mnted to ot>-*^3Hl discountM |(^? gQsoag flC < J oa T 3 f 8 V- iO days, cor- M* 56. etc. payable Syr. terest? liscounted, to Wr 515, etc. NT. lowing notes, ; to 1313.80, $312.62 + . o£176 11 3, M 10 3 + . ig to $618.45, $613.55 + . to je406 7 0, 92 6 6V + . to $8000.00, 7931.69 + . > X270 10 6, 16 10i + . g to $4682.70, 4650.14 + . ; to $2385.30, 2337.89 + , r 2 ioi + . gto$1891.50, 1878.97 + . ''h^^^.J^.iiXi *^ 'dsmtffiJAS PROMISCUOUS BXAMPLE8 IN DISCOUNT.' 217 What were the proceeds, at bank discount, of the following notes, when discounted: — 11. Dated Oct. 9th., discounted June 7th., amounting to £287 6 0. payable in 9 mo., at 6i % ? " Ans. £285 10 1 + . 12. D.ated July 16th., discounted Oct. <12th., amt'g to $626.85, payable m 5 mo., at 4| 51$ ? iliw. $621,225 + . 16. Dated March 2nd., discounted Sept. 28th., amt'g to £422 9 0, Ans. £419 11 01 + , payable in 8 iqo., at 6^ % ? _..„. ^.^ .. «, -r . 14. Dated Jan. 7th., discounted Nov. 3rd., amounting to $2341.60. payable mil mo., at 7i 96? ^na. $2324.062 + . 15. Dated April 10th., discounted Dec. 4th., amt'e to £2340 15 6. payable m 10 mo., at 4| 9$ ? Ana. jE23l8 16 11 + . 16. Dated May 17th., discounted June 22nd., amt'g to $1310.26. payable m 3 mo., at 5i^? ilfw. $1298.439 + . 17. Dated March 14th., discounted Sept. 7th., amounting to $1800. P*(«^ n'?J r^' f u^ V. ^»»- »1786. ' 18. Dated Feb. 9th.. discounted AiJril 13th., amt'g to £2552 14 0. payable m 6 mo., at 7^ ^ ? Ana. £2504 16 8 J + . ' 19. Dated Nov. 11th., disdounted May 4th., amounting to $525.90. ^l^\''ilT''t^^^'^., iltw. $522,306 + . .^0. Dated March 6th., discounted June 9th., amt'g to £2104 8 6. • P*??^'?.'"« "°/' ,^ 7^ ^ ' ^^' *2091 6 51 + . l, T ^"*'° ^2'^-' discounted at a bank, at 6 9g, a note of $705.60. payable June 28th. ; what sum did I receive? Ans. $692,646 + . ^ 22. A bill on 4 months' credit having been discounted at5i, bank oQ"m*L **" 'educed to £37 5 4U ; what was the amt. of the bill ? ji. Ihe contract for a public school was given to a builder on the deduction of 12 95 of his tender. The building being finished, he was ordered to do extra work for $1529. Required the amount of the extras, so that the contractor may receive the $1629, after deducting *Hi^^J . . ^«»- «n37l60. * xt A^^ proceeds of a note, payable Aug. 2nd., and diso^unted May 9th., at the bank, are £39 9^ ; what is the fece of tW^ote. discount being 6 56 yearly ? 4IM. £39 1 2 41 + . •1 oo'oA ^^^ o ^^^ of $514.22 as foUowa : $208.32 payable in lOmo., $123.20 m 18 mo., and the remainder in 22 mo. ; if I can obtain trua discount at 4 %, how much must I pay ? Ana. $488,043 + . 26. A bill amounts to £300 7, and the discount allowed is 21% • to what sum is the amount of the bill reduced ? Ana. £292 16 9|. ' -27. What is the^nresent worth of $769.60, due 3 years and 6 months hence, at 6^? iiiw. $638.67+. iS. Faul invested m business the sum of £1441 10 payable in 3 years, and is at liberty to advance the payment at the rate of i%. bank discount, per month, without days of grace. At the end of 16 months he gave £716 2 6 j in what tim e did>e balance the remainde r. — knowing that he disburwa but £532 T?Aik^ After Wina JO di.~ 29. The sum of $1720 is payable in 1 year, and $10900, in 18- months : but by paying immediately, 6 % true discount, on the first •am, and f i {» on th« Mwnd, md ba obtained } what ia th« dimiootion T .10 . £:^^>»^^k«k^^..u»<'^^£^re^ . 818 8T00KB. 30. For what snm nmst a note, to run 4mo. 15da., at 6 9^, be given that the bank proceeds may be $1954 ? Ans. $2000. 31. A person owes £2250 4|, payable jn 6 months; if he pays ready money at 2 ^ jliscount for the 6 months, how much will he pay/ _ ^ Ans. £2205 4^. 32. Had I bought goods for £875, I would have obtained £120 dis- count: but as I b6jight them for £620, the discomnt amounted to only £98 ; did I obtaih more diminution in proportion to my purchases, and at what % does the eurolus amount to ? Ans. 2^^~ %. 33. A merchant bought $4612.80 worth of oil, on 3 years' credit, and has the liberty of advancing the payment, at a 'discount of | %. After 16 months he gave $2291.60 J at what time did he settle the remainder, knowing that he disbursed bat $1703.52 ? . Ana. 22mo. 20da. after the purchase. 34. What sum discounted for 7mo. 9da., at 6i % per annum, can produce a discount with which may be purchased the makings of 8 oo^«>^ benches, using l|yd. for each, at $1.80 per yd. ? A. $662.79 + 35. Having bought two t^Mcs for $505, on 1 6mo.'s credit, and having paid them before the term of maturity, I obtained $18,05 discount, at 6 56 per annum ; at what epoch did I acquit the debt ? A. 7mo. 3da. aft. 36. In a new building, two iron floors were laid, each floor being 15.36yd. long and 8.26yd. wide. The weight of the irop is 70!b. per yardof superflcie, and after being laid costs $5 per lOOlb. I ask, Ist. >ihe total price of the two floors : 2nd. the discount that can be obtained bxj>»7>ng 68 days before the time, at ^ 5|g discount per month. ^ STOCKS. 888. StOtkf k * general name given to government bonds, and to money eapital invested in corporations. d89. A OOrporftttOB is a body formed and authorized bv law to act as a single person. 890. The legal aet of inoorporation which defines the rights and powers of the corporation is called a OhUTtec. 891. The Oapltal Stock of a corporation is the money oon- tribated and employed to carry on the bosiness of the oompany. Nons:—!. Whn the aapiUl atook has been all paid in; moam may be railed. IfBooMsanr, by lowu, Mound by mortgage upon tnrproptoty. The bonds ifaued fbr these kwni antltle the holden to a flxod rate of tnteresl Thai, bonds drawinc 0^ •oaaaUyaneaBedCperoentitook.orA'fe; Ao. 1. To the bohdi aiW attached what are oalled coupotu, eaoh of which ii • dne ^.£1^'*"'* ^ ^ ^^ ^ *'''°^ '* ^ attae&ed, repreientinz the amount S. ObMAfoli a tenii aUweviated from the •xprenioD "oouolidated annnldei.'* TiMBittiiblOTWUMatbsriag atvarioM tiam bonowed money at diffanat' iSta;;,- STOCKS. 219 %, be given s. $2000. if he pays icb will he 05 4^. id£120di8- mounted to purchases, ars credit, mt of %%. : settle the turchase. ^nnum, can kings of 8 ,$662.79 + and having [iscount, at lo. 3da. aft. floor being is 701b. per Task, Ist. be obtained Qth. )nt bondfli lorized by bhe rights loney oon- lompany. lybenUied, bonds iMUfld Ida drawing oh ii •das the amount ton 'VwXPQOm : annnltiM.'* At diffMwat' t^^s"/suTl%Tlt^^^^^ «t«'ko' bond, '>^'ni-nr>naM/ Zd^^d^Z^LT^ '^^'''^}'!§ ""^^'^ **3^ P«' anna's, P«yaUo aeomed. TKuoflrn,. «f7.; ^o '^ "•" prooeeda of this the otd stock was^re- prettyLurSLrthe^^aennhf ^ ^* P^'P^.'"*' *°°"''»''» »' ~^^. indicate *>ocoLT8Un1id for refeiii! """'^ ""*''""' ""^"^ '"^ "^P'» «««"* «* boi \ fUi??®v: S^Of kJ^oWers are the owners of stock, either by original title or by subsequent purchase. ^ ^ »f.?w*:i-j^?*'m,'^°°®°^*^<'«1"'»^ parte into which capital of oan!^.f r • ^' • ^]^r ''"^ "^ * ^^^^« i° t'^^ o"ginal contributi^ rfifZi "''•'" ^5"'"'"* ««'»P*«ies; in bank" insurance, and railr^d companies of recent organization, it is usually »1 00. vahe S*°«''«'^''«AtPar when they^sell for their original 395. Above Par, at a premium or advance, when they seU for more than their original value. ^ 3»« Below Par, or at a discount, when they sell for le»i than their original value. ^ *®^ ^^'^i ^° Installment is a portion of the capital stock re- quired of the stockholders, as a p^ment on their subicripton. m„?.^?"i^" Assessment is a sum required of stockholders, to meet the losses or the business expenses of the company. nrn^?*; ^ Dlvldeiid is a sum paid to the stockholders from the profats of the business. 400. A person who buys and seDs stocks, either for himself jobber '^ ''^"'' ''' ^ °""'*^ * ^^^^^ *"'®' °' S*"**^-' IXAMPLES POR PRACTIOE, OPERATION. $2700 X .045 = $121.50, premium. $2700 + $121.50 = $2821.50, Arts. Or, $2700 X $1,046 = $2821.60, A»9. AiTALTsra. — Wo oaloulata flrrtly the premiam on the par lat"*. which we find , to be $121.60; we add this to $2700. and obtain $2821.60 whichis the cost. Or, since $1 of the stock "ir.O«, $2700 will cost $2700 X $l-o'45 ^^^^SilM^IuJ^^^^^'*^'''^'^ By proportion, lOO : joo + 4.5 :: 27 x IQO : x hk^i/.^'iWW^ <• * tikdi''6e:ftiik^]l *,*— Al-^,S, rt , -^ »i.l?^V-- 220 8T009Q9. of^^* ?:,?*i?«i*^™ »n "gen* 64 shares of the Ocean Steamers Co. m«!h did I 7*^°*' ^^' ''^"^ he charged me i 56 brokerage ; how OPKBATtOK. $0.16 + .0025 = 0.1625. $1.00 - 10.1526 = $0.8476 proeeeds offl of stock. 6400 X $0.8475 = $5424, Am. ' Ariltbis Adding the rate of brokerage to the rate of dii- ooant, wo have .1525 ; hence |1 will bring ^$1 — $0.1525 = $0.8475, and 64 ibarea dt $0400 will bring 6400' X ,8476 = 15424. 1 Bj/ proportion, 100 : 100 - (15 + 0.25) :: 64X 100 : X. Ex. 3. I pxA $17700 into the h^nds of a broker to be invested in untario Province Bonds when theiir market value is 12 56 below par; how many shares will I receive if the broker charges k% for his services 7 a ^ 1^ OPXBATIOV. . $1.00 -$0.1 2 -$0.88,- market valueof $1. $0.88 + $0.00i- $0,886, cost offl. $17700-t-$0.886te$20000=200 shares, Awt. AvALTsn.— Sinee the ■took is 12 ^ below par, the market value of $1 will be $0.88; adding the rate of brokerage,' we find By proportion. 100 - (12 + .6) : 100 :: 17700 : x -^ 100. ,^^*,f;J^® ftichelieu Company declares a dividend oi \bk%i what will I receire for 24 /shares ? r OPIBATIOV. Akaltsis.— Aoeording to 282, we multiply the $2400 X .16i » $372. S^deSffW "''' "**.' '^^*' ""^ °'"*^ **"* By proportion. 100 : 16i :: 24 x 100: x. Ex. 6. What income can we obtain by inve6ting'$10260 in Quebec Province 6 Id b*nds, purchased at 96 56 ? OPKRATIOK. ' AVALTSis.— We divide the $10260 -i- .96 - $10800, stodk purchased, l^f'^??"*'!'"?*'!'' .V J„^« $10800 X .06 = $648, .^nual iScome. r.S%tSnWnt2 ...... ^.. .^.. . . ">«"** »Ul purchase, (288). veal &Mmt. 1 By proportion, 96 : 100 :: 10260 : xx .06. teamen Co. erage ; how ding the rate ) rate of dii- 25 ; hence 9.1 $U.I525 = area ctt $0400 >< ,847a =3 00 : X. invested in below par ; 1^ for his . — Sinoa the I below par, value of $1 ; adding the rage, we find lollflr of the e |i770« ^ -h 100. multiply the obtain the I ia Quebec re divide the 2fiO, by the tain 110800, I the invest- base, (288). 41, tb». STOCKS. 221 ■&». 6. A person desires to secure $450 annual revenue; what capital must he itivest mo% bonds, when stock is purchased at .80 % ? Analtsis. — Sinee $1 of tlio atock will aeoure |0.06 income, to obtain |450 will reqaire |4&0 -^.05 => $9000, (Bz. 5). Mnltiplying the par OPJERATION.' f450 ~ .05 = $9(000, fitock requiral. f9000 X .80 = $7:300, co8t, or investment. _ , ,., . . . ' . I «»».«;. inuiuuiTin|r luepBI value of the stock by the market price o{$\, wo have f»O0U X .80 «? $7200, the coat of the required *took, or the aum to be inveatod. By proportion. 6 : 100 :: 450 : a? x .80. Ej:. 7. What iper cent, of my investment shall I secure, by pur- chasing Montreal: 7 per cents., at 106 ^ ? QPERATIOK. AKALTSm.— Since $1 of Btoek will cot* $1.06, .07 -f- 1.05 = 6i 56. vo'l'"'^'"^' '''^ ''"^°"' ^ T^ = 6i^ •'^*»» ^• $y proportion. 105 : 100 :: 7 : jf. Ex. 8. A main invested in a Steamboat Company, and received a dividend of 9 56, which was 8 J 95 on his investment: at what price did he purchas^ ? OPERATION. $0.09 -h lO.Osi = $108, Atu. AiuLTSi8.--Sinoo $0.09, the inootto of $1 of the atook, if 8i % of the aam naid for it, we have, $0.09 •*■ fO.Ofi^ a $108, the porehaae prioe. By proportion. 8^ : 100 :: 9 : dr. 9. A perso^ buys 25 shares of the Marine Bank, of $100 each, at 12 56 discounjl j how much must he pay ? Atu. $2200. 10. What fill I receive for 20 shares of the Central Bailroad stock, at 1 35 56, brokerage being 1 1 56 ? Atu. $2665. .11. At 7 i 56 premium, and i 56 brokerage, what will be the cost of 36 shares of tllie Bank of Commerce? ' 4fM. $3879. " 12. A caiikl cost £400000; all expense^ defrayed it brings in £15000 aunuilly. Suppose jt to bave been constructed by means of shares of £50 leach, and that an individual took 25 shares, what dir- idend wijl he rfeceive annually ? ■ Atu, £46 17 6. 