NUMBERS, o AND TO HOW BY A J^ATURAL THE METHOD. PRACTICAL COMPLETE, SELF-HELP AND FOR ARITHMETIC, SCHOOLS FOR THAN THEM: USE OTHER PRIMARY. THE BY JOHN BROWN. F. ""oj"";oo- BosftV)^ PUBLISHED BY MAILED. MASS.: JOHN POSTPAID. F. FOR BROWN, $1.00. " \ Copyright, Bt Ttpogbapht JOHN bt J. F. 8. X$92, BROWK Cushino " Co., Boston. INTRODUCTION. "o" The Natural the presenting and now Method, Science formerly First and I as chosen to differs from Arithmetic, of in employed four numbers foremost have important treated are it, of term methods respects abstract. always as : J There is number nothing in is mathematics of capable The used reference to themselves in whether A phrase relation to ter." mat- and by Wentworth : is employed particular unit, abstract is mathematics pure without Arithmetic," phrase any which quantity again, says as designate 'numbers But 8, 10, 21. all of thing numbered kind the to bers num- is is or mentioned.*' not " are This " And abstractly, Practical A " of species figures." explanations number. without are of following "Abstract abstract quantity or vocabulary the by Webster conception of number. this that " being expressed "congjiders magnitude Hill, has in novel Concrete number. " but concrete, all Second^ fundamental number the of numbers upon numbered Things are deduced the from counting. definitions formal Third meaning. is abstract." operations process without and rules omitted. altogether are J Fourth geometrical principles reserved are the for science J of geometry, where they properly That geometry has no without saying. Yet the filled with Arithmetics and of are ideas, which, For in the instance, in place The problems. has always geometrical result is a the been of cases, perplexity and is that to go otherwise. tions, defini- demonstrations, vagueness majority of ought arithmetic an practice great much belong. confusion grown. out- never in exists " m " " IV INTRODUCTION. - regard apple between the be cut into may but, the foundation employed as at fifteen solved be have in in place no understands is utterly incomprehensible. In and the find within the of fact, if lems prob- problems thoroughly one general principles diflicultywith little came for years, Such by algebra. dent Presi- he fightingover arithmetic. will when two-thirds been hours numbers he for^, simple indignant as that he had few an so found and properly comes of anything of arithmetic. the scope simplifyingarithmetic, geometry and algebra may taught much earlier than they are at present, and students thus By be well in the of use book This treats mathematics, far as people is to treat all be present common after of arithmetic far so in general requiring would come fundamental means powers b, of and for its be rr, y, to about general the kinds to this of units arithmetic plan, lars, (dolbeing and formulae. and algebraic; then, of geometry, beginning principles almost it, head. comprehension, According various ing teach- larger than no principlesof arithmetical of public schools, one difficult of algebraic symbols roots, the age. anything volume, mastery. etc.),the in under more applied magnitude early an know to connection arithmetics, no a, taught expected in at and the ideal method But moderate-sized time fractions, number generalized by be may one more no it is formulae, only. commonly branches school bushels, pounds, would as put into extension principlesof equations and several the It may and in the grounded and so consecu- mere relegated to algebra. are never a pure applying them, that was arithmetic, which in could he says Harvard being tSemeGf^m,^m^ ^ to equal parts; subjects reomring reason, isfactocv treatment Eliot line into number unit numerical analogous an a An piece of the apple and parts of the line are symbols to illustrate \ and ^ ; and numerical comprehended. having been taught, are never a fractions,never For the arithmetical. equal pieces,or a this failure to discriminat to geometrical and principleof tiveness, a piece of Yet directlytraceable is fractions to this After immediately the with INTRODUCTION. development the and interest, arithmetical and topics,would and great condensation If the that the of reception, times Mathematics and reasoning." a therefore Of has rules" the and lesser after not a ; the one fault If the degree. definition should be as the landmarks of knowledge then should we To illustrate safer to say This or avoid having : This that is into do to make in the number to ideas we unite " to not Here text-books Addition Addition.*' 2 " and we take in the abstract and than to What 3 to make conjure up the place a of all The " Or that how of process do unnatural number strict as a and that^ or or schools public like more or counting bers num- these definitions ? count we until addition, taken uniting two when objects, unintelligibleas of in the objectswithout," to is this conveyed by false and by referringit objects at applies,these things are 5 ? of process idea is plete com- of addition, it is Addition largely used is the and possible. as definitions two now sify clas- cannot thorough a serve consideration, and it is to say are is to " We simple example a equivalent number." Unless 5 ? mind " : one together is How first given scientific of Mill, is the reason purpose much as in he is because Stuart under definitions Addition, two Massachusetts numbers things is Addition. respectivelyfrom of the an definition good a reason first have we but formerly,though of scientific classification." things correctlyunless a him. for been has main definitions,in the language of John frame multiplicityof a for this very The thought to framed be frame and tic, arithme- for expected as explained,the deferred. basis reform exists advanced, yet sufficiently as outlined, later. avoid to pupil cannot been subjecthas ** above work a be not Still the same is it must been definitions. indifferent a tendency very- indicate to as Use, comprising definition " allowing such appear may child should A " be General and good definition," and late should geometry, that was other treatment. comprehensive more Schools solid scheme general the present volume and idea the course, the age Percent- geometrical progressions,and simplicityof Common for old due ripe for are algebra,plane The in follow and solids. lines,surfaces,and of measurement V 2 and 3 theory thus in together of " ideas dealing with realityit is only mathematical literal term expressions. INTRODUCTION. VI I as metaphorically, Taken these defined merely Definitions false ideas create phrases than indulged in when should be think, that favorable divert to occupied intended to apply only consideration is are And how of to do to value problem, and rule is deduced, and rule the directions follow rules. occasion to manner, not to make even made, He The the in is repeat be will presented; which with the rules it remarks subject under are so figuringneed to do the own, that he the in but rule pupil by to absurd is made be work under is trained If he to make analysis expected. has rules, nor them the of ready cation. appli- problems certainlyconduce the to different a without for the ing follow- and rule. look is unable vogue, that not to other particularproblem, the to In it. problems coming trained his long where a is first shown general principlesin same of how rules, he works been having recitations; but to These pupil applied to given. Thus, recites apply been that where is told afterwards there the chances has are wrought by easier background. cases then Rules, definitions, and which former metaphysical questions performed by referringthem are prehensive com- are memorizing from rules ? are words, general principlesare a not. or abstruse, like that of number. what a The opportunity and the in kept much mere mind abstruse to be ought that mischief so an the to which likely to requirements not. are The rule. second, it is ; so that nition real defi- a first,being usually inexact, they tend : it is to third, they tend the latter is threefold definitions those know to It is complete? it fulfils the nition defi- true we be to else of the term are general classes,those two exact, and exception,the how is intended less or is, a the connotation And definition a whether sense of are and to like something to trouble the But means. description,more a in the fornler be taken the term thing one It illustrates this. reads which well. very tells what is,it that or than more that ; whether the do qiiasi-definitions may is much likenings of mere desideratum, to glib ciency profi- INTRODUCTION. After becoming definitions hy of And methods of the applying student be may science, the the in a framing exercise. good very understood, and principlesare them be perhaps may grounded fundamental after so, well VU general,specialmodes in in advantageously expressed the of ating oper- the form of rules. That no all but, principle, new before never numbers operations upon been far so carried out (deducingall operations/rom of but obstacles simple and so because numbers the as may say tainted with the concrete of division In teacher, what there w^ere have been question, known instructor I must " high the the to as say school writes sorrow enter with solution of Another a answer have 129 " I grammar number that less and I crete con- the are their treatment by your a the of school high lower the experience as use after course, " grades ? well- a " from less the year abilityto perform unable am to year to get them to pupils the use of our ordinary culations calin reason any problem." teacher repliesin " : say in arithmetic. the with proficiencyin the in course with From " " do you of insuperable whatever evidenced fallacy,as figures of pupils commencing completion And starting-point. ciated, appre- fractions. the to answer the and thought not contrary, all -previousarithmetics writers to been has there ability desir- the because not scheme indeed- is one logicalcompleteness by It is one. to principlehas the aware, its to been as "' am logicala it has the way, in I as reducible are the to pupils schools in my Class There entering this On a different IV, nearly they school they have class had been reviewing percentage examined the weekly marks of the state, partly advance, in arithmetic with for which arithmetic a for four left of whom studied and work all had arithmetic, partly review at part question: same June. last of years. note, in of weekly years my for lesson more. eight weeks our a in This when I assistants sup- Vlii ply INTRODUCTION. I found me. 70 per below that The cent. same four eightweeks for average an class had taken had 37 of them had of 129 out English,with and Some 64 lessons taken The Latin. up fell below 70 per Algebra, 40 English, 20 as is studies. new proportion of cent those given below whose : of 129 out 29 History, each, on eight weeks tory, algebra,general his- up week a for the average "" " " " 129 129 * 4 Latin, Here larger than is much newtstudies in The students. young upon 37 " proportion of failures in arithmetic, an find that the we " which the of age average usually are old study, counted class is severer fifteen near years." It safelysaid be can any time spent upon in the radicallywrong could years which count get him to failure,as made This teach him a to very count experiment. is and young The different and And idea ; but from old, the The to fain child to count Let shouldn't why other is objects. Else why him the it was who was a but doubts be this We the expression? other no four two has is hard are we The all are to easy to teach this make a ever revelation. though It tried and We so? say could complete a one husks. numbers, of day one nature, who thing some- years children. compound. compounded me me eat any Some half a any of In for. result other children would objects. elementary, the objects. and no wander to prone fingers. experiment was alike, young and to or thinking. Noticing to me readily be imagined by can child his count similar a accounted twenty, I called him to about up be plaint com- was teaching. of child of two a The school one any the for return satisfied there methods strikingexperience set a ago to been results meagre with facility the to the contradiction public schools. our I have years adequate no confined It is not locality. For way is arithffieticin general. is one there quarter, that from fear of successful without one the act number ideas, number of INTRODUCTION. the In originalpreface from on numbers Abstract from examples first derived This Pestalozzi Arithmetic Teaching LL.D., Head Master objects. things with such no them upon thing abstract, unless are which be must learned deriving practical as abstract the have been practical." are Colburn. and is followed is reasoning sensible idea of number." operations there ; which those from the which those : all to common sons," Les- : and practicalexamples is First " acquired by observing abstract an Colburn's is this passage quality obtain little further a " is first this acquainted,we are And that observed Having we number of idea The " Warren to published in 1821, first IX the by The from following is in Primary of the disciplesof numerous Methods ** Schools," by Larkin Boston Dunton, School. Normal of Boston, ^1888: When " they children distinguisha can words. it But for numbers what number for a be give to which the to the are of word not a idea definite objectsaround *' him, and of . . him teach mind without abilityto which knowing the words of words In count. number direct his which express take must teacher the of should we words knowledge meaning him, to ideas in the Later the appropriate signifies. The mechanical for the or the know sounds, another this numbers, of of little ; that is, a things by children the or mistake to already partly known numbers." that succession a things one knowledge child as usually count can of similar numbers merely careful real few frequently happens stand should school, they enter the . order words attention their place of objects without." is the This ** good to answer " schools. the the he method And what which has it is of Warren his work, he corrected, and the the must Colburn, the many gross still further in followed now In accomplished ? we question intelligently genius performed which object all order take into eration consid- skill with which in errors fact that teaching it is only INTI^ODUCTION. X owing last ten the during the to present influence but dous We as Method in educational that of succession a counting the perceives used those words in unconnected can words, such with their relation. tell. can His Just I would and that would doesn't the from other be and know what is about to zero it. count ; a then by by he to natural After tens ones, he and to would the that be child three, etc., perceives sense of matter a child takes from not with numbers. place. will He unless doing he one, can count zero, with of is But instinct. great-grandparents taught twenty altogetherout the perceives, nobody child the some sense. two, one, some largely in counted it about to eating or walking. objects,but process know to fact in the it is of of,dog, etc., very will child he of act him, though destitute he much him first have backward any takes A child a jingles,like to words, the repeat how the nonsense to, the, cat, readily is In ? merely " that relation. Thus, utterly grandparents, he readily as him of greater part of their lives, and the or to stupen- saying related, provided are Unquestionably parents, as idea possible evidence best is the the in manner difficulty. The great readily learn so that as This counting. not in be most three, etc., two, one, relation. may this in history. counting out, readily appeal themselves learned that words readily repeat sense words is included there is the really serious one seventy, not Judged fifteen. or its attained back, Arithmetic in sounds,** is these repeats ever ten repeating is any But granted. to our told are only object method, schools, has therefore, go Object failure the that normal must, sixty,or fiftyyears, the the of We sway. manner, fifteen years or will by count fives to you a twos, threes, twenty colored papers Any explanation never imagine tell him know ones hundred, etc. to one all to so, he needs twenty, and and he as to teach backward " INTRODUCTION. Nursery jingles,such XI as One, two, buckle Three, four, shoe ; the door ; my shut etc., conveniently and be may to the aids Those do he is before who things. Number express our rock. regard that to these whether we tens, or man we two or a always two anything general ; two theory is developed " The is then and of method sooner by or the which for the two is mon com- is name same thing, feet,two units, to man, common ; rock, ; is That theory of for it science, " the be here Its is a etc., we name The numbers. can be advocated nothing universallyadopted. various has, simplicitymust still further arithmetical consistentlyexplained. : or simply two, three, etc., is number. method there But later which logicallyand prefacesays pure of tree, tree groups. men, That fully demonstrated. already apparent. must Instead or use. applied to objectsfor practical naturalness trust, been a as spoken name ; object shall represent rock, rock, rock dealing with objectscalls certain precisely the for one, name classifyand represent that which means else. tree, tree, tree man similar or with to a the page. man, to name sim^plytwo, The printed deal which hear we the we see objects origin and which name of have we things. We have say ; to multiplication means the by means all other and by between objecta may have must ThuSf two. else, numbers whenever the written We three. in give tree, rock, man, of the the counting. practicalaffairs one call it to mind man, Our is We it upon two In ideas in nature. it and teach to attempt of number. nature but taught anything of part a discriminate sufficiently not see division,and as addition, subtraction, Thus, he will learn memory. and multiplication, table profitablyemployed, primarily reason also why It is the Colburn I be it only operations can Warren as be in his INTRODUCTION. XU It ifiremarkable " of that which examples involve operations when these all reduce they addition. to principle. And performing them that they the same In this in the impossible According findinghow of is not of Since of the one four one-fourthof 20 As this a it is is the clearly,let have is this follows that (5J + 5| be 5f + cents. either 3 equals of 20 As standpoint 20 the " a tial partioned. men- in of another, ion Thus, divisand sometimes equal parts Fish explained by W. Fish, 1875) cost of 1 cents, (5 + in process of the as his of is here as is one is the cents correctness of another Since 23 5f cents is 5f we of the = : pencil? it follows that 5 20), and 5+5+5= of the been we of the as show cite the of 4 times four say result cents we say obtained explanation an words, for instead taken 23), and have 4 one cents but is to To be cents + method any their way Daniel juggling with mere the cost 5f nor thing method circumstance. 