uirt 
 
 PUBLICATIONS OF SOWER, POTTS &. CO., PHILADELPHIA. 
 
 So 
 
 The 
 this ve 
 ins true 
 and a< 
 vancer. 
 the wh 
 adapte 
 to ma 
 school- 
 and thi 
 
 Rz 
 
 These e 
 and 
 
 Fe 
 Fe 
 
 IN MEMORIAIA 
 George Davidson 
 
 EDUCATION DEPT. 
 
 Professor of Geography 
 University of California 
 
 logical 
 
 I 
 
 
 The uniform testimony of teachers who have introduced these grammars is, that they have been 
 most agreeably surprised at their effects upon pupils. They are easy to understand by the 
 youngest pupil, and the lessons before dreaded become a delight to teacher and pupils. Ex- 
 traordinary care has been taken in grading every lesson, modeling rules and definitions after 
 a definite and uniform plan, and making every word and seatence an example of grammati- 
 cal accuracy. They only need a trial to supersede all others. 
 
PUBLICATIONS OF SOWER, POTTS & CO., PHILADELPHIA. 
 
 THE 
 
 Normal Series of Mathematics. 
 
 BY EDWARD BROOKS, A.M., 
 
 PRINCIPAL AND PROFESSOR OF MATHEMATICS IN PENNSYLVANIA STATE N SCHOOL 
 
 AT M1LLERSVILLE. 
 
 This series, still new and fresh, has had an extraordinary success, an ?d in 
 
 very many of the best Normal Schools, Academies, Seminaries and Pu. Is 
 
 in the country. Wherever the works are known they receive the highe* 
 dations of leading professors and teachers. 
 
 PRICE 
 
 Brooks's Normal Primary Arithmetic . . 25 cts. 
 
 The Primary contains Mental and Written Exercises for very young pupils. Its treatment is 
 very plain, easy and progressive. 
 
 PRICE 
 
 Brooks's Normal Elementary Arithmetic 50 cts. 
 
 The Elementary will furnish a practical business education in a shorter time and with less labor 
 than any other, and is emphatically the work for those pupils who must be qualified for com- 
 mon business in one or two terms. 
 
 PRICE 
 
 Brooks's Normal Mental Arithmetic ... 38 cts. 
 
 The Mental is a philosophical and comprehensive treatise upon the Analysis of Numbers. It is 
 easily mastered by young pupils, and those who accomplish it are ready to grapple with the 
 most difficult problems. It makes logical thinkers on all subjects. All who use it say they 
 cannot be induced to dispense with it. 
 
 PRICE 
 
 Brooks's Normal Written Arithmetic . . 95 cts. 
 
 The Written is a comprehensive and practical work, full of business applications. It is admirably 
 arranged, and progresses step by step until pupils master it almost without conscious effort. 
 Its treatment is novel in many respects and very interesting, and as It is very successful in 
 the school-room, it is very popular among the best educators. 
 
 PRICE 
 
 Brooks's Key to Mental Arithmetic .... 8 .38 
 Brooks's Key to Elementary Arithmetic . .50 
 Brooks's Key to Written Arithmetic ... 1.00 
 
 In addition to the matter usually contained in Keys, the first and last of these contain many valu- 
 able suggestions on the best methods of teaching Arithmetic. 
 

 
 
 I! 
 
 
 1 
 u 
 
 r 
 
 
 
 
 si 
 
 
 t 
 
 -Th 
 
 
 
 
 
 
 7 ■ 
 
 I 
 
V 
 
c?cic(j '<r> 
 
 V 
 
 / tc^ti/ 
 
 £oit ^ 
 
 l 
 
 C^ i 
 
 I 
 
 C 
 
 ^C^^iJ^u Ctt^t 
 
 Wei 
 
 / 
 
/ 
 
 \ Sis* S{i/*f< 
 
 /ist**- 
 
 
 h 
 
 ELEMENTARY ARITHMETIC: 
 
 EMBRACING 
 
 A COURSE OF EASY AND PROGRESSIVE EXERCISES IN 
 ELEMENTARY WRITTEN ARITHMETIC; 
 
 DESIGNED FOR 
 
 PRIMARY SCHOOLS, AND PRIMARY CLASSES IN COMMON SCHOOLS, 
 GRADED SCHOOLS, MODEL SCHOOLS, ETC. 
 
 BY 
 
 EDWARD BROOIvS, A. M. 
 
 FmiNCIPAJ. AND PROFESSOR OP MATHEMATICS IN PENNSYLVANIA STATE NORMAL SCHOOL, AND AUTHOR O* 
 
 NORMAL PRIMARY ARITHMETIC, NORMAL MENTAL ARITHMETIC, NORMAL WRITTEN 
 
 ARITHMETIC, ELEMENTARY GEOMETRY, ETC. 
 
 \ 
 
 \ 1 > v V 
 
 " The highest science is the greatest simplicity." 
 
 PHILADELPHIA: 
 SOWER, BARNES & POTTS, 
 
 530 MARKET ST. and 523 MINOR ST. 
 
 . 
 
Office of the Controllers of Public Schools, ^ 
 
 First District of Pennsylvania. V 
 
 Philadelphia, March 29, 1869. ) 
 
 At a meeting of the Controllers of Public Schools, First District of 
 Pennsylvania, held at the Controllers' Chamber, Tuesday, March 9, 
 186S, rfie following resolution was adopted : — 
 
 Resolved, That the Normal Series of Arithmetics, comprising "Brooks's 
 Nonucii Primary Arithmetic," " Brooks's Normal Elementary Arithme 
 tic." '■ Brooks's Normal Mental Arithmetic," and "Brooks's Normal 
 Writtp-i Arithmetic," by Edward Brooks, Esq., be and the same are 
 hereby nlaced on the list of books to be used in the Public Schools of 
 
 this District. 
 
 From the minutes : 
 
 H. W. Halliwell, Sec'y. 
 
 Entered, according to Act of Congress, in the year 1865, by 
 
 EDWARD BROOKS, 
 
 In the Clerk's Office of the District Court of the United States, in and for the Eastern 
 
 District of Pennsylvania. 
 
 ME*:iS & DUSENBERY, ELECTROTYPERS. SHERMAN & CO., PRINTERS. 
 
 The Board of School Trustees for the State of Maryland, recom- 
 mend Brooks's Normal Arithmetics and Fewsmith's Grammars, for use in 
 all the Public Schools of that state. 
 
 The Board of School Commissioners for the City of Baltimore. 
 hove adopted Brooks's Normal Series and Fewsmith's Grammars, fo! 
 exclusive use in the Public Schools of that city. 
 
- 
 
 
 
 PREFACE. 
 
 The object of this work is to furnish young pupils 
 with an introductory course of Written Arithmetic, 
 Realizing the necessity of such a course in connec- 
 tion with Mental Exercises, the author gave quite a 
 large collection of written exercises in his Primary 
 Mental Arithmetic. So popular was this feature 
 that he was soon urged either to increase the amount 
 of such matter or prepare another work treating 
 exclusively of the elements of Written Arithmetic. 
 Believing that the latter plan would be the more 
 acceptable, and give more completeness to the arith- 
 metical course, the present volume has been pre- 
 pared. 
 
 This work will be found to possess at least five 
 distinguishing features : 1st, Simplicity ; 2d, Gra- 
 dation ; 3d, Practical Character of the Problems ; 
 4th, Variety of Problems ; 5th, Educational Cha- 
 racter. 
 
 Simplicity. — Great care has been taken to make 
 the definitions, explanations, solutions, rules, etc. 
 so simple that the youngest pupils can easily under- 
 stand them. In doing this, however, the scientific 
 character of the subject has not been sacrificed; for 
 it should ever be remembered that the highest science 
 is the greatest simplicity ; and, conversely, the greatest 
 simplicity is the highest science. 
 
 3 
 
 m£89£98 
 
PREFACE. 
 
 Gradation. — The gradation of the work will be 
 found one of its most distinctive and valuable fea- 
 tures. A frequent criticism upon elementary works 
 is their lack of gradation, their sudden transitions 
 from the easy to the difficult, from the simple to the 
 complex. To avoid this fault, great pains have been 
 taken, and, it is believed, with success. For example, 
 see the exercises in Addition, Subtraction, Multi- 
 plication, and Division, where the problems are ar- 
 ranged into classes and cases with respect to their 
 length and difficulty. The same spirit of gradation 
 will be found running through the whole work. 
 
 Practical Problems. — Arithmetics have been 
 criticized for the abstract and unpractical character 
 of their problems. To avoid this error, I have given 
 a large number of practical problems, drawn from 
 the actual events of life. Among these are His- 
 torical, Geographical, and Biographical problems ; 
 problems on the battles of the Revolution; farm- 
 ers', merchants', etc., problems. Such problems will 
 not only add interest to the study of arithmetic, but 
 present much valuable information to the pupil. 
 
 Variety of Problems. — "Variety is the spice of 
 life," in the school-room as well as out of it. It is 
 a great mistake to keep pupils upon Addition for 
 several months, until they have thoroughly mastered 
 it, then upon Subtraction for a corresponding length 
 of time, and so on for the other fundamental opera- 
 tions. The better way is to give them a fair know- 
 ledge of Addition, then take them to Subtraction, 
 and, after they are somewhat familiar with this, 
 give them exercises combining Addition and Sub- 
 traction, and thus through the fundamental rules, 
 
PREFACE. 
 
 leaving each subject before the pupil wearies of 
 it, and returning to it again and again, until it is 
 thoroughly mastered. In this manner tiresome 
 monotony is avoided, and the task of the learner 
 rendered interesting and attractive. This is a fea- 
 ture of the present work which it is believed will 
 commend it to intelligent instructors. 
 
 Educational Character. — This work, like the 
 others of the same series, is characterized by an 
 educational spirit. It is not a mere collection of pro- 
 blems and rules for the training of a pupil to labor 
 like a machine. The spirit of analysis runs through 
 it, making it normal in the broadest sense of the 
 term. Its object is to teach pupils to think as well 
 as to work 'problems, — to develop mind as well as the 
 power of computation. 
 
 Cherishing the hope that it may aid teachers in 
 
 their arduous labors, and become a favorite with the 
 
 little girls and boys of our common schools, I now 
 
 intrust it to the decision of a kind and appreciative 
 
 public. 
 
 EDWARD BROOKS. 
 
 Statb Normal School, 
 
 May 20, 1865. 
 
 1* 
 
SUGGESTIONS TO TEACHERS. 
 
 1. This book is designed to be put into the hands of young 
 pupils soon after they begin the study of Primary Mental 
 Arithmetic. When pupils can add and subtract orally with 
 some facility, they will be prepared for this work. 
 
 2. The Introduction is not designed to be studied by the pupils, 
 but indicates a course of Oral Instruction in the elements of 
 arithmetic ; and it is suggested that these exercises receive the 
 attention which their importance demands. The pupils should 
 have careful drill upon these before beginning the study of a 
 written arithmetic; and such exercises will be found valuable 
 during the entire course, especially upon commencing a new 
 subject. 
 
 3. Pupils should solve the problems upon the slate at their 
 seats, and also be required to work them out upon the black- 
 board, and explain them. In assigning problems at the board, 
 the same problem may be given to the whole class, or each pupil 
 may receive a different problem. Sometimes one method is pre- 
 ferred, and sometimes the other. The object should be thorough- 
 ness and accuracy, and at the same time variety and interest. 
 
 4. In many cases two forms of explanation have been given; 
 one a full logical form, the other an abridged, mechanical one. 
 The object of the first is to present the reasoning process in full ; 
 the object of the second, to give the steps in the method of opera- 
 tion. Though it is important that the pupil should understand 
 the complete logical form, yet for the ordinary recitation, with 
 young pupils, the abbreviated form may be preferred. It will 
 economize time, and better secure that which is mainly aimed at 
 in primary written arithmetic, — facility and accuracy of operation. 
 
 5. Where a pupil has difficulty with a problem owing to its 
 being a little complicated, let the teacher lead him from one step 
 to another, and so on to the end, by a judicious series of questions, 
 leading him to analyze the problem, and thus unfold its com- 
 plexity. This will be much better for the pupil than to pick up 
 his slate and work the problem for him. By attention to these 
 suggestions, and to such other points as will occur to the mind 
 of the intelligent teacher, it is hoped that the progress of the pupil 
 will be rapid and thorough. 
 
 6 
 
INTRODUCTION. 
 
 OEAL EXEECISES. 
 
 The following remarks and oral exercises are designed to illus- 
 trate the manner in which the elementary principles of numbers 
 should be presented to the young pupil. 
 
 LESSON I. 
 
 NAMING NUMBERS. 
 
 Since our first ideas of numbers are derived from visible objects, 
 the child's first lessons in the science of numbers should be given 
 with such objects. These objects may consist of apples, nuts, books, 
 pencils, grains of corn, or any thing the teacher may find con- 
 venient. The German schools for young children are generally 
 supplied with small cubical blocks to be used in the first lessons 
 on numbers. The arithmetical frame is the most convenient for gene- 
 ral practice. 
 
 Counting. — The names of numbers are acquired simultaneously 
 with the idea of numbers. Both of these are given in the process of 
 counting. By counting we do not mean merely speaking the words 
 one, two, three, etc., but that these words should be used in connec- 
 tion with objects, so that the pupil may know what the words mean. 
 I have known pupils who could run off these words very glibly, 
 even as far as fifty, without any definite idea of their meaning. 
 Hence the pupil's first lesson in numbers should be in counting. 
 The method suggested is as follows: — 
 
 The teacher, standing before the class, holding some object, as a 
 book, in his hand, says, "What do I hold in my hand?" Pupils: 
 "A book." Teacher: " How many books?" Pupils: "One book." 
 The teacher, taking up another book, says, "How many books in 
 my hand now?" Pupils: "Two books." And so on, until the 
 
 7 
 
8 INTRODUCTION. 
 
 pupils can number any collection of objects which the teacher holds 
 in his hand. 
 
 After this introduction, he will take the arithmetical frame and con- 
 tinue the exercises upon it. These exercises may be made lively by 
 increasing or diminishing the number by several at the same time. 
 Little counting games, with grains of corn, will also be found very 
 interesting to young pupils. 
 
 Let exercises of this kind be continued until the class can count 
 well. If the pupils can count when they enter school, this exercise 
 need not be continued long. 
 
 LESSON II. 
 
 ADDITION AND SUBTRACTION. 
 
 After the pupils can readily number a collection of objects, — that 
 is, have the idea of numbers and the names of numbers, — they should 
 be taught to unite and separate them. The next thing in order, there- 
 fore, is addition and subtraction. Instruction in these processes 
 should be given in accordance with the following principles. 
 
 1. The first lessons in addition and subtraction should be given with 
 visible objects. This principle is founded upon the law of mental 
 development, and is so evident that it need not be urged. Indeed, 
 if the teacher neglects it, the pupil will adopt it himself, by adding 
 with his fingers, etc. 
 
 2. Addition and subtraction should be taught together in primary oral 
 instruction. This is evident, since the two ideas are logically related. 
 Thus, as soon as the pupil learns that 2 and 3 are 5, he sees that 5 
 diminished by 2 equals 3, or 5 diminished by 3 is 2. Convenience 
 also dictates the same method. In the primary schools of Germany, 
 the two processes are combined, in the manner illustrated. 
 
 Exercise. — The order of exercises will now be given. The pupils 
 should first increase and diminish by one, as far as 12, then by two, 
 then by three, etc. The exercise would be somewhat as follows: — 
 
 Teacher takes one book in his hand, and asks, "How many 
 books have I in my hand?" 
 
 Pupils answer, "One book." 
 
 Teacher, putting another book in his hand, asks, "How many 
 books have I now ?" 
 
 Pupils answer, "Two books." 
 
 Teacher: "How many, then, are one book and one book?" 
 
 Pupils: "Two books." 
 
 Teacher "How many books have I in my hand now?" 
 