13. If 300 shares of the Ottawa Bank sell for $.30112.60, what ia the premium, dach share being $100 ? Atu. 1 56 premium. 14. When thle nominal value of stock is £12 10^ and the diaoount must I pay for 30 shares? Atu. £361 17 6. boat company of tbe Saguehay declares a dividend lall iTeeeive fS>r 66 shaifes the nonfiinatvalae of which te? Atu. t9i6. 3^56, how muc 16. Thestea is $100 per sh 16. Bought s £187 10 0; ho ik at par, and sold it at S 56 premium, gaining tt^nj shares did I purchase 7 Atu. 62^ oharea. / ^ti^iite^A.rWuliui < . •Mfillu'i r ^. v^ ^\^>.». . 222 STOCKS. (pi» 18 in as of 17. An individual bought, at the rate of $168.75, a number of shares in the Pictou coal-mine company, the annual income of which ia $10 per share. With the income he purchases $260 worth of goods ; wliat was his investment, the brokerage being J % ? . Ans. $4398.465. 18. A merchant retires from business with a sum of $34520.50, and buvs with this capital government 6'8, at the rate of $70.45 ; what will be his annual income ? '"^ Ans. $2940. 19. Oritario4i'8are'8oldat therate of£94 17:'wlmt income will I obtain for £3794 ? Ans. £180. 20. Sold $16400 worth of North Bank Stock at 13% premium; " what shall I receive 7 ylns. $18532. 21. A person, having £2250, invests this sum in Ocean Telegraph Company Stock which sells at 17 % discount; what amount of capital does he purchase? iliis. £2710 16 lOih. 22. Bought 36 shares of th6 Western Camper Mine Company, the I»r value of each being $500, at 2 56 premium, and , sold it at 28 % discount; what is my loss? ", Ans. $5400. 23. I have an investment of $15000 in a transatlantic steamship company; how niany shares shall I owit after a dividend of 8 56 decl6tfed and payable in capital stock ? Ans. 1 62 shares of fl 00 each. 24. What should be the rent of a farm, which cost $16992.10, order that the pjirchase capital may produce the same revenue would be produced by the same sum, employed in the purchase 6 i ^ bonds, at 9 1 1 56 ? , Ans. $1 203.80. 26. A former invests £36, the pfice of three oxen, m the pur- chase of 5 ^ bonds sold at the rate of £78 10 ; at what real rate was hie money placed ? Ans. 6^^ %. 26. An exchange agent having $45000 invested in bonds of the Canadian Transatlantic Steamship Company, exchanged them at 88 %, for Capital sto(» in the same company valued at 62^ %. The bonds brought 7 % annually, while the shareholders received two dividends during the year, the first of 3 %, and the second of 3^ % ; how much did the agent gain annually by the exchange? Ans. $968.40. 27. An agent receives $25000, ^vith instructions to deduct his bro- kerage at 1 1 56, and then purchase bank stock for the balance; if the Btock is selhng at 3 ^ discount, what will be the amount of his capital stock ? Ans. $25329.92 -f- 28. An individual desires to invest $11168 \x\. h% bonds. The market value being but $67.35, he waits a few days, when it rises to $69.10. Findj now, what income did he lo^e, and what kicome he would have gamed had the market value lowered to $66.25, brokerage being J 56? Am. Lost $20.95 + income, would have gained $13.73^-}- 29. I have $60500 to invest in bonds, I can purchase 4^^ bonds at the rate of $96.30, and 3 % bonds at^he rate of $69.25 ; Avhich would be the more profitable of the two ? , AHs. The 4 J ^ bonds. 3Q. How much more advantage dividends how much $968.40. ict his bro- incej if the f his capital 25329.92 + inds. The it rises to kicome he , brokerage J$13.7a,+ 4i 5(5 bonds Inch would 16 bonds. 18 in i^% grage being 5.923 + . Company. ^ • PARTNIBSmP. ^ 1 order my agent to buy them when they will rate at 5 J ^5 premium ; how much will the 160 shares cost roe, knowing that the agent will charge me J ^ brokerage ? Aru. $16956.26. 32. A farmer sold corn for the amount of $4134.40. With this sura he buys three 4i % bonds which produce an annual income of $18, at 90f %, and one 3 % bonds, producing annually $20, at 6i^„ %. With V|te^emainder diminished by $1.95, he buys 3 % bonds at 68i ^ ; at «^hat average rate should he purchase 4^^ bonds, to have, for the •price of the corn sold, the same quantity of revenue ? Ana. $98.43 + 33. In buying stock in the Labrador Company for the value of $10425, at 500 per share, and producing $36 for interest and dividend, a farmer secured a revenue of $540. Required the market value of the stock per share, and at what rate he let out his money ? ^ Ans. Ist. $695 ; 2nd. $5.iVv9^ .34. In January 1848, the total amount of British consols waa £378019855; what was the amount of interest piud on them semi- annually? • M»M. £66702971^. 35. A person desires to sell $3500 of Montreal 7'8 ; the market value bemg at 95J ^g, he waits a few days longer when stock rises to ^m!%! what profits did he realize ? 'What loss would be have sus- tamed had the mai^ket value lowered to 94^.^, brokerage, in both cases, being J % ? Atu. $22.75 gain, and $17.50 loss. .36. A mason built 965 sq. yd. of a wall at $21.80 per sq. yd. • He desires to invest this sum in insurance company stock. In the Phoe- nix Insurance Co., the shares are $5000 each ; they produce $200 as mterest and dividend, and are negotiated at 40 % premium. In the Providence Co. the shares are $2500 each ; they produce $60 as in- terest and dividend, and are negotiated at 45 9^ premiam. Which are the most advantageous, and by how much % ? How many shares can he purchase in taking the most advantageous, and what revenue could he secure ? Ans. The first are the more advautageous by $1,478 ^ : 3 shares; and $600 of revenue. PARTNERSHIP. 401' A Partnership is an association of two, or more per- sons in business, eac^i of whom is called a Partner, Such an as- sociation IS called a Company ^ Firm^ or Houte. , ^^T^'T"^' ^"^ Olapfto/ or Stock. IHmiSmd, and At»e$mna, have the same stgnifloation in Partnership ai in Stooki. ^ 402. Oasx I.— To Jind each partner's share 0/ the profit or ^ Joss, when thers is no re gard t o time. ^^ Ea:. Three merchants, A, B, and C, associate together in business : A puts m $276, B $475, and $600. They gained $160 : what part of the profit must be given to each ? ^^W^^^t^'-LxtW^^'-T^Mn'kiji.-i^Si 4h.lLli-nii^'4diA>.'iLi^'SA^\i^^k 224 PABTNIBSHIP. A'8 B'8 C'8 Whole atock, 1275 " 475 " 600 II "fTT50 0PS1UTI0K. ^ $275 X 0.12 = $ 33, A's profit. •476 X 0.12 = 57, B's profit. 60# X 0.12 =- 60, C'h profit. Proof fHo, whole profit. »ptL $160.t>0 4-1250 = $0.12, profit on $1. AlMLTap^— Sln^ Che whola itook h $12S0, and the wholo profit, $150, th« DToflt on eveqr fr of stock lyillbe as many dollars .-u 150 contains time* 1250, or $0.12 on CTarjr f I of stock. I Then, each merchant's stookuiuUipIiod ^y .12 gives his part of the whole proat,| The same result olao may bo obtoiao^, oi^follows :— By proportion. itfo 275) + 475 5 + 500) 408. A ^ f^X - 1250 : 15C X = Ana. ($33, 4^ profit. \ 57, B's profit. ( 60, C'e profit. Proof, $150, whole profit- 'RvLK.'-^The whole profit or lost, divided hy the number denoting the whole $tock, wilt give the profit or loss on each dollar o/ttock; and eacK partner's stock, mtdt^f^d hy the number de- noting the profit on $1, will give his shaffbf the whole profit or loss. As the whoU stock is to each partners stock, so is the whole profit or loss to each partner's profit or loss. V XXAMFL18 rOB PBAOTIOI. 1. With £200, two men gained £50 ; the firet man contributed £125, the second, £75; what part of the gain is each entitled to? * . An$. The first, £31 5; the yeoond, £18 15. 2. Four merchants associated and raised a capital of $45000, to which each man contributed equally. At the expiration of the part- nership, the capital was found to be augmented by ^26877. What ehall be the part of each man, knowing tliat the 1st. ought to have 13 parts; ths^ 2nd., 11 :^ the 3rd., 8 ; and the 4th., 7 7 ^ Ans. Ist., $23959; 2b4., $20273; 3rd., $14744; 4th., $12901. 3. Three men associating together, gained £287 10; the 1st., put in 400 yd. of velvet at £1 per yard ; the 2nd., 350 yd. of cloth at £2 ; the 3rd., 450 yd. of oassimere at I5s. ; what part of the gain should each have? Ans. £80, £140, and £67 10. 4. Four persons havine joined in partnership agree that the 1st. pat in £1250 ; the4{ad^ ^ more than the^ first; tlie 3rd., as mucltAs the two others together ; and the 4th., his industry during the year, which was estimated at £2000 ; what share of the profits, £1525, shall eachVecelve7 Ans. £260, £312^, £562^, and £400. v-'-'-fiv. )foflt. •rofit. rofit. ; profit. it, $ISO, ths inej 1250, or 1 ^7 .12 gives og'followi :— fc 9. Threnarniera bought 148 sheep at $4.1 2 J per head, for the pay- ment of which the Ist. furnished $218:85, the 2nd., $236,321, and the 3r(l. the remainder. They sold the sheep, after havine nurtured them diinn^ 6 months, at a profit of $1.60 per head; how much did '■ each receive of the profits? Ans. $84.88|, $9i.66i, $60,241. 10. Three lumber merchants 4)ought 76500 saplings, on which they realized a profit of £296 8 9. The first man contributed £460 15 7*; the second, £527 6 lOi; the third man's part is not known, but he received, however, £98 16 3 as his share of the profits. Tell us the contribution of the third merchant, the profits of the two others, and the price of the saplings' per hundred ? Ana. Third merchant's 'Jf?o^n^* ^*'^• The profits of 1st., £92 3 li; 2nd.; £106 9 41: £1 18 9 per hundred. 11. Two dealers in furs made a joint purchase of 268 assorted {ox and beaver skins, at £112 10 per hundred j the first dealer advanced £48 10 more than the second, and, together they realise a toroflt of 18 % on the buying price. Required what is due to each, and at what price they sold the skins a piece ? Ana. £149 6 4| due to the 2nd.- £206 10, to the first. The skins cost £1 6 6J a piece. • ' 12. Three students in Astronomy join in raising $698.50 for the purchase of a telescope. The second ftirnished f of what the first gave, and ^le third furnished f of what the two others had advanced; what was tlie contribution of each ? iliw. $277.81 i, $166.68|, $254. 13. Four farmers associated in furnishingaquantityof straw which they sold at $7 per hundred bundles; what did each receive, knowing that the 1st. furnished ^j- of it; that the 2nd. furnished a quahtity not mentioned, and that the 3rd. furnished 600 bundles, which quan- tity equalled the delivery of the Ist. and 4th., who furnished 240 bundles? Ana $25.20, $8.40, $42.00, $16.80. ^ 14. Two clockmakers joiped in the purchase of 120 clock works at ~ th^average price of $7.37* j \n the speculatLont they lost $ 136. TJie ^ *OTSoriRel8t. Btirpasaed thatofth6 2hd.hy$33.605 what were the loss and investment of each ? Ana. let. Inv. $662.30*. loss $84.25. 2nd. Inv. $332.69j, loss $60.76. ^ *«•»«'».««. 10* -.1 jAJSfeA'lisafi^tefcffia^kia^gJi^^^fefe 226 PABTlOBgHIP. . 16. Several person^ agreed to conduct, during^ one year, a paper manufactory., The fint put in f of the Btook ; the Bccond, $40001e8a than tlie flrflt: the third, 14000 less than the second, and so on until the last. If the wveatments had been in sums equal to the hichest, the capital fltOc>irould be Augmented by \. The merchandise sold produced a sum equal to the 4 of what was put in," which was etfi- ployed in buying rags. Ih admitUng that the ,« of the sum proceed- ing trom sales eerre to cover the expanses of fabrication and invest- ment It i« required to ascertain how many, persons there were, how much each one pat in, and what part of the gain each is entitled to? 404. Cask Tl.T-Tofind each partner' a ihare of the profit or hs$, when the stock is employed for different periods of time, Ex. A and B entered into partnership ; A furnished $240 for 8 months, and B $560 for 6 months. They lost $118 ; what was each man's share of the loss? OPSRATIOir. #240 X 8 «$1920. tI920 x 0.025 = $48, A'a loss. 660 X 6 - ^2800. 