5f he number." a find they cents, what of the reverse : of one perfectlysound, is a of is 5 cents." cents accidental an do 4 times equal parts reasoning thing 20 that applying own is contained number is the demonstration division is taken 5 cents such any sometimes how pencilscost of his task, from equal parts Here If 4 Ex. " condemns the Complete Arithmetic," by C* The is And number? a " else. we artificial method an all, division one thing,but one something them to of the findingone use Neither in times many minds, analyze we shall find different ways we Colburn incomplete manner, and or when if explaining division, except of one, ety vari- a different form." a attempted ever perform to Indeed, our only strongest possiblemanner. have disciples an take in are only Warren paragraph them They it is addition. but perform we is able and addition, subtraction, multiplication, operation division, recognizes no he child, although a to what say that may be Now into equals 23 of 23 one-fourth of 1 is more we cents, it equal parts which but what 23 findingequal parts, the f equal parts one fallacystill 20. of cents cents must divided, or, INTRODUCTION. of the 4 1 logically, more either In and so it is case we are itself. which But number expression of an attempting to if with is start we be can or almost one illustrative our 23 in 4 is be abstract The cents. 4 into the of cents stration, demonof the one pencils. Right that is not tained con- division irresistible, and demonstration division that is be not pencilsare enter conclusion means complete and difficulty may to are ure proced- carefullyselect we times many cents explained by a the unscientific " to abstract, this left to try except to find nothing of 23 here, then, is always how cents, and, if equal parts cannot find cannot there else than this most that ion divis- by conception according a universallyadopted so division explain and 3 is divided. performed division, a anything " examples We suggested. into which equal parts result is inevitable, unless is the Xlll numbers essentially a are reversed multiplication. If 5 how men a 2 many the But equal parts but 2 or their have start we be can with is not of we of be that else that to in than a This dead. algebra the which was change the name in piece 2 made correct is in answer. which number is inevitable. result an a not mistake no to of are arithmetical of pieces^and a second abstract. formerly regarded the ^ would men name given. does The not one as ' ^* result, There and that always are the ^. c^v^ ? fraction, is 1 of the a abstract, this collection days days, is wrong. answer obtaining a demonstration a numbers call now have we in 2 obtain we is divided. man for it would piece or a arithmetic, and nature of in 4 conception according a anything demonstration What a succeeded not Here, then, is fraction books, definitions their work, no in the work standpoint this which into amount do the to So, notwithstanding figuring,we if it take of work capable of performing work, men enough. But from do could a directions according to man, be is would men men. man perform can Following 2^ ^ certain part alter the science INTRODUCTION. XIV merely is this be by and, that he If would this shall have do from 4 the we other but out, existing methods. left, 4 and of for which unscientific. them take we there find how away, many 4, 5 are of part a left thus 9, know we ; away, left." teach subtraction to come in and algebra, difficulties ask, has enough For that " its we and, operations illogical give other, what arithmetic, have subtract to ? pointed to is have would apples way when Numerous favor 5 arithmetical should otherwise, or the ? he have is we left many counting he of and of extension algebra apples, 9 how find To into has would apples " carried Charles If " and presentation any reason, cannot 9 outgrowth an it the be and been said to show method Natural judged might be fallacies of inconsistencies fairly and the I but have impartially one upon merits. J. May, 1892. F. B. CONTENTS. PAKT I. NUMBEKS. " PAOB The Expression Numbers op The Arabic System, The French Method, 1 the System Roman the 1, Method English 4 3 .... Addition 6 Subtraction 7 Multiplication 8 10 Division between Relation Division and Multiplication 11, . The two of factors Assimilation of a 11 product and factors multiplying together 12 ... Remainder Divisibility 13, 19 Factors 'Common Greatest Factor Common Least 20 Multiple 22 25 Fractions Fractions What Addition reduce and 29 are and Multiplication To 30 15 Numbers of 30 . Division 31 Subtraction improper an 32 fraction to a whole or mixed number 32 . To To reduce divide whole a by a mixed or number to an improper fraction fraction 33 Cancellation Reducing 33 . 33 a fraction to its lowest 34 terms XV cosTEyrs. XVI PAOB Fractiom Decimal 35 Addition, SabtractioD, Maltiplication.Division reduce To fraction common a to a 35-37 decimal 38 .... PowEBS A5D Boots Sqcabe Root 39 40 Root Cube Roots 45 other thav Square the ahd the Cube 50 .... Tables Practice 52-55 PART II. APPLIED " NUMBERS. Principles General Isterpretatioh Fractiohal of States Uhited 57 Results 58 Mohey 57 Mokey Ekolish Weights aitd 59 Measures 60 Pbrcehtagb 68 Ihterest 74 method, Accnrate 75 Iitterest CoMPOUHD 78 Interest Ahuual Notes, method common Drafts, 82 Checks and 83 Payments Partial * 85 . By United By Merchants* Equation of States Court Method 87 Method 89 Payments 90 Discount True Discount Bank Discount Trade Discount Cash Note. Problems. and and Present Worth 96 Proceeds 97 98 Discount Other applicationsare 98 found under the head of Miscellaneous 1 NUMBEES. I. PAET " ":""o"- EXPRESSION THE Numbers called characters the are numerals. viz. Arabic, either expressed commonly are NUMBERS. OF numerals The words by usually by or employed : 1234567890 characters These used to signify called are number the figures for digits. Digit or which of one these is also character stands. With the only, the the of of two, than three the The counting gives of name name Note. exception idea each of zero ; one, etc. one ; cipher, of number backward. three, of which the than two ; is two, one, Two zero. than one. one. is read two. 0 is read nothing, 10 is read one ten, 20 is read two tens, or zero, nought, ten. or or consequence forward Counting , 2 as same stands. necessary one. less one zero, a character is the character is almost backward, the to characters starting-point counting 1 is read applies ten number a The less these for as which twenty. 1 is one less AND NUMBERS, 3 tens are HOW 5 tens thirty 2"xQJiJty^ ^ 4 tens QXQ forty 6 tens 8 tens are 9 tens are , 7 tens 2A is read two 04 is read no and tens The four, or twenty-four, four, ox four. cipher adds is read ten 200 is read two hundred, 204 is read two hundred and 264 is read two hundred and 1000 is read one thousand, 1004 is read one thousand and 1024 is read one thousand and 2364 is read two thousand three or nothing to the value. hundred, 100 tens, ninety. seventy^ are and tens eighty ^ sixty are ^ or THEM. USE TO one four, sixty-four, four, twenty-four, and hundred and twerUy-threehundred sixty-four^ sixty-four. 00 00 CO 0 d d o o '6 O 00 "'" 00 d " o I S '^ " O 2 ' " 00 rrt OQ fl o -2 T, *-"Sd "H_rj" "- rtCj^ rtflO Sd"^ WehH WehD onaSonatgg'rJ "" 0 Eh ; pq WhS .00 6, 3 7 4, 9 6 5. 0 7 2, 1 3 8 ^ BilUons' Gronp. The names of the groups Millions' Thousands' Qroupi Gronp. beyond s. Units' Group. billions are : trillions. nonillions, quindecillions, quadrillions, decillions, sexdecillions, quintillions, undecillions, septendecillions, sextillions, duodecillions, octodecillions, septillions, tredecillions, novemdecillions, octillions, etc. quatuordecillions, vigintillions, THE EXPRESSION expressionin The hundred three and the is read diagram This is the French method universallyadopted one third of other every each. a sixtythirty-eight. is and A States. is the different usuallywritten each but long number, a and and comma trillion, three half-grouprather than a whole one, clearlyrepresentedif reading will be more constitute method United There, figureof Six naming numbers, the in : hundred hundred one of plan prevailsin England. after every follows as nine billion^ seventy-four fivemillion^ seventy-two thousand, 3 NUMBERS. OP and a and comma, the groups make the omit we of six consist ures fig- figures Thus, 6,374965,072138 V V ' / -^ y Units' MUlions' ^onp. Groapi and hundred In seventy-fourthousand and million, seventy-two sixty-five and thirty-eight. the French trillion,a method thousand million millions; beyond millions billion is a In billions. trillion,a a are the the universallyfollowed, numbers billions are quite commonly read as The hundred English method being, as the names dred hun- millions billion is English, a As billions. ; a a numbers is very methods. two million French running whole Seventeen nine one there used, practically States, although the the United thousand thousand a million often not between little difference In hundred hillion,three Six in method into over England. is as a the Thua, for 1,700,000,000. billion,trillion, etc., logical, the tri-million,etc.)signify, (bi-million, is the second, third, etc.,powers of more a million. 10 units make 1 ten. 10 tens make 1 hundred. 10 hundreds make 1 thousand, etc. 4 NUMBERS, AND USE TO THEM. The base of the system The system is called the deounal system, Another of way ia 10. expressingnumbers The In I HOW Roman is System. this system represents II represents two. one, V represents^ve. X represents ten. L representsj^jfy. C represents hundred. VII D represents ^t;6 hundred. VIII represents eiffht. XIII represents thirteen. M represents CLX one Ill yi represents six. represents IV Vim seven. sixty. I, which the on ished dimin- precedes it. dials of than IIII. rather is used being cloqks represents nine. IX represents/or^y. XC represents ninety. CX represents is used hundred one and rather than Villi. ten. represents /owr hundred. DCC representsfour represents CM DCCCC VI M the watches, IV XL CD The Except represents nine. CCCC MDCCXLII and the lesser value represents four. and IX hundred one represents represents /our,the greater value V by nil represents three hundred. CCC thousand. one represents three. seve7i hundred. ^ ,^ /y - hundred. /:^^ ^^ ^ ^' 2N ^ ^^ ^ represents nine hundred. ^'^~ represents nine hundred. ^^*^-V6"6f^a^ ^^'^^ represents six thousand. represents one use quantitiesover hundred of the vinculum which \X? ^^7) '' million. represents seventeen general y^ it is ( placed are and sand. forty-two thou- ) is to to be indicate taken that together THE as The quantity. one used EXPRESSION in similar a that means but the there expressionof The whole To is Roman used numerals mark to in the ; is read follows used 100, millions : number ; point of Eight ings, writ- in the use and clocks They are in entire ; go 000, number billion ffty-seven Express and Words one ^ : 14002537. 4 205200. Six hundred we thirty-four. 857100000034. 3. as units are The into mentally group below, 34 and hundred off each 857, billions. 17642587. 1. and its dials upon first we Express 7. and chapters,sections, etc. second million^ and 2. This thousands. for occasion any figures each, naming three thousands hundred for are vinculum in old books found are long number, along. Thus, as stands [ ] computation. a of groups longer no also for read is sometimes the Romans, over quantity brackets and numbers. watches, and not Used 5 NUMBERS. parenthesis ( ) manner. of the vinculum use 01^ in 5. 64029530203. 6. 68507332842630. Figures : seventy-threethousand, five hundred twenty-one. 8. and Seventy-fivemillion, two hundred and eight thousand, forty. 9. 10. and four thousand, Seventy-sixtrillion, Eight hundred four 11. twenty-eight. and seventy million, sixty-five thousand, hundred. Five hundred forty-sixmillion, two seven. and and seventy billion,three hundred and hundred eighty-four thousand, and and 6 AND NUMBERS, Express 12. in HOW THEM. numbers the Romans, USE TO from one to further, we one hundred. ADDITION. If 6. We say 3 and If 3 is 6. and 8 do can 2 are we 13. are 3, and with start we this 6, by we count step, one the mean eight 3 and 2 and addition is numbers two count 3 and " second numbers 8 2 number beyond When we than greater 5, The 13. are 5. are get get 13. we 5 is called process Addition. The sign of pliLStwo Sum equalsfive, Sum 5 is called and 324 6 units 465 6 tens 789 4 hundreds 867 6 units 386 4 units and 2 tens and and of 3 and sum 5. = 2. 8 tens. are 3 hundreds 13 are 7 hundreds. are units = 1 ten 1 ten and 8 and tens and 3 hundreds and 6 tens 15 are test and the and result, begin at tens = 5 tens. 8 hundreds are 12 hundreds. carried. The 2 is said to be The 1 is said to be carried. 1961 To Three 9 units. are 7 units hundred Sum the 2 3 + cross. 3 units. 1253 1 hundred upright an the top and add 6, 13, 21 8, 11, 16 3, 12, 15, 19 1961 down. 1 8 AND NUMBERS, hundreds, of the take we tens, the other TO 1 and call It 10 tens. the other take we HOW the 3 units, making 13 Now THEM. have we which is changed form, mentally. process term, the subtracting; reduced 1 ; by come to subtract O's. What this As : we 0, instead we to take have we to add 10 from term to which shall we have to 7 from 3 from from in the which the 1 is of above higher are we course when minuend, 9's in the have a from take we the from one this in place we of the do, then, in the problem justgiven, is take take always numbers through the term and, if O's intervene of 4 the When along,without go to write go cannot we of 3 from and we 1166 to have always we to 4837 is it necessary nor add 9 599is subtract we is unnecessary practice,it units, and 1, call it 10 Leaving In Leaving units. what From USE take 3, we 9 ; instead 6, 4 from 7 from of 8 from All 6. this actuallyperforming we 13 ; instead 0, 8 from can the reduction see 9 ; as even mentally. MULTIPLICATION. Sum This The 3 X 3 is ; 3 is 6 6, 6, and 6 Three 6 Or, 3 times 6 are 18. sixes are 18. are 18. 18 Mnltiplioation. is sign of multiplication 6 6 18. Three an oblique cross. tivies six equals eighteen. called the midtiplier ; 6, the midtiplioand ; 18, and 6, factors of the product. = the uct prod- MULTIPLICATION. 3 Multiplicand 3 Multiplier 437 ' 3 3x6 6x3. = 6 Product 2622 3 3 18 is 6 and of 3. 6x7= 42 6 X 3 = 18, + 4 = 22 18 6 X 4 = 24, + 2 = 26 Multiplicand Multiplier Product of multipleof _3 Sum ten a number For, 10 10 Or as great number Multiplicand Multiplier thus 246 = If each made times ten : we before. was 246 X 10, annex 246 or tens. times ten a if the And as several parts great,the wliole great. as 100x246 For, 100 Product = 246x100 = 246 = 246 X = 246 thousands. hundreds. 24600 Multiplicand Multiplier Product it as are is made 246 X cipher to a number, each digitis removed one place to the left,and consequentlyits value is 2460 times the 246 246 For, 1000 246 X 1000 1000 246000 Multiplicand 7421 Multiplier 6x 40 246 44526 4x 29684 and 14842 Product X 10 X 200 X 7421 44526. 7421 10 7421 29684; 29684 7421 X tens. 14842 hundreds. 4672 3100 4672 14016 Product Multiply 4672 by 31, and 14483200 annex two 7421. 29684 1825566 Multiplicand Multiplier 4 X ciphers. 10 JSUyiBEBSj HOW A!n" USE TO MoldpUcand 436000 Multiplier 2300 THEM. 1306 872 Product Haltiplj by 23, 436 1002800000 and five annex ciphers. 472163 Multiplicand 51002 Multiplier 944326 472163 2360815 Product Disregard the 24081257326 intermediate the ciphersof multiplier. DIVISION. If start we with 7, we subtract can 2 three remainder subtract 2^ Remainder 3 the 5 2) 7 (3 with _2 6 find 3 1 tract, and 2^ Remainder 1 ; or, three 7 Remainder of we can 7 2 X by of times also the subtract no tracting subcan operation, one result. many is called 1 instead separate 2's, we same how times, leaving a That we is,we can remainder longer. The subafter cess pro- Diyision. the dividend; 2, the horizontal line, like that of is called divisor; 3, the quotient; 1, the remainder. The sign of division subtraction,with be omitted place of or a is dot above the dots may the upper short a be and a dot below. The omitted, the dividend dot, the divisor that of the lower. line may taking the * 11 DIVISION. Thus, 7 divided by expressed: be 2 may 7^2, 7:2, |. 12 0 Multiplicationbegins additions of the back run number. same this number subtract to as The it is taken. process division factors of three sixes. taken the even, is the process some product a the Given number are times, and means of times 6, the number find how we 3x6 3 the number taken, and product 18, of similar. not which times many it is by reversingthe process of multiplication ; that is,by a is equivalent to subtracting6 from 18, 6 from which number Now, are as long of 18 and on so as we do it,noting the can of subtractions. suppose, instead required equals 18. may out comes 6 is the number the remainder, and and is division. multiplication. two is taken successive 0. a of reverse proceeds by If, reversingthe process, we have added times as we it,we many Multiplicationreversed Whenever 0, and with appear, How there to find shall is we no the do 6, we number this ? direct method have given which taken Simple by the as which and 18 3 3, times problem it can be performed. have We seen that division reduces to and multiplication subtraction,multiplicationto addition, addition to counting. Every operationin arithmetic and tion subtrac- is reditcible 12 to the and simple process 6 X instead prodoeee the 6 is the factor how we are times for the of purpose equal parts of the the number But, 6 X as 18 as find there are bdchvardj or before as 3 ; and made up six 3's in so for of 3's 18, so seeking. involves always the findingthe to number which taken a number of metaphor, common finding in another, it may certain other a of process is contained certain ; or, ploying em- of findingone number. of equal parts of the three ; one THEM. problem. consider may and a of the two One USE product same amounts convenient a the number one of times number we division many TO By division,we of 6*8. So, while solve not present purpose our HOW of counting^either forward will coQntiDg noticed, 3 he AND NUMBERS, number a is called equal parts, one-third one-half of of the ; one equal parts, one-fourth ; two of the three equal parts, twothirds ; three of the four equal parts, three-fonrths ; etc. four In consider practice,we rather of neither think we 3 X together,and this 6'8 is 6 three as 3's,but nor six 3's,or 6's, or 6 of and 3 multiplied for purposes exact suflficiently computation,though in the final analysisthere multipliedtogether." thing as be can of such no '* It is enabled otherwise being by virtue to have of this assimilation several factors impossible. Thus, be all considered as factors of 2 of of factors we product, which a x that 7 3 X 42, at 42 = ; would 2, 3, and and one are the 7 same time. Divide 2379 by 8. 