INTRODUCTION. 9 
 
 Pupils: "Two books." 
 
 Teacher: "I will take one book away; now how many books 
 remain ?' 
 
 Pupils: "One book." 
 
 Teacher: "One book taken from two books, then, leaves how 
 many books?" 
 
 Pupils: "One Dook.' - 
 
 Let the teacher now take the arithmetical frame, and proceed in the 
 same way, increasing 2, 3, 4, etc., up to 12 with one, and diminish- 
 ing 3, 4, 5, 6, etc., up to 13 with one, each time reversing the addi- 
 tion. Then take one and increase it by two, obtaining three ; then 
 reverse the process and diminish three by tico, and so on until 12 is 
 increased by two, and 14 diminished by two. Proceed in the same 
 way with 3, 4, etc., until the pupils can add and subtract by ones, 
 twos, threes, etc., up to twelves. 
 
 LESSON III. 
 practical exercises. 
 
 The following exercises will be found valuable in teaching pupils 
 to add and subtract with readiness and accuracy. Frequent drill 
 upon such exercises is recommended. 
 
 First. — A valuable exercise to give the pupils readiness in adding 
 and subtracting is the following: Let the teacher name two num- 
 bers, and require the pupils to give first their sum and then their 
 difference: thus, teacher says, "5 and 2." 
 
 Pupils: "5 and 2 are 7, and 2 from 5 leaves 3." After a little 
 practice they may omit naming the numbers, and merely say, "the 
 sum is 7, the difference is 3." 
 
 To vary this, the boys may give the sum, and the girls the differ- 
 ence, and vice versa; or, if the class is all of one sex, a division may 
 be made, one part giving the sum, and the other part the difference. 
 In this and the following exercises care should be taken that small 
 numbers be used at first, until the pupils attain the ability to use 
 larger numbers with ease and readiness 
 
 Second. — Another valuable exercise consists in the teacher select- 
 ing some number, and then giving one part of this number himself, 
 requiring the pupils to give the other part. For example, suppose 
 8 to be the number selected: the teacher says, "five," pupils answer, 
 "three," teacher, "tivo," pupils, "six" etc. etc. 
 
 Third. — The following exercise will also be found valuable in im- 
 parting the art of adding and subtracting with readiness and accu- 
 
one. 
 two 
 three 
 four 
 five 
 six 
 seven 
 eight 
 nine 
 
 10 INTRODUCTION. 
 
 racy. Let the teacher write the words one, two, three, etc., on the 
 
 board, forming two columns as indicated in the mar- ( 
 
 ein Call the first column additive, and the second 
 
 6 • , ,, • n one 
 
 subtractive. The teacher then with the pointer will 
 
 indicate the number, the operation being indicated 
 
 tilTCC 
 
 by the column. When he points to figures in the first 
 
 column, the number which they indicate will be J 
 
 added, but when he points to any figure in the second, J . 
 
 the number indicated will be subtracted from the re- 
 
 . . , . , seven 
 suit which the pupils have previously obtained. 
 
 After the Arabic characters have been given, instead 
 
 of writing the words in columns, the figures may be 
 
 employed for the same purpose. 
 
 Fourth. — The pupils should also be required to add by twos, threes, 
 etc., merely naming the results, as follows; 2, 4, 6, 8, etc., 3, 6, 9, 
 etc., until the additions can be readily given. 
 
 It will be well to commence also with one, and count by twos, 
 thus: 1, 3, 5, 7, etc. ; also commence at 1, and count by threes, 
 thus: 1, 4, 7, 10, etc.; also at 2, thus: 2, 5, 8, 11, etc., continuing 
 the addition as far as it may be thought desirable. 
 
 Let the pupil be exercised in a similar manner in adding by fours, 
 fives, etc., up to twelves. Such exercises should be continued day 
 after day, in connection with the lessons which precede and follow 
 this lesson, until great facility is acquired in the operations. 
 
 These exercises may be conducted sometimes in concert and some- 
 times singly. While one is adding alone, let the others keep careful 
 watch for errors ; a good degree of interest may thus be created, 
 each pupil trying to obtain the largest sum before making a mis- 
 take. 
 
 It is evident that the teacher can give great variety to these exer- 
 cises ; and the author suggests that they will be found of very great 
 utility. 
 
 LESSON IV. 
 
 WRITING NUMBERS. 
 
 The pupil should now be taught how to write numbers. In fact, 
 the writing of numbers should be introduced very soon after the 
 naming of numbers. The following exercises should, therefore, be 
 combined with the exercises of Lesson III. The method suggested 
 
 is as follows : — 
 
 The teacher, standing at the board, with some objects, as two books 
 in his hand, inquires, "What do I hold in my hand?" 
 
INTRODUCTION. 11 
 
 Answer : — " Books." 
 
 Teacher : " How many books ?" 
 
 Pupils : " Two books." 
 
 Teacher (writing two books upon the board) asks, "What hava 
 I written upon the board?" 
 
 Pupils: "Two books." 
 
 Teacher : "Are there two books on the board ?" 
 
 Pupils : "Yes, sir." 
 
 Teacher: "If there are two books on the board, then what are 
 these I hold in my hand?" 
 
 Pupils : "Why, those are two books also." 
 
 Teacher: "Well, if that is two books on the board, and these 
 are two books in my hand, then they must both be the same." 
 
 Pupils: "Oh, no, that on the board is the icords two books, but 
 what you have in your hand are the things two books." 
 
 Thus the important distinction between the thing and the expression 
 of it is attained. 
 
 The teacher will now send the pupils to the board, and let them 
 write the words one, two, etc, and have them solve problems in 
 addition and subtraction with them. If the pupils cannot write 
 (and I presume that will generally be the case at the time such exer- 
 cises are appropriate), the teacher can illustrate by performing the 
 written exercises himself. 
 
 The pupils will soon see the great labor of employing the written 
 words, and will realize the necessity of an arithmetical written lan- 
 guage different from that which is used in ordinary writing. They 
 are now prepared for the Arabic characters; and the manner in 
 which these should be introduced will now be given. 
 
 Characters. — The pupils being now prepared for the Arabic 
 characters, let the teacher give first* in their order the nine digits. 
 They should be exercised in naming and writing these until they 
 are familiar with them and can make them with considerable ease 
 and neatness. They should then be required to solve problems with 
 them in addition and subtraction, the teacher giving no problem 
 at present which involves a number greater than nine. 
 
 Combinations. — When the class are familiar with these characters, 
 they are to be taught to combine them to express the larger num- 
 bers. There are two methods of doing this, quite different in 
 principle and form. We present both. 
 
 First Method. — By this method we give the combined characters, 
 without explaining the principle of the combination. Thus, wo 
 teach that 10 represents ten, 11, eleven, 12, twelve, etc., without any 
 
VI INTRODUCTION. 
 
 reference to tens and units. This method is not quite so philoso- 
 phical as the second method, but will be found preferable in practice 
 with young learners in oral instruction. When pupils study written 
 arithmetic from the book, I would use the other method. 
 
 We would give these expressions as far as twenty, and then drill 
 the pupils in reading and writing them until they are quite familiar 
 with them. We would next give the expressions from twenty to 
 thirty, and drill in like manner, and thus continue as far as one 
 hundred. 
 
 After the pupils are familiar with this method of writing numbers 
 as far as 100, the teacher may then show them the principle of the 
 combination, that the figure in the first place represents units, in 
 the second place tens, etc. When this is understood, we would re- 
 quire the class to analyze these expressions, as follows: — 
 
 Problem. — Analyze 25 [twenty-five). 
 
 A?ialysis. — In 25, the 5 represents 5 units, and the 2 represents 2 
 tens. 
 
 The class should also be drilled upon questions like the follow- 
 ing: — How many units in 2 tens? In 3 tens? etc. How many tens 
 in 20 (twenty) 1 etc. How many units and tens in 24 (twenty-four) 1 
 They should also be required to solve little problems in addition 
 and subtraction with these characters. 
 
 Second Method. — The other method commences by explaining the 
 principle of the combination; that is, that 10 represents 1 ten; 11, 1 
 ten and 1 unit; 12, 1 ten and 2 units, etc., afterward showing that 11 
 (1 ten and 1 unit) is the same as eleven, etc. 
 
 This may be done by making ten marks upon the board, and 
 then commencing a second row with one mark ; these will be repre- 
 sented by 11 (1 ten and 1 unit) ; then have the pupils count them, 
 and they will see that 11 (1 ten and 1 unit) stands for eleven. The 
 same may be done with 12, 13, etc. 
 
 The pupil should be drilled in reading and writing numbers, until 
 he is entirely familiar with the subject. Haste here is "bad 
 speed." A thorough knowledge of Notation and Numeration wil) 
 dispel the usual difficulties of Addition, Subtraction, Multiplication, 
 ind Division. 
 
 LESSON V. 
 
 MULTIPLICATION AND DIVISION. 
 
 After the pupil has become quite familiar with the elementary 
 processes of Addition and Subtraction, he is prepared to take up 
 Multiplication and Division. The first instruction in these processes 
 
INTRODUCTION. 13 
 
 should be given by oral exercises, and in accordance with the 
 following principles: 
 
 1. Multiplication should be presented as a special case of Addition 
 Thus, the pupil should be taught that two 2's are 4, since 2 -j- 2 ==4, 
 or that three times 2 are 6, since 2 taken three times, or 2 -j- 2 — |- 2, 
 equals 6. and so on for the other products. The pupil will then 
 understand the nature of the subject. 
 
 2. Division should be taught as reverse multiplication. Thus, it should 
 be shown that 6 contains 3 two times, since tivo times 3 are 6, and so 
 on for other quotients. In this way the quotients are immediately 
 derived from the products. Division may be taught as concise sub- 
 traction, but the process of reverse multiplication is more convenient 
 and simple. "When thus taught, the pupil will not need to commit 
 a distinct Division Table. 
 
 3. The pupil should be taught to construct the Multiplication Table. 
 The pupil should not be required to commit a Multiplication Table 
 without knowing how it was obtained, or the use of it. He should 
 first be taught to derive the products for himself, by addition, and 
 then be required to commit them, to avoid the labor of obtaining 
 them every time he wishes to use them. In this way he will study 
 them with more interest, and learn them with greater ease. 
 
 4. Multiplication and Division should bu taught simultaneously, or at 
 least very nearly so. As soon as the pupil learns that 2 times 3 are 6, 
 he is able to see that 6 equals two 3's, or that 6 contains 3 two times; 
 and the same is true for the other products. Division, therefore, 
 should not be deferred until the whole Multiplication Table is 
 learned, as has generally been the practice, but should be early 
 introduced and studied in connection with Multiplication. 
 
 We now present the following exercise, which is designed to sug- 
 gest the manner in which the first principles of multiplication and 
 division may be taught. 
 
 EXERCISE. 
 
 Teacher (making two marks on the board, as is indi- 
 | | cated in the margin) asks, "How many marks have I 
 made?" 
 
 Pupils: "Two marks. 
 
 Teacher (making two marks under the former two, aa 
 is indicated in the margin, inquires) : "How many times 
 J j have I made two marks?" 
 Pupils: "Two times." 
 
 2 
 
14 INTRODUCTION. 
 
 Teacher : " How many marks are there in all ?" 
 
 Pupils : " Four marks." 
 
 Teacher: " How many, then, are two times two marks?" 
 
 Pupils : " Two times two marks are four marks." 
 
 Teacher (leaving the four marks upon the board, asks): "How 
 many marks are there on the board?" 
 
 Pupils . " Four marks." 
 
 Teacher: "Are they arranged in twos, or threes?" 
 
 Pupils : "In twos." 
 
 Teacher: " How many twos are there in these four marks?'* 
 
 Pupils : " Two twos." 
 
 Teacher: "Four, then, contains two how many times?" 
 
 Pupils : "Two times." 
 
 The teacher will now write six marks upon the board, 
 
 as in the margin, and ask, " How many times have I 
 
 | | | written three marks?" "How many are there in all?" 
 
 "How many, then, are two times three?" "Are these 
 
 6 marks arranged in twos, threes, or fours?" "How many times 
 
 three marks are there ?" " How many threes, then, are there in 
 
 six?" "Six, then, contains 3 how many times?" 
 
 These, or similar exercises, should be continued up to two times 
 12, each time reversing the process, and obtaining a quotient. Then 
 proceed in the same way with three times, four times, etc., on to 
 twelve times. In this manner the pupils may be led to obtain, and 
 then commit, the products and quotients, usually given in tables, 
 which are always learned with so much hesitation and hard study. 
 
 In practice, it will be well to obtain all the products of "two times," 
 before deriving the quotients. Questions similar to those in the Pri- 
 mary Arithmetic, p. 82, may also be given. The pupil should also 
 be taught to write the table of products upon the slate or board, 
 thus: 
 
 1X2 = 2 4X2= 8 
 
 2X2 = 4 5X2 = 10 
 
 8X2 = 6 etc. 
 
 Pupils generally have considerable difficulty in committing the 
 Multiplication Table; the teacher can lessen the labor in several 
 ways. 1st. By having the pupils make it for themselves, and write 
 it on the slate or blackboard. 2d. By conoert recitation. 3d. By 
 singing the table to some appropriate tune. 4th. Reciting by the old 
 method of "going up," or "trapping." 
 
 To make pupils rapid and accurate in the mechanical processes of 
 
INTRODUCTION. 15 
 
 addition, subtraction, multiplication, and division, the following ex- 
 ercise is practised by some teachers, with excellent results. Let the 
 teacher write four columns of figures on 
 
 the blackboard, as is represented in the (-]-) ( — ) (X) (-*-) 
 margin, the first column being additive, the 1111 
 next subtractive, etc., as is indicated by the 2 2 2 2 
 
 symbols placed above them. The teacher, 3 3 3 3 
 with the pointer, will point out certain 4 4 4 4 
 figures, the corresponding numbers being 5 5 5 5 
 added, subtracted, multiplied, or divided, 6 6 6 6 
 as is indicated by the symbol at the head of 7 7 7 7 
 the column. Care, of course, must be taken 8 8 8 8 
 not to require a division by a number that 9 9 9 9 
 is not exactly contained. This exercise 
 
 may be continued for many recitations, in connection with the follow 
 ing lessons, with great advantage to the pupils. 
 
 LESSON VI. 
 
 TERMS AND PRINCIPLES. 
 
 The following exercises are designed to suggest the manner of 
 giving the terms of the fundamental rules, and also of deriving some 
 of the principles of each. 
 
 ADDITION AND SUBTRACTION. 
 
 Teacher: "What have I in my hand?" 
 
 PuriLS : " Two books." 
 
 Teacher. "What is the difference between the two and the 
 books?" 
 
 Pupils: "The books are the things, and the two tells how many 
 things." 
 
 Teacher: "The two denotes the number of books. What, then, 
 is a number?" 
 
 Pupils : "Why, it is the how many of any thing." 
 
 Teacher : "Very well . remember, also, that a single thing, or one 
 of a collection, is called a unit." 
 
 Teacher: "When I say two apples, what 2 do I mean?" 
 
 Pupils: "Two apples." 
 
 Teacher: "When I say tiro, what do I mean?" 
 
 Pupils : " We do not know.'' 
 
 Teacher: "What 2 may I mean? any two?" 
 
 Pupils: "Yes, sir; any two you choose." 
 
 Teacher: "You see a difference, then, between two and ttn 
 
16 INTRODUCTION. 
 
 books. Very well ; I will give the name which denotes this difference. 
 When I say 2, 3, etc., without telling what 2 or 3, it is called an 
 abstract number; but when I give the name of the objects with the 
 number, it is called a concrete number." 
 
 Tell which of the following numbers are abstract, and which con^ 
 crete : — 
 
 2 cows — three — four — 4 books — 7 hens — eight — 5 — 4 — 10 pigs — 
 8 geese — 7 — 6 — 11 — 14 horses. 
 
 Teacher: How many are 3 and 5? 
 