2800 x 0.025 = 70 , B's loss. #4720. Proof, $118, enUre loss. $118.00 -!- 4720 = $0,026, loss on $1. , Air4X.TBis^It ia eyident that |240 for 8 mo. is thesamo u $240 X 8 c= $1920 tbr 1 mo., sinofl |1920 would Iom as mnoh in 1 mo. as $240 in 8 mo. ; and $660 for £ mo. is the aame a* $M0 X S* »= $3800 for 1 month. The question then ia the aame aa if A had ftamiahed $1020, and B $2800, for equal timea. Then, if ''$1«20 + $3800 = ^4720 loie $118, $1 will loae ^JL- of $118, = $0,026, and $1920 X .025 = $48, A's loaa ; $2800 X >035 1=, |/b, B'l loas. The aame re- aulta may be obtained u foUowa :— ^ By proportion, $^ x 8=1920 ) .„o i $1920) ..i,o.-_.^ 5 $48, A's loss. 660 X 6=2800 \ - *^^^ ' j 2800 J • 11» • ^=^n$. j^ g,^ j^^^^ Proof, tm. 40S. 'RuLT.— ^Multiply bich partner's stock by tlie time it was in trade, and divide the whole profit or loss by the sum of the several products ; by the quotient, multiply the, product of each partners stock and time, and the result will be his share of the profit or loss. \ \ Or, ^ - Multiply^a^portnft^s stochby the time U was m trade f theUy as tlie sum of ihesp products is to each product, so is the whole profit or loss to each partner's profit or loss, f r^ ear, a paper d, $4000 leaa (1 80 on until the highest, haniiise eold ich was eih- »un> proceed- and inveet- e were, how , entitled to? the profit or f time. $240 for 8 at was eacli l'b lose, t's loss, otire loss. X 8 ^ $1920 0. ; »nd $660 estlon then is lea. Then, if : $0,026, and I7he same re- 18, A's loss. 70, B'a loss. 18. time it was urn of the ct of each are of the i ^■■ I the MJhoU * tLvajnasBXB. EXAMPLIB rOB PBACTIOl. 227 1. Two personfl contribute unequal sums towards a capital: the first puts in 12300 for 2 years: the second, $1500 for 18 months. What part of the gain, 11400, should each person receive? • „ ^n». 1940.16, $459.86. ?. Three individuals raised a capital sum with which they gained £I13r 10 : the first contributed £200 for 2) y^afs ; the second, £126 for 25 months^ and the third, £24^4J5 for 35 wonths. What part ' of the gain should each have? Ana. Ist. £382 15 U; 2nd. ;^99 7 0*; 3rd. £556 7 lOi. ^ • 3. A porter associated with a pedler and raised a capital of $16000. After two years they divided the gain, and the pedR^ who had con- tributed $9000, received $1800 : what did his companion r«oeive, knowing that the latter left his share id the business but daring 20 months! Jin$. $1166.66). 4. Four persons agree to form a partnership for 3 years. The first puts in at the beginning $350, and 5 months after $2400 mote: the 8econd.put8 in $8000 at first, and at the end of 20 months wtWlrawe the haM" of his share, and 5 months after withdraws $2400 more: the third puts in $1600 in the beginning, and $6000 at the end of 2 years; the /ourth puts in at first $600, and 6jetj six months aug- ments his portion by a Uke amount; the gain being $80000, what part did each receive? Ana. $14677.36 + , $33336.16-, $19232.39 + , $12764.11 + . , 5. Three merchants joined in business. The first put in £1001 13 for 10 months; the second, £1761 12 6 for 16i months; and the third, £2000 3 9 for 17 mo. and 20 days. B^oired each inerohant'a share of the profits which amount to £360 3 7 Ans. £48 7 6|-s £131 2 4| + } £170 13 U-. 6. Two clothiers associate together; one of them contributed a sum with which could be bought 90 yd. of Broadcloth at $6 per yard, the other put in a sum with which 60 yd. could be purchased atthesama rate. In supposing the 1st. to have had $$ of the profita more than the 2nd., to how much did the profits amount? AJn$. $30. 7. Four far«er8 rent a pasture for $976. The first put 6 beeveson it during 54 days ; the second, 7 cows during 63 days: the third, 8 heifers during 75 days; and the fourth. 6 horses daring fiO days. It was calculated that 1 beef oonsamed 1^ times as aiach as a OMr, or twice as much as a heifer, or \\ times as mucLas a horse : how vooh must each farmer pay ? - /S ^ Ant. $238.45 + ; $269.66-; $264.94 + j $211.9^-. 8. In the working of a mine during 6 years, three partners «uii £21750. The first partner had put in £13437 10 in the begimuna but after 2^ years, he withdrew £3275. The s econd put in his share! which wa8^£4ie00, only it years sitef the commeneeineot'^^^lM — work. Finally^ the third made hia oontribati^ of £63760, bat 3 years after the instalment of the first. Whatputof theptofltsshoald ?ach receive ? Ans\ £35^6 16 0| + j £9867 6 " " . I' 'I rtoftheptofli -: ^68816 V *!-} XI »+. ia.hJi^'' ^J^ -. vw**- '-'>'"* "V-a^* 228 XXOHANai. EXCHANGE. nlal^* ^J?'*?^SL"i^® P"'*^ of remitting money from one place to another by Drafts and Billa of Exchange. Form qf a Drafts iama 4? tnu eiceoant. ^ dr^^' ^^ ^'*^"» °' "tokWi M the person who signs the 408. The Drawte jb the person on whom the draft is made. 409. The Payee is the person to whom the draft is made payable. '^ocepted , ,._ __„, __^ 411. An iBdonMnent of a draft, by the payee, ia made in thesamenuwnerastheindoreemenffofanote. 4ia. A Sight Draft is an order to pay at sight. sp^MltJ*"'^** " " ord«r r«,uiring payment at a 414. A Draft or Bill of Ezchaoge is at a Premium, when the pviue pattl » MM tiiaa itr fuse. aJ^' ^T"*'*!!^^ ^*i^ fachanje, u when both the drawer and dnwreewittde in the same oounSy. / Y from one •M the Com« ma a 4i^. signs the b is made, is made «, topay riting the the draft. made in ent at a irhen the hen t he KSOHANOB. 229 416. Case I.— Given the/ace of a draft, the rate per' cent, of eocchangcy and the time, to find its cost. Ex. 1 . What must I pay in Ottawa for a draft of $640 on Qaebec exchange being 1 ^ % premium ? ^ / OPSRATIOK. 1640 Xf 1.015 = 1649.60, Ans. Akaltsis.— The ooat of ezchanira of $ng $l-f $0,015 =c $1,015, and of $640, 640 X $t.015 = $619.60. TT ?f' ^' ^^^} ™"** ^ P*"^ ''^ Montreal for a draft of $3500 oa MaJilax, at 33 daya, exchange 2J % premium. OPSBATIOK. 11.000 •006 = dlsot. for 30ds. at «%; 9 .994 = oott at par of $1. '022 SB rate of ezebange. 11.016 = cost of $1 of Ihe draft $3600 X 1.016 = 13556, Ans. Akaltsis — The diaoount of $^1 at6 % . for 36 days is $0,006, which being gub- traoted from $1 leaves $0,904, the cost of $1 of the draft, if the ezohango was at par. To this add the premium of $1, ,^0.022, and we have $1,016, the cost of ^l of the draft. Hence the cost of $3500. • the draf^ ia $3500 x 1'016 c= $3556. J '*JT' ?^"-tI- *'o'^ S'gtt draftB.~ifu//ip(y the face of tU draft by Iplus the rate when exchange is at a premium, and by 1 minus the rate when exchange is at a discount. II. For drafts payable after sight.— ifYne? the 'cost of $1 at bank discount for the specified time, at the legal rate where the draft u purchased; then add the rate of exchange when at a premium or 'f tract xt whenat a discount, and multiply the fate of this draft by this result. '' y • « ^ XZAMPUES fOB PRAOTIOl. 1 . A merchant in Toronto wishes to pav in Montreal $7930. and exchange ib | ^^ premium ; what will be the coet of the draft ? Ans 97989 471 2. A merchant in St. John, N. B., wiiihes to pay in Ottawal $980. and exchange u l|5l5 discountj required the (joetof the draft? ' .RJo^vi^^^!:^/'*•'^*' ^? .Kingston, of a draft ^'^^m^'iox $800, payable 60 days after sight, exchange being at a premium of ^, T»yat)le 30 clays S^m^t, at 6 5l^i what did it'^himithi' rata (^exchanae bemg \\ % discount ? ^n,. ^822 78 6. What wjH be the cost of a draft of $4280, tot 60 days, at 6*4L whange beibg IJ % premium ? An$, $4286.06i. ■I t mth the exi ^ L',' tddm^'SSk a, * f. 230 FORnair ixcbancA. 6. A merchant m Quebec receivee from his agent 1200 bushels red wheat, purchiwed in Toronto at 65 cte. per bushel; in payment for ^V/^ \\'.^'"'}^ * ^"^^2"" Toronto, &t^% discount. The tranepor- !^- .olh* ^^^ ^^ *^^- What must he sell it for per bushel to 8*"»»^^S' . ^n«.f0.91t. 418. Case II. — Cfivm the cost of a draft, (Tie rate per cent, of exchange, and the time, to find ittface. Ex. A merchant in Three Rivers paid $6856.10 for a 60 days' draft on Toronto, exchange being 11% premium, and interest 6 56; required the face of the draft. *- 1 tp, AviKms.— By 4ie, Cm« I., Ex. 2, we find the oost of $1 of the draft to be $1.00826. Henoe, tSSM.lO ^$1.00825 « $0800, li the faoe of the draft. OPKBATIOir. fl.OOOO •0^05 =thedl«»iintfdr68d»yi. $ .9895 s= the eost of $1 a^ par. .01876 = the rate of ezehanglkw 100826 = the eoat of $1 of the ' " » 6. A man in Haliiia, has $4800 due him in Montreal} how much more will he realise by making a draft fbr this sum on Montreal and selling it at X^^ discount, than by having a draft on Halifax remitted to him, poichMed ia Montreal for this sum, at 1 9^ premium 7 ilM. $11.73+. JgQKBIQN aX O H A NOB . 4M. i^fortlfB BUI orSlO|ULBf$ is a dnfk jo whioh Uie 0ntwer ana (ura^rw lire in diisnpt 9oaBtri«h bushels red ajment for 3 tranepor- r bushel to . $0.91|. I per cent. \ 60 days' lerest 6 % ; IS, Cm« I., 08t of $1 of B26. Hanoe^ S « $0800, lA. draft for '■&■ he face of » being at 115860. ig at I 9$ r.fl60. ttance by ys. What m? 9 being at lOw much itreal and : remitted a? 1.73 + . F0RBI0I7 IBXOHANOB.' 231 Form of a Foreign Bill of Exchange. .- <^^ y^ o/ e^a. out ^m !me:i!^ o/ SxcfU^ fiieeonc^ one/ i^tiic^ o^ iecetvec/. afd/ onaiae €ne ^afn& {o 'mu account. ..m^!!'™^ '^'J'f •«^"»8*' to preront Ion or delay, two or three drafts of the Kl^mL i^'i^ *" drawn up and sent by different conveyances, or at dif- SSShara'.SL^J'.fa"''" ""•/'^•'-"'"^''o are worthless. Bach draft 481. Foreign exchange is computed as inland exchange, ex- cept that the currency of one country must be reduced to that of another. 422. Rates of Mohange between the Dominion of Canada and Great Britain are oafcmonly reckoned, at a certain per cent, on the old par of exchange, instead of on the new par. ^?!i75' •5:****J!?* °' P'^incW Parliament, it was enacted that £1 sterlinir rJ«;1^^'^'S.V ^"""y-. B°t »>y » *««»»» «' the value of the pound sterlini was flzad at §4.808. Now, the new par is equal to the old par nlus 9i * of the oW par that is. M.444 -|- »« ^ of 11.444. wfiich is .422, equlS to^-See, the new &»kt S?iS2"S'i?iu"i^ of exohang^betwMn the faominion of Canada and Sffnlli tti'n"w*£S?aSr """^•' P«^«™>' »i % before it is at par. ao- Et. 1. A merchant in Quebec wishes to remit to London £660 3 6 eterllng: exchanee being at 1 1 ^ premium. How much must he pay foi: (he bill of exohabge ? *^' OPKBATIOH. fV* X l.U =» $4,931; £560 3 6 :is £560.176 ; i0.m, X i>aa4-gi ni6i.63rAn$, ^ AKALT8U.-.8inoe the old par of £1 sterling =, $4,444, or $V».weninltlplyfybyll^, or f l .l l , the given rate, de'lJ^U x'.i'e^ >,*«« tsiflS^, ii; 232 FOBUON SXOHANQI. Ex. 2. What will be the face of a bill of exchange on livcrpooh purchased ia Montreal fur f 5537.40, exchange being at 10 % premium ? OPERATION. 13537.40 -T- 4.88 J = £1132 13 0. Anutsis. — We find, as in the preceding example, the (xwt oTjEI, at the ({tven rate of exohaSig^ ; tneo we divide $5537.40, the given cost, by the cost of exohange for £\, and obtain ^1132 13 0,Mhe faoe. Ex. .". What is'the cost in Toronto of a bill on Pftris, for 1780 francs, exchange being at 2 J 5i$ discount? ^' . OPERA.TION. Commercial value of the fran (J, = f 0.186 . Deduct 2i % discount, 0.00465 Value ^fl franc, fO.18135 $0.18186 X 1780 = 1322.803, An$. 428. From these illustratioDs we derive the following Rule. — I. To find the cost of a bill, the face being givon.'»»- Multiply the face hy the cost of a unit of the currency in which ilui bill it expressed. II. To find the faoe of a bill, the cost being given. — Divide the given coat by the coat of a unit of the currency in which the bill ia to be expreased. ReDUOTION or THB STSRLINa MONST TO THB QlD OB TO THE New Canadian Cubeenct, new pab. Ejt. ^duce £660 3 4 sterling to Old Canadian 4!)arreno}^ OPESXTIOir. — ^ £660 3 4 + I 6f £560 3 4 =1 112 d + J^ Of 112 8 = 9 6 8| £681 10 81, Ana. And in Decimal Currency, £681 10 8| (233) = |2726.13f. Ahaltsi^. — The jtonnd sterling =i $4.80}, and the Old onrreAoy potand ss $4 ; diff., $0.