8 8) 2379 and 297 3 remainder, in 23 times, with 7 hundreds 77 tens. tens = 8 in 77 tens, 9 tens 50, + 9 = 59, 8 in hundred hundreds, 2 7 hundreds remainder. = times, with 70 tens, + 5 tens 59, 7 times, with 7 tens remainder. 3 remainder. = 5 The remainder operation it with it with the the has the three dividend been not which upon and performed, the so divisor, using the sign of division, and quotient as part For is to be read part of the a division of express Note. is 13 DIVISION. LONG of the present and until divided by eight,and general result. otherwise we put Thus, 297f . explained,the expression } similar expressions in a similar manner. When, as in recorded, the above the LONG 1786929 Divide by DIVISION. 436. 436 436)1786929(409811^ 1744 178. will not By partialmental a readilybe can 3924 in 1786, 4 times. 3689 4 X 3488 1744 201 42 436 in 429 hundreds 436 in that seen 1786 from thousands = 1, in 17, division 436 = 429 = 4290 4292, 9 times. tens, + 9 X 436 2 tens = will in it go 420 42. hundreds, + hundreds. 429, 0 times. = or 1744. = hundreds in go 4292 436 are Short Diyision. is called process final results the example, only = 3924. 4292 tens. 9 14 NUMBERS, AND HOW TO USE THEM. 1786929 PROBLEMS IN ADDITION, AND The to followingproblems the use attained only by the tables furnish If can one things,neither been and made 1; ten, 2; ; the row back ; dividend, a Add the four so next practice the purpose. divisor,or numbers Test the result cardboard, well shuffled ten in were hat, taken for the first digit of the firsthorizontal drawn, for the second each re-shuffled. Table to were digitof drawing, the card factor,is found a of squares squares Practice columns digits, having 100 the whole a of the arrangement small 100 After on. hat, and From 13. The first drawn and Of : etc. number into the purely chance a follows as up the number row be he do the others. can tables present marked is of way perform such exercises readily,everything else in follows easilyand simply. If one cannot do these arithmetic The practice. The kind is required for what constant which DIVISION. given as illustrations of the Proficiencyin figuringcan are Tables. Practice PLICATION, MULTI- SUBTRACTION, A, lines just below it). by adding in the In 1 using reverse the was put tables,if 9 for 0. (Page 52.) 1. to zontal first hori- drawn to be 0, substitute No. the a 6 45459686, (i.e. direction. and a 16 therefore,divisible by 10 and is made number A number In other by 2, both 2. Note. words, It 2 ifits is by 2, mathematically by exact and of tens up 0 by 2. parts, two ible divis- necessitybe units, be divisible the whole hence of tens up is divisible by number 8 is divisible divisible number with ending is 3 units. of 10. It of 10. is, Every divisible are 5. by 8. + is not 00 for 0. factor is divisible 2468 by 5, so 3 Hence, units 2460 = tens, = by 5, a ifits b 0 multiple a its units. 2468 by 2, so is also are number, any number any 0 + by that say There -s- with ending 2468 Any into of must to units. number. any is made 8 is not Every 2. number number divisible are number and of 10. Hence, units the separate therefore,divisible by 10 and A factor its units. divisible are by 0 is divisible number by 2, a part, the tens, One instance, is made Any also + by we THEM. TO If,then, the other part, the parts is divisible of tens is divisible units. and tens up USE HOW AND .NUMBERS, by 2. divisible multiple a by 6. It is, 100. of therefore, divisible by 100, also by 4, a factor of 100. Every units. Hence, is made number A as is divisible number number one of hundreds up are divisible j tens and 8200 + 28. 16100 + 42. ifits ^ by its tens + and unitSy taken together 4. by " 8228= 16142 28 is divisible 42 is not Any by 4, divisible number so = by 4, ending is divisible 8228 so with therefore,divisible by 1000 is not 16142 000 and is a 4. by divisible multiple of also by 8, a by 4. 1000. factor of It is, 1000. and is made number Every is divisible number taken of thousands up togetheras number, one 37000 = + 184. + 188. 184 is divisible 188 is divisible not 6823 = 5000 5000 = 555 20= X 9 + 5. X 9 + 8. 2x9 + 2. 3= by 88 + + 2 5000 500 = 500 tens 50 = 50= 3 of up them), and of 2 + 8 + 8. 500 = tens = 5 tens = 50 nines + nines + 5 nines + 500. 50. 5. the (in this 9's its of sum there case digits(in this 18). Hence, = is divisible number A For by 3. is made number any 5 + case, + divisible is not 371*88 so 8. by 3. Thus, 555 20 + is divisible 37184 so by 8, 800 + 88 = by 8, units, 8. by divisible are 37188=37000 are hundreds, tens, if its hundreds, tens^ and S by 37184 800 its + Hence, units. A 17 NUMBEKS. OF DIVISIBILITY ^ by ifthe sum of its digitsis divisible 9, 5823 is divisible divisible 3746 number by is not not divisible So, from A byS. the number of its digitsbeing 18, a ber num- by by 3. of sum its digitsbeing 20, a 9. of the sum will number 9 is divisible by 9, the by If the remainder dividing the sum 9. divisible Corollary. by 9, the by 9, the be the digitsof same a as number the be divided remainder after 9. number Any of 9*8 are divisible by 3. previous demonstration, is divisible by d if the su7n of its digitsis divisible 18 is divisible 28146 number divisible HOW by 3, for number 2 and Since A divisible not number Since 3 4 Take of the sums its of sum digitsbeing 10, a by 6 ifit is by 12 divisible 2 and by by 3. 12, = such alternate 0 is (Note that a the factors of 6, are number, any digitsis 21, of its sum 3. by is divisible number A by 3, the is divisible 3 X the THEM. 3. by divisible is not 1342 USE TO AND NUMBEKS, that digitsis by multiple of 11. 0x11= 3 and by between difference the some multiple of a divisible ifit is 4. the 11. 0.) Take, for instance, 8263817. 7 + 5 + 8 + 825381 7 8 1 + 28. = 7 + 10+800+ = 3 + 2 28-6 6. = = 22 3,000+50,000+200,000 2x11. = +8,000,000. 7 7= 10 10= 800= 8 ^x99+ Wx99+ 3,000= " 30 " ' 50,000= 500. mx99'\%"my99+ 200.000= 2,000. 8,000,000-^0,^X^^+80,000. the Strikingout 5x99, we etc.), and being 8 such and bdng the ?0xR^+ 20 (8 5 + of alternate 20 5 ^0Px^jr+8OO. = 11 ^x99+ be 7 + a 8 + 3+2+10 + 10 + 30 + 20 multiple 11, 5 + 8 + + 30+20 is also 10 + 99, 30 X = 30 + other the so 1 + 30 + by We set. 20 11. by X 99, 20; 7,8,5, the took a tens' ber num- of the alternate sums 3 + 8 ^x^jr+ 99, 500 digitsof 8253817, be divisible divisible 800 X 10 + 8 + that the diflference between diflfersfrom if 1 + 80,000 left 7 + 8 + set one digits should and 2,000= multiplesof have digitsof 10, 30, 500= 2 + a 10 + 30 + 20 multiple of 11, by 11, 7+8+5 But 1 + 3 + + 8 2 + 10 + 30 7 + 20 + 8 + 5 + = 11 + 33 + 22, 10 + 30 20 8 + parts into which by 11, Put A sums in is alternate of sum by 11, in connection If the If be the of sum same after subtracted remainder the so two ible divis- each 11. after after difference 11, this latter the number Practice numbers between Which of the the two taken be these be tens, two the the by sums number units the will by 11. less than and sums the same by each if the by 11, as the 11. Table a-g these the with remainder by present, and of dividing digits beginning dividing the 11. digitsbeginning with digits beginning with alternate of the the by 2) 5 is 5. = digits beginning with alternate remainder 6 " divisible dividing the difference the dividing 11 3 + (6 + 151. 271, page the 22346 is 6. for alternate the 11. is not are following: 2, 3, 4, 5, 6, 8, 9, 10, 11, 9, and The are by ^y omitted the of sum after as from Which 24. 11. by digitsof 22346 be of sum From 23. by 11, if the differencebetween (4 + 2) set Problem the alternate of the remainder, divisible 8253817 divisible therefore the remainder, sum be other with be greater than 11, will divisible 11 by following corollary may tens, the is of alternate set one divisible units quantity is divisible digitsis of the Corollary. the divisible sum later up a general terms, of its The , separated 8253817 therefore is 11, the not have we number The + 19 NUMBERS. OF DIVISIBILITY first five numbers No. 1. divisible of the 12? a-m are divisible by 3, 11, respectively? FACTORS. 2x6 3 X No and 4 = 1 2. 2and6 -= 12. 3 and other 1. two 4 numbers are factors of 12. are factors of 12. are factors of 12, except 12 itself 20 AND NUMBERS, 2x3 2 and 6. = 2x2 3 There 2 are the numbers 12 3 is the are 3 10 12. ; the same 3. 2 30 5. 3 15 10. = 2 X = 2x Or the factors are Common The prime factors of 98 are 2, 7, and One 2 and one Find These and the 7 are of 56 is the another 12. cannot factor method be so may 7. 7. and greatest common greatest common numbers 3, except both. to common common/actors 14 or Factor. 2, 2, 2, and 14. before. required. are = 2 60 6 X of 56 7 of 2 = factors 2 X as prime factorsof prime are 12, = result factors are The 7 4 = factors of 60 ? Greatest 2 and 3 numbers. the are prime 6 5 12. 2 X compositenumber. a 60 2, 2, 3, and X 1. prime are 2, 2, and What and themselves 2 and = 2 which numbers no 12, 2 = 2 for 4 in 3 X x factors of 12 are 6 X 2 X 3 X 3, 2, and THEM. USE of 12. Substituting2 4. = factors are TO factors of 6. are x' 3 for 6 in 2 Substituting2 2, 2, and 3 HOW of 611 98. factor of 56 and and readilyresolved be 98. 1363. into tors, their fac- advantageously employed. 21 FACTORS. The 141)611(4 564 of 564, 564, it is their difference,141. a of 141. ; the Hence, 47 In above numbers 141 common 611 same is the by of remainder, the required is 611 47, it is of it is 47. tor fac- of 47 is factor of either 47 564, a a or multiple of their factor factor of 1222, a and of 1222 of factor a and that 47, of it is 611 two numbers. tiple mul- a factor a after the are factor of these two numbers. affectingthe result. we required to omit may that 47 itself is first division, we of 611. not 47, is the greatest a com- factorrequired. and 611, factor of 1363, and 611* and a and example, and or greatest common factor of 141, but of 141 of factor proved now factorof a factor 1363. factor of the factor being a ; is factor of 564 being a. have common 3 is a by the last remainder, required i^ a common factor of 141 a being a of their sum, mon given numbers number Therefore, the of 611 We number The being 611; sum, of of 47. 47 141 factor difference,47. the last divisor,141, 141. factor But two of their factor of and of a a ; and 1222, requiredbeing a factor of 141 is also a multiple of 141 ; being a factor of 611 and a a Divide 47 being 122^, it is of number The factor 611 factor a 1363 of the the smaller of fac- requiredbeing a is also multiple 141 Divide number of 611 tor 47)141 (3 a by the less. 1222 141. divide the greater number Mrst 611) 1863 (2 141 47)611(13 47_ 141 Not can being a common the greatest readilysee the factor 3 from Thus, 141 We find the have that tor fac- 141 without 22 AND ITUMBERS, Find HOW TO THEM. USE the greatest common factor of 1547, 1729, and 91 is the greatest common factor of 1547 13 is the greatestcommon factor 13 is the greatest common factor of From 25. What are prime a-(7 ? and 1729. 1677. 1547, 1729, and Table Practice the of 91 and 1677. No. factors of each 1677. 1. of the first thirteen a-b? 26. Of 27. What 28. Of t-w 29. Of a-Cf 14-25, is the and ac-z LEAST 2 X 2 X 280 = ji!x^x2x each COMMON 7 X = X 5 2 X multiple of Multiple. = 3640 13. 5x7. 364 of the first ten lines? is a S'lso contains 280 not multiple all the of 364 prime found in 364 therefore,a multiple of 3640. both yz? MULTIPLE. of 364 and wx ? d-g^ and h-l^ in 364 factor of greatest common and the smallest 280. It is number their factors ; it 280 that Least ; it is, and is Oommon a 24 AND NUMBERS, Find 91 the least is the = some 1547 = 17 X 17 TO USE THEM. multiple of 1547, 1729, common factor of 1547 greatest common 1729 1729 HOW and 1677. and 1729. 91. multiple of 91. X 29393 = = least common multiple of 1547 and multiple of 1677 and 1729. Similarly,3791697 29393, and hence of From 30. lines 31. What 16-25 Of = is the least common 1677. 1547, 1729, and Practice least common Table c-e, and/-A, 1. multiple of ? a-b, No. in lines 6-15? ah and cd in 25 FRACTIONS. FRACTIONS. .1 is read one tenth. .01 is read one hundredth, .001 is read one thousandth. .0001 is read one ten-thousandth. .0024 is read twenty-four .0637 is read six .4208 is Tea,d "10 is read hundred and tenth, one thirty-seven, ten-thousandths. hundred forty-two . ten- thousandths. and hundredths. ten or The eight, ten-thousandths. cipher adds nothing value. the to 00 OQ OQ 00 a OQ a ^ o o " " 2 a -^ CD a ?:: " r4 00 a "=" 7 O ITS ^ no '^ a oc a """ fl 2 Billions' Millions' (}ronpi The whole part number The ; dot Thonsands' CJTonp. of the the 2, part is called at the "" a Q .2 a O 13 :72 (3 0 "" "V n ^ P4 138.03620759 Units' (}ronp. number TS 'C hWehhWJ^ W"hD 5, 07 "is -^ rP "D WhpqWhSWhh ^ ^" g 2 '=' " o ^ 00 4, 96 a 0 o .2 -2^ 6, 37 "!-" 0 O o " OQ 00 00 a o 4Q " r5 ^ "4^ TS " 00 CS (}ronpi at the the left right, decimal a of the decimal point. dot is called fraotioni or a mal. deci- 26 Al^fD NUMBERS, THEM. USE TO READING IN EXERCISES HOW WRITING AND DECIMALS. Express To last read long decimal, a digit,read to it the give of each the hundredths, The first find decimal digitis succession of the found from : last whole a digit. the by naming the decimal second in the billion^six hundred twenty-nine thousand, and three one hundred column number, The tion denomina- denomination of etc. below is read 7nilUon, five hundred and and point. Thus, tenths, thousandths, ten-thousandths, first decimal of the denomination the though as denomination last digitin the Words in : teen nineand trillionihs. sixty-tivo, 27 DECIMALS. EXPBESS WOBDS: IN 34. 35. 719601529362. .019601629362 71960152936.2 .09601529362 7196015293.62 .000601529362 719601529.362 .0061529362 71960152.9362 .01529362 7196015.29362 .000529362 719601.529362 .029362 71960.1529362 .9362 7196.01529362 .00000362 719.601529362 .00062 71.9601529362 .0000002 7.19601529362 .719601529362 Express Seventy-one billion, nine 36. hundred one six and two decimal one hundred One 38. three and : hundred fifty-twothousand, nine and sixty million, and hundred thirty- tenths. Seventy-one thousand, 37. three Figures in nine million, five hundred and and and sixty and twenty-nine thousand, sixty-two,ten-millionths. and hundred thousand, hundred hundred nine and twenty-two and decimal seventy-one, twentyhundred- thousandths. Six hundred 39. nine Seven three one, and hundred hundred decimal and one hundred thousand, three 40. and and and million, five hundred and twenty- sixty-two, trillionths. nineteen five hundred and thousand, and sixty-two,millionths. six hundred twenty-nine thousand, 28 AND NUMBERS, hundred Five 41. and TO HOW THEM. USE three hundred twenty-nine thousand, sixty-two,billionths. and 42. Nine thousand, three hundred 43. Nine billion,six hundred twenty-nine thousand, and and and three sixty-two,millionths. million, five hundred one hundred sixty-two, hun- and dred-billionths. Twenty-nine thousand, 44. three hundred and sixty-two, millionths. hundred Sixty-one million, ^ve 45. three hundred 46. Decimal 47. One forty thousand and sand, twenty-nine thou- sixty-two,ten-billionths. seventy-sixmillionths. and million, five hundred hundred three and and and twenty-nine thousand, sixty-two,hundred-millionths. ten-millionths. 48. Two 49. Three, four decimal and hundred and two, ten- millionths. thousand, five hundred and three Sixty-two million, 50. and and hundred eleven, and twenty-eight thousand four decimal six, ten-thousandths. 61. Sixty-two hundred-thousandths. 62. Decimal five and hundred and billion,six hundred nineteen twenty-nine thousand, three one m^lion, hundred and sixty-two, trillionths. 63. Twenty-four 64. Three 66. One hundred hundred ten-thousandths. and and sixty4wo, hundred-millionths. twenty-eight thousand, and decimal eight hundredths. 66. hundred Twenty and thousand fifteen hundred-millionths. and forty-eight,and thousand, three hundred decimal and eight sixty-four, 29 FRACTIONS. See As tenths 10 divided by 10, divided is 248 .248 248 1000, : fraction. neither 10 the a nor number which the very If any will decimal have be form 76 = numerical numerator .76, for 3 number. 300 = hundredths, or of fraction. a number by and the a 1000 3, .003 X 3. = its denominator, whole number. decimal a uct prod- Hence, the its fraction is the product would only may a particular be expressed fraction in the common decimal. hundredths, .76. Hence, and any 300 hundredths fraction has a value. numerical generalnam,efor fraction is Note. = form, and a more being numerator value, for otherwise into being value. general. Any changed fractionis a The as fractional a as scale, it follows that decimal the decimal fraction in = the multiplied by But ; the 1000, or ; is ; the numerator ^. or nature fraction 1 or fraction. .03 X 10, of two product fraction,it numerical a numerical a be Thus, I must the common may of 100 be value. of the the of the 3, = the : 7 is divided decimal a is the denominator nature decimal has no case .3 is the is called general same decimal a 10, 1 -h by which is not as fraction has 10 X a which terms are the decimal A number it is divided From A the number divided denominator -^-4 But 1 by 10, expressed 248-^ be may the is of general nature same expressed and by 1000, multipliedtogether,J The in be may -j^^^. i or decimal lO's and divided tenths .1 is 10 equal 1, 13. Page Note, It is a be established numerical is quantities number. number. only by Being qv/intity. an a process extension and of evolution of primary developed before that a fraction has become ideas, underlying principles this is possible. 30 NUMBERS, of idea The division may 14 number TO considered 3 -5- out 3 X THEM. been now even, the as 4f - USE having to come decimal first three The a as be now may HOW be made hereafter division and fraction a AND and multiplication so of each reverse 4| developed,any other. 14. = placesbeing tenths,hundredths, and thousandths, .3 is read tenths; .03, three three hundredths; .003, three thousandths. .3,.03, and But .003 ^^ ^^^^ Y(ftnF-S^" A and the are Y^^3 three thousandths. Similarly,f is read three tenths three thirds; f, three halves, 2 ; or, denominator 24 . ^ is read three fifths;|, read also be may fourths; f, by and half; denominatarf one three 3, numerator divided half divided by three one and two numerator^ or f "^, y^, as three hundredths; j^-^, ; already explained,3 as and two three fractions same two three 2. thirds, and two thirds. ^ ^ fraotions. oomplez ^ are ^ 3 f ^ is simplefraction. a is 2^ f 3| of mixed a 4- is nmnber. not it has fraction,as a not a and numerator a denominator. - " - IS fraction. a 6x7 Note. The this evident. this idea, the their size numbers a Fraction means compared the Applied is of denominator as fragment fraction pieces. or quantitiesthat is just as much a fragment denominates with Whole the means or the entire a used make piece. Following numerator thing,as out pieces to indicate the names whole; an or terms numerates or from distinguished piece. to nitudes, and geometrical origin. The the numbers, can be cut piecesare an entire these into have terms piecesare no or geometricalquantities, geometricalfractions. quantity as a significance.The whole A number numerical is. only m^g- fraction 32 AND NUMBiyElS, Addition iH The least multiple of m 5 7 MH- = m 4_3x4_12 Fractions. of 5x7 if H m - and 364 uii = + Ml* ' is 3640. 