 Teacher: When we unite two numbers into one, in this way, the 
 result is called the sum, and the process is called addition. 
 
 Teacher: What is the sum of 2 and 3? 4 and 6? 7 and 8? 
 
 Teacher : What is the sum of 3 cows and 5 turnips ? 
 
 Teacher : Why can you not add them ? 
 
 Teacher: If they were all the same, could you add them ? 
 
 Teacher: Numbers which express the same kind of objects are 
 similar concrete numbers, and those which denote different objects 
 are dissimilar concrete numbers. 
 
 Teacher : What kind of numbers can be added, then, and what 
 kind cannot be added ? 
 
 How many remain when we take 3 apples from 5 apples ? 
 
 The process of taking one number from another is called sub- 
 traction. 
 
 The number which is taken away is called the subtrahend, the 
 number from which it is taken is called the minuend, and the result 
 is called the difference, or remainder. 
 
 If you subtract 4 from 9, which is the minuend, which the subtra- 
 hend, and which the remainder? 
 
 If you add the remainder and subtrahend together, will it pro- 
 duce the minuend ? 
 
 If you subtract the difference from the minuend, what will it 
 equal? 
 
 Can you subtract 3 apples from 5 potatoes ? 
 
 Why can you not subtract them ? 
 
 Are these similar or dissimilar concrete numbers? 
 
 If they were similar, could they be subtracted ? 
 
 What kind of numbers, then, can be subtracted, and what kind 
 eannot ? 
 
 PRINCIPLES OF MULTIPLICATION AND DIVISION. 
 
 1. When we find the result of a number taken any number of 
 times, the process is called multiplication. 
 
INTRODUCTION. 17 
 
 2. The number taken a certain number of \imes is called the 
 multiplicand. 
 
 3. The number which denotes how many times the multiplicand 
 is taken is called the multiplier. 
 
 4. The result obtained is called the product. Each of these three 
 is called a term. 
 
 5. What is the product of 8 apples multiplied by 4? 
 
 6. In this problem, which is the multiplicand, which the multi- 
 plier, which the product? 
 
 7. When we take 8 apples 4 times, is the result apples, or some- 
 thing else ? 
 
 8. Can the product be any thing else than apples? 
 
 9. The product, then, is of the same denomination as what term? 
 
 10. Can we take 8 apples 4 peaches times, or simply 4 times? 
 
 11. Is 4 an abstract or a concrete number ? What kind of a num- 
 ber, then, must the multiplier be ? 
 
 12. When we find how many times one number is contained in 
 another, the process is called division. 
 
 13. The number which contains the other is called the dividend, 
 the number contained is called the divisor, and the number denoting 
 how many times the divisor is contained is called the quotient. 
 
 14. If we divide 8 apples by 2 apples, is the result apples? If 
 not, what is it? 
 
 15. Are 2 apples contained in 8 apples 4 peaches times, or 4 
 apples times, or simply 4 times? 
 
 1G. What kind of a number is 4, and what kind of a number, 
 then, must the quotient always be ? 
 
 17. How many times 2 equal 8 apples ? Is 2, or 2 pears, contained 
 any number of times in 8 apples ? 
 
 18. What 2 are contained any number of times in 8 apples? 
 
 19. The divisor, then, must be of the same denomination as what 
 term? 
 
 LESSON VII. 
 
 TABLE OF FUNDAMENTAL RULES. 
 
 We now present the tables of the four fundamental rules, for such 
 teachers as wish to use them. The author suggests that the ele- 
 mentary sums and differences are better- taught by the exercises 
 which have been already suggested than by the study of these 
 tables. The Multiplication Table, however, must be thoroughly 
 committed, and then the elementary quotients may be derived from 
 these products, or by the study of the Division Table. 
 
 2* 
 
18 
 
 INTRODUCTION. 
 
 ADDITION TABLE. 
 
 2 and 
 
 3 and 
 
 4 and 
 
 5 and 
 
 are 2 
 
 are 3 
 
 are 4 
 
 are 5 
 
 1 " 3 
 
 1 " 4 
 
 1 « 5 
 
 1 " 6 
 
 2 " 4 
 
 2 « 5 
 
 2 " 6 
 
 2 « 7 
 
 3 " 5 
 
 3 " 6 
 
 3 » 7 
 
 3 " 8 
 
 4 " 6 
 
 4 " 7 
 
 4 S* 8 
 
 4 " 9 
 
 5 " 7 
 
 5 " 8 
 
 5 « 9 
 
 5 " 10 
 
 G " 8 
 
 6 " 9 
 
 6 " 10 
 
 6 " 11 
 
 7 " 9 
 
 7 " 10 
 
 7 " 11 
 
 7 " 12 
 
 8 » 10 
 
 8 " 11 
 
 8 " 12 
 
 8 " 13 
 
 9 " 11 
 
 9 " 12 
 
 9 " 13 
 
 9 " 14 
 
 10 " 12 
 
 10 " 13 
 
 10 " 14 
 
 10 " 15 
 
 11 " 13 
 
 11 " 14 
 
 11 " 15 
 
 11 " 16 
 
 12 " 14 
 
 12 " 15 
 
 12 " 16 
 
 12 » 17 
 
 6 and 
 
 7 and 
 
 8 and 
 
 9 and 
 
 are 6 
 
 are 7 
 
 0- are 8 
 
 are 9 
 
 1 " 7 
 
 1 " 8 
 
 1 " 9 
 
 1 " 10 
 
 2 " 8 
 
 2 " 9 
 
 2 " 10 
 
 2 " 11 
 
 3 " 9 
 
 3 " 10 
 
 3 » 11 
 
 3 » 12 
 
 4 " 10 
 
 4 " 11 
 
 4 « -12- 
 
 4 " 13 
 
 5 " 11 
 
 5 " 12 
 
 5 " 13 
 
 5 " 14 
 
 6 " 12 
 
 6 " 13 
 
 6 " 14 
 
 6 » 15 
 
 7 " 13 
 
 7 " 14 
 
 7 " 15 
 
 7 " 16 
 
 8 " 14 
 
 8 " 15 
 
 8 " 16 
 
 8 " 17 
 
 9 " 15 
 
 9 " 16 
 
 9 " 17 
 
 9 » 18 
 
 10 » 16 
 
 10 " 17 
 
 10 " 18 
 
 10 " 19 
 
 11 " 17 
 
 11 " 18 
 
 11 « 19 
 
 11 " 20 
 
 12 " 18 
 
 12 " 19 
 
 12 " 20 
 
 12 " 21 
 
 10 and 
 
 11 and 
 
 12 and 
 
 13 and 
 
 are 10 
 
 are 11 
 
 are 12 
 
 are 13 
 
 1 " 11 
 
 1 " 12 
 
 1 » 13 
 
 1 " 14 
 
 2 " 12 
 
 2 " 13 
 
 2 " 14 
 
 2 " 15 
 
 3 " 13 
 
 3 " 14 
 
 3 « 15 
 
 3 " 16 
 
 4 " 14 
 
 4 « 15 
 
 4 »« 16 
 
 4 " 17 
 
 5 " 15 
 
 5 " 16 
 
 5 « 17 
 
 5 " 18 
 
 6 " 16 
 
 6 " 17 
 
 6 « 18 
 
 G " 19 
 
 7 « 17 
 
 7 " 18 
 
 7 " 19 
 
 7 " 20 
 
 8 " 18 
 
 8 " 19 
 
 8 " 20 
 
 8 " 21 
 
 9 " 19 
 
 9 " -20 
 
 9 " 21 
 
 9 <l 22 
 
 10 " 20 
 
 10 " 21 
 
 10 " 22 
 
 10 » 23 
 
 11 " 21 
 
 11 " 22 
 
 11 » 23 
 
 11 " 24 
 
 12 » 22 
 
 12 " 23 
 
 12 " 24 
 
 12 " 25 
 
INTRODUCTION. 
 
 19 
 
 SUBTRACTION TABLE. 
 
 1 " 
 
 1 from 
 
 
 2 from 
 
 
 3 from 
 
 
 ■ 
 
 4 from 
 
 1 leaves 
 
 2 
 
 leaves 
 
 3 
 
 leaves 
 
 4 
 
 leaves 
 
 2 « 
 
 1 
 
 3 
 
 1 
 
 4 
 
 1 
 
 5 
 
 1 
 
 3 * 
 
 2 
 
 4 
 
 2 
 
 5 
 
 2 
 
 6 
 
 " 2 
 
 4 < 
 
 3 
 
 5 
 
 3 
 
 6 
 
 3 
 
 7 
 
 3 
 
 6 < 
 
 4 
 
 6 
 
 4 
 
 7 
 
 4 
 
 8 
 
 4 
 
 6 < 
 
 5 
 
 7 
 
 5 
 
 8 
 
 5 
 
 9 
 
 5 
 
 7 < 
 
 1 6 
 
 8 
 
 6 
 
 9 
 
 6 
 
 10 
 
 6 
 
 8 « 
 
 4 < 
 
 9 
 
 7 
 
 10 
 
 7 
 
 11 
 
 7 
 
 9 < 
 
 8 
 
 10 
 
 8 
 
 11 
 
 8 
 
 12 
 
 8 
 
 10 < 
 
 9 
 
 11 
 
 9 
 
 12 
 
 9 
 
 13 
 
 9 
 
 11 ♦ 
 
 < 10 
 
 12 
 
 » 10 
 
 13 
 
 " 10 
 
 14 
 
 " 10 
 
 12 < 
 
 < 11 
 
 13 
 
 " 11 
 
 14 
 
 " 11 
 
 15 
 
 « 11 
 
 13 < 
 
 < 12 
 
 14 
 
 " 12 
 
 15 
 
 " 12 
 
 16 
 
 " 12 
 
 51 
 
 rom 
 
 
 6 from 
 
 
 7 from 
 
 
 8 from 
 
 5 lea 
 
 .ves 
 
 6 
 
 leaves 
 
 7 
 
 leaves 
 
 8 
 
 leaves 
 
 6 « 
 
 ♦ 1 
 
 7 
 
 1 
 
 8 
 
 1 
 
 9 
 
 1 
 
 7 < 
 
 2 
 
 8 
 
 2 
 
 9 
 
 2 
 
 10 
 
 " 2 
 
 8 « 
 
 3 
 
 9 
 
 3 
 
 10 
 
 3 
 
 11 
 
 3 
 
 9 « 
 
 4 
 
 10 
 
 4 
 
 11 
 
 4 
 
 12 
 
 4 
 
 10 < 
 
 5 
 
 11 
 
 5 
 
 12 
 
 5 
 
 13 
 
 5 
 
 11 < 
 
 6 
 
 12 
 
 6 
 
 13 
 
 6 
 
 14 
 
 6 
 
 12 < 
 
 7 
 
 13 
 
 7 
 
 14 
 
 7 
 
 15 
 
 7 
 
 13 < 
 
 8 
 
 14 
 
 8 
 
 15 
 
 8 
 
 16 
 
 8 
 
 14 « 
 
 ■ 9 
 
 15 
 
 9 
 
 16 
 
 9 
 
 17 
 
 9 
 
 15 < 
 
 < 10 
 
 16 
 
 " 10 
 
 17 
 
 " 10 
 
 18 
 
 " 10 
 
 16 « 
 
 1 11 
 
 17 
 
 " 11 
 
 18 
 
 " 11 
 
 19 
 
 " 11 
 
 17 « 
 
 1 12 
 
 18 
 
 " 12 
 
 19 
 
 it ^o 
 
 20 
 
 " 12 
 
 91 
 
 rom 
 
 
 10 from 
 
 
 11 from 
 
 
 12 from 
 
 9 lea 
 
 ves 
 
 10 
 
 leaves 
 
 11 
 
 leaves 
 
 12 
 
 leaves 
 
 10 « 
 
 1 
 
 11 
 
 " 1 
 
 12 
 
 1 
 
 13 
 
 " 1 
 
 11 < 
 
 2 
 
 12 
 
 2 
 
 13 
 
 " 2 
 
 14 
 
 2 
 
 12 < 
 
 3 
 
 13 
 
 3 
 
 14 
 
 3 
 
 15 
 
 3 
 
 13 < 
 
 4 
 
 14 
 
 4 
 
 15 
 
 4 
 
 16 
 
 4 
 
 14 « 
 
 5 
 
 15 
 
 5 
 
 16 
 
 5 
 
 17 
 
 5 
 
 15 ■ 
 
 6 
 
 16 
 
 6 
 
 17 
 
 6 
 
 18 
 
 " 6 
 
 16 • 
 
 
 17 
 
 7 
 
 18 
 
 7 
 
 19 
 
 7 
 
 17 « 
 
 8 
 
 18 
 
 8 
 
 19 
 
 8 
 
 20 
 
 8 
 
 18 ' 
 
 9 
 
 19 
 
 9 
 
 20 
 
 9 
 
 21 
 
 9 
 
 19 « 
 
 * 10 
 
 20 
 
 " 10 
 
 21 
 
 " 10 
 
 
 " 10 
 
 20 ■ 
 
 ■ 11 
 
 21 
 
 " 11 
 
 22 
 
 " 11 
 
 23 
 
 " 11 
 
 21 
 
 1 12 
 
 22 
 
 " 12 
 
 23 
 
 " 12 
 
 24 
 
 " 12 
 
20 
 
 INTRODUCTION. 
 
 MULTIPLICATION TABLE. 
 
 Once 
 
 2 times 
 
 3 times 
 
 ■■■■■- ■ — - n 
 
 4 times 
 
 1 is 1 
 
 1 are 2 
 
 1 are 3 
 
 1 are 4 
 
 2 " 2 
 
 2 " 4 
 
 2 « 6 
 
 2 " 8 
 
 3 " 3 
 
 3 " 6 
 
 3 " 9 
 
 3 " 12 
 
 4 " 4 
 
 4 " 8 
 
 4 " 12 
 
 4 " 16 
 
 5 " 5 
 
 5 " 10 
 
 5 " 15 
 
 5 " 20 
 
 6 " 6 
 
 6 « 12 
 
 6 " 18 
 
 6 " 24 
 
 7 " 7 
 
 7 " 14 
 
 7 " 21 
 
 7 " 28 
 
 8 " 8 
 
 8 " 16 
 
 8 " 24 
 
 8 " 32 
 
 9 " 9 
 
 9 " 18 
 
 9 " 27 
 
 9 " 36 
 
 10 " 10 
 
 10 " 20 
 
 10 " 30 
 
 10 « 40 
 
 11 " 11 
 
 11 " 22 
 
 11 '* 33 
 
 11 " 44 
 
 12 " 12 
 
 12 « 24 
 
 12 " 36 
 
 12 " 48 
 
 5 times 
 
 6 times 
 
 7 times 
 
 8 times 
 
 1 are 5 
 
 1 are 6 
 
 1 are 7 
 
 1 are 8 
 
 2 " 10 
 
 2 " 12 
 
 2 " 14 
 
 2 u 16 
 
 3 " 15 
 
 3 " 18 
 
 3 " 21 
 
 3 " 24 
 
 4 " 20 
 
 4 " 24 
 
 4 " 28 
 
 4 " 32 
 
 5 " 25 
 
 5 « 30 
 
 6 " 35 
 
 5 " 40 
 
 6 " 30 
 
 6 " 36 
 
 6 " 42 
 
 6 " 48 
 
 7 " 35 
 
 7 " 42 
 
 7 " 49 
 
 7 " 56 
 
 8 " 40 
 
 8 " 48 
 
 8 " 56 
 
 8 " 64 
 
 9 " 45 
 
 9 " 54 
 
 9 " 63 
 
 9 " 72 
 
 10 " 50 
 
 10 " 60 
 
 10 il 70 
 
 10 " 80 
 
 11 " 65 
 
 11 » 66 
 
 11 " 77 
 
 11 " 88 
 
 12 " 60 
 
 12 " 72 
 
 12 " 84 
 
 12 " 96 
 
 9 times 
 
 10 times 
 
 11 times 
 
 12 times 
 
 1 are 9 
 
 1 are 10 
 
 1 are 11 
 
 1 are 12 
 
 2 " 18 
 
 2 " 20 
 
 2 " 22 
 
 2 " 24 
 
 3 " 27 
 
 3 « 30 
 
 3 " 33 
 
 3 " 36 
 
 4 " 36 
 
 4 « 40 
 
 4 " 44 
 
 4 " 48 
 
 5 " 45 
 
 5 " 50 
 
 5 " 55 
 
 5 " 60 
 
 6 " 54 
 
 6 " 60 
 
 6 » 66 
 
 6 " 72 
 
 7 " 63 
 
 7 " 70 
 
 7 " 77 
 
 7 " 84 
 
 8 " 72 
 
 8 " 80 
 
 8 « 88 
 
 8 " 96 
 
 9 « 81 
 
 9 " 90 
 
 9 " 99 
 
 9 " 108 
 
 10 " 90 
 
 10 " 100 
 
 io « no 
 
 10 " 120 
 
 11 « 99 
 
 li " no 
 
 11 " 121 
 
 11 " 132 
 
 12 " 108 
 
 12 » 120 
 
 12 " 132 
 
 12 " 144 
 
INTRODUCTION. 
 