^}. Th«n £1 sUr. KT £1 - eurrenej. ber=J number. Hence • nom- of ^ of th«;t the ^iQidRTET.~Td MatiicFsi _ money to Old Caha3iSi Garrenoy, new par, — Add to the given aum ita Afth plua one twelfth o/thejtfth, . ■i;;., ^J,^^i^h£ii^&,^JLS^^ai^-A^^ > J(i.'it^riSi^.ui^BKiWU&i,i> i^-g^ii ^■- Liverpook premium ? 1, as in the le (Mst of^I, ihaSige; tneo le given cost, e for £1, and le faoe. ^ for 1780 ing . in which Divide ihe the bill it D OB TO aoy.' The jtonnd B}, and the iind s $4; i«n jeiater. - £1^ old ^of annm- of ^ of tlu;t the Canadifo plus one ^BQUATION OF PATMINTS. BXAHPLKS FOR PaAOTIOB. 233 ^^In^y^^T^^^l^i^® *'**'"'"*''<"'»' ^*'"« in Ottawa, of a bill of ex- Chan« on Lonaon for £390 10 ster., at 9% prem. ? Jlne. 11891.75* the cnni! J T^ in AmBterdam, a bill on Montreal for $681^34, the course of exchange bemgat $0.38 per guilder ? Ant. $717.20. ' rJ'mPl ^uP*'^ in Kingston for a bill of exchange on Paris for 3000 franca, exchange being at 2 % above par ? ^ at i ^TpreraiUri 7 ^°^*' *° ^^^^^^ *'*'* *>^' ^n Boston, for $2000, costin^Oniw" * ^i'lff ^cl^ange on Hamburg, for 6000 i^'rcs^biSco, cost m Quebec„at 1 ^ above par, the marc banco being equal U> -tptim^J"""^ 'J^Ji"^?"^ * ^'" of exchange on Glasgow for X675 2 6 Sterling ; wUa did it cost me at 8^ % premium ? ^^. $3255.60 + . franVs tW r ^^'r"" Halifax a bill of exchange on Rouen for 66245 irancs, the course of exchange being 6 fr. 64 centimes per dollar ? r- a rxru i. 11 u .!„ - ' ^n». $10152.63 + . MMtr£u,'^^^'f''^^^''^''^^r^ that may be bought in juontreal for »J1 25.60, exchange being at 9^ % premium ? 9. Paid in Quebec £2170 16 7, old Canadian currency, for a bill excS^: bTrp^r^ '^^^^ ^- '' ""^^^^-'^^^^^si^'^^ £m e'^oln V^T^ ""''rj^ * Co., London, abin^f exchangtVor tlA ^ ^' °" J; Chalmers A Co., Quebec; what is ita value in deci- mal currency of Canada, at 9 % premium 7 Am. $1846.94 + ! EQUATION OP PAYMENTS. *«^*** ^^?*^»» Of Payments is the process of finding the S^tJZwItten^^^^^^^^^ °^-«-». --' ^- •^<^if- b^^^due.' ''"" "'°'*^* ^ *^« time toelai»ebefyreadebt 487. The Average Term of Credit is the time to elapse before several dates, due at different times, may aU be Daid^ once, withouVloss to debtor or creditor. ' ^ ^ ^«^*®' ^^«*!9»ated Time is the date at which the several debts may be cancelled by one payment / "^vorw. 429. 0a8« I.— To find the average or eguitable time of pav- ing teveral debte due at diferenUimet. 60 days ; the thiM for $300 payable 'in 90 days. Whit wm Xe al° erage term oforedi^ and what the equated time of paymeT? t- J KiU^l. i.-^U^W,lf. ,*-^>.i H* ^ « 234 EQUATION OF PATM1NT8. OPEBATIOM. $250 X 30 = 7500 200.x' 60 = 12000 300 X 90 = 27000 ;^ $750 76500 if^^? T.^^LT 62da.. average credit. March IV 62da. = May 3, Aps. Akiltsib;— The intenat of $260 for 30 days ia the same- ai the interest of $1 for 7600 days ; • and of fSOfl fo#«0 days, the samo as. of $1 for 12000 day*; 'and of $300 for 90 days, the same as of $1 for 27000 days. Henee, the interest of all the sums to the the interest of $1 for 7600da. + 12000da 4- 270oTd^'-^*Sn !? ""' T' f, credit; an.l'. March 1. the date^at which the credite S rtf2r^ S« « the equated time of payment. *^' ^ ""• '=' '^^ '' Ea:. 2. Bought of D. I. Lyons several Bills of goods at diffpront tMuee and on various terms of credit, as by the fo Wing fetatemS What ,8 the equated time for the payment of the whored ^'*^'""'*- Jan. 1, Feb. 7, March 16^ April 20, a bill Amounting to $300, on 4 months. ;; " 185, oa 6 months. " " 280, on 4 months. (I u It II II It 210, on 6 monjhs. Due II ii> It May July July Oct. 15, 20, OPEBATIOK. 1300 185 X 67 280 X 75 210 X 172 9975 12395 aiooo 36120 69515 days. . . 69515 ~ 975 := 71-^y days. . " ' . ^ May 1 + 71 days = July 11, Ana. AifALTSis.— We first find the time when nnAli nt »k. mi. _•» t. Then, since it Wll. shorten the oZaU^n aVS chaSgl S.^^'Zl ^Ze^^t firtt tttne whenanjf btUT>eco,net due, instead of its da/e or ♦L^il.* 7 *'^J'^ to eompnte the average time. No;, since May 1 Su.epSriocTfrom'^^^ averse time i. computed, no time will be reclconed on thK bil? buTffti™. for the payment of A« second bill extends «7 days bevond M« i .^ IT '.?* crerfu, and diuide the sum of the products by the sum of the pay. ■ NoTB.— If the date of the nverajre time of payment lUreauInMl n» \n v, » Khiehmjtum become* due. JP/kii proceedaein tisrlu^^t. ^ ****?.. T ^^tTATIOIT OP PATjif NTS. » EXAMPLES FOB PRAOTICfi. 235 nai* /".rttr^^ltL* P"'-cha8ed'£4750 worth of cloth, and a^Teed to WMiwuibe the amonnt of e§ch payment? Arts £950 2. A man ^Wes $15960 payabl^as follows- i in cash a in fi rn'« 6 mo^nH .K ^^;f ^7\0 '8 to be parf in two installments, viz:: I in firft fi!.V.nn^'''-"^v.f*"^""'-^' ^««8' a merchant ga4"thrIeT;tea^';he .SSF -^^i''i« \JS.l^^^^^^ Ana. 8mo. 24da. theexprationofS moblh,, ,h»ll I o.e the win'e?' ^ '' *^' OPERATION. 30 X 4 = 120 40 X 2 = 80 TO 205 $180 -$70 = $110: 200-4-110 = IbIo. 25da., W A ^''■*^"»— Tije interest oh tho $30 for 4 months 18 equal to tho interest of |l for 120 months and the int.of $40 for2 months iuT^u^ to that of 11 for 80 months; and thus the mt on both partial pavmeirtB. at the expirauon of thq^g montJis, i, oqaai to the int. of $1 for 120 -f 80 = 200 Shfc Ik!'^"?:'- *u " *'!L^^i' "f '"'•• t»»« bal. of the -uin unpaid, .a« th. 8 month.. ^ o» ^oLtreJ itta f^^' ^VLK.— Multiply, each payment hy the time it loitmads Wore tt become, due, and divide the ^mo/the p^du kiads-^ the mean erent qua- in. lixed, the rith Icwt. i Icwt. of iwt. is worth , and 2owt. t. 4- lowt. + $29—. Ilan aa 4ft 437. Alligation Alternate is the process V finding the proportional quantities to be taken of several articles or ingredients, whose prices or qualities are known to forma mixture of any given rate or quality. / « I ^ 48S. To find the proportional ftuxntiti/ to be used Ijf each ingredient, when the mean price or ^lity of the mixture iigiirn. Ex. 1. What relative quantities of timothy seed worth $2 ahushel, and clover seed worth 17 a bushel must be used to form a iiixture worth 16 a bushel ? . F / Ahaltsis.'— Smc« on «v«iy ingredient used whose price ^w quality i« Cat than Ana. tlie mean rate thei« will bo a gtiin, and on every ingredient wliose price oi| quality . , , . L is greater tiian the mean rate toon wiU M a tot*, and sinee the gains mad losses mast be exaotly equal, theirelatiw qnantities used of each should be such as represent the unit o{ value. BiselUiut one bushel of timothy seed worth $2, for $6, there is a gain of $3 ; and to gidn Z v., 'f1"'"* * ^f * bushel, which we placO opposite the 2. By selling one bushel of clover seed worth $7, for $5, there ia a loss of $2 : and to lose SI wookl J^a^'raj of a h'J 'h eU which Jt> p la o e oppo s it e th e^? . _^_-— ^^_„ ...^ ■.. In every ease, to find the unit of value we must divide $1 by the gain or leM par boshal or pound. Ac. Henoe, if, every time we take } of » ]mnh«lot timothr seediwe take i of a bushel of clover seed, the gain and loss wlU be ezaotly eaoaJ. and/we BhaU have J and! for the i»-oportw>iwryi»ai«»W«t. ^ . OPERATIOK. III 1 i I m m^.H '. >>S*'jit,'j, ' Vdi ; i&i,A^ / ALLIGATION. in^^'i'i-^" ^''"^ proportions of coffees worth respectively 3, 4, 7 and lings a JTnd ?^" ' ""'''' ^ ''^'" "^ ^'""' * "'^^^^ wor'th'c sbU f OPBBATION. 1 2 3 4 8 h 4 4 4 1 1 1 2 2 1 < sJ 3 Analysis — To preieire the eqaslity of game and losses, we mast alw«y« com- pare two prioes or simples, one greater and one Uit than the mean rate, and treat each pair or couplet as a separate exam- ple. In the given example we foirm two couplets, and may compare either 3 and 10, 4 and 7, or 3 and 7, 4 and 10; " We find that i of a lb. at 3s. must be JJ-iJ^'r J?.'?""''^^''"" * »f »'"• at'j't^gl.f'l^sffffgf'a^n'J 1 TifV^' ewo^.l'J"'i:^«-. H?" P^Portional numbers, obtained by comparing tho two couplets, are placed in columns 1 aad 2. If, now, we reduce the jialbcrs obtSnTJilZtS "^ a common denominator, Ui i- their riuieraU,^ we obtain the integral numbers in columns 3 atfd 4, which, being arranged in column 8, give the proportional quantities to be taken of each. * »""»80« »«> «>»»«>n whi«ir!i' '^!?°° *^'!i'i° """paring the simples of any pair or couplet, one of Tn.n "K™"'"' and the other less than the mean rate, the proportional number offi."''"" TK *^"' •'""""■ ^"^ " '*•• difference between the mean rate and the Li 4 whi.^'.-, fh"!lff <"""P»f'?« 3 and '0. 'he proportional number of the former Is 4, which 18 the difference between 10 and the mean rate «; and the propor- faonal number of the latter is 3, which U the difference between 3 tad thSXn the mean rate are integers, the intermediate steps taken to obtain the rfnal pro- ^/, L"" iiiT*?'" '^i." ?"'H°">8 1. 2. 3, and 4, may bo omitted, and Uie sSmo results readily found by takmg the difference between each simple and the mean rate, and placing it opposite the one with which it U compared. From the foregoing examples and analyses we derive the fol- lowing 43». Rule.— I. Write the several prices or qualities in a column and the mean price or quality of the mixture at the left. II. Form couplets hy comparing, any price or quality less, with one that ts greaOr than the. mean rate, placing the part which must be uted to gain 1 of the mean rate opposite the less simple, and the part that must be used to lose 1 opposite the greater sim- ple, and do the same/or each simple in every couplet. III. If the proportional numbers are fractional, they may he reduced to integers, and if two or more stand in the same hori- zontal hne, they must be added; the final results will he the pro- portional quantities required. tna°h^n'eoM ^^ '"""'^" '"^ '^^ couplet or oolumn bars » oommon fiwtor, it 2. tVe may also .... . ' - »o multiply the numbers in any eonptet or odiimn by any mul- Swot flnillSSt "" "^ °°' otithioh. being taken wiU giv« a 60 r 3, 4, T and vorth 6 8b il- ' the e^ality It altfayaoom- I, one greater rate, and treat iparate ezsm- I we fonn two either 3 and nd lo; " i 3s. mutt be d i of a lb. at 1 1 lb. at 7i. amparing the tbq ABmbcrs ineralon, we ged in oolumn aplet, one of tional nambor rate and the of the former the propor- iod the mean simples and bhe final pro< nd the same and the mean re the fol- ilities in a ! the left. f lets, with *art tohich sa simple, eater tim- ey may he ame hori- be the pro- toD fiwtor, it > y any mul- ALUOATION. KXAMPLKS FOE PBAOTIOB. 239 1. A grocer has sugars worth 10 cents, 11 cents, and U cents per pound ; m what proportions may he mix them to form a mixture worth o ^'" • ' "*"*• ^ '^' *^ ^^ <'*^'' *"*^ 2 lb. at U^and 14 cts. 2. What proportions of water at no value, arid wine worth $1.20 a gallon, must be used to form a mixture worth 90 cents a gal- o * ^ t. J ^ ^"*- ^ sal. of water to 3 gal. of wine. 3. A farmer had sheep worth $2, $2i, $3, and $4 per head ; what number could he sell of each, and realize an average price of $2* pep J J . , ^'**- ^ of the Ist. kind, a;nd 1 each of the 2nd. and 3rd., and 3 of the 41 h. kind. ' ' 4. What relative quantities of alcohol 80, 84, 87, 94, and 96 per" cent, strong mast be used to form a mixture 90 percent, strong? Am. 6 of the first two kinds, four of the 3rd., 3 of the 4th. and 16 of the 5th. J 440. To%nd the proportional guanHty to he used of each ingredient, when tJie quantity of one of the simples is limited. Ex. A miller has oats worth 30 cents, corn worth 45 cento, and barley worth 84 cents per bushel j he desires to form a mixturf worth 60 cents per bushel, and which shall contain 40 bushels of corn • bow many bushels of oats and barley must he take "7 OPKRATION. ANALTSia. By the same 4 4 I 20 ^ process as in Case I we find . the proportion*! qnaatitiei ^ns. of each to be 4 bushels of oats, 8 of com, and lo' of 40 bushels of com. which U 5 times the proportion.l^Sbertind St^el^e ^^hT^^l " h*'*"' ^V"' ^t """* ^^ ^^'O »>»• proportional qa£X^ each of the other simples, or 5 X 4 = 20 boshels of oatTand 6 X 10 = M bushels of barley. Heme the fbllowing -"»«»«« a lu =- o« ^"^^•^^^^"-^J^ind the proportional quantities As in 438, Lhmdethe gwen quantity by the proportional quantity of the same ingredient, and multiply each of the other proportional quanttliea by the quotient thus obtained, * A tXAMPLIS FOB PBAOTIOX. hn^ L'^^^^'^i ^ ^^ ^"^"^^ *^' ^^' ^S' and »0 cents per pound ; how many pounds of each must he uw with 20 pounda of that worUi 75 cents, to form a mixture at 80 centa ? fm^Sf: ^" ^^' ^^fa»^the firqt three kinds, and 130 Iba. of the t i r^ M, Ind iKoi will give a buvifa^i^^«^"f*]*f?