280 mi - Utt- = a^v = 35 t 4x^, "x4=^-^, For THEM, ^^. common 3 USE TO Subtraction and and Add HOW and = 7 7 seventh one of ^^ 5 5 5 _3x4 "5x7* A fraction whose is termed less than To an improper fraction its denominator Reduce This the division is the is a Improper an OR Perform equals or numerator ordinary Mixed ; case proper ; one Number. = whose 2^. of division. to nator denomi- numerator fraction. Fraction thus, -1 the exceeds a Whole is 33 FRACTIONS, Keduce To Whole a Mixed or Improper Reduce is the This the 2J-to divisor The the 2)^2 = expressed,f Reduce The 2 to divisor dividend Or, since 1 + is 2 l 2 X 22 = to the 4. = both 5 having quotient 2, and divisor 2 2 for its denominator. with no remainder, Ans. f = |. . fraction. ^=l Multiplying given 1. ' 2xf = simple a Dividend fraction improper an have the remainder 5. = We case. . being 2, and rl Reduce previous 2, the quotient 2, and dividend thus are fraction. of the reverse an to Fraction. improper an Number 2j_j., of the terms fraction ^- by 3, have we 5x3 2 5 X __ 11 So, to divide denominator and by a 15 3 _ 2x11 22 invet't the fraction^we fraction^taking the the numerator for a numerator, for a denominator, multiply. Cancellation. is the What Instead we will the of product of performing merely write the V-, "" the 27, |i, and several various operationsafterwards ; ^ operationsas numbers thus, as ? we go along, factors,and form per- 34 AND NUMBERS, ;g gg X USE TO THEM. 3 3 ^ HOW X S!jr X 11 X 1287 13 . 7x^x;^X^^ dividing Since does number same of the in 16 goes by drawing 4 a the 16, and writing 4 27, 3 times ; in cancels is Reducing the factor 16. a convenient a Fraction 4. 321 We 13 16. 4 factor 4287 the cancel the 36 3, 3, are denominator. Lowest its a remain ; 9 in of method and that the through is left of 7 in the to line which and way cancel we 39, 3 times factors 4 is so in The the 4 of the denominator the 4, another The by inspectionthat see cancel the by fraction,we times, 4 through 36, 4 times. the to both. numerator above denominator of common in the numerator, 13 Cancellation We are line the 4 left of the 11, and and alter the value that ^' 7 numerator not all factors strike out factor both Qog _ _ Terms. both of both are factor terms, so divisible 3. 107 we 3. by be cannot " 107 divided /Ju 11 X by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 121, = number a greater than 107 "jmf^ A^^ If 107 = had factor a -lAnQ corresponding factor than 1429 11. has 107 no greater than would factor If the be all the factors readily common of the found not and numerator and the 11, its less less than 11. and of 1429. factor being a the cannot greatest denominator, and have we the denominator by inspection,find divisor of the numerator common divide both this number. by To the divide second one number is contained tte comparative value is factors,107 107. to be have Hence, 107 is a prime number, cancelled 11. or f of 5. by another in of the the is to find how first. numbers. The 3 many quotient -^ 5 = f, times indicates and so 3 2 is what What To number what which 3 times We to part of 7 ? is part off find is the given find the other Ans, ^? part Ans. number one factor. What the and This by is The of what ^ number = required division. 12h-3 = 4. ? product being 16, ^ ? are Ans, 16 the part. number is the ^. |f. = divide we product,and do we |xt = it is the of which one is 12. factor one |-f-f of another^ is part by the number certain a have 35 FRACTIONS. DECIMAL and factor one 16xj ^, the other factor is ip-22|. = T ^ of number of ^ -J-fof a certain number is What 18^. is the ? 11 4 11 j;gX 18^ X 17 8228 ;g_ X _ _ A-XttXII" 3x3x^x;^ 3Q. ^' og 27 3 DECIMAL Decimals numbers, be may FRACTIONS. and added subtracted the merely observing the positionof 16.437 Minuend 29.081 Subtrahend ^^^"^ same as the decimal whole point. 840375.3007 64068.470361 Remainder 776306.830339 .03914 391.65714 Sum In whole a similar number, manner, or divide we a may decimal multiply by a a whole decimal number. and a 36 24 X 12362 thousandths 12362 HOW AND NUMBERS, s thousandths = 125 thousandths. 296688 = THEM. USE TO thousandths. 515j^ 24 We may continue wish, thus we the division to as many decimal places a^ : 256)5172.14(20.20367 512 521 512 940 768 1720 1536 1840 1792 48 Divide 42307) 12.03 12.0300 by 42307. We (.0002 84614 until 35686 to contain obtain the divisor, point off the dividend, and as continue get we many the the ciphersto annex the a the number large enough divisor first dividend at least once, significantfigure decimal places as there division as far desired. as are of in Multiply 32.142 1.23. by 32.142 1.28 32.142-^%^. 1.23 32142 96426 and for the 32142 123 them multiply 64284 37 FRACTIONS. DECIMAL the being numerators, of the x\U- we 3953466 together, obtaining numerator ^^^ -f^. product. WMu'- 39.53466. = 39.53466 Thus, in multiplication, we as both of the Divide " 36.4728 number the right. dividend head being by of We We number same there as the a decimal, the point must of also decimal are places in number necessary the it into point number. same the into point 1223.) 3647.28(2.98 ^^^^^.J^JyJ^,.^y^^^^.u^ has As affect the the then instead 1201^ and the places dividend, we has as as and used, and " Divide There there and whole are then .6 by decimal four in the divisor,we numbers. to ;i desired, as or, been done in the less those this the last until been has in brought " point off then " quotient as in the as there divisor. .03275. being divide places the dividend places in " ;i is divide may 274 decimal to division " ^ digit of the dividend many come we 9784 down the shifting the point in divisor of example, under quotient. continue a decimal mauy shown TVhen ^ and divisor quotient - ^ been places dividend the of whole a of places,thus multiplying both not put change may move fractions, this does may we the it, and are point off to 12.23. by by moving the product have always factors. divisor The to the places in many shall as though places less annex four dividend " in the ciphersto and dividend the divisor than dividend, were both 38 AND NUMBERS, HOW USE TO THEM. (18. .03275).60000 3275 27250 26200 1050 To Reduce Reduce Common a ^^ to Fraction to Decimal. a three-placedecimal. a 167) 42.0 (.251+ 334 We 860 and 835 perform 250 Jhe 167 mainder. the ciphers to annex the division. indicates + numerator, that there is a re- 83 PROBLEMS 1-13 ; ^, Reduce when 1-13; a-c",m ef terms, and 01 r- and lowest to terms and to the the fraction is an Reduce improper io lowest the fraction . p-7' mixed number 14-25, find the to an and sum improper also fraction. difference the a In lines a one. -" 0 60. then lines In 59. 1. No. n-p number 58. Table . ij mixed FRACTIONS. Practice From 57. IN 14-25, find the " sum of ^, \ gh kl and ^. n 40 AND NUMBERS, 2x2x2 third root, 8 is the 8. = cube or .2 X .2 X fourth .2 is the .2* 8. = .2 The 4^ ; The a If of the show to a root = 2*. ="- is the ; 2 2. fourth is 4he .0016^ of power .2 ; .2. = fifth power root is to' be = I a (^)i or of -I;lis = |. expressed V4*, "\/4^or 4^. of 4 is little above a both, is called or fractional the exponent. another index The or to the left of the = sign -y/, 2^ (2^)^ =^ 2=' X root. 2" = 2". V2^-2'\ 2" process ,of findingpowers The process of findingroots to nent. expo- radical taken, is the index of the The or taken, is called the radical sign. and 2"x2'=2x2x2x2x2 - .2, or rightand is to be above written what is the fifth power, root, a or fraction,it is number or power at the written indicatingthat The = A^ fourth of the root power a .0016 "^/^ ^^- fifth root indicate cube, of 2 ^. = number 8* 2, = :^A- = the fourth -^/S A/J0016 |X|x|Xfx| (F or THEM. of .0016. root the fifth root of USE third power, .0016. --= .0016. = TO root, of 8. 2" .2 X HOW is called is called Involution. Evolution. 59049 998001 sign, SQUARE Each above of the considerations To the we digits in 100. than number the four annex number less than one will good, as either twice contains powers twice as This many. the following from appear as : smallest 100, square or holds principle always Take second its root, digitsas many 41 ROOT^ number digitsin of and O's ; 1000, square so of digitsthan we To less 10000, contain can of O's, one square sqioare number number the three annex No the twice less than 1000. on. digits,viz., 100. three has O's, one two annex To that a we less root, less in its square one. Now take 999. 999 1000 greatest number 999 is has 999 X 10000 X the number every number made of digitsthan Hence, ever^ its squxire or less than separate figures each, by putting same the as the with a of twice that as 999 And contain as But 999. X in its square a has. so for greater root. digitsas many many. over periods of into every number digits in digits,viz., has. can number dot units, the number twice any 1000 number square either contains one If, therefore, we beginning the the number twice square root, No three that 9999 number of 9's. up has than twice only six digits, has 'twice the 9999 X number smaller a that periods will of the alternate square root two figure be thp of the number. If we multiply 42 by 42, settingdown we have We 40 X 40, 40 2, 40 X represent it thus may X 2, and each partialproduct, 2x2. : 42 42 40* 40 X 2 1 " 40 ^ X 2 the = square . twice " the J 2* the - square of the tens. of product ^ of the tens units. by^ units. 42 AND NUMBERS, If there all but consider The 2 + than more are the of square last the The the 83 period,49, the of the root i-equired. The the tens contain of for twice the than in 249 goes be the units the We If 3 a be product of will by twice the tens and so number of the second the in Also, in 18 of tens, and so root. is the 18 is be must greatest square in 18 units tens 16, and the twice for the add than the Hence, the of the root its root tens 40, or 4 is of the the nothing already found 80, for trial 249, If 249 square. and We 3 to the 3x3 1849, leaves units times. units by units, and + by units, three root, and correct tens the hundreds, from tens take little over of the first greater square tens by of product units. 80 83. of tens that, dividing 249 give the 16 square, product the twice were in required. root Subtracting the the some greatest square digits. part of this square. no no the of hundreds, be of two is contained of root greatest square of the tens' of up consist square be can must can square square of square hundreds 18 made must root for the number 16^ 249 the of tens some 3 is therefore periods,its root 1849(43 249 may of 1849. period,18, 80 we tens. as root two of square the number, units*. square being of 1849 digitsin two one THEM. USE TO number any (tens X units)+ Extract HOW more would trial divisor, a will 3 to assume divisor,making root, 3 square X 80 is twice of the units. If 3 is 3 be 83 X 2, add 3 to 3 X is 83 Extract this If right. reduce equal digits. 2 instead root consists of 3 to 80, and first leave out 4 tens, and 86, of is in this remainder plus 2 tens the first of the 36589 of 6 as will we hundreds, 8 illustration, last 8 of the the 684 tens, period, next of 68486. root product before, Con- have we already of tens by units find the units' we root square of 68486 is part of the root As units. units' is twice of plus figure, 261, with next tens find we find of the is the the tens, by tens There divisor So trial divisor. This already found. is twice (twice 20 or complete trial divisor. root before, of to units units' der. remain- no complete. 26 first product 36589. = proceed the square out the complete divisor, 46, of the third taken for twice the and period,89, have units + X to the add tens We is the trial divisor we the last square 2617 The which of 365. leaving figure. The 1. case 6848689. the the 68486, 67600, the square the As units. bring down 2610', if the the 26 as of out for twice leaving 886 of the in as find sidering26 square root but 4 units. will we taken We present Now, bringing down 36589 a all leftover. 5227)36589 which four get 26 for the root, with 521 the the Proceeding tens, of of consideration for regard 521)886 26 2. its root periods, 684, two 6848689(2617 we must we multiply 82 by periods,and four 276 249, required. root of 284 that know shall we of 6848689. first 46) 249, greater than be is the the square will than product 43 We less or just 249. number This to 43 ROOT. SQUARE is 20 + + The 6. twice 6) 46, its units also, 521 simpler than + 1 tens 6, = we the next ; so get tens doubling the 4:4 If the power same AND NUMBERS, contains HOW decimal, a USE TO proceed we the of merely noting the position, way, THEM. ni exactly the point ; thus, 372.14623(19.291+ 1 29)272 261 The 382)1114 764 decimal many Notice 3849)35062 but 34641 that 30. may places as the There are period except 36581)42130 We complete. not that the root indicates + it to period is two figuresin not one. 3549 the square root of 95986 to decimal two 95986(309.81+ 9 609) 5986 5481 6188) 50500 49504 61961)99600 61961 37639 2 2 ^ 3 Extract the 2 X 3 square V4 V4 /4 /4 4 2 _ _ 3x3 root V of _ 121 -x/Q V9 -^j. V49 49 7 _ Vl2l 2 _ \9" ^^9 9 11 as 3* 3, every 38581 Extract is wish. we last first the carry found places. * CUBE Extract the of root square 45 ROOT. -Jf . 43) 16.0 (.37209302 .37209302 129 (.6099+ 36 310 1209)12093 301 10881 . 90 12189)121202 86 109701 400 11501 387 " The 130 be 129 8 evidently 9. or We YQQ^ figure would next call the root may .6100-. 86 14 From Extract 68. places. Prove Of 69. the Table Practice by squaring the 14-19, ^ of root square No. 1-13, and root to five decimal 1. a-d to the adding decimal two remainder. places. " " cd Of 20-25, -5L 70. to eight decimal places. . ; " i b"d ROOT. CUBE Let To 100 the times us take the by itself,we multiply 100 again,we cube as many two annex of 100. as number smallest The the cube root. of three two annex ciphersmore, contains Take the two viz., 100. digits, O's ; to multiply by making 1,000,000 for digitsless than three smallest number of four 46 NUMBERS, AND The digits,viz., 1000. cube has of cube USE 1000 is three less than two digits, ten TO HOW THEM. '1,000,000,000. times as The the as many root. cube No take Now 999 X The latter number times X This contain can cube rootf or in its cube one The + 2 digitsin square or 45' = 45 = 40 + of nine smaller a product three digits, is times of we (40'x5) = tens' + 3 X = tens' + (3 X 3(40' as periods third of the digitsas its many. If, of three ures fig- figure beginning be the same as the number. already shown, equals tens^ as multiply we + 5" + 5 (40' X 5) + (40 40" + three times into root many the square of 45 get 40 = as periods will If units'. three of digitsthan times every number, 2(40x5) 2 three over the third any 5, 40' + 40" + 45" is latter twelve number separate any (tensX units)+ by 45, has largestnumber 9999 X 999. X root. less than two or units, the number of number number The 9999. has either each, by putting a dot the 9999 1000 X the greater numba^ a contains therefore, we with Take x number 1000 999 9999. times the number cube 10000 three times digits,viz., This 999. as 10000 as Every than 999,000,000. 9999 cube No three of number many X of digitsthan two. digits,viz., 9999. than many is as 999,900,000,000. as smaller a product three digits, of four is 999 root, less largest number the X less number a in its cube the number 999 contain can X + 5) + tens' X tens' + 5') X + 5" (40 X 5') + 5" 2(40x5') 3 units 3 X + tens 3 X X tens X units + units' + units' units')X units. "18 AND NUMBERS, Extract cube the of root TO HOW THEM. USE 55480717144. The cube with remainder a We 55480717. taken tens* (3 + units*) X 3 -- we 1140 1 + 3 the out 4= 3 cube tens of the units. which 4* = 3 16. divisor. root required. X root tens, units X 43548300, the = X cube + we tens 43548300 43594036 X 3810*, leaving units*) for use units 3 = 45720 + X 4 = X + divisor, 16 174376144. = = 433200 the plete com- cube root with We X 1 X a mainder re- 174376. tens* X 4 = have for divisor, 3810 3 X 380 X 381, = 3 trial a 433200, units. 174376144 units. X units X ^ The 55480717144. of tens 608717, 434341, =- 55480717 of the 27,872,- trial a the = divisor. find + Unit8*=l*=l. 1140. Now, the 380* X for 1 units X of tens X for use obtaining of already units. X tens* which 3 + of root remainder, the = + the 27,000,000 = 000), tens 608. cube the 38, is Having out 3 X 55480 of find now (380* of root = taken (3 3 X X obtaining 3814 3810* 4 for Units* 45720. 43594036, tens* the is the plete com- cube Extract the cube of 6918 root 49 KOOT. CUBE to two decimal 6918(19.05+ 4707375 Extract the cube of root 3^^. 216* 216 \i /216Y_ \^133iy 1331* Extract the cube of root |f|f 6 11 21-f||^. .242301-f. = 21.24230i(2.76+ 1200 8 420 49 13242 1669 11683 218700 4860 36 223596 1559301 1341576 217725 places. ' 50 and numbers, From Extract how Table Practice the cube root to 71. Of 1-13, a-g 72. Of 14-19, 73. Of 20-25, decimal one use to them. 1. No. place. hi decimal to three -" places. decimal to four places. 7n"t Roots Vl6 than other V4 4. = ^J^l25 VT5625=125. V6561 V81 ""/5l2 ^8 8. = the fifth root We first must similar to Representing tens' X + 3 X + u', and Note. tens When tens t' + letters are Considering used ("+ w)" no understood. number any = "" + as by units' 2 tw + gether It will be alto- fifth power. u, the formula units',becomes + cubes, tens' for ^ + 3 ^u to represent numbers, if two So, of sign made letters,or sign of a operation between means 2 x of tens up the t X letter 3""M"4- u. -f- units, 5"*u - ^ 4- {5"* 4- + lOt'w* 10""u "V tt* 3"V + 3"u* + 4- lOtV + 5"u* 4- u^ e"u"+ "5 + 3 iw' 4- lO^v? 4- 5 iu' 4- plication multiand a plication them, multi- 3 "'u 4- 3 "u" 4- v? = + w*. 2 tu eB + 3t*u+ (e + u)* a cubes. units x together,with always and omitted. commonly come is t, and by for /ormu^a for squares units + 3 X for squares, is number, those ^6561. = of 5030919566507. the produce 3 3. = 2=^512. 2. Extract V'9 9. = = Cube. the 5="/l5625. 5. = 81. = and 2=\/l6. 