 21 
 
 DIVISION TABLE. 
 
 lin 
 
 2 in 
 
 3 in 
 
 4 in 
 
 1 1 time 
 
 2 1 time 
 
 3 1 time 
 
 4 1 time 
 
 2 2 times 
 
 4 2 times 
 
 6 2 times 
 
 8 2 times 
 
 3 3" 
 
 6 3 " 
 
 9 3" 
 
 12 3 " 
 
 4 4" 
 
 8 4" 
 
 12 4 " 
 
 16 4 " 
 
 5 5" 
 
 10 5 " 
 
 15 5 " 
 
 20 5 " 
 
 6 6" 
 
 12 6 " 
 
 18 6 " 
 
 24 6 " 
 
 7 7" 
 
 14 7 " 
 
 21 7 " 
 
 28 7 " 
 
 8 8" 
 
 16 8 " 
 
 24 8 " 
 
 32 8 " 
 
 9 9" 
 
 18 9 " 
 
 27 9 " 
 
 36 9 " 
 
 10 10 " 
 
 20 10 " 
 
 30 10 " 
 
 40 10 " 
 
 11 11 " 
 
 22 11 " 
 
 33 11 " 
 
 44 11 " 
 
 12 12 " 
 
 24 12 " 
 
 36 12 " 
 
 48 12 " 
 
 5 in 
 
 6 in 
 
 7 in 
 
 8 in 
 
 5 1 time 
 
 6 1 time 
 
 7 1 time 
 
 8 1 time 
 
 10 2 times 
 
 12 2 times 
 
 14 2 times 
 
 16 2 times 
 
 15 3 " 
 
 18 3 " 
 
 21 3 " 
 
 24 3 " 
 
 20 4 " 
 
 24 4 " 
 
 28 4 " 
 
 32 4 " 
 
 25 5 " 
 
 30 5 " 
 
 35 5 " 
 
 40 5 " 
 
 30 6 " 
 
 36 6 " 
 
 42 6 " 
 
 48 6 " 
 
 35 7 " 
 
 42 7 " 
 
 49 7 " 
 
 56 7 " 
 
 40 8 " 
 
 48 8 " 
 
 56 8 " 
 
 64 8 " 
 
 45 9 " 
 
 54 9 " 
 
 63 9 " 
 
 72 9 " 
 
 50 10 " 
 
 60 10 " 
 
 70 10 " 
 
 80 10 " 
 
 55 11 " 
 
 66 11 " 
 
 77 11 " 
 
 88 11 " 
 
 GO 12 " 
 
 72 12 " 
 
 84 12 " 
 
 96 12 " 
 
 9 in 
 
 10 in 
 
 11 in 
 
 12 in 
 
 9 1 time 
 
 10 1 time 
 
 11 1 time 
 
 12 1 time 
 
 18 2 times 
 
 20 2 times 
 
 22 2 times 
 
 24 2 times 
 
 27 3 " 
 
 30 3 " 
 
 33 3 " 
 
 36 3 " 
 
 36 4 " 
 
 40 4 " 
 
 ■14 4 " 
 
 48 4 " 
 
 43 5 " 
 
 50 5 " 
 
 56 5 " 
 
 60 5 " 
 
 54 6 " 
 
 00 6 " 
 
 66 6 " 
 
 72 6 " 
 
 63 7 " 
 
 70 7 " 
 
 77 7 " 
 
 84 7 " 
 
 72 8 " 
 
 8 " 
 
 88 8 " 
 
 96 8 " 
 
 81 9 " 
 
 90 9 " 
 
 99 9 " 
 
 108 9 " 
 
 90 10 " 
 
 100 10 " 
 
 110 10 " 
 
 120 10 " 
 
 11 " 
 
 no n " 
 
 121 11 " 
 
 132 11 " 
 
 108 i ! •• 
 
 120 12 " 
 
 132 12 " 
 
 144 12 " 
 
THE 
 
 NORMAL 
 
 ELEMENTARY ARITHMETIC. 
 
 SECTION I. 
 
 1. A Unit is a single thing, as a book, pen, apple. 
 
 2. A Number is one or more things, as two books, 
 
 three pens. 
 
 3. Arithmetic is the science of numbers and the art 
 
 of using them. 
 
 4. Mental Arithmetic is the solving of problems 
 without the aid of written characters. 
 
 5. Written Arithmetic is the solving of problems 
 with written characters. 
 
 6. In studying arithmetic we first learn to name 
 numbers, and then learn to write them. 
 
 NUMEKATIOK 
 
 7. Numeration is the art of naming numbers, and 
 of reading them when they are written. 
 
 8. We will give the names of some of the numbers 
 up <o one hundred. 
 
 six ; 
 
 one ; 
 two ; 
 three; 
 four ; 
 five ; 
 
 seven; 
 eight; 
 nine; 
 ten; 
 
 23 
 
24 
 
 NOTATION. 
 
 eleven, or one and ten ; 
 twelve, or two and ten ; 
 thirteen, or three and ten ; 
 fourteen, or four and ten ; 
 fifteen, or five and ten ; 
 sixteen, or six and ten ; 
 seventeen, or seven and ten ; 
 eighteen, or eight and ten ; 
 nineteen, or nine and ten ; 
 twenty, or two tens; 
 twenty-one, or two tens and one; 
 
 twenty-two, or two tens and two ; 
 thirty, or three tens; 
 thirty-one, or three tens and one; 
 forty, or four tens ; 
 forty-one, or four tens and one ; 
 fifty, or five tens ; 
 fifty-one, or five tens and one ; 
 sixty, or six tens ; 
 sixty-one, or six tens and one ; 
 seventy, or seven tens ; 
 etc., etc. 
 
 Note. — The pupil should be drilled upon these equivalent forms 
 of naming numbers, as a preparation for Notation. The teacher or 
 pupil may fill out the omissions. 
 
 ABABIC NOTATION. 
 9. Notation is the art of writing numbers. 
 lO. Figures. — In writing numbers we use the follow- 
 ing ten characters, called figures. 
 
 1 expresses 
 
 one. 
 
 6 
 
 expresses 
 
 six. 
 
 2 « 
 
 two. 
 
 7 
 
 u 
 
 seven. 
 
 3 
 
 three. 
 
 8 
 
 « 
 
 eight. 
 
 4 
 
 four. 
 
 9 
 
 u 
 
 nine. 
 
 5 
 
 five. 
 
 
 
 a 
 
 naught 
 
 11. Combination. — By these figures and their combi- 
 nations all numbers can be expressed. 
 
 The method of combining them is as follows : — 
 
 1st. A figure standing alone expresses units or ones. 
 
 2d. When two figures are together, the one in the first 
 place at the right expresses units, the one in the second 
 place expresses tens. 
 
 3d. A figure in the third place expresses hundreds, in 
 the fourth place thousands, etc. 
 
 1*2. Thus, in 25, the 2 expresses 2 tens, and the 
 expresses 5 units. We will illustrate this by the follow- 
 ing table. 
 
NOTATION. 
 
 25 
 
 10 is one ten. 
 
 20 
 30 
 40 
 50 
 60 
 70 
 80 
 90 
 100 
 
 two tens, 
 three tens, 
 four tens, 
 five tens, 
 six tens, 
 seven tens, 
 eight tens, 
 nine tens, 
 one hundred. 
 
 11 is 1 ten and one. 
 
 12 
 23 
 
 34 
 47 
 
 58 
 65 
 79 
 
 86 
 105 
 
 1 ten and two. 
 
 2 tens and three. 
 
 3 tens and four. 
 
 4 tens and seven. 
 
 5 tens and eight. 
 
 6 tens and five. 
 
 7 tens and nine. 
 
 8 tens and six. 
 
 1 hundred and five. 
 
 13. The pupils will now write and read the follow- 
 ing numbers : 
 
 13 
 15 
 17 
 19 
 21 
 
 24 
 26 
 
 27 
 29 
 30 
 
 33 
 
 36 
 38 
 39 
 40 
 
 41 
 43 
 45 
 
 50 
 52 
 
 57 
 60 
 68 
 63 
 
 70 
 
 74 
 76 
 82 
 95 
 126 
 
 14. The pupils will now learn the names of the first 
 twelve places, as represented in the following 
 
 NUMERATION TABLE. 
 
 CO 
 
 Names. 
 
 Places'. 
 Periods. 
 
 03 
 
 a 
 o 
 
 © 
 
 
 
 
 03 
 
 
 
 S3 
 c3 
 
 
 
 
 
 
 
 
 o 
 
 •** 
 
 
 
 03 
 
 © 
 
 — 
 
 -*- 
 
 • 
 93 
 
 
 
 
 
 ■ 
 03 
 
 a 
 
 
 1 
 
 03 
 
 o 
 
 
 
 od 
 
 
 
 
 o 
 la 
 
 03 
 
 © 
 
 © 
 
 T3" 
 
 
 T. 
 
 a 
 o 
 
 1 
 
 © 
 
 u 
 
 •73 
 
 o 
 
 r3 
 
 a 
 
 33 
 
 © 
 
 QQ 
 
 © 
 
 u 
 
 T3 
 
 
 ■/' 
 
 a 
 
 I— I 
 
 a 
 M 
 
 a 
 
 r— I 
 
 a 
 
 d 
 
 a 
 
 
 
 © 
 
 
 © 
 
 
 3 
 
 © 
 
 J3 
 H 
 
 
 © 
 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 1 
 
 4 
 
 P=T 
 
 A 
 
 
 
 
 
 
 
 
 
 
 -*- 
 
 © 
 
 ^3 
 
 ^a 
 
 .a 
 
 ^ 
 
 ^a 
 
 .a 
 
 M 
 
 pi 
 
 +A 
 
 f— H 
 
 -^-> 
 
 -*-* 
 
 h_> 
 
 
 -^j 
 
 
 T3 
 
 ^3 
 
 03 
 
 I— 1 
 
 i—i 
 
 Ci 
 
 CO 
 
 i- 
 
 o 
 
 >-o 
 
 •5 
 
 CO 
 
 <N 
 
 
 4th. 
 
 8J. 
 
 2d. 
 
 1st. 
 
 15. The pupil should now be drilled upon questions 
 similar to the following. 
 
 3 
 
26 NOTATION. 
 
 Required the names of the following places : 
 
 1. First. 
 
 2. Third. 
 
 3. Fourth. 
 
 4. Second. 
 
 5. Fifth. 
 
 6. Sixth. 
 
 7. Eighth. 
 
 8. Seventh. 
 
 9. Ninth. 
 
 Required the places of the following : 
 
 1. Tens. 
 
 2. Hundreds. 
 
 3. Ten-thousands. 
 
 4. Thousands. 
 
 5. Millions. 
 
 6. Hundred-thousands. 
 
 16. Periods, — For convenience in writing and read- 
 ing numbers, we arrange the figures in periods of three 
 places each, as shown by the table. 
 
 The first three places make the first or units, period, 
 the second three places make the second or thousands, 
 -period, etc. 
 
 Required the period and place of the following : 
 
 1. Hundreds. 
 
 2. Thousands. 
 
 3. Millions. 
 
 4. Ten -thousands. 
 
 5. Tens. 
 
 6. Ten-millions. 
 
 7. Hundred-thousands. 
 
 8. Hundred-millions. 
 
 17. The combination of figures to express a number 
 forms a numerical word. Thus, 25 is the numerical word 
 which means the same as twenty-five. These numerical 
 words may be analyzed. 
 
 PROBLEMS. 
 
 1. Analyze the numerical word 324. 
 
 Analysis. — The 4 represents four units, the 2 represents two tens, 
 the 3 represents three hundreds : hence the numerical word is thrct 
 hundred and twenty-four. 
 
 Analyze the following : 
 
 2. 426 4. 652 6. 853 8. 395 10. 1234 12. 43762 
 
 3. 357 5. 785 7. 687 9. 785 11. 5678 13. 85967 
 
NOTATION. 
 
 27 
 
 IS. "We will now give some exercises in Numeration 
 and Notation. 
 
 RULE FOR NUMERATION. 
 
 I. Begin at the right hand, and separate the numerical 
 word into periods of three figures each. 
 
 II. Then begin at the left hand, and read each period as 
 if it stood alone, giving the name of each period except the 
 last. 
 
 1. What number is expressed by 3254789 ? 
 
 Solution. — We separate the numerical word into 
 periods of three figures each, as in the margin. The 
 third period is 3 millions, the second period is 254 
 thousands, the first is 789 ; hence the number is 3 
 millions, 254 thousands, 789. 
 
 Eead the following 
 
 OPERATION. 
 
 3,254,789 
 
 2. 
 
 2384 
 
 6. 
 
 64327 
 
 10. 
 
 6321456 
 
 3. 
 
 7428 
 
 7. 
 
 52105 
 
 11. 
 
 78535217 
 
 4. 
 
 6321 
 
 8. 
 
 43246 
 
 12. 
 
 852106721 
 
 5. 
 
 8357 
 
 9. 
 
 785625 
 
 13. 
 
 12345678935 
 
 19. Having learned to read numerical words, the 
 pupils are now prepared to write them. From the 
 principles we have given we derive the following 
 
 RULE FOR NOTATION. 
 
 Begin at the left, and write each figure in order towards 
 the right, giving each figure its proper place, and filling 
 the vacant places with ciphers. 
 
 1. Write the number four thousand three hundred 
 and- seven. 
 
 Solution. — We write the 4 in thousands' place, the operation. 
 3 in hundreds' place, the 7 in units' place, and, since 4307 
 
 there arc no tens, we write a naught in tens' place. 
 
28 NOTATION. 
 
 Write the following in figures : 
 
 2. Three thousand and seventy-five. Ans. 3075. 
 
 3. Five thousand six hundred and fifty. 
 
 4. Seven thousand eight hundred and four. 
 
 5. Twenty-three thousand four hundred and ninety. 
 
 6. Twenty-five thousand three hundred and seven. 
 
 7. Two hundred and six thousand four hundred and 
 six. 
 
 8. Four hundred and eighty-six thousand nine hun- 
 dred and eight. 
 
 9. Seven hundred and forty-three thousand four hun- 
 dred and ninety. 
 
 10. Two millions, three hundred thousand four hun- 
 dred and eighty. 
 
 11. Four millions, five hundred and six thousand and 
 twenty-five. 
 
 12. Six billions, six millions, six thousands, six hun- 
 dred and six. 
 
 Remark. — Pupils should be drilled in exercises like those given, 
 until they can read and write numbers readily. 
 
 ROMAN NOTATION. 
 
 The Eoman Method of Notation employs seven let- 
 ters of the Eoman alphabet. Thus, I represents one ; 
 V, five; X, ten; L, fifty ; C, one hundred; D, five hun* 
 dred ; M, one thousand. 
 