*f? ** «2ahead} how maoy must he r2.r.fL i^^J^-^' *?**^« ">»y «e" the whole at an>avera«e .^ce of 14 a head, without loss ? 4«#. ?4 at $8, aad ?2 at JS!^ X /■ ^ <1 <^s'A £^%liSi^isL^ ^^iJiSkj 4^-^'ifD.^aH -i^i'*'- ^rf'i-j- *..: 240 ALLIGATION. 3. How mucl) alcohol worth 60 cents a gallon, and how much water, must be mixed with IHO gallons of rum worth $1.40 a gallon, that the mixture may be worth 90 cent« a gallon ? „ v^»- 60 gallons each of alcohol and water. 6. How many a^jreS of land worth 35 dollars an acre must be added taa farm of 75 acres, worth $50 an acre, that the average value may be $40 an acre? Atu. 150 acres? .f» ^ merchant mixed 80 pounds of sugar worth 6i cents per pound with some worth 8 J, cents and 10 cents per pound, so that the mixt- ure was worth 7i cents per pound: how much of each kind did he user ^ 442. To find the proportional quantity to he used of each tngredient, when the quantity of the whole compound i$ limiUd. i7ar. A grocer has sugars worth 6 cents, 7 cents, 12 cents, and 13 cents per pound. He wishes, to make a mixture of 120 pounds worth 10 cents a pouMl ; 4iow many pounds of each kind must he tue? •o{ OPERATION. 6 7 ll3 k 3 3 30 k 2 2 20 4 3 3 30 k 4 4 12 40 120 the to be 10 times u mnoh at „ „„ „, quantity of each «implon8ed most be 10 times aa portional, whioh would required 30 lbs. at « ot»., „ 12 otf ;, and 40 IDr. at IS ota. flenoe we deduM the foUowing Ahaltsis. By Case 1 we Bnd the proportional qnantitiea of each to be 3 lbs. ^ oti., 2 lbs. at 7 ots., 3 lbs. at 12 (As., and 4 lbs. at 13 ots. By^ding the proportional quan- tities, w« find that the mixture would bi but 12 lbs. while the required mixture is 120, or 10 times 12. If the whole mixture is ,%*J>'opo'*>oi"^ quantities, then the ' "^ " ' muoh as its respeotiTe pro- 20 lbs. at 7 eta., SOlbsVat 448. RuLB.— jPiW the proportional nunU>en a$ in 438. Vivide the given quantity by the um of the proportional quan- tities, and mukipty each of the proportioncU quanHtiet bu the quotient thm obtained. KXAKPLES FOR PSAOTIOB. 1. A fwrmer sold 170 sheep at an average price of 14 shillings a head: fokvsome he received 9s., for some 12b., for some ISs., and for others^Os. ; how many of each did he sell 7 / Ans. 60 at 98., 40 at 128., 20 at ISs., and 60 at 20e. 3. A Jeweler melted together gold 16, 18, 21, and 24 carats fine, 80 as to make a oompottnd of 51 ounces 22 earats fine; how much of each sort did he take? Ana. 6 ounces each of the first three, and 33 ounces of the last 3. A man bought 210 bushels of oats, corn, and wheat, and paid fig tb s wbol s $ 17B.50 1 fortLauulfrh epH dfifoftlrtcoro, " ^ ^ t ushe ftnr the wheat 91^ per bushel; how many busKels of each kind^id he b«jFt An§. 78 ba. each of oats aad corn, imd 64bai ofwhcAtt ; f fe and ind did h( ■^^#^. how much 40 a gallon, knd water, sre niuat be the average 160 acres. « per pound at the mlxt- kind did he $ed o/each t limited. cents, and 120 pounds id must he M 1 wo Bnd ititios of each lbs. at 7 ots., Ibi. at 13 Ota. rtional (jnaa- th« mixtare I. wbilo tho 120, or 10 le mixtare is »8, kh«D the speotiTo pro- ■., SO Iba. at I in 438. ituU quan- tita by the shillings a 18s., and at 208. arats fine, w much of >8t three, and piud cind did he ifwheatt J. ^ IVOIUTION. 241 4. A, B, and C are under a joint contract to furninli aaA u i. • of corn, at 48 6ts. a buSl,el ; K\ corn is mirth 4fi .?- S?" ^"^^^^ andC'8 64 *^« "^««« «f Involution. ^sa' it S?° • » * ^'"'2*^'' ^ °°« <>^ »*■ «q«a» factors. ^ttO. The First Root of a dumber is the number itself. ] \ rf^iw-*»j*f 242 1' BQUABB BOOT. 451. The Second Root, or Square Root, of a number, is one of its two equal factors. Thus, 4 is the square root of 16 = , ^ ^' 452. The Third Root, or Onbe Root, of a number, is one of Its three equal faotors. Thus, 4 is the cube root of 64 = 4 X 4X4. 458. The Radical sign is the character, V, which, placed before, a number, indicates that its root is to be extracted. 454. The Index of the root is the figure placed above the radical sign, to denote what root is to be taken. When no index M written, the index, 2, is always understood. 455. The names of roots are derived from the corresponding pp^wers, and are denoted by the indices of the radical sign. Thus, V 36 denotes the aqmrerootof36 ; ^"36 denotes the cuhe root of 36; V^ denotes the /owr^A root of 36 ; etc 4^59. A Rational Root is a root which can be exactly obtained. 457. A Sard is one which cannot be exactly obtained. ')■■-■ SQUARE ROOT. The roots of the first ten integers and their- squares are : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. ' 1, 4, 9, i6, 26, 36, 49, 64, 81, 100. Nons.— 1. It win Iw obterred that the leoond power or iqaare of each of the auoibon jpootaiiu twice aa maqj figures aa the root, or twice as maoy wanting •ne. Hence, to ascertain the number of figures in the square root of a giren nnmher,— Beginning at the right, pQint it off into at many perioda at pottiMr, of heofigwrtt tach; and thert toiU be at manyfiguret in the root at there are periodt. 2. When the given number eontaini an odd aamber of figures, the period at the left can oontain bat on* figure. Ex. What is the square root of 4096 ? OPXBATIOK. 40961 64, Ant. 36 124 496 496 AMALTSm. — Beginning at the right, we separata the number into periods of two figures each, by plac- ing a point (.) over the right-hani figure of each period. Now, the greatest square of 40, the left-hand priod, is 30, the root of which is 6. Placing tho 6 on the right of the number, we subtract its square from the period 40, and^ the right of the remainder bring down the next period. We then double the 6, the part of the root already found, and, placing it on the b<>fihadividaB«lferapartiai^y}tof>i»e u e i ue i t a - H la eMtalnad in the dividend, (omitting its right-hand figure), 4 times. Plnoing the 4 cA Ilia right of ttaa not, alao on the right of the partial divisor, wo mnltipiv the divisor thus eemplatad bgr ^ Md aojitniet tha product from the dividand. Vhai«ot«r«9nrwia64e ^^ X. aj!UV.^££>','iii>-^c -u&> > . f^i-"-"?^ SQUARE ROOT. 243 ;r, IS ono 4aS. RuLB. — I. Point off the^iven number into periodi of two figures each, counting from units' place toward the left and right. II. Find the greatest perfect square in the^ left-hand period and write its root on the right for the first figure in the rcot. ■ III. Suhtrad the square of the root figure from the left-hand period, and to the remainder annex the next period for a dividend. IV. Double the part of the root already foxind for a trial divisor, and see how many times it is contained in the dividend, exdusite of the right-hand figure, and write the quotient as the next divisor of the root, and also at the right of the trial divisor. V. Multiply the divisor thus formed by the figure of the foot lastfound, and subtract the produtt from the dividend. , ^ V I. To this remainder annex the nes^t period for a neu> div- idend, and divide the same by twice the root already found, and continue in this manner until all the periods are used. NoTiB.— 1. When any dividend, ezolutire of its right-hand figure, does not contain the divisor, a cipher must be placed in the root, and also at the right of the divisor ; then, after bringing down the next period, this laat divi«>r must be used as tho divisor 6f the new dividend. 2. When there is a remainder after all the periods have been brought down, penods of ciphers may be annexed, and tho figures of the root thus obtained will be decimals. 3. If the given number is a desimal, or a whole number and a decimal, tha '* manner as in whole numbers, except, in pointing off the decimals, either alone or in conneottoo with the whole number, #e plaoo a point over every snspnd figure toward the right, from the leparatriz. filling the last period, if inoonqSretti, with a cipher. 4. The square root of a common fraction may be obtained by extracting tbe square roou of the numerator and denominator separately, provided the tcnna are perfect squares ; otherwise, the fraction may be reduced to a decimal. 0. Mixed numbers may be reduced to Xht decimal form bef215? of 927748? of 977137? of999999? iln«:^47, 674, 711,798, 805, 859, 899, etc. 4. What 18 the square root of 234.09? of 5.4750? of 17.3056? of 256.6404? of 0.1024? of 0.120409? of 0.000088.^6? of 609151. 76100? Ans. 15.3, 2.34, 4.16, 16.02, 0..'^2, 0.347, 0.0094, 780.481. 5. What/ " " ° " - - ~ Am. Ans. ia.3, Z.34, 4.16, 16.02, 0..'^2, 0.347, 0.0094, 780.481. What *8 the equare root of J ? of 4^^ ? of uVfr ?• of 60 A ? of of|?of28|i?ofif?ofm|?of])5,V? no. 0.86602 + , 2^, /y, 1%, \, 0.7745 f, 5|, 0.858 + , 4|, 9|. APPLICATIONS OF THE SQUARE ROOT. i Wliatisthelengthofone a^e of a square farm containing 90 acres? ilns. 120 rods. 2. A certain general has an army of 141376 men ; how many must he plaoe in rank and Tile to form them into a square ? Ans. 376. 3. A company of person* spent $75.69 ; each spending as many cents as there were persons in the company. How much did each expend? ^n«. $0.87. 4. Bought 200 yards of carpeting 1 J yards wide ; whatia the length ofcone side of the square room which this carpet will cover? A. 45ft. 5'. A man owns three pieces of land ; the first is 125 rods long, and 63 wide) the second is 62^ rods long, and 34 wide ; and the third contains. 37 acres : what will be the length of the side of a square field whose area will be equal to the three pieces ? Ans. 121.11 + rods. 6. Purchased 2 house-lots ; the first is 242 feet equare, and the second contains 9 times the area of the first ; how many feet equare in the second ? Ans. 726 feet. 7. Kequired the sides of a rectangular court-yard huviog an area of 432 rods, and whose breadth is only the % of the length? 8. A certain field contains 48020 square rods ; the length exceeds the breadth by 49 rods : what are the sides ? Ans. 245 rods long; 196 rods wide. 9. A school-master says that the number of his pupils multiplied by s of itself is 2523 j how many pupils has he? Ans. 87. 10. How much will it cost to rouglicast the walls of a garden, having a surface of 8100 yards, at 87^ cts. per yard, the walls being 2A yd. high? ^fw. $1449. 1 1. The greater of two numbers is 40, and the sum of their squares 1625 ; what is the smaller number ? Ans. ,6. 12. A clock-maker sold three watches whose ptices are as 5 is'to 6, and as 6 is to 9; the a)im of the squares of the prices is $3550. What is the price of each watch ? Ans. $25, $30, $45. 13. What is the price of a raking machine, knowing that the price added to its square gives $186 for result? Ans. $13.13j. 14. In d ividi ng the equare of the number of ao llarH thnt T have ty J"bTlhe number itself I obtain |?6 for result. How many barrels ^f codfish, at $4 per barrel, can I buy with the money I possess ? Ans. 6 barrels. OUBB ROOT.' 245 ? of CUBE ROOT. The roote of the first ten integers and th^r cubes are :— ■ 1' «2,^ 3, 4, 6, 6, 7, 8, 9, 10. 1, 8, 27, 64, 125, 216, 343, 512, 729, lOOO: l««°Inr^' T; " ''?.