2. = Square the w*)u. BOOTS OTHEH THAN ThU SQUARE OR 61 CUBE. 5030919566507(347 5th = root required. _(5^*+ Extract (t+ uy the seventh = 1''+ 7^u = i^ + (7^ Thus, whatever + + ^u" 21 ^i^ + to deal with any + 3ot*u^ 35 iV be the root principleapplies. roots In 10^"w" + of 24928547056768. root 21 10^w to 35"3w* + + 35 ^^3 be + + 21 "^u^ + 21 "V + extracted, the practical problems except the square and we 7iu" 7tu^ same + + u' w*)u. eral gen- rarelyhave the cube. 52 NUMBERS, AND Practice a b cdefghijh 1 4546968663625382717 2 0839988693515916262825 3 59948740583160670179140320 4 436392145137 6 2212254303198160680681072 6 12855886264315323923815302 7 67581903586705777467838781 8 76918628642497807490474739 9 46978743989590524278074627 10 84364458030376770984321954 11 04890223684505101402206478 12 9285900800771967220196643 18 19110769407458521478884803 14 88284560361341478032610794 16 05978151908701279270398788 16 2278347499530974 17 32592101337785572228748558 18 82805120942314062544411878 19 78931290301228894683716725 20 03501925068680097704213363 21 48514670584843619553826028 22 7846356781231472902444576 23 86884246898972648751274910 24 67376004732305941727677807 26 53453347556728796015788643 TO HOW Table T tyi THEM. USE No. 1. nopqrstuvwxy z 648427 1 746 1101033815365 1 0 7 7 '4 52684861 2 1 PRACTICE Practice abcdefghijk 1 96121582183995511822622838 2 51163756015082783412609635 3 61331054800986883735679874 4 66298706480301418337028432 6 02090827188646294387519878 63 TABLES. Table I vi No. n o 2. r p q 9 7785 s t u v iv 0 44698 x y 6-35877712601471699552139644 7 123683296045764 8 05443994748766404154772024 9 23849820629876446007177304 10 69460752955933142342204 2 66 11 88366228134448646905416 2 33 12 01012411521700252869576384 18 77521422777944332998543392 14 63937835099468872278549310 15 97641566345292657319833774 16 68741577973349285417266697 17 6 18 29264390047844461683642738 19 13583593990741807180578333 20 9218744797246 21 88491503546174080173771742 22 60335489026115614276018856 23 40626672341408701128835937 24 45566925732848866433608098 25 27557798794796916166482142 1236692108778216755984692 8 423650012983 z 54 NUMBERS, AND Practice ahcdefghijk 1 52968823828287473979264754 2 40794194185114171000960639 8 46810532633584507677718434 4 64426872344769601043723827 6 76807441642338679801467886 6 19936091860897753600766911 7 27977987926170461188611015 8 48917697329370139471870239 9 36249408632686262296313003 10 42370149220246213231724685 11 35889664471351336868944095 12 37780388872833287676325857 18 34008376619651176920339685 14 32652756561030796150970207 15 3016814154617 16 328041438030 17 14852816360503871656818158 18 18275847880490863384819364 19 87690777456899725032657162 20 5418 21 70436376089213778781704006 22 40452642326326403225317749 28 40328411898259262174981420 24 81762463452230799205876354 25 44137486524829585761525309 HOW USE TO Table No. THEM. 3. Imnopqrs 3 4^4 4963604929063582733588 716574259533 963810181617 t u v w x y z PART APPLIED IL NUMBERS. " States United The followed is dollar. the is applied $2347.863 and forty-seven, States United to number by the money, and which unit it is of which hundred Twenty-three means, hundred eight the that $ indicates character Money. and thousandths sixty-three dollars. hundredth One a mill. tenths The worth If the one We pencil is three worth one a is cent and three mills. is called cents ten of eighty-six read, and eagle ; an dime, a A coin dollars twenty to If 1 20 1 cent. 10 cents = 1 dime. 10 dimes = 1 dollar. 10 dollars = 1 = is a Ans. 20 20 = applied and to not upon pencils This 4x5 means the operation cents. The being factors cents, how many pencils can be ? Operation: ^ = ? cents. cents. cents, of 4 cost formed perare 4 20. 5 costs 20. eagle. is the what cents, cents 20 = product pencil cents = numbers upon 5, the mills 4x5 say cents 5 X 10 5 costs 4 : may applied for of be may tenth One oent. a eagle. Operation and value dollars ten double $.863 is eighty-six cents or of coin dollar a decimal cents, A of 4. 67 bought 58 NUMBERS, If 4 Operation: -2^ 5. Note. problems, and = The last two In general head. same and the factor is what $14f how men a be not of 3 true answer, for 4 to men, and cannot be 2 as Or, if 1 A fraction there boy, There stated, 2^ being all the work numbers. pure in 2 in 4 days, days? men. 3 Yet men. 3 would men require the not labor entire imaginary quantity as applied of the problem the conditions an We 1 working man could boy a cost? of work with. working is,however, but do 4 2 days, would the work half days, would to answer one that readily see can of also 2 just a be man, just problem the men. is also no by dividing the 1 ton amount take that days, and 4 does to do 2^ is 2^ fullycomplied and enough. Ans. know we the work. men factor, one __. certain a for it would days. so working men do we performedupon are it would sense men and product a the 1265 it take 2J^. = certain a under $6. arithmetio would men Operation: f In ? following,come This factor. $4, what Ans. perform can many pencil cost = costs 6. = t operationBin If 5 1 115x11 of coal Operation: All does given ^=_^^ ton a other THEM. given. 14f ,. Operatum: -Jof the USE the two have we TO part of $371^? ^ If each find required to are product by do cents, what 20 pencilscost HOW AND coins be numbers may 6J cents each, the of computation. an imaginary quantity as applied to cents, of cent applied. numerical 4 X 6^ = to which denominations If handkerchiefs quantity 25, 2 X 6^ 6 -: J are advertised is taken 12^. fractional the basis as If at we buy 4 handkerchiefs, 13 can The cents. first case, be we 25 pay the however, applied; in cents is the price ; if buy 2, we in same result the 59 NUMBERS. APPLIED both obtained second instances. by the In pay the multiplication it cannot case, must we be, without modification. $1014.10 27. $3871. 1924.67 234.86 Sum = $284.62 $1275.96 Difference = 5 $1946.33 72)$6722.64 ($93.37 Product =$1423. 10 Quotient. = 648 242 216 266 216 504 504 Money. English farthings(far.) 1 penny 12 pence = 1 shilling(s.). 20 shillings = 1 pound ("1 4 = (cf.). or 1 L). A coin worth 5 is shillings a orown. A coin worth 20 is shillings a Bovereign. A coin worth 21 is shillings a guinea. guinea is no longer coined, but meaning 21 shillings. The Farthings are expressedas fractions of the a term penny. is still used, Thus, life?. 60 16s. "42 7 13 Sum AND NUMBERS, "63 = 11 14 6 14 l". + nd. USE THEM. + ^d.==21d, 14".+35. + "13 "1 16 7 = TO + "7 + 16s. ls.9d. = 34s. = "114s. = -="63. + "42 9d. s. "42 Difference ed. + ^d. 3 HOW 3 s. Product 13 "105 = llrf.("7 3s. 5i|rf. s. Quotient. = 161 12 inches (in.) yards 40 rods 8 fur. or or 5280 or 16i feet 320 144 square 9 square BOi square square 160 square square 640 acres 1 rod = 1 | (rd.). furlong. ^ ^.^^ 1 league. = 1 square foot = 1 square yard (sq.yd.). j ^^^ (sq.in.) feet yard (yd.). Measure. Surface yards or 272* _ ^^^^ (sq.ft.). (^ J or 43560 ) i __ feet = (ft.). j feet feet rods 1 = or inches = rds. ) 3 miles Square 1 foot = 3 feet 5} 3 c?. 6 bd. s. 19s. 23)"164 12 "17 11 12 "35 4rf. s. ^^-g ) = 1 square mile. ^^ ) 6d, Cubic 1728 1 cubic foot feet = 1 cubic yard (cu. yd.). feet = 1 inches 27 cubic 16 cubic feet cord (cu. ft.). = cubic 8 Measure. Solid or 61 MEASURES. AND WEIGHTS or (cu. in.) 128 (cd. ft.). foot cord ) cu- i -i __ ' bic feet ) Weight. Avoirdupois 16 (oz.) ounces pounds 2000 28 4 20 1 = 1 ton quarters or 112 hundred-weight pounds or quarter = 1 hundred-weight ) i for weighing /rn i. ^ ^^ v '^ Weight. gold, silver, and precious grains (gr.) = 1 carat. 24 grains = 1 20 pennyweights = 1 ounce 12 ounces = 1 pound 4 (cwt.). J pounds Troy Used (qr.). 1 _ 2240 (T.). = " pounds (lb.). Table. Ton Long pound = stones. pennyweight (oz.) (lb.). (dwt.). 62 AND NUMBERS, Reduce 7 lbs. 11 oz. HOW 13 dwt. TO 18 USE THEM. grains. gr. to 7652 3826 45912 18 7 lbs. 11 45930 Reduce oz. 13 dwt. 18 gr. grains to pounds, = 45930 ounces, gr. pennyweights, and grains. 24)45930(1913 dwt. 24 20) 1913 219 12}95 oz. 216 " 13 dwt. 7 lbs. 11 " oz. 33 24 "90 72 18 gr. Ans, 7 lbs. 11 oz. 13 dwt. 18 gr. 64 NUMBERS, AND HOW USE TO THEM. Time. A by 100, will is year not a if its number if its number or be leap year is divisible The leap year. a is divisible year by 2000 units 12 dozen ^ - year 1 dozen. 24 sheets 1 gross. 20 quires or 1 gross 1 great gross. units 1 score. 480 -. sheets j Books. A book formed of sheets folded in 2 leaves is in 4 leaves is a 8 leaves is an ti a folio, quarto. octavo n 12 leaves is a duodecimo n 16 leaves is a 16mo. n 18 leaves is an n 24 leaves is a 24mo. n 32 leaves is a 32mo. n 64 leaves is a 64mo. 18mo. or quire. ) ~ 20 1900 Paper. _ 12 not will be. Numbers. 12 The 400. 4 and by 8vo. or 12mo. ream. WEIGHTS The of sizes AND taken commonly most paper 65 MEASURES. the as basis are the Cap i4 inches by 17 inches. Crown 15 inches by 19 inches. 16 inches by 21 inches. 19 inches by 24 inches. (de-my') Demy . . Royal Thus, each we a sheet being have may standard, and books being depending can still printed page. sizes adopted by York) 7^ Large 8vo Small Svo dimensions variety of used was forms and of dimensions following lisher pub- though the definite idea of the size of the of the type page mon com- in 7 X 4 J inches. 6f X 3| .... . . . " " 18mo 5 (( the margins: type pages, for instance, x3 an inch a bound or two 12mo must book be added is about inches, etc." blank still used, books, the though the always correspond. terms names cap and quarto, crown the ; sizes, the etc., leading publishing house a the as standard, printed 5|x3f the For name 12mo being 5 Similarly, : "These X other no longergive any . for when used, which, instead paper the are 17 in., by taken formerly was endless an used, no of the Following royal Consequently, 8vo, 12mo, are (New the sizes,is of whatever terms " in. 14 leaves. 16 in desire. may make be said to be any the upon of sheets hardly made established certain to as understood was there now so made quarto, a demy folio,a royal 8vo. crown a book a folded printed books, For but is 16mo cap page 8vo, etc., are dimensions do not 66 and numbers, how to use them. Problems. The 74. ending profitsof net June 30, 1891, certain a were follows as 1887 1888 265,147.08 1889 i243,854.87 1890 265,448.52 268,960.87 ..... At 76. the was 35 : $269,862.33 1891 What for the five years company profitper year? average what hundred, cts. per is the of cost 655 lbs. of ice? Express the 7 operationsand cancel thus, ; ^^^y^ thus, Or 131 229i. = 6.55 -35 $2.29. ^725. 1965 4 2.29 is the value 76. What 77. If coal 78. oranges is the 28 cts. cost a long each profiton 14 cts. per is sold ton for dozen ? $6.25 per lbs. ? 2000 is the dozen, what a pint of peanuts rate same Callingthe of 81. How 82. What Answer per at eggs cost of 14 doz. and 7 ? at the 80. the At If half 79. cost costing $4.10 ton, what short of 1004 a many of to three of the a bushel pound sterling$4.8665, what peanuts in United inches square decimal would l^c?.,what ? value bushel cost of decimal a cubic States in 1 square yard places. is 13 cu. is money? mile? ft. 1124^ cu. in. ? 1472 3 tons 83. pounds, How 86. 1900 1891 year many how many grains? be there will leap years many followingthe equivalent of is the oz. drams, scruples,and ounces, How 84. lbs. 13 67 MEASURES. AND WEIGHTS in the 2000 years ? seconds in the nineteenth seconds does century (1801 to ? inclusive) How 86. many revolutions about How many 87. $2.25 what How there in 27 of each is the cost half-pintbottles many to make the earth 100 ? sun pencilsare gross, per 88. the it take doz. ? 11^ gross At pencil? be filled from can head hogs- a containing 68 gals.2^ qts.? 89. How 90. What I 91. buy If 92. in of a 13 cord of wood? of acres land 2^ at cts. per 16 a invested 7 cts. per of note what I net Taxes $1800. $654.92. to I sell 12^ cts.,and ft. at foot, 1322 profit? 6 weighs paper must for amount is my What cts. ream quarter of land a charge per of pages lbs., and quire to costs double 18 my ? money How 93. from at acre at and money pound, per 14 leaves 94. many additional If the steps in a a 12mos man what in second,-how 164 measuring to each paper pound, If cap of paper reams cts. per 95. value acre on an the balance cts. an interest half one is the inches foot? square and cubic many for waste book weighs is the X lbs. to the cost of the paper takes will 32 it take in. him be made ? and ream, per at can in.,allowing 8 28 leaves 32 walking long 17 in. each a to go book costs 12 ? 2 step, and 14 miles ? 68 wheel If the 96. the tire,how 48 HOW AND NUMBERS, of a USE THEM. make the wheel will in. around 94.248 measures wagon revolutions many TO in going miles ? PERCENTAGE. The terms centum, 6 per per and cent meaning, by cent, or 6 %, percentage from the Latin per is the base, 9 is the come the hundred. .06 of 150 or 9. = 6 is the rate per cent, .06 is the rate, 150 peroentage. (1) troods which is the What 14%. $162.50 cost (2) Goods 14%. sold at which a loss of 14%, $162.50 cost is the What the are loss is sold at $22.75. advance an sellingprice? 57 325 ' IM^xiM=. ^ $185,25. r Goods is the which $162.50 cost are sold at a loss of 14%. What sellingprice? 325 43 IM#_Xi^ $139.75. = r (3) is the Goods which of $22.75. = are advance an 7 "M2li^ they at ? profit 325 If sold are cost $162.50 are ? profit $185.25 162.50 $22.75 sold for $185.25. What of 69 PERCENTAGE. which Goods $162.50 cost sold are $139.75. for What is the loss ? $162.50 139.75 $22.75 Goods (4) $185.25. The is the What profitis ^^ cost, the cost the sold at are is sellingpriceis ? profit sellingpriceis \^^ of sellingprice,the profitis ^^ cost, the of the of the ^| The profitof 14%. a the of sellingprice. 325 ^^"^'^^ X li $22.75. = Goods sold are $139.75. .What loss is The the cost at 14%. loss of The sellingprice is is the loss? y^^ of the cost, the of the -y^ a sellingprice^^ sellingprice,the loss ^ of of the cost, the selling price. 7 325 $Mlixi^ $22.75. = (5) what Goods was are sold for $185.25. The profitbeing $22.75, the cost ? $185.25 22.75 " $162.50 Goods was are sold for $139.75. The the cost ? $139.75 22.75 $162.50 loss being $22.75, what 70 AND NUMBERS, Goods (6) of the 14% sold are at What cost. HOW a TO profit(or loss) of $22.75, the was 325 the was sold is 50 1162.50. -. are which cost? mM"im (7) Goods THEM. USE for $185.25. The profitis 14%. What cost? 325 m?^ 50 . Goods the sold are $139.75. for The 14%. loss is What was cost ? 325 50 ?lM^"li22 = (8) Goods profitis that what per $162.50 cost cent of the are $162.50. sold 7 per $162.50 that cost cent of the $162.50 139.75 The 2 %%n^m^Y\ 162.50 what $185,25. cost? $185.25 Goods for cost are sold for $139.75. ? 7 2 mi2"M.^u The loss is 72 AND NUMBERS, Goods (12) sold at are is the wm Problems bbls. of be sold per barrel A 98. much How I 99. sell How cost. I 100. will he more house a buy and shoeing the horse, $7.50 whip, sell the and per week than is raised 5%. formerly ? 20% carriagefor $355. than more $450. for $2.25 for I pay is the per What it ? repairingthe carriage,$1.50 for outfit has his wages the transaction by they must ? day $4200, which I make horse a per receive for do much $164.32, for what cost earning $2.80 man Percentage. yield a profitof 28% to is $139.75. = in apples which 43 mMj"M $186.25. = If 90 THEM, 325 m X USE sellingprice? 57 325 97. TO profit(or loss)of $22.75, a What of the cost. 14% HOW for cent a of profit? A 101 of $1232.16, which I 102. market What buy value per 103. for 152 An of $.62 grain at I compelled am was of lot at What cost. dealer is 22% sells more a lot of than they grocer sells flour at of the cost. architect a profit a the cost? was bushel. per The sell at to potatoes cost. for $.57. 60 What profitof $1.10 What charges 5% is his cts. did he the house the commission including commission ? a per barrel. sellingprice? commission plans and superintending the building of cost and bu. ? profitis 18% 105. bu. produce pay His of the 16f % house a do I lose ? bushel, which A sells declines, and cent A is 1000 per 104. dealer real estate . house. for making The being $6240.21, total what 73 PERCENTAGE. An 106. He I send He coffee. A 108. the buying, and A 109. at a of the par from The in commission, and America ? 98 at 30,800 lbs. J-(per cent much J, how (of of profit is the value) par selling? stock of broker's The much value, how 10%. to invest I receive bought 113 at shares 13J%. for his of goods ? $4000 being \% for same sells 28 of the broker 110. paper the broker premium it is sold for the coffee. is commission broker's made, the for If commission a in South $1000 railroad bond value). par of the money pound per on America transportingthe in the cost was in South 4% reserves goods did he receive agent my expends $144 What What $152.47. remits 107. sells auctioneer is the for $3171, which is being ^% commission principalentitled to receive ? from following is advertisement an in daily a : Vases worth $2.00 for 73c. Vases worth $1.00 for 44c. Vases worth 50c. Flower Globes for worth 19c. 75c. for 29c. .... Jars Rose Bonbon Trays worth 49c. 75c. for 32c. .... and Finger Bowls Water Pitchers, $3.98, Creams worth 75c. for 42c. . worth 50c. . . for 22c. 79c. now Pitchers,98c., now Cream 39c. Pitchers, 98c., now 24c. Sugar Bowls, $1.98, now 49c. Teapots,$2.29, now 49c. is the buys entire $1.25 for Sugars Milk What worth one per cent of discount article of each purchase ? on each kind, what article? is the per If cent a on chaser pur- the 74 AND NUMBERS, In 111. the same Pears in Choice is the paper Yellow Choice TO HOW THEM. following : Peaches, 12c. per^doz. can, a glassjars,qts. 35c., L. USE 2 qts. 65c. . . ^. . Raisins, 9c. lb., 6 lbs. for M. 60c. . Best Mince Pure Meat, Lustre 25c. 31c. . . . . . Starch, 9c., 3 for and Mocha . 25c. 25c. . . Best . box, 3 for Salt, 5-lb. bag 6c., 5 bags Table Electric 9c. Tartar, ^ lb. 18c., 1 lb. Cream Best Java . . $1.00. Coffee, 35c., 3 lbs. . What saved is cent per $1.20. . each on . by buying article the in larger quantities? INTEREST. paid Money loaned money is the for each much If ; ^uL^ $575 is termed money The at interest is certain a interest. a rate The percentage per of i.e, annmn, year. for 2| at years 6% interest, $500 is the 3 5 rate principal. $500 be loaned principal: of use is reckoned principal,and the so the for is the ^^ =$75, " is the interest; 6% is the amount. Thus, The number The What interest is the product of the principal^ the raie, and the of years. is the amount is the 9, 1891, at 5% interest ? of the principal sum on $642.71 from and Dec. the interest. 13, 1888, to Sept. 75 INTEREST. Method, Accurate Dec. 13, 1888, to Dec. 13, 1890 Dec. 31, 1890 13, 1890, Dec. to 2yrs. . 18 31 1891, Jan. dys. . Feb. 28 Mar. 31 Apr. 30 May 31 June 30 July 31 Aug. 31 9 Sept. 1-9 2 yrs. 2 yrs. 270 dys. 2fH = 7^. = ^^W^ dys. 270 yrs. 10 $642.71xgx;000_^33Q^ 73 This is observed a in greater degree computing the interest for month, and 360 days days of interest. is reckoned a accuracy By as than is commonly ordinary methods, a though 30 days made the year. Oommon 13, 1890, 2 yrs. Dec. Dec. 13, 1888, to Dec. 13, 1890, to Aug. 13 to Aug. Aug. 31 to Sept. 9 Method. 