 To express other numbers these characters are com- 
 bined according to the following principles : — 
 
 1. Every time a letter is repeated its value is repeated. 
 
 2. When a letter is placed before one of greater value, the 
 difference of their values is the number represented. 
 
 3. When a letter is placed after one of a greater value, 
 the sum of their values is the number represented. 
 
NOTATION. 
 
 29 
 
 4. A dash 'placed over a letter increases its value a thou- 
 %and fold. Thus, VII denotes seven thousand. 
 
 ROMAN TABLE. 
 
 I 
 
 II 
 
 III 
 
 IV 
 
 V 
 
 VI 
 
 VII 
 
 VIII 
 
 IX 
 
 X 
 
 XI 
 
 XIV 
 
 XV 
 
 XIX 
 
 XX 
 
 One. 
 
 Two. 
 
 Three. 
 
 Four. 
 
 Five. 
 
 Six. 
 
 Seven. 
 
 Eight. 
 
 Nine. 
 
 Ten. 
 
 Eleven. 
 
 Fourteen. 
 
 Fifteen. 
 
 Nineteen. 
 
 Twenty 
 
 XXX 
 
 XL 
 
 L 
 
 LX 
 
 LXX 
 
 XC 
 
 c 
 cc 
 
 D 
 
 DC 
 
 DCCCC 
 
 M 
 
 MM 
 
 MCLX 
 
 Thirty. 
 
 Forty. 
 
 Fifty. 
 
 Sixty. 
 
 Seventy. 
 
 Ninety. 
 
 One hundred. 
 
 Two hundred. 
 
 Five hundred. 
 
 Six hundred. 
 
 Nine hundred. 
 
 One thousand. 
 
 Two thousand. 
 
 One thousand one hun' 
 
 MDCCCLIX1859. [dred and sixty 
 
 The Eoman Method is named after the Romans, who 
 invented and used it.. It is now employed to denote 
 the chapters and sections of books, pages of preface and 
 introduction, and in other places for prominence and 
 distinction. 
 
 LUMBERMEN'S NOTATION. 
 
 Lumbermen in marking lumber employ a modifica- 
 tion of the Roman Method of Notation. The first four 
 characters are like the Roman ; the others are as 
 follows ; 
 
 A Al All AIM X X XI 
 
 5 6 7 8 9 10 11 
 
 A Al All AW XIX X X\ M 
 
 15 16 17 18 19 20 21 
 
 XII 
 
 12 
 
 22 
 
 80 
 
 90 
 
 28 
 
 m 
 
 100 
 
 ///TtTTi 
 200 
 
 XIII 
 
 13 
 
 XIII 
 23 
 
 ^\ M m JK\III XMIII Ifk X W^ "K 
 
 25 26 2" 28 29 30 40 60 fin 
 
 60 
 
 14 
 
 w 
 
 24 
 70 
 
30 ADDITION. 
 
 SECTION II. 
 ADDITION. 
 
 20. Addition is the process of finding the sum of 
 two or more numbers. 
 
 21. The Sum is a number which contains as many 
 units as the numbers added. 
 
 22. The sign of Addition is -j-> and is read plus. 
 The sign of Equality is = , and is read equals, or equal 
 to. Thus, 4 -j- 5 = 9, is read 4 plus 5 equals 9. 
 
 Case I. 
 
 23. To add when the sum of* a column is not more 
 than nine of that column. 
 
 24. CLASS I. —Problems of one column. 
 
 1. What is the sum of 2, 3, and 4 ? 
 
 OPERATION. 
 
 Solution — We write the numbers one under an- 2 
 
 itther and commence at the bottom to add. 4 and 3 3 
 
 are 7 and 2 are 9. Hence the sum is nine. 4 
 
 9 
 
 EXAMPLES FOR PRACTICE. 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 (5.) 
 
 (6.) 
 
 (7.) 
 
 (8.) 
 
 2 
 
 4 
 
 6 
 
 1 
 
 7 
 
 3 
 
 1 
 
 1 
 
 1 
 
 
 
 5 
 
 
 
 4 
 
 2 
 
 3 
 
 3 
 
 3 
 
 3 
 
 2 
 
 2 
 
 6 
 
 (9.) (10.) (11.) (12.) (13.) (14.) (15.) (16.) (17.) 
 324623553 
 512014212 
 143262234 
 
ADDITION. 31 
 
 18. What is the sum of 2, 0, 3, 4? 
 
 19. What is the sum of 3, 1, 0, 2, 3? 
 
 20. What is the sum of 2, 2, 3, 0, 1 ? 
 
 21. What is the sum of 4, 1, 0, 2, 1? 
 
 22. What is the sum of 3, 0, 2, 0, 1, 3 ? 
 
 25. CLASS II.— Problems of more than one 
 column. 
 
 1. What is the sum of 21, 32, 43 ? 
 
 Solution. — We write the numbers so that units 
 stand under units, and tens under tens, and com- 
 mence at the right to add. The sum of the units is 
 3 and 2 are 5 and 1 are 6, which we write in units' 
 place. The sum of the tens is 4 and 3 are 7 and 2 
 are 9, which we write in tens' place. Hence the sum 
 is 96. 
 
 OPERATION. 
 21 
 32 
 
 43 
 
 96 
 
 
 EXAMPLES 
 
 FOR 
 
 PRACTICE. 
 
 
 (2-) 
 
 (3.) 
 
 (*•) 
 
 (5.) 
 
 (6.) 
 
 31 
 
 20 
 
 34 
 
 12 
 
 15 
 
 23 
 
 14 
 
 20 
 
 23 
 
 40 
 
 24 
 
 25 
 
 15 
 
 54 
 
 34 
 
 78 
 
 
 
 
 
 (7.) 
 
 (8.) 
 
 (9.) 
 
 (10.) 
 
 (11.) 
 
 121 
 
 214 
 
 610 
 
 234 
 
 361 
 
 213 
 
 312 
 
 156 
 
 432 
 
 215 
 
 432 
 
 153 
 
 213 
 
 123 
 
 123 
 
 766 
 
 (12.) 
 
 (13.) 
 
 (14.) 
 
 (15.) 
 
 (16.) 
 
 612 
 
 314 
 
 712 
 
 416 
 
 201 
 
 105 
 
 212 
 
 150 
 
 141 
 
 305 
 
 271 
 
 271 
 
 137 
 
 222 
 
 281 
 
 # 
 
32 
 
 (17.) 
 
 2021 
 
 18.) 
 5234 
 
 (19.) 
 6141 
 
 (20.) 
 7124 
 
 (21.) 
 6214 
 
 3514 
 
 1321 
 
 1213 
 
 1321 
 
 322 
 
 2361 
 
 2141 
 
 2032 
 
 2042 
 
 1211 
 
 (22.) 
 34123 
 
 (23.) 
 41210 
 
 (24.) 
 50273 
 
 (25.) 
 1234 
 
 (26.) 
 23071 
 
 14310 
 
 13025 
 
 17202 
 
 4012 
 
 12303 
 
 20341 
 
 21613 
 
 21310 
 
 3701 
 
 20413 
 
 11111 
 
 12030 
 
 10101 
 
 1020 
 
 21210 
 
 Kequired the sum 
 
 27. Of 2031, 1234, 3122, and 1010. 
 
 28. Of 1207, 3040, 2430, and 2112. 
 
 29. Of 2051, 3027, 1500, and 1320. 
 
 30. Of 21021, 2712, 12032, 102, and 21. 
 hi. Of 12201, 23021, 2142, and 12012. 
 
 Case II. 
 
 26. To add when the sum of* any column is more 
 than nine. 
 
 27. CLASS I.— Problems of one column. 
 
 1. What is the sum of 7, 6, and 8 ? 
 
 Solution. — We write the numbers one under 
 the other, and commence at the bottom to add. 8 
 and 6 are 14 and 7 are 21. We place the 1 
 under the column, and the 2 in tens' place. 
 
 OPERATION. 
 
 7 
 6 
 8 
 
 21 Ans. 
 
 13 
 
 EXAMPLES FOR PRACTICE. 
 
 (2.) 
 
 (3.) 
 
 (4-) 
 
 (5.) 
 
 (6.) 
 
 (7-] 
 
 3 
 
 7 
 
 8 
 
 6 
 
 8 
 
 3 
 
 4 
 
 2 
 
 2 
 
 3 
 
 2 
 
 9 
 
 6 
 
 5 
 
 7 
 
 7 
 
 6 
 
 7 
 
ADDITION. 
 
 33 
 
 (8.) 
 
 5 
 4 
 3 
 2 
 
 (9.) 
 
 7 
 3 
 8 
 5 
 
 (10.) 
 
 6 
 3 
 
 8 
 
 7 
 
 (ll.) 
 4 
 3 
 6 
 5 
 
 (12.) 
 
 7 
 3 
 8 
 5 
 
 (13.) 
 6 
 7 
 3 
 8 
 
 (14.) 
 
 (15.) 
 
 (16.) 
 
 (17.) 
 
 (18.) 
 
 (19.) 
 
 8 
 
 3 
 
 7 
 
 2 
 
 4 
 
 7 
 
 7 
 
 7 
 
 1 
 
 
 
 5 
 
 8 
 
 3 
 
 2 
 
 3 
 
 7 
 
 6 
 
 9 
 
 2 
 
 5 
 
 5 
 
 8 
 
 7 
 
 6 
 
 4 
 
 5 
 
 6 
 
 9 
 
 8 
 
 5 
 
 Bequired the sum 
 
 20. Of 6, 7, 5, 3, 2, and 4. 
 
 21. Of 3, 2, 7, 4, 6, and 7. 
 
 22. Of 3, 4, 5, 6, 7, and 8. 
 
 23. Of 4, 5, 6, 2, 3, and 5. 
 
 24. Of 2, 7, 3, 1, 4, and 6. 
 
 25. Of 3, 5, 4, 3, 2, and 4. 
 
 26. Of 3, 6, 7, 2, 1, and 4. 
 
 27. Of 1, 3, 2, 7, 8, and 5. 
 
 28. Of 8, 2, 4, 6, 5, and 6. 
 
 29. Of 3, 6, 5, 3, 7, and 2. 
 
 30. Of 4, 3, 2, 5, 6, and 7. 
 
 31. Of 6, 2, 7, 4, 5, and 8. 
 
 2S. CLvdSS II— Problems of more than ona 
 column. 
 
 1. What is the sum of 65, 46, and 32? 
 
 Solution 1 . — We write the numbers units under 
 units and tens under tens, and commence at the 
 right to add. 2 and 6 are 8 and 5 are 13, units, 
 which equal 1 ten and 3 units : we write the 3 
 units under the column of units, and add the 1 
 ten to the column of tens. 3 and 1 are 4 and 4 
 are 8, and 6 are 14, tens, which equal 1 hundred 
 and 4 tens ; we write the 4 tens in tens' place, 
 and the 1 hundred in hundreds' place, and we 
 have 143. 
 
 OPERATION. 
 
 65 
 46 
 32 
 
 143 Ans. 
 
34 ADDITION. 
 
 Solution 2. — After the pupil is familiar with the above solution 
 he may abbreviate it thus: 2 and 6 are 8 and 5 are 13; we write 
 the 3 and add the 1. One and 3 are 4, and 4 are 8, and 6 are 14, 
 which v T e write. 
 
 EXAMPLES FOR PRACTICE. 
 
 (2.) 
 43 
 
 (3.) 
 
 27 
 
 (4.) 
 37 
 
 (5.) 
 43 
 
 (6.) 
 58 
 
 (7-) 
 76 
 
 38 
 
 56 
 
 25 
 
 49 
 
 36 
 
 24 
 
 81 
 
 
 
 
 
 
 (8.) 
 23 
 
 (9.) 
 28 
 
 (10.) 
 34 
 
 (11.) 
 44 
 
 (12.) 
 
 82 
 
 (13.) 
 18 
 
 36 
 
 51 
 
 47 
 
 56 
 
 17 
 
 71 
 
 47 
 
 35 
 
 22 
 
 31 
 
 45 
 
 49 
 
 (14.) 
 247 
 
 (15.) 
 462 
 
 (16.) 
 442 
 
 (17.) 
 756 
 
 (18.) 
 361 
 
 (19.) 
 826 
 
 358 
 
 379 
 
 867 
 
 482 
 
 484 
 
 108 
 
 (20.) 
 317 
 
 (21.) 
 424 
 
 (22.) 
 365 
 
 (23.) 
 813 
 
 (24.) 
 
 678 
 
 <25.) 
 725 
 
 452 
 
 536 
 
 407 
 
 791 
 
 123 
 
 146 
 
 324 
 
 817 
 
 324 
 
 142 
 
 414 
 
 234 
 
 (26.) 
 463 
 
 (27.) 
 282 
 
 (28.) 
 365 
 
 (29.) 
 216 
 
 (30.) 
 417 
 
 (31.) 
 318 
 
 217 
 
 187 
 
 149 
 
 418 
 
 282 
 
 182 
 
 345 
 
 208 
 
 372 
 
 732 
 
 479 
 
 479 
 
 (32.) 
 729 
 
 (33.) 
 321 
 
 (34.) 
 242 
 
 (35.) 
 813 
 
 (36.) 
 183 
 
 (37.) 
 815 
 
 538 
 
 467 
 
 517 
 
 916 
 
 517 
 
 581 
 
 212 
 
 213 
 
 343 
 
 732 
 
 648 
 
 186 
 
 400 
 
 457 
 
 525 
 
 145 
 
 422 
 
 307 
 

 
 ADDITION. 
 
 
 
 (38.) 
 
 (39.) 
 
 (40.) 
 
 (41.) 
 
 (42.) 
 
 (43.) 
 
 361 
 
 217 
 
 678 
 
 678 
 
 489 
 
 289 
 
 163 
 
 721 
 
 321 
 
 910 
 
 201 
 
 303 
 
 725 
 
 548 
 
 473 
 
 112 
 
 232 
 
 132 
 
 643 
 
 918 
 
 258 
 
 814 
 
 425 
 
 333 
 
 146 
 
 172 
 
 345 
 
 756 
 
 267 
 
 456 
 
 (44.) 
 
 (45.) 
 
 (46.) 
 
 (47.) 
 
 (48.) 
 
 (49.) 
 
 4567 
 
 1718 
 
 2526 
 
 3343 
 
 4243 
 
 1525 
 
 8910 
 
 1920 
 
 2728 
 
 5363 
 
 4546 
 
 3545 
 
 1112 
 
 2122 
 
 9303 
 
 7389 
 
 4748 
 
 5565 
 
 3456 
 
 2324 
 
 1323 
 
 4041 
 
 9505 
 
 7585 
 
 (50.) 
 
 (51.) 
 
 (52.) 
 
 (53.) 
 
 (54.) 
 
 (55.) 
 
 5960 
 
 7374 
 
 8163 
 
 8124 
 
 2185 
 
 6215 
 
 6162 
 
 5789 
 
 2738 
 
 1792 
 
 6727 
 
 8372 
 
 3646 
 
 2100 
 
 2543 
 
 8547 
 
 9858 
 
 5728 
 
 5666 
 
 4731 
 
 7342 
 
 3218 
 
 2832 
 
 6217 
 
 7869 
 
 2578 
 
 1856 
 
 4002 
 
 1479 
 
 1234 
 
 
 (56.) 
 
 (57.) 
 
 (58.) 
 