*''«e'^ed *•»»' t'le cube or third power of each of the uum- wwfn^ "':''''T''"""°*"y ^S""» " '»'<' '"Ot. 0' 'Arw tirae/L many 7uhlrirr "' ■""' "^ ""J'- "„«"'«' '° determine the number of figure" irt"e cube root of a gryon number,-/?.j^«„,„j; at the right, point it off Lo o. L»v i7a-. he cube root of 157464 ? OPERATION. 58 = X 503 = 7500 X 4 = 600 42= 16 True divisor, trial div., H 3 X 50 157464(54 125 8116 X 4 = 32464 32464 . PROOF. 54 X 54 X 54 = 157464. AKAiTais,— Begin- ning at the right, wo separate the given number into period?, by placing a point orer the units' figure. then over thousands, Binoo the number of periods is two. the root will con^i.-t of two fig- ures, tetu and unitt. Then 157464 =r: the cube of tens, ptusthreo times the square uf the tens into tne units, plus thro, times the tons into the square of the units, plus «bo cube of the'uni'S' tieVutw ''^h'" "'T'.'"^^ and\ust therefore be' found in the thousands of the number. The greatest number of tens whose cube does not exceed la? thou- sands is 6, which we write as the tens figure rf the root. We then subt^t the Ji fh^°»f'^'^''' ^•""/"'^ "^■^^^ t *•""• <■"«" *« "' thousands, and tbereVmain 32 thousands; and. annexing the next neriod, we have as the entire remainder. 32464, equal three times the square of tVe tens into the units, plus throe tiDiei: ;h™^»"■J"^^ "* ■<1"»™ »f 'he units, plus the cube of the units, or the product of three times the square of the tens, plus three times the tens into the nnjS^, pluil the square of the units, multiplied 6y the units. By dividing this remalWw by''* three times the square of the tens of the root, we obtain the units, or a niMal»r unite. We thorofore maltthhtM times tte square of the ten of the root. = 76 hundreds, a trial divisor. with7?&oh w. dividT. the 3S« hnndredi of thrroi^in- fhI'.«Ti'"?* K*.**"" I* "."J"' '^""^ "">' ownotform any part of th. piSd^ of the square of the tens by the unite. The quotient Igon obtained, 6. mlist bTth^ oomnlete the divisor on the supposition that A la the trtie nnite Bgnro of the S)ot riHei *.l'r?"'.'r '".«• f«r tfiTromainder. We theroforo t«lw 4. a nSiXr ^»Sr«?^f h ^? ",'>"»i"'»« of tha tiUl divisor thraa times the 3 tens of the 5^l.nr flinft yi?"%P''"."'2'*»1."*"'"'« * '"»""5 W'* multiplying Uie true divisor. 81 116, thus formed, by the units, and sabtraoting the produot7MSl4 ^ OVBB ROOT. 4S9. Bulk.— I: Point off tht given number into periods of three /igure$ evu^, counting from units place toward the left and right. II. Find the greatest cube that does not exceed the left-hand period^ and write its root for the first figure in the required root ; ' suhtrat^ the cube from the left-hand petiod, and tojhe remainder bring down the next period for a dividend. ' y Illrf At the left of the dividmd write three times the square of the first figure of the root, and annex two ciphers^ for a trial div- iior ; divide the dividenJU^^ the trial divisor, and write the quo- tientfor a trial figure it^ie root. '., IV. Add to the trial divisor three ttihes the product of the tens ■figiire of the root by the u^its figure with a cipher annexed, and the square of the last figure, for a true divisor. V. Multiply the complete ditisor by the trial figure ;' subtract the product from the dividend, and to the remainder bring down the next period for a new dioidend. ^ VI. Multiply the square of the root figures already foundy.by 3, And to the product annex two ciphers for a new trial divisor; ' and proce^.as before until all the periods^ are brought down. NoTi.->Tl)pob8enratioiuinadein Notes 1, 2, 3, 4, and 5, under the rnle for the eztraotion of the square root (468), are equally applicable to the extraction of the onbe root, except that two ciphers must be plaeed at ^he right of a true divisor when it is not contained in its corresponding dividend; and, in pointing off deoimals/ each period must contain three figures. EXAMPIiES FOB PBAOTIOE. 1. What is the cube root of 12326391 ? 2»« iTrial divisor, 3 x 20» = 3 X 20 X 3 a 3«'= True divisor, • OPEBATIOtr. 1200 180 9 12326391 8 1389 X 3 = 4326 4167 231 Trial divisor, 3 x 230» = 158700 '8 X 230 X 1 = 690 - l« = 1 True divisor^ 169391 X 1 = 169^91 15939 1 a. W hat liTEecnBerootbftaait 6r337frYof-t2Wtf of 327€6^ of 110692 T Ana. 11, 15, 23, 32, etc. 3. 'What is t^ie cube root of 186193? of 272144? of 456533? of 704969? of 970299? Atu. 67, 64, 77, 89, etc. \ . i' ''- J C k>'»f , »' CtTBE BOOT. 247 4. What ig the cube root of 1367631 ? of 9938^7fi? nf ii'raiQ99<> r''^}^lrf^:^^^^^^ ^♦W. 2.091, 37.244, 2.22. 803 2^45 84t'^f^V4r„,7fftnfyt^' °' " °' 3'»S"''«?Hi?-«f .W. 3i".§f + , 3*, f, 4.334 + , 6i, 2i, i^. APPJilOATIONS IN CUBE BOOT. 97!; "^ [!?T".''if^^^ *^ ™'** * C'^bical cistern that shall contain 2744 cubic feet of water; what must be the length of one of its sfaS? whatfs ^e dept'onhf bJi^^ *^* '"' ^^^'^ ''' ^-^^« ^'^"-t 5 3. Wh^t quantity of paper will be required to make a cubical hn. which shall contain fj Va^olid foot ?^ j^. J of a Vd 4. A carpenter has a plank 1 foot wide, 22M' feet loL and 21 ' inches hi^j^and wishes to make a bo± whie^idrehalf'bJ twij seed, bought>t 35 cents per lb., knomng that the 4 of^ the cabb of the,nuinbei^ual ?650916ff? , ^ * , °* 6. A mattress-raaker purchased 84 lb. of hair, for which he cave a sum such that the ^f of the cube of the price, d minidSd by WSit same cube equal $0.6591. How much £d the 84 Ib'S hhpY^ 7. Required the valuVof the articles contained in 2rb!ies*®l.hV confining as many articl,^ which cost a^Slny into •s'fheJel^S ^v«e-9l';^;;:^ie?r'"^'' ^^- *' *''-^ * -4W|er, n/mbe^rT^jqt^^^^^^^^^ ^ there are boxes ; and each orange posts twice as many cento w^eS areboxe^. Required the numb^ ^ boxes and oijnges. ' 10 Individin^thecubeofao^n^rmitfefte^^^^ of the same numW, we obtain rft for quotientf what ii thfsTu*.^ 11. A reservoir, whose length iiCto ito breadth as 13 is toT and ' dlmenirions of the reservoir ? *^«*»»*^* uf waterj what are tti9~~ . 12. Somemerchant^oia^iJitn^;^^^ invested 1000 aama^ydoUar.aa&'^^'lJu^ravin^^ ' i^ NJ u^ "^ ^ \ht> Uf ■J I »k ^^ *. .^^'''iHti. 248 -W" ■ARlTHMBnOAL PROGRESSrON. a profit of 12560, they find that they have gained the half as much per cent, as there are associates. How many partners were there in the company ? Ans. 8. IjL An inlayer hoaght a certain quantity of pearl-shells ; by paying 11.35 per lb., and multiplying the square of the sum he laid out by the ft of itself, it gives a product of 59049. Required the number of lbs. he bought ? ' , Ans. ^5^j]h. • 14. How much must a merchant pay, «t 65 cent! per lb.,, for a certain niAnber of bales of wool, each bale containing 145 lb., the number of bales being such that in multiplying together its |, ^, and I, the product will be 8640 ? ' Ans. $'6828. PROGRESSIONS. ARITHMETICAL PBOQRESSION. ' i . - ' 460. An Arithmetical Frogiesslon is a aeries of num- bers increasiDfi; or ditoreasing by a constant difference. 461. The Terms of a series are the numbers of which it is formed. 462. The |^remes are the first) and last tetms. 463. The IJte&LIiS are the intermediate terms. 464. The Oommon Difference is the number added or sub- tracted, in order to form each successive term. 465. An Ascending Series is produced by adding the coniinon difference to each term successively ; as, 1, 3, 5, 7, 9, 11,13, 15, and 17. 466. A Descending Series is produced by subtracting the oommon difference fr(»u each term successively ; as, 17, 15, 13, 11, 9, 7, 5, 3, and 1. 467. The sum of the extremes is equal to the sum of any two terms equally distant firom them, or to double the middle term. Thus, '1 3 5 7 9 1_7 1^ li Li _? 18 18 18 18 18 46S. The following are the Jive quanUtiea considered, three of which being given, ^e other Uoo may be found : — 1. The first term, donoted by a. 2. The last term, " " 1. ~S. The (ommon differencK^ ;^fc =^r « « 4. ' The number of terms, • 5* Thetumofall the terms, <' " •• Noii^Hitlf the ran of any two namben !■ called their ArithmtHcal Mew%. ?»-\: "', 15, 13, , ArITHMBTIOAL PEOORESSrON. 4 1 «nI^'^~^'"' ""' *•"" '» ^ the aocond term = fatterTi : •^7"" ^'fferonoe. etc. Therefore the 249 18 = 19 - 1 54 58, the last term, N0T..-If the series is descending, subtract the product from the flrit fhn. EXAAIPLB8 FOR , PRAOTIOB. 3. Tlie first term of a descendinir «>ripq in in oj.j *i. ■^"*" *^^^' ference 5 ; what is the 13th tS ' *^*^ ^''^ *T™°" *^'''- . 4. A board 2| inches wide at the narrow end and 10 f!f; 'i^' - — • ~ Ans. 17Jin. I: 5. If the first term of kn aso(>n(i;n « «U4 *i. u . .**•• '• old ; their ages increase n VriCSl J^^ -^^ *''^^* ^* y**" diflrJrence of theit a-eT? ''"'^'"**''»' Progtesaion. Required the -.41 .!" '^ ,^^ij<(fo»-a;'*A'.A!'i , i^ ."t-iifci I .-.«.%,. v&IT^ii ^1'^^ 230 ABITHHKtlOAL PR0ORB8SION. 4. If the extremes are and 2 J, and the number of terms is 18, what is the common difference ? An8. -^f. 473. Casb III.— Oivm the extremes, and the common' dif- ference, to find the number of terms. AnALTais. — Since, a + (n— ^C^l, n= — ^ h 1. Honoe, the 474. Rule,— Dtrufe the difference of the extremes by tJie common difference, and increase the quotient by I, EXAMPLES. 1. The first term is 8, the last term 203, and the common diiTerence 6 : what is the numtei^ of terms ? Ans. 40. 2. A man going a jdwrney travelled the first day 7 miles, the last day 51 miles, and each dav increased his journey by 4 miles; how many days did he travel? J .4n«. 12. 3. The extremes are 2 J And" 40, and the common difference is 7^ ; what is the nui]|iiber of terrts? ^ns. 6. 4. -In what tirtie can a debt be discharged, supposing the first week s pavment to be f I, and the payment of every succeeiljng week to in- prease by $2, till the last payment shall be $103 ? Ans. 52 weeks. 475. Case IV. — Given the extremes, and the number of terms, to find the sum of all the terms. AUAfcYsra.— Since, the asm of the extreme! of an arithmetioal progression te equat to the «uin of any two terms equally distant from them, it follows that the terms must average half the sum of the extramesi Hence, g » i (a + 1) n. 470 B.VLZ.— Multiply half of the sum of the extremes by the number of terms, EXAMPLES. 1. The extremes of an arithmetical series are 3 and 19, and the number of terms 9 ; what is the sum of ihe series ? Ans. 99. 2. A man bought 16 acres of land, giving $1 for the first acre, and $121 for the last acre ; the prices of the successive acres form an ar- ithmetical progression. How much did the 16 acres cost ? Ans. $976. 3. A gentleman wishes to discharge a debt in 11 annual payments such that the last payment shall be $220, and each payment greater than the preceding by $17 ; what is the amount of the d^bt, an I the first payment? Ans. Ist. payhient, $50. 4. A merchant bought 20 pieces of doth, giving for the first, $2, and for the last $40 ; the prices of the pieces form an arithmetical ,:,:jeriefrv^w much did the doth cost ? Ans. $i20. 5. If 100 oranges are placed in a line, exactly T yards from ea6F other, and the first 2 yards from a basket j what distance must a boy travel, starting from the basket, to gather them up singly, and return with each to the basket? •'^■^ ?M* lU.,*tt^I^^. )>i_ 'j<'.^ . ■J^'k^i j^ .X'llta^t'w 57 "^Ji , »«•,- ^ *','!I^ , tlBOMOTBlOAL WWMaMfllON. " 261 and the > M. 99. icre, and m an ar- .$976. )ayment8 t greater an I the it, $56. first, $2, Ibmetical . $420. om each' ~ GEOMKTRIC^PROGRESSION. . 477. A Geometrical ProgHlsaloil is a series of numbers' increasing or decreasins by a ooJsLhatio. «» «' °a°»'>e" U2* A «*'**'". /^ ^'^^ constant multiplier or divisor. 1. The fir8t>rm, depoted by a. ^. 1 nejast term, « « j 3. 5^h:§Tatio, * « << ^ . , The number of terms, «' « n* 5. The sum of alHhe terms, \" « g' thSTr^jS.* """^^^ ^ batw..n two nibM. I. th..,u« «attf 482. ^^^^-f^fn^jhefi^^^^^ the ratio, and the nun^ oer 0/ terms, tqfind the last term. whfusKrS?'