8 Aug. 13, 1891, mos. 18 31 ... . 9 dys. " .... 2 yrs. 8 mos. 27 dys. 76 AND NUMBEES, Interest $1 on " " HOW for 2 yrs. at $1 " $1 "27dys. 8 " TO 6% " -= 2 X = ^ " mos. THEM. USE of 6 cts. =$.12 8 cts. = Iof 27 " = mills .04 .004^ = $.164^ 642.71 .164i 257084 385626 64271 32135 6)$105.72579 17.62 " '" " seems 27 mos. but the time directlyfrom in " long is the when the dates the time, July 18, 1891, interest at 8% the so on 1^. " " " " " " "5%. written out in detail in this method given,and the we interest this write on $1 operationis reallyquite short. $243.84 from April 22, 1891, ? Accurate Method. 8 31 30 18 87 dys.,at 6%. " performing a problem by manner, What for 2 yrs. 8 = process directlyfrom $642.71 on = $88.10 The Int. = days. 1812 to 77 INTEREST. Method. Common 243.84 "014i 97536 2 26 mos. $.014J dys. Int. = the time. = oaqqa $1 on the for 6% at 8128 time. given 3)3.49504 = Int. at 6%. = Int. at 8% 1.165 $4.66 What is the interest $10,000 on for 24 6% at ? Method. Common Method, Accurate days = $10000 20 ?00 .004 $^0000X6X24^^3^^^ ;00 " * m X 73 Problems In of a the following interest month, months stated,consider What the rate is the interest problems, consider 12ths as Interest. in to be of a 6%. on $1 for each number of 113. $1 for each number of months 114. $1 for each number of years 115. $146.24 for 4 yrs. 6 mos. 18 116. $64 for mos. 19 dys.? 117. $2245.86 for mos. 26 dys.? 118. $862 for 9 mos. 2 Unless year. 112. 1 yr. 3 days 19 days dys.? from 1 to 30? from from dys.? 1 to 11? 1 to 10? as 30ths otherwise 78 at at 8 $352.46 120. $1226.83 for 121. $864.41 for 2 yrs. 122. $256.30 for 11 mos.? 123. $22.98 124. $684.37 for 11 What at for 7 yrs. 119. is the 9 dys.? 7 15 dys.? for 1 yr. 3 mos. mos. ? mos. 23 mos. dys.? of amount 17, 1886, 125. $624.84 from June 126. $146.22 from July 4, 1890, 127. $1042 128. $6234.16 6^% Feb. from 28, 1888, June from 130. $324.72 from 131. $1216.66 ? 10% ? 30, 1888, at Sept. 19, 1891, to Nov. July 18, 1889. to 30, at Aug. 4, 1888, to 1890, Aug. to 7^^% ? 5% ? at 29, 1891, ? Jan. from $623.50 133. $2241.62 16, 1891, Dec. from to Dec. 1887, 29, 14, 1891, 24, Aug. to at 3|% ? 1889, ? June $26.50 from 30, 1868, interest is considered When due of time, the interest unpaid, may principalto as compound be form well interest a as new on as after regarded to Nov. at 7^%? as the a payable at specifiedintervals lapse of new loan principal. Thus, the 29, 1891, Interest. Compound interest 7%? at Sept. 18, 1890,-"it8% Nov. to April 6, 1887, from 132. 12% to Aug. 7, 1889, ? $24.98 from Jan. 14, 1876, 4f % to 11, 1891, 129. 134. if USE ? mos. 4 yrs. 3 THEM. TO HOW AND NUMBERS, every to such interval, be added interest originalprincipal. This is to the paid on is termed 80 AND NUMBERS, What interest is the of amount compounded TO HOW USE $10,000 THEM. 1 yr. for 6 at mos, 4|%, semi-annually? 10000. 1.02| 10237.5 = amount of $1 = amount of = amount = amount for 6 4f %. mos. at $10,000 for 6 mos. of $10,000 for 1 yr. at of $10,000 for at 4f %. 1.02f 204750 102375 12797 25594 10480.64 4f %. 1.021 2096128 1048064 131008 262016 $10729.55 Problems What 135. mos. at 4|%. Interest. of is the amount $613.72 Compound in 18 for 3 yrs. 7 mos. 16 dys.,interest compounded annually? 136. $221.32 for 4 yrs. 11 at mos. 5%, interest compounded annually? 137. $950 for 1 yr. 7 at mos. 8%, interest compounded semi-annually? 138. What July 14, 1891, is the interest interest on $617.25 compounded from July 1, 1889, semi-annually? to 81 INTEREST. different In regard to simple interest Savings be can In due, the taken immediately back states, where some it is allowable of to interest by as a and in only unpaid law. credited Being interest* to is, in effect,paid and loan. new be interest cannot compound other rule is that due Interest at prevail usages general interest charge simple ; in and action compound pay depositor when the The charged. be collected banks laws interest. compound however, may, different states interest on deferred each states, compound collected, interest ment pay- may be collected. What is the interest 6%, interest paid until allowed 2 yrs. 4 the ^expirationof the deferred 16 mos. being payable semi-annually, but .the on $416.12 for on not dys. at actually time, simple interest being payments ? 416.12 .03 $12.48 Ist deferred paym't of = 6 interest. mos. $12.48 bears int. for 1 yr. 10 mos. 16 dys. 4 mos. 16 dys. 10 mos. 16 dys. 4 mos. 16 dys. 4 yrs. 6 mos. 4 dys. 1 yr. (( It (' It = $3,378 interest $12.48 for on = interest on $12.48 = interest on principal. = interest on interest. = total interest. for 4 yrs. 6 416.12 .142f 59.366 3.378 $62.74 mos. 4 dys. 82 AND NUMBERS, interest Simple is sometimes What called annnal is the TO USE principaland the on HOW annual THEM. each on year's interest interest. interest for $10,500 on 3 yrs. 4 mos. 24dy8. at6%? 2 yrs. 4 1 24 mos. 4 24 4 24 4 yrs. 2 dys. dys. 12 mos. m 630. = interest $10,500 on for 1 yr. .252 $158.76 10500. .204 2142. 158.76 $2300.76 Problems What is the annual = interest on principal. = interest on interest. = total interest. Annual in interest 139. $462.17 for 140. $1226.82 for 141. $617.25 for 4 yrs. 3 What is the amount at 142. $1654.26 for 143. $891.82 144. $1682.36 from 146. $642.50 from from 7 3 yrs. 2 yrs. mos. mos. annual 2 yrs. 3 June on mos. 6 Interest. mos. 17 dys.? ? 12 dys.? interest of at 19, 1887, to Apr. 30, 1888, Nov. 16, 1887, 7% to ? Jan. 6, 1891, at Dec. to Feb. 6, 1890, 28, 1891, at at 5J% ? 5% ? 7^% ? NOTES, AND DRAFTS, NOTES, Two months Jordan, Marsh after date d One 17, 1891. I promise order the to to pay of Co. hundred and -^5^dollars. thirty-two received. An obligationof made payable of a time after days termed the days of at this note time or specified. These Unless grace. until notes without interest after a Notes that time The ment pay- until three days notes specified, due. bear are are do not maturity, After interest the at legal differs in different states. New At Seventh National days sightpay three Bank, York, Aug. 25, 1891. received,with William current to the order of NY. Eighty-six To note. additional three become $86.42. Value ^-^^ demand. on otherwise they until called j^^^ legallydemanded be cannot interest rate, which descriptionis spegifiedtime a bear The CHECKS. Boston, Nov. $132.47. Value 83 CHECKS. AND DRAFTS, dollars. ^^ of exchange rate on York. New F. 8. Francis. Jmies, Boston, Mass. This William of is a bill of Jones $86.42, and him. exchange being the indebted account or draft,as to F. it is S. Francis being due, the usually termed. to the creditor amount draws on 84 above The draft is indorsed National Seventh J. D, Bank . Boston, of York. of New Cashier. Orady, W. : drawer, F. S. Francis, leaves the draft with the Seventh The of Bank National Bank National 28. he National debtor The presents it Bank National Seventh The collection. for draft to the Traders' the Traders' Jones, Aug. the draft, York, New sends The Boston. and Bank, follows as order,for collection,account or of the back upon National Traders Pay THEM. USE TO HOW AND NUMBERS, writes Bank liam to Wil- face of the across " then in stands similar a the position to maker of a note. Drafts certain On drafts The three is made in draft a specifiedtime, sight,or at a sight. York it.New draft at days' grace sight drafts, and on grace allows the after time time allow payable made are does are do not. some The not. payable always allowed. law of Some Massachusetts the and where state governs. question being presented Aug. 28, days' sightexpire Aug. 31, states the draft is the three payable three days later, Sept. 3. bank A by one to pay check. check who a has is a a depositat specifiedsum Being without speciesof drawn grace. to draft. the bank, and the order upon It is a of the calls upon the payee deposit,it is draw^n presumably named payable on bank in the tation presen- in calculating in Although the determining not We so. 1 If must month the last business than days is the of of Jan. grace next the grace, 1888, How note? month, drafts it is Feb. legal holiday a note or In preceding. day 28. next draft is (Sunday, payable counting days following the on other holiday taken. one demand has 1 as and notes expires upon etc.), the business count days. 1891, falls days of actual 31, PARTIAL A 30 maturity the Christmas, day interest, of count day Thanksgiving, the time from 85 PAYMENTS. PAHTIAL note the much for following shall be PAYMENTS. $6240 at payments paid July 6% interest, indorsed 14, dated the upon 1890, to take June back up 6, : the 86 The natural most payments is the HOW AND NUMBERS, and obvious following TO USE THEM. in procedure of case partial : 6240. 1.0l^ 24960 6240 6240 1040 4160 6332.66 for 2 = amount of = amount due Sept. 4, = amount due Dec. 22, 1888. = amount due June 7, 1889. = amount due May 14, 1890. = amount due July 14, $6240 mos. 446.21 5886.35 1888. 1.018 5992.304 800 5192.304 1.027J 5335.958 110.25 5225.708 ' 1.0561 5519.218 500 5019.218 1.01 15069.41 1890. 29 dys. 88 AND NUMBERS, Problems A 146. note is indorsed under United for |1642.71 with follows as TO HOW USE THEM. Method. Court States Feb. interest from 17, 1888, : 6, 1888, $164.42 July ; Sept.17, 1889, $55.12; 15, 1890, $350. Dec. is the amount What A 147. Partial due $1500, for note payments June ? 7, 1891 is dated 6%, at Sept. 6, 1890. : Dec. 12, 1890, $45 Dec. 31, 1890, $65.50 ; ; Apr. 16, 1891, $655. What due is the amount A 148. 14. July months' four Payments 16, 1891 Nov. for note $540, ? interest,is dated 5% at : Aug. 17, $100 ; Aug. 30, $50. What due is the amount A 149. maturity ? $10,000, for note at is dated 6%, at : Apr. 16, 1888. $750 The United by ourselves some with Dec. States of these principle,local 31, 1891 Court the ; 4, 1891, $5000. Apr. is due ; 22, 1888, $2500 Dec. much 21, 1886. ' Indorsements How Nov. state ? Method courts, modifications. usages can has but If been we one be followed fied variouslymodi- need not understands without concern the general diflSculty. PARTIAL Paetial this By about time Payments a year, and on by if the method, Merchants' the does account interest is allowed each until the than either of the Method. for run the the entire it is made time favorable more than more principalfor from It is is settled. not the on partialpayment account 89 PAYMENTS. to the debtor already given. methods w $1240 is loaned made are June follows as 4, 1890, 28, 1890, $462.20 Feb. 20, 1891, $112. due in 7 $462.20 " $112. " ; Dec. is the amount $240. Payments : Sept. 6, 1890, $240 What interest. 6% at April 10, mos. 3 " 1 " 4 1891 ; ? dys. amounts $248.56 to 13 " " " 21 " " " 470.134 112.952 $831.65 $1240 in 10 6 mos. dys. amounts $1303.24 to 831.65 Amount due If the account calculated end of 1891 for much $471.59 = than more the on made payment till the runs April 10, principalfor during the year, 1 the balance The year. and year, from a the year, on time is taken interest each it partial made was for is a new principal. Problems 150. A Sept. 18, How much demand 1890. is due note Method. Merchants* under for Indorsements $600, at 6% : Dec. 12, 1890, $45 Mar. 16, 1891, $240. Aug. 14, 1891 ? ; interest, is dated 90 AND HOW $4000 at NUMBERS, A 151. How for note is due much PAYMENTS. is due 1246.10 1460. 131.84 3, 1889. May ? OF EQUATION is dated 6% 30, 1891 Mar. THEM. USE TO in 1 month. " " 2 months. " " 3 months. 11837.94 When the may loss to either without We will which the the as first amount becomes date what made were lose the interest date for due. is assumed, the at be amounts paid all at once, ? party first assume, convenience payment these of sum time It is any the payment, date a matter mere will assumed, the date of If answer. debtor on would on $1460. 131.84 fori month for 2 months =, at 6%, $7.30 =, at 6%, 1.32 $8.62 The date assumed, assumed so be required must that interest to the date the enough on required,will later $1837.94, be $8.62. than from the the one date EQUATION The interest ffj 30 X The 28 payment If had we have would instead of As amount have is the rate the rate may be may be done of interest in the of ^ that. The per taking followingmanner are interest day. problem into account. rate any final tables interest mill we $8.62, and this rate, the At a $8.62 of affect the result,the does not without performed Unless same. and days. instead fifth less than usually taken. month, per 28 fifth less than one one the been 5%, at amount assumed. in 1 month is due =|9.19. 6% at date interest an $9.19, an is 1 cent for 1 month $1837.94 obtained used, 12% $1 of 91 PAYMENTS. after the days = calculated result would on $1837.94 on = OP It : 246 1460 132 X 1 = 1460 X 2 = 264 By paying the at the debtor time sumed, as- would lose 1724 1838 equal amount an 30 $1724 on 1838) 51720(28 $51,720 3676 est for 1 for day $1838 for on est to the inter1 month, or the inter- = 28 days. 14960 14704 In Arts, Average the followingaccount Mr. a Debtor 1 month 28 days. : to B. Mr. Credits, Due April 17, 1891, $240.38 "' July 2, " " Sept. 26, " 100.46 56.20 $397.04 360. $37.04 May 17, 1891, by cash, $160. June 14, on 2 mos. " ** mdse. credit, 200. $360. 92 AND NUMBERS, Interest on $100.46 HOW for 2 56.20 dys. at 12% 9 " THEM. USE 15 mos. 5 " TO " $2.51 = 2.98 "" " = $5.49 Interest on $160 for 200 " 1 3 " 12% at mo. 28 " dys. " $1.60 = 7.87 " = $9.47 5.49 $3.98 Interest on $37.04 for 1 at mo. 37)398(10 12% =$.37. dys. 23 mos. 370 "28 30 840 74_ 100 The assumption full amount of $5.49 $200 in gain,the for this beforethe $3.98 is the for mos. on on less 10 of the June time, to the of these making loss to the May on gained $3.98. In of the account must debtor 17, and payments the on Striking the interest. in the which on order to make up be considered as assumed. interest payment Suppose has a by paying $160 gained $9.47 balance date April 17, 1891, 2 But instead balance, he the date as due, involves was interest. date, he has interest time April 17, 1891, $397.04 Aug. 14, on assumed due of of $37.04 for 23 mos. dys. balance 14, 1891, B amount of 10 = 23 mos. May 25, dys. 1890= equated $37.04. of had $300. merchandise The account of now A on stands, EQUATION OF 93 PAYMENTS. Dr. Or. April 17, 1891, $240.38 July 2, "" Sept. 26, ** 17, 1891, $160. May 100.46 300. " Aug. 14, i^ 66.20 397.04 $397.04 $62.96 Without is counting interest,there Interest $100.46 on 66.20= "( Interest $2.61 = 2.98 A. $62.96 due $160 on " $1.60 = 300= " H.80 $6.49 $13.40 6.49 $7.91 Interest for $62.96 on " There being balance of the the balance gain of be deferred may 12% $.629. = 17dys. =$7.91. " being account at mo. 12 " net a 1 interest due the debtor, and to the to the of debtor, payment long enough to make for this up gain. April 17, date for B If on are if the the account of assumed balance There " In balance is fact, the of Equated $62.96. and the and the account really nothing 6% in is rate partialproducts methods The = balance of be are the assumed balance the date will interest of earlier interest than the date. Note. rate, the 4, 1892 May = oppositesides^the equcUed on of 6. dys. the balance A of 17 mos. side,the equated date will be later than same ; 12 + to pay the balance the date 1891 are interest,Part question that of the account to be gained by taking 12% instead altogetherpreferable. smaller and easier to the With handle. See smaller cut short- III. usually arises is,not due, but how much at date what is due at some is the par- 94 NUMBERS, ticular date. may be, how it under the HOW AND Thus, in the much does B general head just case THEM. considered Oct. 1, 1891 A owe of USE TO the ? question This brings partialpayments. $405.15 Or, Amount i( of "c $160 300 for 4 1 " 14 mos. dys. 17 " = $163.57 302.35 " = $465.92 405.15 Balance due A Oct. 1, 1891 = $60.77 TABLE Showing the number of days date from in any any other date in month. one month to the same 96 AND NUMBERS, HOW THEM. USE TO Or. June 1, 1891, by balance Oct. 4, " " cash, Oct. 16, " " mdse. account, $142.76 ol ; 251.76; 150.42. dys., 30 Dr. Jan. 156. ; 44.88; " 60 " 30 " 321.84; " 936.90. May 6, " Aug. 4, " " 16, " ** Nov. dys.,$120.60 60 to mdse. 17, 1891, 90 Cr. Apr. 16, 1891, by cash Oct. "... 14, . " . $640. . 500. DISCOUNT. True debt A is of required considered The 2 mos. the 6% 18 dys. = 6% at 2 at 18 mos. dys. the once, use principalwill amount true for the $1 between $246.10 given of the to 158. $1000 159. $164.76 due due $246.10 in time. $242.94 mos. account. present worth $1642.80 due and for 2 $246.10 on 157. being money Am. discount is the of amount ? ""242.94. difference What annum? per of amount in debt the pay present worth What is due question is,what ^1Sir is the to worth $1,013 The $246.10 Discount. in 4 mos. in 2 yrs. 5 in 3 mos. of 3 mos. 17 dys.? ? dys.? 18 is $242.94. $3.16. dys. This $242.94 is 97 DISCOUNT. 160. $500.25 due 161. $1600 due 162. $5647.28 163. $223.50 due in 30 due 10 mos. dys.? in 90 in 60 dys. at 10% dys. at 9% due in 30 dys. } $700 due in 60 dys. 90 dys. $247.62 due I in in 2 I hold suppose 15 mos. writing my name to who one any its due, provided I checks provided they of indorsement Wanting dys. to 18 mos. for 2 mos. avails What or ? of all be the not the failure to by th" order is blank a particularperson names no use, I take me it when pay transferred payable by payable note makes and does order my the note legal owner, just mentioned it the money discounted. 2 month to pay. ment, indorse- of. An ment, indorseto whom is to be made. payment 18 makes if the maker made are called because so This may kind the per I indorse legallynotified am % $246.10, payable to its payment Notes, drafts, and 1 at interest. become may responsiblefor for note a the back. on ? Discount. without dys. ? . Bank Now dys.? $250 f 164. in 2 to the three Adding bank The run. the note days'grace, deducts the the to a note interest to be bank has 2 mos. $246.10 for on dys. $3.20. This is the bank discount on $246.10 18 dys. I receive $3.20 $242.90, the $246.10 = - proceedsof the note. is the bank discount in 30 and the proceeds of 165. $1560 due 166. $1860 due in 167. $520.94 due in 168. $257.42 due in 4 mos. at 7% ? 169. $320.40 due in 2 mos. at 8% ? dys.? 60 dys.? 90 = dys.? a note for 98 AND NUMBERS, HOW TO USE THEM. _ * Trade the list or Frequently price reallyasked, 40% sellingprice being the list $60. is If sold this is reckoned and 40 Goods listed at $110, discount 171. Goods listed at $560.50, discount 172. 5 doz. articles at 173. 16 174. 150 bu. at 175. 325 bbls. at $14.40 at per commercial As may of goods listed $100, which have been sold 3% in for cash be is the cash 176. 16 6% Note. in 10 on 30 and 60 5? 6? ? 4 may a were time, with mos. dys. be on cash discount sold at 40 a further this basis, the On tion time, in addi- 30 and also. 10, discount dys. price $52.38. What and at 50, 10, and ? commonly are the would or transactions there Thu9 ? Discount. discount a 35% 10% trade to The 50. 