 
 
 48721 
 
 32 
 
 173 
 
 67321 
 
 
 
 32578 
 
 82573 
 
 73214 
 
 
 
 41625 
 
 21 
 
 289 
 
 84366 
 
 
 
 78321 
 
 47020 
 
 92785 
 
 
 
 47856 
 
 21832 
 
 12346 
 
 
 35 
 
 59. Find the sum of 2185-f 6357+4832+6719+4324. 
 
 60. Find the sum of 4344+4647+4849+5051+5253. 
 
 61. Find the sum of 6432+7253+2187+6730+5087- 
 
 62. Find the sum of 2426+3275+8397+2547+8037, 
 
 63. Find the sum of 234+6721+853 + 8762+3739. 
 
 64. Find the sum of 834+6737+8321 + 123+9207. 
 
 65. Find the sum of 3246+2109+465+3712+2573 
 
36 ADDITION. 
 
 66. Find the sum of 8213+123+6785+3282+7654. 
 
 67. Find the sum of 123-f 456-f7821-f 6731+1234. 
 
 68. Find the sum of 622+8763+1234+5678+910123. 
 
 69. Find the sum of 23456+12345+70205+21846+ 
 31082. 
 
 PRACTICAL PROBLEMS. 
 
 1. Mary has 15 apples and John has 23 apples: how 
 many have they both ? 
 
 Solution. — If Mary lias 15 apples and John operation. 
 
 has 23 apples, they both have the sum of 15 15 
 
 apples and 23 apples, which is 38 apples. 23 
 
 Note. — Very young pupils may say, they both 38 Ans. 
 
 have the sum of 15 apples and 23 apples, which is 
 38 apples. 
 
 2. There were 25 robins on one tree and 36 robins on 
 another tree; how many robins were there on both 
 trees ? 
 
 3. AYiiiie has 36 cents in one pocket and 45 cents in 
 the other; how many has he in both pockets? 
 
 4. A little boy had 37 walnuts, and then picked 56 
 more ; how many walnuts did he then have ? 
 
 5. Emma's doll cost 95 cents, and a little cradle for it 
 cost 225 cents j how much did both cost ? 
 
 6. There were 48 roses on one bush and 39 roses on 
 another bush ; how many roses were there on both 
 bushes? 
 
 7. A little girl read 146 words one day and 178 words 
 the next day j how many words did she read both 
 days ? 
 
 8. Harry had 246 cents in his money-box, and his 
 uncle gave him 175 cents; how many cents had he 
 then? 
 
 9. Peter's kite arose 436 feet, and Andrew's kite went 
 58 feet higher ; how high did Andrew's kite arise ? 
 
 10. Edward took 692 steps in going to school, and 
 
ADDITION. 37 
 
 Mary took 742 steps ; how many steps did they both 
 take ? 
 
 11. Mary's garden contains 47 roses, 39 pinks and 
 52 lilies; how many flowers are in Mary's garden? 
 
 12. Sallie spelled 25 words correctly, Jennie 36 words, 
 and Maggie 28 words; how many did they all spell 
 correctly ? 
 
 13. Charlie wrote 346 words last week and 378 words 
 this week; how many words did he write in the two 
 weeks ? 
 
 14. Minnie saw 46 swallows in a flock, and Maggie 
 saw 54 swallows in another flock; how many swallows 
 did they both see ? 
 
 15. Frank says he took 627 steps in going to school, 
 and only 596 steps in coming from school ; how many 
 steps did he take in all ? 
 
 16. My father has 6 horses, 13 cows, and 46 sheep ; 
 how many animals has he in all? 
 
 17. Emma's new reader contains 46 pictures, and 
 Ella's contains 78 pictures; how many pictures are 
 there in both of these readers ? 
 
 18. Edward's top cost 25 cents, his whip cost 43 
 cents, and his ball cost 75 cents; how many cents did 
 they all cost ? 
 
 19. Albert's father owned 27 little pigs, and Peter's 
 father owned 34 little pigs ; how many little pigs had 
 they both ? 
 
 20. My father gave 215 cents for my cap, 365 cents 
 for my vest, and 625 cents for my coat ; how many cents 
 did he give for them all ? 
 
 21. One old hen had 17 little chickens, another had 
 15 little chickens, and another 16; how many chickens 
 did the three hens have ? 
 
 22. Henry learned seventy-five words one week and 
 eighty-four words the next week ; how many words did 
 he learn both weeks ? 
 
88 ADDITION. 
 
 23. Maria has fifty-seven cents in her money-bank, 
 and her aunt put twenty-five cents more in the bank ; 
 how many cents did she then have ? 
 
 24. There were sixteen robins in a tree, twenty-four 
 on the barn, and thirty-nine in the meadow; how many 
 robins were there in all ? 
 
 25. Julia gave a poor old soldier ninety-six cents, 
 Annie gave him seventy-seven cents, and Carrie gave 
 him one hundred and seventeen cents ; how much did 
 the old soldier receive ? 
 
 PRACTICAL PROBLEMS. 
 
 1. A gave 27 dollars for a cow, 45 dollars for an ox, 
 and 150 dollars for a horse ; what did they all cost? 
 
 2. A has 120 acres of land, B has 310 acres, C ha? 
 516 acres, and D has 715 acres; how many acres have 
 they together? 
 
 3. There are 31 days in January, 28 in February, 31 
 in March, and 30 in April ; how many days in these 
 four months? 
 
 4. A man travelled 215 miles one week, 195 the next 
 week, 273 the next, and 378 the next ; how far did he 
 travel ? 
 
 5. A weighs 127 pounds, B weighs 215 pounds, C 
 176 pounds, D 184 pounds, and E 234 pounds ; what is 
 the sum of their weights ? 
 
 6. A farmer raised 576 bushels of corn, 918 bushels 
 of oats, 3149 bushels of wheat, and 2785 bushels of rye; 
 how many bushels did he raise in all? 
 
 7. A owns 214 acres of land, B owns 719 acres, C 
 owns 2136 acres, and D owns 372 acres ; how many 
 acres do they together own ? 
 
 8. A bought a horse for 168 dollars, and a carriage 
 for 376 dollars, and sold them so as to gain 89 dollars; 
 what did he receive? 
 
ADDITION. 39 
 
 9. A drover had 327 sheep, 496 cows, 819 pigs, 123 
 oxen, and 216 horses in his drove; how many animals 
 had he in the drove ? 
 
 10. There are 39 books and 929 chapters in the Old 
 Testament, and 37 books and 260 chapters in the ]S"ew 
 Testament; how many are there in both? 
 
 11. In an orchard 87 trees bear apples, 26 bear 
 peaches, 38 bear plums, and 17 bear cherries; how 
 many trees are there in the orchard ? 
 
 12. Mr. Wilson's farm is worth 3720 dollars, his bank 
 stock is worth 1250 dollars, and he has 7257 dollars in 
 money ; how much is he worth ? 
 
 13. A man bought a farm for 7500 dollars, paid 6550 
 dollars for building a house and barn, and then sold it 
 so as to gain 725 dollars; what did he receive for it? 
 
 14. Harvey bought a knife for 37 cents, a hoop for 
 75 cents, a book for 68 cents, and a top for 87 cents, 
 he sold them at a gain of 23 cents ; what did he receive 
 for them ? 
 
 15. William lends his brother 3275 cents, his sister 
 4287 cents, his father 3851 cents, and has 4892 cents 
 left; how much money had he? 
 
 16. In one book there are 725 pages, in another book 
 there are 327 pages, and in another book there are as 
 many as in both the former; how many pages in all ? 
 
 17. A merchant bought cloth for 756 dollars, silk for 
 859 dollars, muslin for 367 dollars, and calico for 255 
 dollars ; how much did they all cost? 
 
 18. A paid 325 dollars for a span of horses, and 248 
 dollars more than this for a carriage; for how much 
 must he sell them both to gain 275 dollars? 
 
 19. A gains in one year 465 dollars, B gains 136 
 dollars more than A, and C gains as much as A and 
 B both; how much did B gain? how much did G 
 gain ? how much did they all gain ? 
 
40 
 
 SUBTRACTION. 
 
 SECTION III. 
 SUBTEACTIOK 
 
 29. Subtraction is the process of finding the differ' 
 ence between two numbers. 
 
 30. The Difference, or Remainder, is the number 
 of units more in the greater than in the less. 
 
 31. The Minuend is the number from which we sub- 
 tract. The Subtrahend is the number to be subtracted. 
 
 32. The sign of Subtraction is — , and is read minus. 
 It denotes that the number after the sign is to be sub- 
 tracted from the one before it. 
 
 Case I. 
 
 33. To subtract when no figure of the subtrahend 
 expresses more units than its figure in the minu- 
 end. 
 
 34. CLASS I.— When the subtrahend is one 
 figure. 
 
 1. Subtract 4 from 9. 
 
 Solution. — We write the 4 under the 9 and 
 Bay, 4 units from 9 units leave 5 units, which 
 w; write beneath. 
 
 OPERATION. 
 
 9 
 
 5 Ana. 
 
 EXAMPLES FOR PRACTICE- 
 
 (2) 
 
 (3) 
 
 (4.) 
 
 (5.) 
 
 (6.) 
 
 (7.) 
 
 5 
 
 7 
 
 5 
 
 6 
 
 8 
 
 7 
 
 2 
 
 3 
 
 3 
 
 2 
 
 5 
 
 5 
 
 (8) 
 
 (9.) 
 
 (10.) 
 
 (11.) 
 
 (12.) 
 
 W 
 
 8 
 
 7 
 
 6 
 
 9 
 
 8 
 
 9 
 
 3 
 
 4 
 
 3 
 
 6 
 
 2 
 
 7 
 
SUBTRACTION. 
 
 41 
 
 14. Subtract 3 from 9; 6 from 17; 7 from 19 ; 8 
 from 19. 
 
 15. Subtract 2 from 14; 4 from 9; 8 from 18; 6 
 from 19. 
 
 16. Subtract 7 from 18; 5 from 17; 6 from 18; 5 
 from 16. 
 
 17. Subtract 8 from 19; 5 from 16; 9 from 19; 4 
 from 17. 
 
 35. CLASS II— When each term is two or more 
 figures. 
 
 1. Subtract 24 from 67. 
 
 Solution — We write the 24 under the 67, units 
 under units, and tens under tens, and commence 
 at the right to subtract. 4 units from 7 units 
 leave 3 units, 2 tens from 6 tens leave 4 tens; 
 hence the remainder is 4 tens and 3 units, or 
 forty-three. 
 
 OPERATION. 
 
 67 
 24 
 
 43 Ans. 
 
 EXAMPLES FOR PRACTICE. 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 (5.) 
 
 (6.) 
 
 (7.) 
 
 58 
 
 86 
 
 72 
 
 53 
 
 46 
 
 76 
 
 35 
 
 24 
 
 41 
 
 22 
 
 15 
 
 24 
 
 (8.) 
 
 (9.) 
 
 (10.) 
 
 (11.) 
 
 (12.) 
 
 (13.) 
 
 49 
 
 67 
 
 85 
 
 97 
 
 86 
 
 99 
 
 27 
 
 26 
 
 52 
 
 25 
 
 73 
 
 25 
 
 (14.) 
 
 (15.) 
 
 (16.) 
 
 (17.) 
 
 (18.) 
 
 (19.) 
 
 625 
 
 456 
 
 763 
 
 617 
 
 767 
 
 896 
 
 312 
 
 215 
 
 512 
 
 215 
 
 123 
 
 432 
 
 (20.) 
 
 (21.) 
 
 (22.) 
 
 (23.) 
 
 (24.) 
 
 (25. J 
 
 872 
 
 725 
 
 857 
 
 907 
 
 840 
 
 876 
 
 161 
 
 413 
 
 654 
 
 205 
 
 320 
 
 345 
 
 4* 
 
42 
 
 
 
 SUBTRACTION. 
 
 
 
 (26.) 
 
 (27.) 
 
 (28.) (29.) 
 
 (30. j 
 
 (31.) 
 
 279 
 
 807 
 
 796 736 
 
 967 
 
 875 
 
 136 
 
 502 
 
 452 432 
 
 234 
 
 345 
 
 (32.) 
 
 (33.) 
 
 (34.) (35.) 
 
 (36.) 
 
 (37.) 
 
 8763 
 
 9076 
 
 3769 5076 
 
 4872 
 
 7659 
 
 4321 
 
 4054 
 
 1546 3075 
 
 2342 
 
 3237 
 
 (38.) 
 
 (39.) 
 
 (40.) (41.) 
 
 (42.) 
 
 (43.) 
 
 8769 
 
 4876 
 
 8275 8799 
 
 8591 
 
 6857 
 
 3257 
 
 2142 
 
 3251 2542 
 
 7230 
 
 1234 
 
 (44. J 
 
 (45.) 
 
 (46.) 
 
 (47.) 
 
 (48) 
 
 82345 
 
 57596 
 
 72578 
 
 27397 
 
 67385 
 
 22121 
 
 21321 
 
 41362 
 
 22315 
 
 24123 
 
 (49.) 
 
 (50.) 
 
 (51.) 
 
 (52.) 
 
 (53.) 
 
 57897 
 
 67858 
 
 87578 
 
 96754 
 
 81296 
 
 21472 
 
 32721 
 
 21335 
 
 21423 
 
 20135 
 
 ■ 
 (54.) 
 
 (55.) 
 
 (56.) 
 
 (57.) 
 
 (58.) 
 
 253786 
 
 472589 
 
 87695 
 
 56728 
 
 98785 
 
 213123 
 
 212423 
 
 23542 
 
 21306 
 
 21342 
 
 (59.) 
 
 (60.) 
 
 (61) 
 
 (62.) 
 
 (63.) 
 
 373967 
 
 873972 
 
 72587 
 
 95837 
 
 89976 
 
 212851 
 
 132421 
 
 51234 
 
 51321 
 
 32742 
 
 Subtract 
 
 64. 314 from 678. 
 
 65. 425 from 658. 
 
 66. 561 from 789. 
 
 67. 254 from 576. 
 18. 437 from 869. 
 
 Subtract 
 
 69. 1235 from 376$. 
 
 70. 3726 from 4969. 
 
 71. 2532 from 8748. 
 
 72. 4720 from 87856. 
 
 73. 12345 from 68799. 
 
SUBTRACTION. 43 
 
 Case II. 
 
 36. To subtract when a figure in tlie subtrahend 
 expresses more than the corresponding figure in 
 the minuend. 
 
 37. CLASS I.— When the subtrahend is one 
 figure. 
 
 1. Subtract 8 from 12. operation. 
 
 Solution. — We write the 8 under the 12 ; then 12 
 
 8 from twelve is four. 8 
 
 4 Ans. 
 
 EXAMPLES FOR PRACTICE. 
 
 (20 
 12 
 
 (3.) 
 12 
 
 (4.) 
 13 
 
 (5.) 
 14 
 
 (6.) 
 10 
 
 11 
 
 (8.) 
 
 10 
 
 (9.) 
 13 
 
 9 
 
 7 
 
 8 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 4 
 
 (10.) 
 15 
 
 (11.) 
 15 
 
 (12.) 
 16 
 
 (13.) 
 13 
 
 (14.) 
 16 
 
 (15.) 
 17 
 
 (16.) 
 16 
 
 (17.) 
 14 
 
 7 
 
 8 
 
 9 
 
 7 
 
 8 
 
 9 
 
 6 
 
 5 
 
 (18.) 
 10 
 
 (19.) 
 17 
 
 (20.) 
 13 
 
 (21.) 
 11 
 
 (22.) 
 10 
 
 (23.) 
 19 
 
 (24.) 
 14 
 
 (25.) 
 16 
 
 3 
 
 G 
 
 5 
 
 4 
 
 2 
 
 9 
 
 8 
 
 7 
 
 3S. CLASS II.— When each term is tivo or mors 
 figures. 
 
 1. Subtract 45 from 82. 
 
 Solution 1. — We write the 45 under 82, operation. 
 placing units under units, and tens under tens, 82 
 
 and commence at the right to subtract. We can- 45 
 
 not subtract 5 units from 2 units; we will there- _ . 
 
 fore take 1 ten from the 8 tens, leaving 7 tens; 
 1 ten equals 10 units, which added to 2 units 
 equals 12 units; 5 units from 12 units leave 7 units; 4 tens from 7 
 tens leave 3 tens; hence, the remainder is 37. 
 