*^^"^'"**^"^'^"*^^*' ^^^^ «»«o-3j ^"^^"-?t f^inl'S,. : 1' Jt^- *^« -t- «^ the series, The fourth term = 4 x 3» 4 x 3« = 26244. An$. and 80 on. Hence, the last term, 1 a a X r"~*' 488. RvLE.^Multiply the fint term, hy thai power of the ratio denoted hy the number of terms, leu one, ^ ' BXAMPLKS. is L^h'^Sri^T '''^^'^'^^ ^^ •« 2, "^^^h^ 2. T he first tgrmof a aeries is I 458^th« MmbMLoLteS; ^ t l ad die ratio 4 5 whatw ^ia^WiiS^^^^^^^^^^'*^^^^''^^^ 2 t\\t rX" ^"^J* ^ ^S^"' *8^'°« *o pay 1 ™i" for Ae first ek 2 mills for the seco^Kl, and so on ; what difthe last egg coat her T^ Ai#.f0.266. . I i- .siU ^ k 4-5w Art^V /fi^-. «*t • 252 aiOMRTRIOAI. PBOaBKKICflf , 4. If the first term of a series is 30, the ratio 1.06,' and the number of terms 6 ; what is the last terni ? An$. 40.-14^767328. 6. A person traveling goes 2 miles the first, 4 miles t)>e second, 8- miles the third day, and so on, increasingin geometrical progression .for 10 days. How far did he travel the last Mf 7 Ans. 1024 miles 6. Bought a lot of land containing 15 acres, agreeing to pay for the whole what the lai*t acre would (jonie to^ reckoning 5 cts. for the first acre, 16 qts. for the second, and so on, in a threefold ratio. What did the lot cost me ? AAs. $239148.45. I'- '^ 484. Cass 11.-^ Given lie extremet and ratio, tb find the turn of all the temu. and the .ratio 4 ; re- Ex. The first term is 2, the last term quired the sumnif all the terms. ^ OPIRATIOK. 8 + 32 + 128 + 612 = 4 X sum of the series. 2 + 8 + 32 + 128 s= 1 X sum of the series. ■ ■612 — 2 = 3 X sum of the series. Hence B^^g" » ^ 170, the sum of the series. Analtbis. — Since 512 Hence, the ir, 2 = a, and 3 = r — 1, s Ir-A 489. Bulb. — Multiply the last term hy the ratio^' subtract the.firat term from the product, and divide the remainder by the •ratio leu one. Norn.— 1. If the rtfio la 1«m than 1, the prpdaot of the laat term, mnltiplied by the ratio, must be anbtraoted from the fint term ; and, to obtain the divisor, the ratio jnust be gubtraoted from the unity, or 1. 3. When a dewending aeries ia continued to inflnitvit becomes what is called an Iiinirm Siwis, whose last term mnst be regarded as 0, and its ratio as • fraction. To find the sam of an Infinite Series,— X)*eMb tktfint ttm bjf a tmit dminuhed by tlur/rtntim dmu^ (As ntio. IXAHPLIS. 1. The first term ofa series is 4, theijlast term is 62500, and the ratio 6: what is the sum of all the terms ? Ans. 78124. 2. If the first term of a series is 12, the ratio 3, and the number of terms 8 ; what is the sum of the series 7 Ans. 39360. 3. The first term of a decreasing series is 102, the last term 4, and the ratio ( ; what is the sum of the series 7 Ana. 151. 4. If theflrstierm of a seriea ia 6j.,tLa xatio |^i terms 6 ;. required the sum of the series. Ana. ISlff 6. fhe first term of a decreasing serie? is 106, the last te^iiTIo, and the ratio ^; required the sum of tne terms. ■ Ana. 130. .>1.1s'Wl*''' tj *Mi. Jh^-* rji>f5p- r-^* klASURBHENT OF LUHBSR. 253 6. In what time will A certain debt be discharged by monthly, pay- ments in geometrical progreaeion, if the firat and last payments are ^1 and $2048, and the ratio 21 Arts. In 12 months. 7. A young man agreed to sen'c in a store for 6 months. For the nrst inontb he was to receive $3, and eacl\ succeeding month's wag63 were to be increased by J of his wa^ps for the month next"^ preceding ; what sum did he receive for the 6 iqbntha ? Ans. $91.95 + . 8. A gentleman wishing to purchase a piece of ground, measuring 10 square rods, thought $1 per sq. rod too high a price ; he, never- theless, agreed to give 1 cent for the first sq. rod, 4 for the second, 16 for the third, and so on, in a fourfold ratio : how much did that ground cost him? ilns. $3495.2?. MEASUREMENT OP LUMBER. 4S6. Boards a)e usually measured by the square foot. The board is considered lo be 1 incl^ in thickness. 487. Flanks, ipeams, Joists, etc., are usually measured by board measure. Round timber is sometimes measured by the ton, and sometimes by board measure. A 488. To find the contents of a board. Bulk. — Multiply the length of tJte hoard, taken in feet, hy ita width in feet, and the product will be the contents in sq.feet. Or, Multiply its length in feet by its width in inches, and the prod- uct divided by 12 will give the contents in square feet. Non.— If the board ia tapering, take half the anm of the width of itt enda for the width. ^^ Ex. 1. Vhat are the contents of a board 36 feet long, and I^ f«et wide? Ant. 64 sq./eet. 2. What are the contents of a board 24 feet lone, and 15 inches wide ? Ans. 30 sq. feet. 3. What are the contents of a tapering board, 20 feet long^ whose ends are, the one 24 inches, and the other 13 inched wide 7 489. To find the contents of planks, beams, joists, etc. Ri^LE. — Multiply the width, taken in inches, by the tJiicknesSy in inches, and this product byjhe length, in feet; and the last product divided by 12 willgive the oontentt in feet,, board measure. .^^ Jj^tiiB plan*, twain, «10. irapiiffig ffiwiaiBTKSe half thelomWiKr width of the ends for tbo width : and if the taper be both of the width and th« Uiicknesa, the common rnlo of obtaiping the oontents in cubic feet ia, to mtdtiplu A«ry<*« »im of the arm» nftKe two etuh 6y th» length, and divide theprodiM 6y ti 'i-fjf^-ii I. :m^.. ^ «• -. * -I-J. 254 MWOeIv.ANKOUS AXAMPLB8;., 2 How many feet are there i:. 9 joifits, wl.icli are 15 fcet'7onjr'. 6 i"che« wuJe, and 3 inchea thick? Ans. ICHi S. xvnh'h .r ""r"^ '^i '° ^ **^*"'« 24 feel ]0Dg, 10 inches thick, whoee w!.Hh tapers from 18 to 16 inchee ? Ana. 858 feet. 4»«e girth is 60 incites ? Ana. 781 !n,b flT f «;■ „ J" • ?r^n'- ''*?"^o'^ *''^* ^'''^^ «f *""ber whose leh-th is 30 feet, and girth 40 mches ? ^„, 2047661 MISCELLANEOUS EXAMPLES. sum ^n t en^' ^ *^** *° ■"^'°^' '^^°" '^** * of AlTnS^t'he 3. A gentleman bought 96 Vards of cloth, | of ayard wid^fo; $100. and gave the aame an(r$26 for cloth of the elme quality, 1 yaS wide How many yards did he buy ? An$ IS^yd 4. A fS|ther devised ^ of his estate to one of hi* sons,' an? jJ of the residue to tne other, and the remainder to his wife/ The dffler- ence of his sons' legacies was found to be £267 ' 3 4. What moner did he leave for his widow? Ana. £635 lOM. ^,?\ How many bricks 8 inches long, 4 inches wide, and 2 inches fShick ?'' ^ ^"'^'* * """^ ^^'^^ ^°"«' 20 fe'et high, and 1 —if. Ifa nTaircsrpainl^ square ^S^s in one hour, and is 31 h. the widl ? ^^°* "* ^"*^"^ ^"^^ ^^^^ <^^* ^*" ^ ^«et high ; how lang is Ans, 80 ft. ,h^ / . ■V'"'-^ /."^'h^fW^'- mSOXLLANEOUS EXAMPLB8. 255«k 7. By selling wheat at 12 e: 6 d. per bushel, I gain £30 on the quan- tity purchased J but if I sell it for 13 s. 6 d. a bushel; I shall gain £42 on the same quantity. How^^tany bushels were bought? Ana. 240. 8. A grocer bought a hogshead of wine for $28.35 ; how much water must be added to reduce the first to 35 cts. per gal. ? Ana. 18 gal. 9. A father, dyjng, left his son a legacy, { of which he spent in 8 months ; f of the remainder lasted him 12 months longer, after which he had 6nly f410 M. What amount did bis father bequeaUi t>ni? j«i!i|P^' ^n«- f956.66|. 10. A man had \ of a yar*f^y|il^ and 60 cents for 11 yards of cassimere. What did the <^|||eC^l'>>n P*"" yd. ? A. $2,661. 11. How many dollars, Cm^smM^^y, afe equal to $160 linked States currency ? , T ^g aaip^ 12. A grocer wishes to mix t^ptherbrandy at 80 cts. a gal., wine at 70 cts., cider at 10, cts., and.ifrater, in such proportions that the mixture may be worth 50 cts. a gal.; what quahtity of each must be used ? Ana, 3 gal. of water, 2 of cider, 4 of wine, and 6 of brandy. 13. If the longitude of Boston is 70" 4' west, wjjat will be the time in that place when it is 3 h. 35 niin. A. M. in London ? i4n». 10 h. 54 inin. 44 sec. P. M. of the previous day. 14. A merchant sold goods to a certain amount, on a commission of 4 5(5 ; and, having remitted the net proceeds to the owner, received 4 % for prompt payment, Avhich amounted to $15.60. What was his commission ? ^n^, $260. 15.1 purcliksed railroad stock to the amount of $2356.80, and found that the sum invested wfis 40 % of what I had leftr what sum . had I at first ? Ana. $8248.80. 16. If 1 3i bushels of wheat make 3 barrels of flour, how many bush- els of wheat will be required to make 401)arrel8 of fiour ? Atia. 180. 17. The capital ofan insurance company is $250000; its receipts for one year are $58760 ; its losses and expenses — jgNfljITr What rate of dividend can it declare? ^^mSna. 1h%. 18. By selling a lot of books for $438, a booksellerlo^lO^; how much should the books have been sold for, to gain 12^%? 19. What is the difference between the interest and the discount of $540 at 6 %, for 6 years 10 months? . 20.. I own 25 shares of $50 each in the Gosford Railroad Co., which*has declared a semiannual dividend of 3J %. How much do I receive? ilna. $43.75. n 21. If 12 boarders eat $25 worth of bread in 2 mo., when flour is $9.50 per bbl. ; in how many months will 16 boarders eat $60 worth of bread, when flour is $1 2 per bbl. ? Ana. 3^ mo. 22. B hired si, house for one war for $300 ; at the end of 4 months he takes in C as a partner, and at the end of 8 months he takes in D. At the end of the year, what rent must each pay ? Ana.B $183i ; C $83i ; D $33fc. ~4Wr- A~p«Wft miieA M ^jwl. of sugar at fiO, -witE^S^cwt Trt f8|,~ and 8 cwt at f 7i ; how much was 1 cwt. of the mixture worth ? iln«. $8||. .\- ^. k\A.£:. . ^ijE) ' , \ "-■Y. ^56 MISOELLANEOirs EXAMPLES. 24. Afihipnientofwheat was insured at 2»%, to cover I of its value ; the premium paid was $44.07 ; the wJieat being worth aO eta per bushel Jiow many bushels were nhipped ?■ Ans 2825 bush. 2.J. A stack of hay will keep 24 cows or 18 horses one week. Hon many days will it keep 5 cows a-nd 5 horses ? Ans U^ da .26. 0, of Montreal, remits to D, of Quebec, a bill of exchange on Liverpool, the avails of which he wishes to be invested in goods on his !nH Vi r ' l'a^'."g reserved for himself 4 r^ on the sale of the bill, fnvlri forcommission, he invests the remainder. Whatis theamount invested, and for how much was the bill drawn ? 9-7 w. ♦ ''*"*:^pvestmeTit,$9461.68Ai the bill was £2025. 27. What per cent, is gained by buying oil at 80 cents a gallon, and selhng It at 12 cents a pint? j„g 20 «^ 28. A merchant oays $10050 for a stock of goods : he, sells them at an advance of 34 jg ; the expenses connecte(T with the business are 29 What o clock is it wlten the time from noon is X of the time ,torn,dnight? ^n*. 5 o'cl. 2Ymin. P. M. ^0. A merchant receives on commission three kinds of flour ; from C he receives 20 bbl., from D 25 bbl., and from E 40 bbl. He finds that C^ flour is 10 ^ better than D's; and that D's is 20 56 beSer S Es. He sells the whole at $6 per bbl. What in justice should each man receive? ^n«. C receives ll39W; D, $1581Jf ; B, $21U«. 31. For wMt sum must a note be drawn at4 moT, that the proceeds of It, when discounted at bank, at 7 %, shall be $875.50 ? *. * A^cf'^^°/'Vf'"°,'?y'''"^«^'''^e«os*^3.37f, whaf^ill be the cost ot 36^ yards 1 i yards wide? Ans $62779 33. What must be the face of a note at 60 days, the proceeds' of which, when discounted at Bank, at 6 %, are $100 ? Ana. $101.06 + 34. A merchant sold a piece of cloth for $24, and thereby lost 25 9f. • what would he have gained lia(i he sold it for $34 ? Ans. Gi % ' 35. A bankrupt compromises with his creditors fot 374';^; how much- will he pay on a claim of $3056 ? Ans $137r 36. A man, dying, left $3565 to be placed at intei»*st for his son who was 16 yr. 6 mo. 16 da.' old; how much will he receive when he iS 21 years old, allowing 7 % interest? Ana. $4698.37 + 37. A garrison, consisting of 360 men, was provisioned for 6 months: but at the end of 5 months they dismissed bo many of the men that the remaining provision lasted 5 months longer,: how manv men were sent away? jins 288 38. What sum must I invest in liie New Brunswick 6 % stock, selling 8, 4 and 2 inches? Ans. 32148 bricks. 