10? discount 5% $5.20, discount Cash 60 and gross, $1.35, discount counts. dis- goods make doz., discount per count dis- a $60. on 170. $1.36 at more do not certain a or and 10 the termed, the above-mentioned sellingprice of gross it is two are is the What of sold are off,as there $100, the 10 on 40 is not 10, the 40 case $100 at price,or goods discount trade a Sometimes at In bring $54. 40% there from of catalogue price Thus, if goods listed percentage. of Discount. doz. price of articles at off for spot cash ? Cash a dys. or is 30 flexible dys. $.85 term. per It may doz., discount mean payment 40 and 10, immediately, 99 DISCOUNT. 4 6 177. cash for gross $35 at and drafts, as maturity. Nominally, become and in due two the before bears debt any E. invoice species or bill of Atwood, : 30 dys. 2% bear 10; and 2% after interest interest Boston, 10 dys. discount of goods after it has are illustrated : Conn., Mar. 30, 1891. Mass. To Terms and payable. last-mentioned following stated, always Norwich, F. 60 ? Notes The discount gross, per CHASE MFG. CO., Dr. PART CONTRACTIONS III. AND " EXPEDIENTS. -odi^oo- BY ADDITION in Suppose, 24, and easily This the next is much 27. and 4 tens by 1, The the the thus is smaller 3 8 51299 1 1 5"47 12 8 7"9 7^'0 5 7"0 9"4 5 7 2"5"4^"3 5 6 3 of the of time 34. and then 3, 7, increase the operation. one the form to can we group can digits in the grouping, easy united expressed sums idea, first disregard the get the add smaller ; 8" these get expressed by 3, 12, 10, 6, figures (9, 11, 7, 6; small the latter the take Now etc.). to numbers the columns, instead of so group and figure numbers, numbers columns those add the in columns. 5" 16 9 2 4 9 6 3 After Never one can single digits, he niaterially by make added. we same 7, 24 to by practically digits, and as 0 6 24 to 3 4, reached 24, making to add be the at example rpo 0 2* 9^*2"1' 2 24, and 10 to 7 beyond 14 an be and 3 are add it would next to added has sum figures : 11 9"3 3 4 figures;our and 10 being mentally 51051471141156 4 be adding following columns digits to than digit 14, add into of 7 into easier If the and 3 column a two combine 7 to by adding GROUPING. skip simple a digit add readily can increase and smoothly his practising grouping. combinations and return that to it can speed very At be by first easily afterwards, 101 bijt 102 clean everything up the process, to into a Most and the three much grouping. and of three should be put digitsenough to little A 10. over sum used becoming After rapid figuringgive directions on columns do THEM. along. go the make at practical. Unless can you USE TO practice produce astonishingresults. will books as HOW general rule to group day every he AND NUMBERS, better The ^o be work columns be may a method added of indication following general at not two very lightningcalculator,*' " the by adding is This, however, once. happens one for is once singlecolumn the of given two way for the sake completeness: 161 The 2190 process 42 + 57 = 50 99, + If while So, if carry. 15486 91805 43 45172 46 47114 82878 653103 to 46 38 we 161. columns to necessary how out perform a first,and for any column, the process rupted, be inter- go back to the there were many get a required,and addition still further must we unnecessary 61 full amount 653103 add work, of we each as difierent a alreadygiven. reason the the second of the is ADDITION. COLUMN find Thus: combinations. by = adding long column 83930 31135 LONG it becomes 91799 98214 shortened 149, + 12 = IN CARRYING 65570 be may may to a test footing addition back for To avoid this express the go column, partialamounts. vious pre- and then \ 104 We add may the remainder number, the remainder third number, and USE 1 with the Also, on. THEM. digitsof the second with the digitsof the obtained thus so TO HOW. AND NUMBERS, may omit Drop the we 9*8 as we go along; thus, 12, 19 1 (6 + 6=12,+7 drop 4 + 4 5, 13, 24-6(l 12, 13, 23-5 10, 12, (6+ 6 5 + 19. = 9 t)f 19, also 9). = 13,+ ll = 5, + 8 = 12, etc.). = 2+4 24. = = 6). 15-6 9, 8, 14 5 - 12,19, 9-0(5 + 7 1 + 12, 15 7 12,+ = 8 19. = Drop the 9 of 19. 9). = 6 10, 16, 21-3 5 8 9, 6,12-3 number Any Let r be may resolved 9's + be X 9's + being made 9'8 X up into 9*s + ^"^" ^" made r + 9's X up of 9's + 9's,each that the 9 + r^ Xr first one 4 3 = 3 2316656167293 . = have we of of 9's. Product 12 remainder except the term 543639 = number remainder. a a multiplyingout, 4261387 rV other Any resolved rV ; for + r' der. certain remain- (read r prime) represent this remainder. The product of the two num- rV 9's + a Let r' r 9's + r' = 9's + represent this remainder. may Product into 3. The product product obtained To = 9's + . 3. The . inference is correct. to division,we apply the principle that a dividend quotientX divisor = 3 12 . have + only to remainder. ber remem- is OUT CASTING 105 NINES. THE 3075 and 9's is the the Casting out best possibletest for multiplication division. Application of the 38" = 54872 38 = 9's + Application op the 9's = Involution. to 8. 9's + 2. 9's 2^ 8. = Evolution. to 176413(420.015+ 16 420015 82) 164 = 9's + 164 3 3 8400iyi30000 84001 9-0 399775 = 9's + 4 840025)4599900 4200125 399775 4 176413 = 9'8 + 106 Perform the Add 1-13; 179. Add 14-25 180. Add 1-25 181. Add the what by adding Table Practice 178. of USE TO THEM. and following problems, From record HOW the test results by the 9*8. castingout No. a-",f-j,k-o, p-t, a-A, i-q, whole there r-z. Table Practice of is to each carry instead downward u-z. n-z, ; a-m, ; 53.) (Page 2. of No. 3. the Test time. and upward, Keep also the result by casting the 9's. out Multiply Or-jin 182. No. 2 by the same same same 186. in a-h 187. in a-o eight first lines of Table each line from 9 to 16 by i-7n in line each from 17 to hj 25 p-um line. a-j in each the of eight lines by first h-m in line. Divide a-h in Divide a-o each line line What is the square 189. What is the cube 190. What is the fourth 191. What is the fifth power 192. Extract 534533475567. by i-^m in the from 17 to 25 by jh-u in the places. 188. the 9 to 16 from places. in each line, to 2 decimal same the of line. same line, to 3 decimal same each line. Divide 185. the in the Multiply 184. the A-m Multiply 183. of AND NUMBERS, of 61341? of 627? square of of power root of 4585? 243 47? of 4231406? of ? of of 65? 828051; 621 of ? 227834749; 193. Extract 194. Extract 407458 ; of 23140625 the of sixth and To " denominators the WHOSE 1. A for the numerator the denominator the for difference the ; product is the What also the iandi? iandi? ^^andl? is the product 99 " and 6749 60741 ; sum. for of ^ij-and^? 99 ? 674900 6749 denominators sum SUBTRACTION. BY of 6749 the of the difference ^andi? MULTIPLY of denominator. iandi? TO A"- = the between for the and sum A + iandi? What of NUMERATORS A- denominators numerator 196. A- i = subtract, take the place, ^. the multiply decimal 1 to root, FRACTIONS 1 Add places, of 75567287960. TWO J decimal 2 root, to ARE Add numbers. three same 3259210133. ; ADD of the root fourth the Extract 195. TO the cube 107 MULTIPLICATION. AND ADDITION 668151= = = x 6749 1 X 6749 100 99x6749 60741 668151 By annexing 6 figuresless two than O's and is subtracting,we required by the do the work ordinary method. with 108 AND NUMBERS, The O's need well just as not thus be HOW TO The written. USE THEM. subtraction be may done : 6749 6749 668151 To multiply by 999, multiply by multiplicand ; twice to the 1000 subtract and multiply by 98, multiply by and 100 tract sub- multiplicand. Multiply 197. 3216 by 99. 3007 by 999. 32167 5648 by 98. 4067 by 997. 6750 WHEN THE MULTIPLIER EITHER 2467 7345 16 31 14802 OF We one HAS WHICH let can of the the by TWO IS by 998. 9994. DIGITS, 1. multiplicand stand partialproducts ; thus, 7345 7345 2467 2467 22035 14802 22035 39472 227695 39472 227695 198. the Multiply 57 13. by 287 142 by 16. 2612 307 by 19. 479 1286 540 by by 41. 71. by 31. by by 140. 107. 1642 by 112. 3756 by 321. 1635 by 13^. 1503 by 610. 329 2287 by 109. 1641 by 107^. by 301^. for THE THE 67436 In 11 67436 the ^^^" ^^ ^^" Then do this result the multiplicand for tens come thousands, and well write and of the tens product. as have partial products, we and and just can these units hundreds 67436 LOCK-STITCH. adding add 109 LOCK-STITCH. on, so mentally without as and to the hundreds, pairs. in we go along, the down putting We partialproducts. 27 X 11 X there 11 737. = we may of the to put the of sum being 13, the digitis 3 ; 6 to 7. raise we intermediate the 47 by 11. 2284 by 11. 15641 41 by 11. 1365 by 11. 141089 by 11. 97 by 11. 2289 by 11. 365414 by 11. the of each add the be must X 14 carry and that = the digit of the left-hand 605696. 4 X 4 = 16 29 1 + 4x6 + 4 = 2 + 4x2 + 6 = 16 1 + 4x3 + 2 = 16 1 + 4x4 + 3 = 2 + 4= next 20 6 is 1, the units lower digitof of the the tens multipliedby 11. first of which multiplicandby the digit of multiplier,and to digits,the two by Principle. Lock-Stitch the multiplierhas multiply remembering 43264 merely Multiply Extension When 7 6 + 1 to carry, being 199. have them. digitsbetween 67 We 297. = order, tiplicand the mul- multiplier. 110 When we multiply each the multiplicand the be must first of which multiplicand by the and to carry, THEM. digits,the double add USE TO two digitof multiplierand order, remembering the HOW multiplierhas the may of AND NUMBERS, digitof the the the multiplied by 2, units lower next left-hand that* the is digit of tens of the multiplier. 67431 X 26 1753206. = 1= 6x 6x3+ 2 = 20 2 + 6x4+ 6 = 32 3 + 6x7+ 8 = 53 5 + 6x6+14 = 55 = 17 5 + 2341 654 by by 13. 82403759 Total units of the AND 19 teens of the X annex units. a 26. 28. by NUMBERS 24. by 23. BETWEEN INCLUSIVE. (15 + 3) + (3 X 5) mentally, add smaller, by the product 10 by 254037925 17. TWO = by 5671543 16. ANY product multiply 3251 by MULTIPLY 11 18067 19. by 314065 To 12 Multiply 200. TO 6 to the cipher,and larger number increase this the result 112 ' AND NUMBERS, yds. @ 44 yds. @ 33 $1.75? $1.62^ ? HOW $88. 11. = "77! = $33. 4.13 yds. @ $1.80 ? = = 1.80= yds. at $1.33 ? $133. $1.34? 1.75. @ $1.00. " i. " f 1.62^. @ cost $2. " tVo^2. = " 1.80. = cost of cost i of 100 [email protected] 1yd. 49 yds. of 100 " i = 99,75= yds. @ " " " = 33.25 50 J. " $88.20= 75 " = $98. $90. Or, $2. = = $88.20 THEM. cost == $53.63 9.80 USE cost" = 16.50 49 TO of 100 75 " yds. @ $1.80. " 1.80. " 1.80. $1.33. [email protected] $1.33. yds. " 1.33. 2) $134. 67. yds. @ 55 $1.38? $69. 6.90 $75.90 Multiply 202. 165 by 33^. 168 by 25. 175 by 54. 186 by 55. 218 by 150. 16| by 15^. 76i by 75. 347 by 180. 86207 281 by 45. 674 by 112^. 162 by 225. 164 by 37f 1567 169 by 66| (= i^). by by 95. 75. MIXED MULTIPLYING NUMBERS MIXED MULTIPLY TO is the cost 'of If 17^ 3f of tens and The fraction of l + of 17 + , products taken to the I of 17 nearest 11 = 17^ ? cents numbers consisting + be may 3x17. omitted, and other the cent. 17 = 86 ; i of 85 11). = 1 Sxi = 3x17 =51 is the cost @ two 3x^ (5 X 63 What THE shall have fraction a would we units, we I of^ cloth Sf yds. of multiply as we TO UNIT. NEAREST What 113 NUMBEKS. of 13 cents. lbs. 11 oz. 10 (3x13 = 39; 3 (4x11 = 44; of sugar iof89 @ = T^of44= 4f cents ? 10). 3). 52 65 cents. Multiply 12| by 6^. 12^ 11 6 72 89 6 X 5J. f product, which for this = But is the fractions dropped, will by calling6 X 6. -J^, be both being large, considerable. We make their up 114 AND NUMBERS, Multiply 203. to TO HOW the nearest MULTIPLICATION 227374 MULTIPLES. BY in 14 147 THEM. unit 16241 113687 USE the 7, we twice product by 7. 7, then product by 14. by multiplierbeing multiply by first this partial product 2. 2387427 should Care make 67.034 come 52.15 mistakes 134068 335170 1005510 3495.82310 product by 2. product by 5. product by 15. the digitsof in to avoid footingthe when Put figure product under figureof its of the each the to order same line,in order products. hand taken be tial par- rightpartial right-hand multiplier. CRISS-CROSS 115 MULTIPLICATION. Multiply 204. 1647 279. by 953 by 30535. 34681 by 27381. 866. by 20371 23146 by 6024. 134437 by 43619. by 48612. 857032 by 360724. 165432 346281 by 56728. 8562371 12406 by 213363. by 314065817 8643. CRISS-CROSS 972648. by MULTIPLICATION. Analysis. 21 units, 3x1. (3x2 43 tens units X units = units. 1 units X tens = tens. J tens X units = tens. tens X tens = hundreds. , * 63 84 (4 tens X 1 4 tens X 2 tens, multiply 21 by 43 hundreds, 903 If the we partialproducts total and 63 Units product tens. These write the at tens of the little the written ordinary method, tens, which find we add we by unitSy so we write may Units by the what total tens may the tens and is to units tens. the the give mentally, and Tens carry, by which by tens the units figure of combine product. there unit's for have we and obtain we only partialmultiplicationof once. Adding product, and only. tens partialproducts we hundreds. a 84 and tens is the units, is product After by units by total units the this process, product. Analyzing by units, with by by we tens express give this multiplicationis complete. practice it is just about partialproducts,and as express easy to do away the final result 116 KIJMBEBS, AKD HOW USE TO THEM. 903 The foregoing process, should and not be as given other The 43x1 Multiply the reason For way. = 67 43; by as indication an performed. for that some is of by 37. 93 by 48. 62 by 51. ' of a general particularproblem purposely made multiplicationcould be done instance, 43x2 = 86, +4 48. Multiply 54 this how digitsare 3216 205. illustration an 86 by 43. 29 by 26. 87 by 54. = 90. Total, 903. small, easier Multiply 121 by 117 MULTIPLICATION. CRISS-CROSS 243. Analysis, 121 3x1 243 3x2 tens 363 3x1 hundred 484 4 tens X 1 242 4 tens X 2 tens 29403 4 tens X 1 hundred 2 hundreds X 1 2 hundreds x 2 tens 2 hundreds X 1 hundred Or, differently arranged. units, 3x1 3x2 { tens tens 4 tens hundred 3x1 I hundreds 2 hundreds 3 xl-3 thousands, 1 X 4 tens X 2 tens 4 tens X 1 hundred 1 thousands ten 1 X 2 hundreds X 2 tens 2 hundreds X 1 hundred 21 121 12 X I X 43 243 24 3x2 = 6 4x1 = 4 10 3x1 = 3 1 tocarrj 2x1=2 4x2 4x1=4 to carry 2x2 = 8 2 4 2x1=2 1 4 9 14 29403 Notice written that but the the work is final result. entirely mental. Nothing is 118 AND mJMBERS, HOW Modifications One who performed has read some employ modifications digits,and are simple,and apparent. to fall back THEM. Cbiss-Ceoss. the the Any fairlyapt one with the method are special cases of shortening of the a after be of USE previous pages understandingly,and how to examples, will readily remember multiply criss-cross. may TO a little However, upon. advantage. the always has use The criss-cross process. practice the one in the figures following with They short-cut the of will are two very usually general principle 120 NUMBERS, When AND Tens the aee HOW TO the same ADD 36 4x3 tens hundreds, 34 10 X 3 tens In all cases + 3 tens like units X for = Units the of the 4 tens multiply the hundreds This 3 tens. X of the units 3 tens and 10 = product this,we higher number next tens and be the tens will 1224 the so THEM. 10. TO 6x3 + TTSE units makes (4x6 = 24) answer. X 12 hundreds. 3 tens= the number of by the Units by tens of the product. completestheprodiLcL 68 7 62. X 6 42. = Annexing 2 X 8 = 16, we have 4216. 4216 To 35 35 1225 210. number a square oj tens number by the 48. 65 by 65. 24 by 26. 79 by 71. 15 by 15. 35 by 35. 83 by 87. 92 by 98. 55224 6, inultiplythe higher number and annex ^ Multiply by 234 next with 25. 42 236 ending Considering annex 4x6. 23 as tens, we multiply 23 by 24, and THE 121 CRISS-CROSS. Multiply 211. 547 by 543. 996 by 995. 872 by 878. 635 by 635. 425 by 425. 659 by 651. 406 by 401 438 by 432. 174 by 176. 745 by 745. 885 by 885. 351 by 359. 723 by 727. 125 by 125. 238 by 232. 375 by 375. Tbe whole *^ principleapplies same is alike number both in numbers mixed to and factors the wben fractions tbe add ^- 3f 15| 240H 12f Multiply 212. 14^ by Uf 12i by 12J. 17| by 17^. 62f by 62|. I^byl8}i. 225iby225i. 36^ by 36f 27^ by 27f When Units the are the ADD TO same . and Tens the 10. " This 67 47 6 tens add we digits. This givesthe rest, 63 43 but a slight modification of the previous principle. 3149 -So is The X 7 + 4 tens the number gives of the hundreds product of cipher occupiesthe units lens* x7 = units of by the 100x7 to the 7 hundreds. = product of the Units answer. units place in being the X less than answer. tens* units 10, a 122 AND NUMBERS, 167 47 16 Considering of hundreds the TO HOW USE tens, the as the answer THEM. add tens have we 4 X So 20. to 16 for 7. twice + 7849 214. Multiply 163 by 43. 186 141 by 61. 59 by 159. 102 118. 65 by 145. 87 by 98 When Digits the of THOSE 44 73 are OTHER the number the Make Factor one OF 74 26. by whose the same, and 10. the digits are 3 X 10 X 4 + 7 X 10 X 4 = 10 = 4 hundreds. ' 7x4 Multiply of the 127. the same multiplicand. 3212 than 102. by by TO ADD 134. by the number the rmraber answer. hundreds of tens Units + 4 hundreds the in in multiplier. by units 10 8x4 = of tens the X X 4 = gives the rest. X hundreds. multiplicand by This 100 one more gives hundreds 4 MULTIPLICATION MULTIPLICATION DECIMALS OF EEVEESE What is the BY THE METHOD. and of 1246.329 product 123 DECIMALS. OF to two decimal Reverse Method. 1.435 places? 1246.329 1246,329 1246.33 1.435 5341 1.435 i 6231645^ 623165 I S 3738987 U 373899 4985316 124633 00 "3) 49853 bO 623 124633 00 " 3739 498532 1246329 CO 00 CM 1788,482115 There multiply places decimal three decimal the usual manner in the If, in order thousandths 6 over 9 thousandths .01. The best we saves only do can four and in figures,and simple Hundredths hundredths in order one, by ; units units hundredths. under the So, if the the left instead of to the each the time with we hundredths remaining digttsof had what right,and ; digitof the tenths it is a by tenths, the of in a do. to units by the that sandths, thou- plier multi- multiplicand, and multiplierreversed, But through go ; tens the of call the it is. hundredths write of to the error particularcase produces hundredths by hundredths, is to way of drop an multiplieras this in find out to of have we be would off the error an should we sort the retain preliminary calculation,though very If this be two don't we strike would multiplicand,there the multiplicand 1246.33, this obtained. product of should figures,we save that we off six get the to more multiplier,there the of the in to four get we point to order is, in factor, if each have shall we want, in places That product. figures we want. 5 being in 1788.48 1788.48355 the is,4, 3, 5 to multiplying,commence multiplicand directlyover the 124 of one lower the each of the two decimal What THEM. using, disregarding the with multiplicand, is to carry partialproduct partialproducts the next from be will exception lower ination, denom- be will the and hundredths, in obtained thus the to correct places. is the decimal there USE TO then are the of in what sum HOW multiplier we denominations adding of AND NUMBERS, product and of .23047651 35.12043 three to places? .23047651 2153 6914 1152 23 6 8.094 Write the units of the multiplicand,and To be the rest the of sure be well to one really wanted, But if look we out of the last and then sharplyfor 216. the product, it further place omit the what there usually be exactlyright without of the multiplierreversed. figure of decimal one go thousandths multiplierunder than is to carry, this the last figure obtained. last taking times some- may shall we precaution. Multiply 12.491 by .36 to two .17653 by 122.5 60.31654224 by 340.6582716354 to decimal one 521.2 by decimal to four places. place. decimal 3.7012407853612 places. to three .1243 by 3.256 to the nearest thousandth. 