 Solution 2. — We cannot take 5 units from 2 units; we will there- 
 fore add 10 units to the 2 units, making 12 units; 5 units from 13 
 
44 SUBTRACTION. 
 
 units leave 7 units. Now, since we have added 10 units, or 1 ten, to 
 the minuend, our remainder will be 1 ten too large; hence, we must 
 add 1 ten to the subtrahend; 1 ten and 4 tens are 5 tens, 5 tens from 
 8 tens leave 8 tens. 
 
 Note. — In practice we solve thus; 5 from 2 we cannot take, but 5 from 
 12 leaves 7, 4 and 1 are 5, and 5 from 8 leaves 3. 
 
 39. From the preceding explanations we have the 
 following general rule. 
 
 Eule. — 1. Write the smaller number under the larger, 
 with units under units, tens under tens, etc., and commence 
 at the right to subtract. 
 
 2. Take the number denoted by each figure of the subtra- 
 hbnd from the number denoted by the corresponding figure 
 of the minuend, and write the result beneath. 
 
 3. If the number denoted by a figure in the subtrahend is 
 greater than the number denoted by the corresponding figure 
 in the minuend, add 10 to the latter and then subtract, and 
 add 1 to the next left-hand place in the subtrahend. 
 
 40. Proof. — Add cne remainder to the subtrahend ; 
 the sum will equal the minuend if the work is correct. 
 
 EXAMPLES FOR PRACTICE. 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 (5.) 
 
 (6.) 
 
 (7.) 
 
 73 
 
 64 
 
 32 
 
 41 
 
 53 
 
 62 
 
 25 
 
 27 
 
 14 
 
 26 
 
 28 
 
 28 
 
 (8.) 
 
 (9.) 
 
 (10.) 
 
 (11.) 
 
 (12.) 
 
 (13.) 
 
 75 
 
 31 
 
 57 
 
 63 
 
 87 
 
 95 
 
 26 
 
 18 
 
 29 
 
 45 
 
 28 
 
 59 
 
 (14.) 
 
 (15.) 
 
 (16.) 
 
 (17.) 
 
 (18.) 
 
 (19.) 
 
 87 
 
 75 
 
 63 
 
 77 
 
 87 
 
 94 
 
 39 
 
 38 
 
 25 
 
 48 
 
 59 
 
 49 
 
 (20.) 
 
 (21.) 
 
 (22.) 
 
 (23.) 
 
 (24.) 
 
 (25.) 
 
 72 
 
 84 
 
 70 
 
 81 
 
 90 
 
 97 
 
 27 
 
 48 
 
 17 
 
 18 
 
 39 
 
 79 
 
(26.) 
 342 
 
 (27.) 
 573 
 
 (28.) 
 692 
 
 (29.) 
 545 
 
 (30.) 
 826 
 
 (81.) 
 357 
 
 124 
 
 245 
 
 457 
 
 328 
 
 252 
 
 183 
 
 (32.) 
 573 
 
 (33.) 
 
 748 
 
 (34.) 
 835 
 
 (35.) 
 968 
 
 (36.) 
 839 
 
 (37.) 
 538 
 
 248 
 
 375 
 
 573 
 
 675 
 
 584 
 
 394 
 
 (38.) 
 659 
 
 (39.) 
 839 
 
 (40.) 
 547 
 
 (41.) 
 658 
 
 (42.) 
 
 735 
 
 (43.) 
 848 
 
 475 
 
 583 
 
 284 
 
 372 
 
 373 
 
 539 
 
 (44.) 
 524 
 
 (45.) 
 752 
 
 (46.) 
 845 
 
 (47.) 
 307 
 
 (48.) 
 456 
 
 (49.) 
 450 
 
 356 
 
 387 
 
 579 
 
 138 
 
 387 
 
 382 
 
 (50.) 
 854 
 
 (61.) 
 
 943 
 
 (52.) 
 607 
 
 (53.) 
 500 
 
 (54.) 
 
 704 
 
 (55.) 
 403 
 
 396 
 
 765 
 
 309 
 
 325 
 
 507 
 
 285 
 
 (56.) 
 726 
 
 (57.) 
 
 857 
 
 (58.) 
 
 735 
 
 (59.) 
 792 
 
 (60.) 
 
 807 
 
 (61.) 
 650 
 
 387 
 
 389 
 
 558 
 
 295 
 
 328 
 
 357 
 
 (62.) 
 
 3876 
 
 (63.) 
 6385 
 
 (64.) 
 6735 
 
 (65.) 
 
 4076 
 
 (66.) 
 4070 
 
 (67.) 
 4135 
 
 2379 
 
 3527 
 
 2547 
 
 3128 
 
 2137 
 
 1216 
 
 (68.) 
 8672 
 
 (89.) 
 
 5283 
 
 (70.) 
 8175 
 
 (71.) 
 2534 
 
 (72.) 
 6735 
 
 (73.) 
 7219 
 
 3728 
 
 2426 
 
 2836 
 
 1235 
 
 5376 
 
 1972 
 
 (74.) 
 8522 
 
 (75.) 
 7135 
 
 (76.) 
 6347 
 
 (77.) 
 8135 
 
 (78.) 
 7345 
 
 (79.) 
 4372 
 
 6243 
 
 1872 
 
 2503 
 
 £453 
 
 2876 
 
 2583 
 
 (80.) 
 35672 
 
 (81.) 
 43763 
 
 (82.) 
 87253 
 
 (83.) 
 73875 
 
 (84.) 
 63527 
 
 (86.) 
 
 53413 
 
 23828 
 
 24235 
 
 34365 
 
 38376 
 
 14238 
 
 28401 
 
 45 
 
46 
 
 
 
 SUBTRACTION. 
 
 
 
 (86.) 
 
 73285 
 
 (87.) 
 20307 
 
 (88.) 
 87004 
 
 (89.) 
 76500 
 
 (90.) 
 20500 
 
 (91.) 
 37201 
 
 43836 
 
 15231 
 
 34523 
 
 43654 
 
 37254 
 
 23534 
 
 (92.) 
 83030 
 
 (93.) 
 90304 
 
 (94.) 
 50310 
 
 (95.) 
 60204 
 
 (96.) 
 70000 
 
 (97.) 
 100000 
 
 76513 
 
 40372 
 
 30311 
 
 30205 
 
 32463 
 
 1 
 
 PRACTICAL PROBLEMS. 
 1. Mary had 25 roses and gave Sarah 12 of them; 
 how many did Mary then have ? 
 
 OPERATION. 
 
 25 
 12 
 
 13 Ans. 
 
 Solution. — If Mary had 25 roses and gave 
 Sarah 12 of them, Mary then had the difference 
 between 25 roses and 12 roses, which is 13 roses. 
 
 Note. — Quite young pupils may say, Mar} 7 then 
 had the difference between 25 roses and 12 roses, 
 which is 13 roses. 
 
 2. Willie had 34 cents and gave James 18 cents; how 
 many cents did Willie then have? 
 
 3. A little girl had 54 pins and gave her cousin 27 
 of them- how many did she have remaining? 
 
 4. Fifty little robins were sitting on a tree, and 23 of 
 them flew away j how many were then left ? 
 
 5. There were 96 peaches on an old peach-tree, and a 
 gust of wind blew 37 of them off; how many then re- 
 mained on the tree ? 
 
 6. Henry's top and ball cost 120 cents; how much 
 did the top cost, if the ball cost 75 cents ? 
 
 7. Emma's doll and its little cradle cost 320 cents, 
 and the doll cost 95 cents ; how much did the cradle cost ? 
 
 8. There were 87 roses on two rose-bushes; how 
 many roses were there on the second bush, if there were 
 39 roses on the first bush ? 
 
 9. A little girl read 324 words in two days ; if she 
 read 146 words one day, how many did she read the 
 other day ? 
 
SUBTRACTION. 47 
 
 10. Edward and Mary together took 1434 steps in 
 going to school; how many steps did Mary take, if 
 Edward took 692 steps ? 
 
 11. Minnie had 372 cents in her money-bank, and took 
 out 164 cents to give to a little beggar-girl j how many 
 cents remained ? 
 
 12. Andrew's kite arose 494 feet, and this was 58 feet 
 higher than Peter's kite went ; how high did Peter's 
 kite fly ? 
 
 13. Charlie wrote 724 words in two weeks; he wrote 
 346 words the first week; how many words did ho 
 write the second week ? 
 
 14. Mary's new reader contains 76 pictures, and 
 Fanny's contains 92 pictures; how many does Fanny's 
 contain more than Mary's ? 
 
 15. Two little girls picked 74 quarts of blackberries 
 one summer; if one picked 37 quarts, how many quarts 
 did the other pick ? 
 
 16. Thomas said he counted 283 crows in his father's 
 cornfield ; he threw a stone and scared 126 away ; how 
 many then remained ? 
 
 17. Floy and Eugie together took 3000 steps one 
 day ; if Floy took 1786 steps, how many steps did Eugie 
 take ? 
 
 18. Effie and Eddie counted their chestnuts and found 
 they together had 1232 ; now, if Eddie had 675, how 
 many had Effie ? 
 
 19. Mary's mother bought her an arithmetic and 
 slate for 125 cents ; if the slate cost 45 cents, what did 
 the arithmetic cost ? 
 
 20. Herbert's father bought him a cap and coat for 
 850 cents; he paid 125 cents for the cap; how much 
 did he pay for the coat ? 
 
 21. Mr. Xelson's horse and carnage cost four hundred 
 dollars ; what did the horse cost, if the carriage cost 
 two hundred and twenty-five dollars ? 
 
 
iS SUBTRACTION. 
 
 22. Two little girls picked seventy-four quarts of 
 blackberries one summer; if one picked thirty-seven 
 quarts, how many quarts did the other pick? 
 
 23. Mr. Barton raised two thousand bushels of wheat 
 and rye ; how much rye did he raise, if he raised five 
 hundred and sixty-five bushels of wheat ? 
 
 PRACTICAL PROBLEMS. 
 
 1. A man had 78 cows and sold 24 of them; how 
 many cows remained ? 
 
 Solution. — If a man had 78 cows and sold operation. 
 24, there remained the difference between 78 78 
 
 and 24, which we find by subtracting is 54. 24 
 
 Note. — Quite young pupils may merely say, there 54 Ans. 
 
 remains the difference between 78 and 24,which is 54. 
 
 2. A boy had 150 cents and spent 75 cents; how 
 many cents then remained ? 
 
 3. A man had 325 apples and sold 180 apples ; how 
 many had he then ? 
 
 4. Henry had 1735 dollars, lent his brother 854 dol- 
 lars ; how man}'' dollars remained ? 
 
 •5. A bought 570 horses and sold 295 of them; how 
 many remained unsold ? 
 
 6. A and B together had 7256 acres of land; how 
 many had B if A had 3627 ? 
 
 7. Two men have 8570 bushels of grain, and the first 
 has 2846 bushels ; how many has the second ? 
 
 8. Washington was born 1732 and died 1799; how 
 old was he at his death ? 
 
 9. John Adams was born 1735 and died 1826; how 
 old was he at his death ? 
 
 10 Jefferson was born 1743 and died 1826; how old 
 was he at his death ? 
 
 11. Madison was born 1758 and died 1836; how old 
 was he at his death 1 
 
SUBTRACTION. 49 
 
 12. Monroe was born 1758 and died 1831 j how old 
 was he at his death ? 
 
 13. John Quincy Adams was born 1767 and died 1848; 
 now old was he at his death ? 
 
 14. Jackson was born 1767 and died 1845; how old 
 was he at his death ? 
 
 15. A has 5480 bushels of oats, which is 975 bushels 
 more than B has ; how many bushels has B ? 
 
 16. In an army of 50000 men 628 were killed and 
 2596 wounded; how many remained unhurt? 
 
 17. A farmer had 234 hens and bought 367, and then 
 sold 489 ; how many then remained. 
 
 18. A merchant sold goods to the amount of 7580 
 dollars and gained 1396 dollars; what did the goods 
 cost? 
 
 19. A and B have each 1840 acres of land ; A sells B 
 895 acres ; how many has each then ? 
 
 20. A farmer has 1346 sheep and 849 lambs; how 
 many more sheep has he than lambs ? 
 
 21. Mary and Eliza have each 789 cents; if Eliza 
 gives Mary 247 cents, how many will each then have ? 
 
 22. Subtract six hundred and seventy-eight from nine 
 hundred and four. 
 
 23. Add seven hundred and fifteen to five hundred 
 and seventy-three, and subtract the sum from two 
 thousand. 
 
 24. Find the sum of one thousand and ninety-six and 
 five hundred and forty-five, and subtract it from three 
 thousand. 
 
 25. Frank solved four hundred and sixteen problems, 
 and Fanny solved five hundred and three problems; 
 how many did Fanny solve more than Frank ? 
 
 26. A farmer had 2346 bushels of wheat ; he sold one 
 man 687 bushels and another man 1560 bushels; how 
 many bushels did he sell ? how many remained ? 
 
 5 
 
50 PRACTICAL PROBLEMS. 
 
 PRACTICAL PROBLEMS 
 in Addition and Subtraction. 
 
 1. If I have 75 cents in my money -bank, and my 
 uncle puts in 26 cents, how much will be in it then ? 
 
 2. If Willie reads 125 words this week and 187 words 
 next week, how many words will he read in all ? 
 
 3. My father had 236 little chickens, and a mink killed 
 48 of them ; how many remained ? 
 
 4. If I have 438 dollars and give my sister 246 dol- 
 lars, how much will I have remaining? 
 
 5. Mary's father had 360 acres of land and sold 125 
 acres ; how many acres did he then have ? 
 
 6. Peter had 467 dollars and lent his brother 185 dol- 
 lars ; how much did he then have ? 
 
 7. I have 365 cents in my money-bank ; how many 
 must I put in that there may be 400 cents in it ? 
 
 8. Sallie had 72 cents and her brother gave her 
 enough to make her money 134 cents ; how much did her 
 brother give her ? 
 
 9. Carrie's brother teased her because she couldn't 
 tell how many she must add to 245 to make 400 ; can 
 you tell ? 
 
 10. Matilda had 120 cents, her mother gave her 236 
 cents, and then she lent her brother 248 cents ; how 
 many cents did she then have ? 
 
 11. Fannie picked 236 chestnuts, her little brother 
 gave her 78 chestnuts, and she gave 95 to her school- 
 mates ; how many chestnuts remained ? 
 
 12. Mary cried because she couldn't tell her teacher 
 how many she must add to 367 to make 500 ; tell me 
 how many it is. 
 
 13. One morning in going to school I took 726 steps ; 
 how many more would I have taken if I had taken 
 1000 in all ? 
 
 14. My kite was up in the air 436 feet, it then fell 
 185 feet, and then arose 260 feet ; how high was it then ? 
 
BUSINESS PROBLEMS. 51 
 
 BUSINESS PROBLEMS. 
 
 1. I went to a store mid bought a book for 87 cents 
 and a slate for 35 cents ; what did I pay for both of 
 them ? 
 
 2. My mother took me to a store and bought me a top 
 for 15 cents, a cap for 75 cents, and a knife for 45 cents ; 
 what did they all cost ? 
 
 3. William's slate cost 26 cents, his arithmetic 55 cents, 
 his reading-book 48 cents, and his spelling-book 37 
 cents ; what did they all cost ? 
 
 4. I went to a store and bought a knife for 56 cents 
 and gave the storekeeper a dollar bill (100 cents) to pay 
 for it ; how much change did he give me back ? 
 
 5. Mary bought a flower-vase for 375 cents, and handed 
 the storekeeper a five-dollar bill (500 cents) to pay for 
 it ; how much change should she have received ? 
 
 6. Mr. Barnes paid 75 dollars for his watch, and sold 
 it so that he gained 12 dollars ; what did he receive 
 for it ? 
 
 7. Martha's new shawl cost 875 cents ; if she should 
 sell it so as to gain 125 cents, what would she receive 
 for it ? 
 