44. A woman buys apples at the rate of 5 for 2 cts., and sells them at the rate of 4 for 3 cts. ; how many must she buy and sell to make a profit off 4.20 ? ' - Ans. mo. 45. Sent $12300 to my agent in Toronto, with which to purchase flour at $10 per bbl., after deducting hia. commission of 21 %. How ' many barrels of flour did I receive ? Ana. 1200. 46. Borrowed of A $150 for six months; afterwards I lent him $100 ; how long shall he keep it to compensate him for the use of the ^^mhe\eatmt? Ans. 9 mo. 47. A broker charges me 1 J % fbjr purchasing some uncurrent bank bills at 2o^ discount; of these bills, tlireeof $10 each, andoneof$50 became worthless ; I dispose of the remainder at par, and thus make $520. What was the amount of bills purchased ? Ans. $2500. 48. A grocer mixed 6 lbs. of sugar, at 8| cts. per lb., with 80 lbs., ^t 7, cts. per lb., and 60 lbs. at such a price that the mixture was worth DJ cts. per lb. Required the price per lb. of the last kind of suga;- , , , . Ans. U\ cts. 49. A gentlenmn's garden w 234 rods long, and 134 rods wide, and 18 surrounded by a good fence 1^ ft. high. Now, if he shall make a walk around lus garden within the fence 7/, ft. wide ; how much will remain for cultivation t Ans. 1 A. 3 R. 7 p. 85uk ft. 60. A certain principal, at compound interest for 5 years, at 6 9^ will amount to $669.113 ; in what time will the same principal amount to the same sum, at 6 $6 simple interest ? Ans. 5 yr. 7 mo. 19 + da. 51. I invested I of my money in R. R. stock, which depreciated CJ % I the remainder I invested in real estate, whiclv advanced 15 % and thereby I gained tl600. How much did I gain in both invest menta ? Ans. $250. 52. What % in advance of the cost must a merchant mark his j^oda, so that, after allowing 5 ^ of his sales for bad debta, an average credit of 6 months, and 7 96 of the cost of the gooda for Mb expenses he may make a clear gain of 12^ 56 on the flrat coat of the goods! money being worth 6 56? iliw. 29.66 + *. 63. What 18 the greatest possible number of hills of rye that can he planted on a square acre, th^hills to occupy only a mathematical point, and no two hilla to be nearer than 3^ feet? Ana. 4166. 64. I wish to line the carpet of a room, 6^ yd. long and 6^ yd. wide, with duck, I yd. wide. How many yards of lining must I purchase if It will shrink 4 96 in length and 6 ^ in width ? Ans. 43*j yd 66. A man bequeathal \ of his estate to hia son, and i of the re- mainder to hia daughter, and the residue io hia wife ; the difl'erence between his son and daughter's portion was $100 : what did he give ^^'^*1 . t Ana.%m. ^58 MISCELLANEOUS EXAMPLES. to eLu f' Z ioh"'-^^^""" ^ ^T' ^ *^*y «>' ^S days, were able Le by workfnVo i '" ^»«^^ "'^"^.^aya can they complete the res- number? ^ ''^ * '^^y' '^ * workmen be Jded to their nn^^n ?"«'!' '"ercliandise as follows : July 3, $35% TlTm65 on 30 da. ; Aug. 17. $6 48. Spnt 19 «>;n >*"'"■/'.">•'"'/*> •*0''>i>j .. count Oc.: I2,iia .iVi^;"'- ''■ *'"■ "•* "ir ,U2 60"°" ^fi/ P.?r.rt' money being worth 6 56? aL$US6I2^ incrLS^? TllT>'^"f"*'''^P'" ^5 M*^ 1 ^^ value had i Sf <•; ,?L^""® ,^ '^ '''''"® ^^ 30 51$ more thaa May 1 ; July 1 I sold t for 16 56 less than its value June 1, receiViS in Dayment a ^Sfin^' °'>te, which I got discounted at k bank a! 7 56rSv ng U iri±i .^r Tt ^^^r, Proflton theoats? ^ni. $'3238.52 ^ of ?ee*t ofl^ ^t\h2it" t^'^^ri^ 1^ ^^^ '«l">'«i the number oi leei ot lead mpe that can be made from 80 lb. of lead, the caliber of the pipe to he 1 inch, and the thickness of it i of an inch onef;u?tr;o^°?i'^"*°"*^ ^^g°^« -«« ^ol'^ toi:i«'"a'cet^n-56, rwhTS/ii 56r^rnri^^T tix^^iAr 66. A merchant in Kingston haa'soOO fran^'due him on' fJctnt to re^t id.S»^n h'*"".' *"" ^^\«a» adviae his co^rrespondent in Paris on r^La ^ °° Ffu*^** purchased with the sum duVhim, exchange on Canada being at the rate of 6 fr. 20 centimes oer doUar WhS sum will the merchant receiye by each metW ? ^ **** Aru, By draft ou Paris, $970 ; by remittancrfrom Paris, $961.63. ill h.tj, ; r " 'T'"^ *? «''''"^ ^^^ bushels of proyenS, worth ' I Qft ^ !?®^' ^'*"" ?*i^ "^"^^ *-^0, corn worth $.80, barW worth f.90, and rye worth $1.10, and wheat worth $1.30 per hS How many bushels of each kind may he take? *^ " ^^ fis w« 1, » '*!f*'3 20, 20, 60, and40, respectiyely. Of ?Lp!°aTfTfa ^g^tn? ^'^^^ ^ ''•' ""^^ ^ «^^«" ^^J-^- cai^'co1irnZ\trio''*r-"""" ?a9hday,for 20 days, K a caiuc containing 10 gallons of wine, each time supplying the defloiency by the addition of agallon of water ; and'-then, to^Ssca^ detS, hJ 'MIS0ELLANE0U8 EXAMPLES, 25» worth Xfnf Hnw n Lf •' '"PP^^ .°^ the deficiency each time by a gallon ^'"^ How nmch water etill remains in the cask ? ^ f . 10 Amerchant hLl';'^ 7 T'" *''*".'* ^'^''^ half a pint. t\.l 'a f- merchant has JilG due him, to be paid ig 7 months- hnt wan^ one to fall due Mfyp^,Tvh:; ^5^ IVrTe mS' ^^^ nrJ^^i^^l^. ^"Sht merchandise as follows: AnHr*8 ^1150 22- May 23 $55.64, on 30da. ; June 2, $82.60, on 2 nfo^anl JulJ u' $90.^ What was due on the a^fcount Sept.' 26, n,on;y bei-ig worth exlcute^f rS h'°"" * '''' *^^^*--' 12 mtVere able to E^?l ■ if i •^' K^'T '"*"y '"^" "'»J^ be withdrawn, and the refr waltfwe^nTtdt' wTr " ' 'f ' ^l^''^^^' »^« "IttThand the precise time when the hands were in the first position ? ZaTf^i'^^^T"^:^'^" """'ber of dollars which E puUn wm Spitll 1 i"™^' "^'^*^' ^' ^'*« •""P>^^«* '° 9d«- Wh'S w^ E^ 77. If stock bought at 8 45 discount will r.„-^ t "rl .i.^""' ^^^^' at what rate shoul J it be bo^l to ^Iri" ^96 ? ^^ as'fi'^'^Illf "*» Ihr-h^,'^ 'T^"^,' -'i «'«th% a w^ioVsalfdealel^MO-^td'rn;.; the who eaale dealer sold it to a clothier at 124 <^ advTnce. ShJS^ 80. A merchant sold i of his good, at an advance of 25 ^J 1* of them at a loss of 8 96 ; J)y of them at a profit of 30 ^ and 1 of Sr«m tu ^"^T'ff^ '^k 5^ "'^«^' 56 «f ^^« cost n^ustThe remlindeJ K Bold m order to lose 5 ^ on the whole ? AnT^^ltl ■ \ I/,1°«'^«J *" .8 % dividend on Montreal city railroad ^tock and orcMed to $13750, .what was the amount of my dividead? A. $1000 82. A tador bought 40 yards of brdadcloth, 21 y^wid^* but on ?ponging It, ,t shrunk in length upon every 4 yd. half a quarter and in width, one nail and a half upon every II yl To irnfthL Moth he bought flannel 6 quarters wiS;, whic^. fcwet J^nk t". t& •> • « x\=«^ --%3r.r_ 1 260 MISOELLANEOFS EXAMPLES. width on every 20 yards in length, j^nd in\ width it shrunk half a clotix '^""■^'^ ^^^ number of yards of flannel used in lining the «w■i'''M^P"'■*'^^^'^ at 6% premium payaW on~the ^^vSment, fti' ^A "^ '^P^V^ purchased at 1 5 ^6 discount ? Ans. 7 J, %. m. A mprchiant failing in business can gay T^ cts. on a dollar. He otJers, to pa,y his whole itidebtedness withSut interest in 5 years if his ZoTa r" ^''^^^l^''" to g« on with his business ; his offer being ^ccepted, how much wiJI .hia creditors lose in tjie 5 years, money being worth 7 96? ■ ■, ^„,. $.026 on a dollar" ^ I^-'^l^f^^'^Vit^»^ityofm^e for $676.32 J, at 85 cents per g&l^n ; but a part having leaked out, the remainder was sold^t leaked*our?°^' °"^ °°* '^** realizedi Whilt quantity 86. A owes B J600 due in 4 months, and $840 diie i^'e inonuls • T<-?^®li.^'''^.'* *^"^ '" ^ "months. If A should make bresent pavment r ef ^ ' ''^'^" ''""''* ^•'" J"8t'«« to P^y ^ ? ^n*- Irt 2mo. 10 Ida. . \, 87. How many pounds of sugar at «, 13, and 14 cts. per pound, ZZ\T^ ^"^ ^^^.- ^* ^* cts., 21b: at 8 J cts., sTd 4 Jb. a I4ct8.*lb., soastogam 16^ by selling the mixture at HA cts. •^'^HS wv, ♦ .. A-^ '^"f' ^ '•'• *f ^ 5 ^i '^- *t 13 ; «lb. at i'4. /•Qnnn Vf tJ>«,d£T"''^ ^**^^" ^he true and banl^, discount of ^$3000, payable in 120 flA ?*" °^^^ * ''*"'* ^ ^ P*'*^ '•* ^ ^"*' installmenta at 4, 9, i^v am.! 20 months, respectively ; discount being allowed at 6 %, heiinds that $750 ready money will pay the debt ; how much did he ^"■oV n. T.. ' ^n«- 1784.74 + . ^ T? 1 J ^ ""^"^y ^*^ '^ E's as 2 to 3 ; when D had spent $40, and t. had Ppent 40^ more than D, D's money, minus $20, was to E's money, plus' $2, as 4 to 9. How much had eacli at first ? 03 wui- , - ^n«. D,$108;*E, $162. r .1 nnP*'*i^- *^* *'<*** <*^* ^^ '^^y*^ bi" on Montreal, to the amount QQ n?.' *' • ^ premium, and int. off at 6 ^ ? Ans. $95«75. '' . J Dfilf!f^'"^"*"g*g*^ '" '^« lumber trade: A furni8hedl4000. I and B $6000 ; they gained $1 680, of which C's share was $840. Be- quired C's stock and' A'a and B's gain. ^ns. G's stock, $10000 ; A'a gain, $336; B's $504. V ./.i^'"?" having lost | of his money, found he had remaining ony »672 ; how much had he at first ? Ans. $1792. 101. A speculator invested a certain amount in railroad stock'^, by selling these stocks at a deteriorated price he lost i of hid investment : by investing the remainder he cleared $240, and afterward lost I of the money he had remaining, . which left him possessed of $480; how much did he invest? vln«. $3600. 102. Bought a certain number of horses for $2600 ; had I bou'^ht 8 more at $10 less each, all would have cost $3560 : how many horles did I buy? Ans. 29. 103. Louis can do a piece' of work in 8 days, and John in 12 daj's ; m how many days can both dp it?-. Ana. 4i days. 104. A grocer bought U bushels of chestnuts at $3 a bushel, and retailed them at 3 cents a half pint. What per cent, profit was his , gam? Ans. 2815^ 105., The head of a fish is 12 inches long, the tail is as long as rfJe head + i of the body, and the body is as long, as the head' and / together ; what is the length of the fish ? Ans. 96 inched 106. A consignor stfndsSOd^Mifcels of flour to a commission mei chant, with jnstrudtions to selfit ahd remit the net proceeds by draft. 1 lie consignee pays $120.40 for freight and expenses, sells the flour«t $8.40 per bbl., charges 2^56 comiuiasfbh, and pays l^ premium for draft; how much does the cpnsignor recaptj: Ans. $4008.31 + . 107. How many horses could be kept ^^5 acres' of lahd^ if for every 3 horses there is of the 25 acres, 1 »cJ*e of plowed ladHj^d for every 2 horses, 1 acre of pasture ? / Ana. 3OTlpes. 108. Purchased 24a=*ushel8 of e(at8 at the ratiftof 18 bushels for $22.50, and sold it at the rate of 22^ bu. for $331 : how much did I gain on the whole? . ^n«. $60. 109. I'paid£93 16 0, at the rate of 2 i^,fcr insurance on a shoe factory ; for what amount was the policy given ? no. Exchanged 75 iPailroad bonds of $500 each, at 36 % below par. for bank stock at 5 56 premium, how many shares of $100 each did I receive? . 4^. 2284. HI. Invested £868 in Government bonds at 106%, paying II56 brokerage, and afterward sold the stock at 12 9j premium, brokerage H%- What waa my gain ? ^ ' - - my gain : dnt . £U^ ^ # 262 k Consj^nti- wbat time ]D80ELLANE(HrS(^ 112. Ithe longitudiof Paris Is #^20' 2 nople, 8° 39' E. Wlij^tisl A|,fti^t the.Mter pli , . , is It a|the former? .' \5r Ataqf^^n. 25^^3!§. past midnight. " ■ ■ rd^ of a i^DUector, w 8,56 cpp&nBll^6n,|,|)j(^ 113." Having pl^ice^fllW of $775|!j;lhe haild^ of a i^DUector, who succeeded in o)btaitiW|ff^ bf it, ahrf (Jharl • *» * v ^ receiy*? tha^ the suft-ninga orsilie were $472240, which wfts sam^tHonth in 1869. ninga 0: I* 7224;^* ijnig.hrandy and>ater,';i Of the whole +3 the whole +2 gal. is water; requfred the 4n8. 43 gal. Wndjr, 17 ^1. water: -, Lane,.an(i Gameau arewiltners ; Hamel takes ises; Pecry. J, Lar^e J, an4 Qare|?au the remainder. ,y^.,.«™viv,. .,..,/ear, the resourtjes of the firm afeii CaSh $10312.50, ^ ,JeKjhandiae $13447.50, BQnda add Mortgages $Hi75, Bank Stock »t^. '<46d0i Hamel has drawn from th^ business $9«yPerry 1525, and ^-, . Lane ^85 ) the liabilities are : Notes Outstanding wl60 ; Balance in in London jr, at 30 5(5, price per «. 12.16. el; A puts i: C, 1640 A received d€, $1040 if 15 IDO. t business ; tal. They not being the year, K85; bap. ►O-j wheat. they owed rere $750, i what was ' M 4 ■if H*" ..i^&r.:. «* r. 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