1243. by 325.6 to the nearest thousand. 33546.86 by 5728.6 to the nearest million. decimal " places. CONTRACTED CONTRACTED The DIVISION DECIMALS. OF be principle may same 125 DECIMALS. OF DIVISION applied the to division of decimals. Divide will not go in quotientwill the As soon be found is be tens, and bringingdown what last dropped. may sometimes number two there of each To be be 1764 well 3681 the delay of divisor,we the last digitof the the decimal 3681) 1764 (.48 29 figureof quotient,it until contraction in the divisor. 147 stop In quotient remaining to two figureof multiplying,we of the digit multiplication of the last to the the number by drop partialdivision. sure the first in the number from quotient. five. figures, quotientremaining to the figuresof the is to carry figuresot less than Divide than ; so total number the dividend, and from the divisor before add of the places. will be in the 18, but will go in 189 less one decimal figuresthere many the number as to three 32.467 by first find how We 32 1897.5431 places. to be found the is 126 AND NUMBERS, HOW TO USE THEM. Divide 217. by 286 176.241 by METHODS places. decimal four 3.7542134 by 132674 Reverse to decimal two to 378.2 by 41 13.47 9321765 places. six decimal to three to decimal CALCULATING OF places. places INTEREST. multiplicationis especiallyapplicable to figuring interest. is the What $1642.27 of amount for 2 yrs. 3 16 mos. dys. at6%? It is the just as multiplierin and good a saved to. write easy order, reverse figures are many by multiplying that Note manner. denomination same this in f is as 7. to Dec. of 1868.35 is the What at 6% interest $1642.27 on from July 26 19 ? July 26 to Dec. 19 = 4 dys. 23 mos. 1642.27 It is easy |32 of the to see where multiplier the first should be figure placed. 3285 Unless want we the result to mills, 492 unita 55(|-i + the of the i) two 39.14 of multiplier go under dredths hun- multiplicand; hundredths, placesbeyond. 128 There the be AND NUMBERS, are great majority of is the amount $1246.11 for 4 $245.81 for 63 dys. at 8% ? for 2 yrs. 3 1 yr. 6 for 93 for 8 is the interest $3,214 .16 3.37 .01 of by on 1.18 for 22 ^ .1 of int. for 20 = interest for 22 = = i for 90 dys. dys. any for 90 time the time is not practicalexcept into dys. = interest for 20 dys. = interest for dys. 2 dys.? of int. for 60 for int. for 3 dys. dys. dividingup above. dys. for 60 interest = dys. dys.? interest for 60 interest for 60 dys. = = ? dys. at 6%? interest for 63 interest very for 63 interest = ? Parts. Aliquot = amount 1.607 dys. at ^% $321.44 ? ? ^ of .1 of int. for 60 dys. same $3,214 15 dys. at 6% =^ = .11 4 mos. principal amount $1.07 The = same $4.82 cannot ? dys. at 6% 14 mos. dys. at 5% mos. Interest the dys. at 6% 3 mos. ? $782.53 On in of dys. at 7% $2347.62 the just illustrated method for 33 $1242.57 for On figuringinterest,but for $642.67 $2644.32 What THEM. upon. What 218. the cases USE TO specialrules many improved HOW = int. for 30 dys. dys. be may convenient in a found in this way periods,but the few simple cases by method like the STATES UNITED AND All specialinterest rules are formula, Interest principalX = there is no shortening the of way already been shown. FIND THE TO founded rcUe United as States of the value ; further than VALUE and has OF MONEY. ENGLISH The general of years any STATES the upon number X process UNITED 129 MONEYS. ENGLISH be taken pound sterlingmay ent at differ- $4.8665, though the proportionvaries somewhat times. first reduce We of As 20s. pound, a shillings, pence, and then It, 28. =.11. = and multiply So we farthingsto this number divide' the number the 4.8665. by of mal deci- shillings if there pound, callingthe odd shilling, there be one, .05. Pence reduce to farthings. As are we 960 farthingsin a pound, 1 farthing .001 of a pound, and a by 2 for tenths of a = little pound, add we exactly equal to .0121 of a farthings are 36 farthings to .0371. So, if there are 12 farthings sandths. thou2 extra extra thousandth; if 36 farthings, 12 over. and one As to a there 4 are and shilling, pence farthingsto a and penny, farthingstogethermake 12 not farthings. What is the United "2.75 States ="2 49 $13.62 = value 15s. llfrf. of "2 15s. llfrf.? pence over 47 130 AND NUMBERS, What is the value HOW $13.62 of USE TO in 48665)136200(2.799 THEM. English ? money pounds. 9733 .75/. 3887 M9L 3406 = = 15 shillings. 47 far. 11|";. = "481 438 43 219. What is the "13 175. 6^. "22 lis. 4cf. value Ans. in United States "47 ? "1127 ? "2 of money 18s. Hid. 3s. llfd lbs. ? 7c?. ? , "241 "56 220. Os. 19s. What is ll|c?. S^d. the "462 17s. Sid. "286 12s. 4d ? ? value in English money ? ? of $46.87? $3342.76? $3,7.41? $216.34? $207.29? $286.53? $706.51? $361.42? $541.89? I . PROBLEMS. MISCELLANEOUS -^:*?" 221. Complete by casting out the following table, proving tlie 9's the divisions : MASSACHUSETTS. 131 132 NUMBERS, SIMPLE AND HOW INTEREST ONE DOLLAR. TO USE THEM. TABLE. SIMPLE SIMPLE IKTEBE3T INTEREST ONE DOLLAR. TABLES. TABLE. 133 134 NUMBERS, AND SIMPLE 360 Days to HOW TO INTEREST Year. 8 Months USE THEM. TABLE. Days. 4%. $9000 136 NUMBEES, AHD COMPOUND HOW TO USE TABLE. INTEREST Amoubt of THEM. $1.00. INTEREST COMPOUND TABLES. INTEREST COMPOUND Amount of TABLE. $1.00. 137 I' . 138 Complete 222. The AND NUMBERS, calculations HOW the interest 12) .04 = $1 on 1 third previouspages. : for 1 yr. at interest = THEM. the six follows as interest and .003333 USE tables oh be made may TO for 1 mo. 1 3333 .006666 2 3333 " " " " 2mos. = Jl .010000 0 " 3 = etc. 30).003333^ and .000111 1 ninth Ill 1 .000222 2 .000333 interest for 1 = " " 2dys. " " 3 = '3 dy. = " etc. The different last amount additions, if the ones in depended be may having amounts been each upon, found by series is except successive right,the for the last decimal figure. Proof. .055 = interest on $1 for 11 mos. at 6%. 2 3).110 .036667 .005 = = interest interest on on $1 $1 for 11 for 30 mos. at 4%. dys. at 6%. 3) .01 .003333 = interest on $1 for 30 vious pre- dys. at 4 % . INTEREST If we now last decimal the over go work verifythe figuresin and be shall place,we 139 TABLES. the results obtained sure the are absolutely accurate. 3)40. 13.33^ 26.66| interest = lll= " $1000 '* . 26.77 and 7 ninths 11 for 8 $1000 on 4%. Idy. "' interest = at mos. for 8 " , 1 mos. dy. 1 etc. 8 X 2677^ and 21422 = 88 2 ninths = $8000 int. on for 8 mos. 1 dy. 8 mos. 2 dys. 8 mos. 3 dys. 8 " $8000 " " $8000 " " $8000 " 8 mos. 4 dys. " $8000 " 8 mos. 5 dys. = = = 21777 etc. Proof. .045 = interest on $1 for 9 mos. at 6%. 1000. 3) 46. " " " " = $1000 for 9 mos. at 6%. 9 mos. at 4%. 15 30. = $1000 " 140 AND NUMBERS, Verify ^/y ^J? 73 2 the figures in the I.I30137 = decimal last laOia? = THEM. USE TO HOW place before. as cents day = accurate int. on $1000 for 1 = accurate int. on $1000 for 151 at 4|% 13.0137 151 1965.0687 cente dys. at 4f %, 13.0137 1978.0824." dys. at 4|% " " $1000 " 152 " " $1000 "' 153dy8.at4f%. = 13.0137 *' 1991.0961 = etc. 90 ^ 73 % 1.03 = of amount $1 for 1 yr. at 3% compound interest. 309 1.0609 " $1 "2yrs. at3% " $1 "3yrs. at3% " $1 " " $1 " = 31827 1.092727 = 327818 1.1255088 = 4 yrs. at 3% yrs. at 3% 337653 1.1592741 = 5 4" (I INTEREST 141 TABLES. / 1.036 = amount of $1 for 1 yr. = amount of $1 for 2 yrs. = 1.035 = amount of $1 for 3 yrs. = amount of $1 for 4 at 1.035 6175 3105 1035 1.071225 5301 3i^%. at reversed. 10712250 321368 53561 1.1087179 53 1 11087179 332615 55436 1.1475230 yrs. etc. Take Proof. 2 year, the first etc., also we make obtained similar 3 years, the bears the from amount a from manner. to the first the the form column first, and the second, below. series. the term preceding. next one of amounts series, each a steadily increasing the The column. etc., years, ratio same get 1^% third second These $1 for 1 which of Subtracting the from numbers The second from the third, should column second, is in a 142 AND NUMBERS, HOW THEM, USE TO 15225 22 Q 15920 ifl ^l 5 16159 ^l 4 16402 ^l 2 16647 16898 ;^V o^5 2 17151 f.^ 4 17408 17669 ;^{ i^i 17934 ;^^ 5 18204 ;;" 2 18476 ^'^ 5 15686 1 6 4 4 18753 5 QQo 19320 i%i ^T 6 19611 o^i 2 19035 3 293 19904 Suppose had we of the twelfth year, mistake a arid had We 1.195628. year made in the obtained should then fifth decimal the as place for amount that had have 17408 271 17679 245 17924 280 18204 and^ the A would error slighterror by in have the this process. originalfiguringshould been apparent. last decimal To be make gone place might .of the last sure far as over not as be discovered digit,the necessary for the purpose. The use of interest examples What at 4% be illustrated by the following : is the ? tables may amount of $642.37 for 2 yrs. 6 mos. 14 dys., 144 AND NUMBERS, What is the compounded 9^ years 1^% at TO $1324.83 of amount HOW USE for 9^ THEM. years, 3%, at semi-annually? 3% semi-annually is equivalent to 19 annually. 1.32695 384231 132695 39809 2654 530 106 4 1757.98 Compute 223. the What followingfrom is the $756.42 $29.60 for 4 yrs. 12 the completed for 1 yr. 3 $190.99 for 8 for 2 yrs. 4 dys.,at 4% bank discount 30 days' note for $480.12, 90 days' note for $2500., 60 days' note for $542.84, What for is the interest $856.41 for 8 mos. dys.,at 7% 11 is the note ? dys.,at 10% mos. on a 6% at at 7% at ? ? 10% at ? 8% ? on dys.,at 4% ? ? dys.,at 8% $3486.22, 17 ? ? 25 mos. What 4 months' dys.,at 4% 16 mos. 29 mos. tables on dys.,at 4% for 93 $564.80 $1560. 225. interest for 3 yrs. 4 $2845.75 224. est inter- ? ? : years at INTEREST for 8 $1652.50 I mos. 3 $582.40 for 8 mos. 15 for 3 yrs., at $266.50 for 18 yrs., at for 4 yrs. at for 3 yrs. at ? annually, wljat is the ? 5% ? 7% ? 4^% ? 7 mos., $761.48 for 11 $876.29 for 5 yrs. 2 mos., $342.17 for yrs. 2 yrs. 3 at 14 mos. 5% 6% at ? ? dys.,at 2^% . semi-annually,what being compounded Interest 227. 3% $762.41 $92.56 is the of amount for 7 yrs. 6 mos., $2284.39 $854.17 $84.40 for 5 yrs., at $100. for 2 yrs. for 20 (First find,by take for 10 to the the 3 of the for a 3% ? ? at 5% 13 mos. 3% yrs., at at ? dys.,at 6% ? table, the for amount principaland new ? 10 years, find the amount more.) $322.50 his use this amount years value 7% for 3 yrs. 6 mos., $1231.84 form dys.,at 4% being compounded $1241.62 As ? of mount then dys.,at 4% for 8 ? dys.,at 4% 20 mos. $342.17 Interest 226. 146 TABLES. for 35 best form in the own yrs. 8 for interest solution tables As dys. at 5% tables, and of interest conclusions. compiling of 19 mos. an as to their problems, the exercise is invaluable. ? in the tical prac- reader use of can ures, fig- 146 AND NUMBERS, the Complete 228. 1885) HOW TO USE following table THEM. (Mass. Classified Details AND Capital Capital op (2,366 Establishments). Percent- Invested. ages. Amounts. $896,310 Buildingsand fixtures 3,256,603 Machinery 4,613,370 and Implements tools 696,281 capital 20,354,644 Supplied by partners stockholders or . payable, accounts long time, on 1,215,355 . 3,380,858 etc. Total 100.00 Classified and Land, buildings, Summaey. fixtures . Machinery, implements, Cash and . . . . . . $4,152,913 tools 5,209,651 capital Credit 20.354,644 capital 4.596,213 Aggregate A 229. payable bank each 100.00 for note in 30 bearing the be there A date same is dated second is the 20, 1891, and Dec. for the note payable of each on in same What month. 1 notes, and above amount when made and is the must paid ? The followingis interest Rule. $12,000 days. discount 230. as of SHOES. Land Bills Census : BOOTS Cash State given general rule as a as many for calculating : Multiply are often the days, and principalby then divide as follows one-hundredths : 4 5 6 7 8 9 10 12 90 72 60 52 45 40 36 30 Percent, Divide by how Show 147 PROBLEMS. MISCELLANEOUS is deduced rule the the general formula yr. 9 mos. 6 yrs. 7 mos. 27 dys. was 12 dys. was from for interest. The 231. $29.68. interest What The 232. $93.15. interest What The 234. dys. was for $584 2 the rate ? interest What $295.89. 1 ? rate on was The 233. the was for $240 on $340.10 on the was interest for 8 yrs. 8 mos. ? rate on $682.41 on $1421.95 7% at was $371.80. What $29.15. What the time ? was The 235. interest the time was A 236. rate A 3 for 1 yr. what 239. more Method discount what dys. long had days' note 60 a for the $128.13 The the amount What Why on how the at bank yielded the note to run? $8.01. The was the amount? was 14 was discounted being 6%, rate pays mos. rate of the principalwill use of of interest a certain sum charged being loan? amount to in 6 $684.67 mos. 12 ? Merchants' is the favorable ? $2250 borrower dys.,at 10% 240. for bank being 6%, 238. 8%, The The 237. was ? note $2237.62. 6% at debtor the to Illustrate $1000 bears interest is the amount due by the from Dec. Method 1 ? than the of Partial United following problem Aug. 1. $500 is paid Payments States : A Oct. Court debt 1. What of 148 AND NUMBERS, If 4 241. men of 10 hours do men The the work same of number 42 days, 7 of do it in 4 times required by analysis. the put how men do the work can as 1 in smaller men days would 7 be in 7 number of in the in many days. days, 1 to require less in certain work a go time would of days through this than'^ the numerator, (If4 man ^ the number isn't necessary men number the ^^as men will answer it in do days, 7 that do can given a It man. days, the many can many Notice each 9 hours of in 42 ? If 4 days. of work amount days many THEM. USE TO certain a each, in how question being terms men do can HOW so men, the larger in denominator.) hours 10 If, working 9 hours days, working number certain a it will take day a it takes day, a -^ as of days. many 2 ^^i^ ? = = 26|days. 3 If 9 boxes 242. at the each rate same weighing 90 each weighing lbs.,cost $61.20, 80 will be the cost what pound, per of 14 boxes, lbs. ? of flour cost barrels If 56 243. of soap, $317.52, what bbls. will 126 cost? If it costs 244. and 18 high, for ton, as high, with feet will it cost to heat the former, and many to If a marks $.193, and 36 room will of of 40 room long, 25 time, with coffee oz. $.238? ft. ft. wide, long, 22 per ton, how ft. wide, and coal much 17^ costing$4.75 coal heat dimensions 7 lbs. 13 mark ft. latter that the their a costing $6.50 coal the pound a to heat length that proportionalto 245. a same assuming the $48 gives out -^ required for the much as two ft. per heat rooms is ? costs cost, half, how a franc and a franc being equivalent a If 12 246. in dollars and At 247. When minute 60 12 be next In 60 the minutes gains 11 minutes of a being 39.37 watch are inches ? together. together? the hour-hand spaces, metre hands is the cost francs, what 28 cost 3^ yards, a o'clock, the they Solution, of cents 12 will of cloth metres 149 PROBLEMS. MISCELLANEOUS 0 minute-hand one How spaces. The space. 12 goes five- minute-hand it take long will in it to gain spaces ? 60x12 '" n.s 65^ = " " " mm. 1 o'clock An8, At 248. The 249. numbers of sum ratio three other each to hands the of apart will they far How apart. same o'clock 6 be watch a 7.45 at is numbers that 3, 4, and 30 are ? 161. minutes 8.15 at ? bear They What 7 do. min. 5^ the the are ? 250. i + i + i of 251. On a is 165. number the Centigrade thermometer, the number is the What ? of freezing-point , is marked water these 68. What hang should What should be the tigraderegisters10 252. What 253. What boiling-point100. marked points are thermometers two the 0, and side be and by side. the reading of below zero is the square is 32 the square of heit, the Fahren- respectively. The 212 The reading On Fahrenheit of the the Fahrenheit ters regis- Centigrade? when the Cen? ? 17^^ ? root of 17-j^ to three decimal places? 254. Perform the operations indicated expressions,results being required to vTr, three in the decimal following places: ^/3^ 14*, 5^ (16|)*,.653', .OOOOli 150 NUMBEBS, AND 266. Simplify the 266. What and ? -J- 257. What 268. How subtracted 260. Shopworn What sold at advance of A and bbls. 150 are ? f 4|^ be fractional, may 35% when been an Goods If mos. . For cost, bring they $500 be paid what had been an at and once, paid ? bbls. at $4.80 bbls.-at he sells 100 per $2.90, ance he sell the bal- price must What 40%. be 1000 buys sell at to down if profitof 8% average marked received the balance flour dealer marked below ? should $3.40. at 35% at portion being damaged, to make 263. sold have in 4 wholesale A barrel. cost would in 2 mos., 262. and , denominators.) common multiple of 3^, ^, and common goods, is due $2240 $1000 12|^,^, f part of 12f is 11| ? $164.16. 261. having of ? 16f What divisor common times, whole many 259. an fractions to THEM. ^ greatest is the least from USE TO fraction is the (Change HOW ? cent per 65% of advance of the on this profitdoes leave ? by sellingcloth 264. If 25%, what of A 265. 6%. at 266. 4 months' What I owe are note the $1.40 per sellingprice be for $1000 yard there in order is to is a loss gain 25% discounted at a ? bank proceeds? $1000 due to settle the pay the must at in 4 How mos. much account, interest being allowed should cash at the rate I of 6%? 267. cash A 4%. bill of How goods much amounts cash to $1000. Terms will settle the bill? : 4 mos. or 152 NUMBERS, AND HOW TO USE THEM. List Priee. Price Extensions. Net Price Eztensioni. MISCELLANEOUS 153 PROBLEMS. $201.33 Verify actual of cost price per net dozen ; gross is otherwise, the Example. at kind $5.40 of goods, and per given, find cost and the Less The first article in gross, 60%, 10%, and price price or sellingprice per bill.No. 2538, is listed the 5% off. = 5.40 reversed. == cost of No. 14 $1.85 .36 Net list 171 .1=4 of .1 the and cost 45 "lo ^ When sellingprice .342 .60 Less also the the sellingpriceper singlearticle. 1.00 Less footings. Ascertain yield a profitof 40%. to necessary each the and price extensions the = = 18 ^dd .342 of list pr. .4= .74 12)2.59 $.22 2538 per = sellingpr. " = gross. per " gro. doz.

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