 8. Mr. Taylor's new house cost him 3675 dollars, and 
 he sold it for 565 dollars more than it cost him ; what 
 did he receive for it ? 
 
 9. My father bought a cow for 38 dollars and sold 
 her for 52 dollars; how much did he gain on the cow ? 
 
 10. Robert Stewart had a coat which cost him 45 dol- 
 lars ; he sold it to Edward Taylor for 37 dollars ; how 
 much did he lose ? 
 
 11. Harry Hartman sold his watch for 67 dollars and 
 lost by the sale 15 dollars ; what did the watch cost him ? 
 
 12. Mary's papa gave her a 5 dollar bill to go a shop- 
 ping; she bought a fan for 75 cents, somo silk for 165 
 cents, and a pair of gloves for 125 cents; how much 
 change did she bring home ? 
 
32 SUBTRACTION, 
 
 PRACTICAL PROBLEMS 
 in Addition and Subtraction. 
 
 1. What is the value of 675 + 432 + 285 + 672? 
 
 2. What is the value of 362 + 486 + 721 — 367 ? 
 
 3. What is the value of 473 + 325 + 604—1206? 
 
 4. What is the value of 3072 + 4861 + 2075 — 6785? 
 
 5. Subtract 1678 from the sum. of 985 and 863. 
 
 6. Subtract the sum of 265 and 381 from the sum of 
 281 and 678. 
 
 7. Subtract 218 + 318 + 418 from 379 + 279+479. 
 
 8. A having 475 dollars earned 220 dollars and then 
 spent 567 ; how much remained ? 
 
 9. Newspapers were first published in 1630; how 
 long have they been published ? 
 
 10. Quills were first used for writing about the year 
 636; how long is it since? 
 
 11. Cotton was first planted in the United States 
 about the year 1769 j how many years since ? 
 
 12. Glass windows, it is said, were first used in Eng- 
 land in 1180 ; how long is it since then ? 
 
 13. A sold his farm for 12450 dollars, which was 1680 
 dollars more than it cost; howjnuch did it cost? 
 
 14. A gave 6500 dollars for his farm and 2560 dollars 
 for his house, and sold them for 12000; what was the 
 gain ? 
 
 15. A farmer had 5600 bushels of corn, and sol'd 1850 
 bushels to A and 2810 to B ; how much remained? 
 
 16. The area of Maine is 30000 square miles, and of 
 New York 46000 ; how much larger is New York than 
 Maine ? 
 
 17. The area of Massachusetts is 7800 square miles, 
 and of Pennsylvania 46000 square miles ; how much 
 larger is Pennsylvania than Massachusetts ? 
 
 18. How much larger are Pennsylvania and Maine 
 together than New York and Massachusetts together? 
 
MULTIPLICATION. 53 
 
 MULTIPLICATION. 
 
 41. Multiplication is the process of finding the re- 
 sult of taking one number as many times as there aro 
 units in another. . 
 
 42. The Multiplicand is the number to be multiplied. 
 
 43. The Multiplier is the number by which we 
 multiply. 
 
 44. The Product is the result obtained. 
 
 44£. The sign of Multiplication is X > and is read 
 multiplied by: thus, 4 X 3 = 12 means 4 multiplied by 3 
 equals 12. The 4 is the multiplicand, 3 is the multiplier, 
 and 12 is the product. 
 
 Note to Teachers. — If the pupils are not familiar with the Mul- 
 tiplication Table, let them now turn to page 20 and learn it. 
 
 Case I. 
 
 45. When the multiplier is one figure. 
 
 CLASS I. — When no product exceeds nine, 
 1. Multiply 34 by 2. 
 
 Solution. — We write the multiplier under the operation. 
 multiplicand, and begin at the right to multiply. 34 
 
 2 times 4 ufcits are 8 units; we write the 8 units 2 
 
 in units' place. 2 times 3 tens are 6 tens ; we qq Ans. 
 
 write the 6 tens in tens' place. 
 
 (2.) (3.) (4.) (5.) 
 
 32 24 14 41 
 
 2 2 2 2 
 
 (6.) (7.) (8.) (9.) 
 
 21 12 23 32 
 
 3 3 3 3 
 
 40. CLvdSS II— When some of the products ex- 
 ceed nine. 
 
 1. Multiply 56 by 4. 
 
 5* 
 
51 MULTIPLICATION. 
 
 Solution 1. — We write the multiplier under operation. 
 the multiplicand and begin at the right to multi- 56 
 
 ply. 4 times 6 units are 24 units, which equal 4 _4 
 
 units and 2 tens. We write the 4 units in units £24 Ans. 
 
 place, and reserve the 2 tens to add to the next 
 product. 4 times 5 tens are 20 tens, plus the 2 tens equal 22 tens, 
 which equal 2 tens and 2 hundreds, which we write in their proper 
 places. Hence the product is 224. 
 
 Solution 2. — 4 times 6 are 24; we write the 4 and add the 2 to 
 the next product. 4 times 5 are 20, and 2 added equal 22 ; hence 
 the product is 224. From this we have the following 
 
 Eule. — Write the multiplier under the multiplicand, 
 draw a line beneath, begin at units, and multiply the num- 
 her denoted by each figure of the multiplicand by the multi- 
 plier, carrying as in addition. 
 
 (2-) 
 25 
 
 (3.) 
 36 
 
 (4.) 
 47 
 
 (5.) 
 73 
 
 (6.) 
 
 28 
 
 3 
 
 2 
 
 3 
 
 2 
 
 4 
 
 63 
 
 (8.) 
 
 75 
 
 (9.) 
 
 36 
 
 (10.) 
 27 
 
 (11.) 
 43 
 
 5 
 
 4 
 
 5 
 
 6 
 
 5 
 
 — 
 
 — 
 
 — 
 
 — — 
 
 ■ 
 
 (12.) 
 
 75 
 2 
 
 (13.) 
 
 86 
 3 
 
 (14.) 
 92 
 4 
 
 (15.) 
 
 76 
 5 
 
 (16.) 
 ,84 
 6 
 
 (17.) 
 73 
 
 (18.) 
 47 
 
 (19.) 
 76 
 
 (20.) 
 
 85 
 
 (21.) 
 
 73 
 
 5 
 
 6 
 
 7 
 
 8 
 
 8 
 
 — 
 
 — 
 
 — 
 
 — 
 
 ■ 
 
 (22.) 
 234 
 
 (23.) 
 425 
 
 (24.) 
 673 
 
 (25.) 
 723 
 
 (26.) 
 351 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 (27.) 
 425 
 
 (28.) 
 314 
 
 (29.) 
 421 
 
 (30.) 
 636 
 
 (31.) 
 
 854 
 
 6 
 
 7 
 
 8 
 
 7 
 
 3 
 
MULTIPLICATION. 
 
 (32.) 
 
 (33.) 
 
 (34.) 
 
 (35.) 
 
 (36.) 
 
 256 
 
 375 
 
 873 
 
 358 
 
 725 
 
 4 
 
 6 
 
 7 
 
 8 
 
 7 
 
 (37.) 
 
 (38.) 
 
 (39.) 
 
 (40.) 
 
 (41.) 
 
 581 
 
 809 
 
 394 
 
 908 
 
 765 
 
 7 
 
 8 
 
 6 
 
 9 
 
 8 
 
 Multiply 
 
 42 
 
 3124 by 4. 
 
 43. 
 
 2856 by 5. 
 
 44. 
 
 7863 by 6. 
 
 45. 
 
 2185 by 7. 
 
 46. 
 
 4182 by 8. 
 
 47. 
 
 3075 by 8. 
 
 48. 
 
 4107 by 9. 
 
 49. 
 
 7685 by 6. 
 
 Multiply 
 
 50. 13257 by 2. 
 
 51. 36072 by 3. 
 
 52. 85761 by 4. 
 
 53. 35167 by 5. 
 
 54. 84307 by 6. 
 
 55. 30754 by 7. 
 
 56. 21836 by 8. 
 
 57. 35168 by 9. 
 
 Case II. 
 47. Wlien the multiplier consists of two or more 
 figures. 
 
 4S. CLASS L— When the multiplier consists of 
 two jig ures. 
 
 1. Multiply 64 by 23. 
 
 Solution 1. — We write the multiplier under 
 the multiplicand, placing units under units, and 
 tens under tens, and begin at the right to mul- 
 tiply. 3 times 4 units are 12 units, which equals 
 1 ten and 2 units ; we write the units under the 
 3, and reserve the 1 ten to add to the next pro- 
 duct. 3 times 6 tens are 18 tens, and 1 ten 
 added equals 19 tens, or 1 hundred and 9 tens, which we write in 
 their proper places. Multiplying 64 by 2 in the same manner, we 
 have 128, and since the 2 is 2 tens we have 128 tens, which we write 
 m its proper place ; then, adding the two products, we have 1472. 
 
 Solution 2.— Three times 4 are 12; we write the 2 and carry the 
 
 OPERATION. 
 
 64 
 23 
 
 192 
 128 
 
 147li Ans. 
 
56 MULTIPLICATION. 
 
 1 : three times 6 are 18, plus the 1 equals 19 ; which we write. 
 Then, 2 times 4 are 8, which we write uuder the 2, and 2 times 6 
 are 12, which we write beside the 8. 
 
 Bule. — 1. Write the multiplier under the multiplicand, 
 placing units under units, tens under tens etc., and begin at 
 the right to multiply. 
 
 2. Multiply the multiplicand by the number denoted by 
 each figure of the multiplier-, writing the first figure of each 
 product under the figure of the multiplier used. 
 
 3. Add togethej the partial products, and their sum will 
 be the entire product. 
 
 Proof. — Multiply the multiplier by the multiplicand ; 
 if the two results ag*'ee, the work is probably correct 
 
 (2.) 
 38 
 
 (3.) 
 43 
 
 73 
 
 (5.) 
 81 
 
 (6.) 
 29 
 
 ( 7 -.) 
 
 57 
 
 23 
 
 24 
 
 35 
 
 67 
 
 82 
 
 75 
 
 (8.) 
 87 
 
 (9.) 
 39 
 
 (10.) 
 
 87 
 
 (11.) 
 29 
 
 (12.) 
 123 
 
 (13.) 
 245 
 
 28 
 
 43 
 
 52 
 
 92 
 
 37 
 
 32 
 
 (14.) 
 436 
 
 (15.) 
 534 
 
 (16.) 
 427 
 
 (17.) 
 426 
 
 (18.) 
 534 
 
 (19.) 
 672 
 
 43 
 
 43 
 
 35 
 
 43 
 
 45 
 
 46 
 
 (20.) 
 725 
 
 (21.) 
 634 
 
 (22.) 
 807 
 
 (23.) 
 475 
 
 (24.) 
 
 709 
 
 (25.) 
 493 
 
 42 
 
 47 
 
 37 
 
 54 
 
 88 
 
 82 
 
 (26.) 
 756 
 
 (27.) 
 762 
 
 (28.) 
 675 
 
 (29.) 
 467 
 
 (30.) 
 
 762 
 
 (31.) 
 
 812 
 
 93 
 
 48 
 
 39 
 
 37 
 
 62 
 
 45 
 
MULTIPLICATION. 
 
 57 
 
 (32.) 
 
 (33.) (34.) 
 
 (35.) 
 
 (36.) (37.) 
 
 1234 
 
 2341 6724 
 
 6357 
 
 7138 2536 
 
 28 
 
 35 42 
 
 35 
 
 52 25 
 
 (38.) 
 
 (39.) (40.) 
 
 (41.) 
 
 (42.) (43.) 
 
 6347 
 
 8192 4736 
 
 4825 
 
 3121 4073 
 
 46 
 
 73 63 
 
 72 
 
 37 46 
 
 
 Multiply 
 
 
 Multiply 
 
 44. 
 
 6538 by 83. 
 
 49. 
 
 4175 by 28. 
 
 45. 
 
 7384 by 45. 
 
 50. 
 
 7186 by 85. 
 
 46. 
 
 2185 by 67. 
 
 51. 
 
 8391 by 94. 
 
 47. 
 
 3407 by 82. 
 
 52. 
 
 2187 by 89. 
 
 48. 
 
 3584 by 46. 
 
 53. 
 
 6543 by 98. 
 
 49. CLASS II.— When the multiplier consists 
 of three figures. 
 
 (1.) 
 
 4126 
 
 (2-) 
 5731 
 
 (3.) 
 7351 
 
 (4) 
 1375 
 
 (5.) 
 5379 
 
 234 
 
 243 
 
 432 
 
 342 
 
 423 
 
 (6.) 
 6725 
 
 (7.) 
 2183 
 
 (8.) 
 7321 
 
 (9.) 
 8193 
 
 (10.) 
 2147 
 
 345 
 
 544 
 
 265 
 
 4/o 
 
 813 
 
 (11.) 
 2143 
 
 (12.) 
 8192 
 
 (13.) 
 2435 
 
 (14.) 
 4167 
 
 (15.) 
 8246 
 
 227 
 
 426 
 
 146 
 
 245 
 
 642 
 
 (16.) 
 7346 
 
 (17.) 
 7516 
 
 (18.) 
 8927 
 
 (19.) 
 4928 
 
 (20.) 
 2076 
 
 643 
 
 571 
 
 352 
 
 816 
 
 437 
 
r>8 
 
 MULTIPLICATION. 
 
 
 (21.) (22.) 
 
 (23.) 
 
 
 (24.) (25.) 
 
 4752 7385 
 
 8492 
 
 
 2937 6473 
 
 185 218 
 
 537 
 
 
 439 567 
 
 Multiply 
 
 
 
 
 Multiply 
 
 26. 14651 by 283. 
 
 
 
 35. 
 
 28352 by 345. 
 
 27, 31251 by 625. 
 
 
 
 36. 
 
 41678 by 287. 
 
 28. 36782 by 234. 
 
 
 
 37. 
 
 34073 by 435. 
 
 29. 43678 by 452. 
 
 
 
 38. 
 
 40735 by 628. 
 
 30. 36507 by 634. 
 
 
 
 39. 
 
 29304 by 789. 
 
 31. 40725 by 365. 
 
 
 
 40. 
 
 90705 by 897. 
 
 32. 32107 by 681. 
 
 
 
 41. 
 
 43445 by 678. 
 
 33. 25697 by 329. 
 
 
 
 42. 
 
 37436 by 835. 
 
 34. 42046 by 456. 
 
 
 
 43. 
 
 88888 by 789. 
 
 50. CLASS III.— When the multiplier consists 
 of more than three figures. 
 
 4137 
 
 (2.) 
 3642 
 
 (3.) 
 6724 
 
 (4-) 
 4183 
 
 (5.) 
 3645 
 
 (6.) 
 4526 
 
 2185 
 
 2531 
 
 3625 
 
 2426 
 
 2841 
 
 2182 
 
 3482 
 
 (8.) 
 2846 
 
 (9.) 
 
 3707 
 
 (10.) 
 4172 
 
 (11.) 
 
 2882 
 
 (12.) 
 8567 
 
 2534 
 
 2528 
 
 2851 
 
 2174 
 
 2773 
 
 3178 
 
 (13.) 
 
 5185 
 
 (14.) 
 9187 
 
 (15.) 
 
 4785 
 
 (16.) 
 
 8197 
 
 (17.) 
 4376 
 
 (18.) 
 8765 
 
 8763 
 
 2567 
 
 7372 
 
 1846 
 
 5273 
 
 5678 
 
 Multiply 
 
 19. 28751 by 3146. 
 
 20. 17346 by 2435. 
 
 21. 21307 by 3147. 
 
 22. 85276 by 3452. 
 
 Multiply 
 
 23. 72509 by 3167. 
 
 24. 85216 by 2431. 
 
 25. 73519 by 4735. 
 
 26. 81897 by 3456. 
 
MULTIPLICATION. 
 
 59 
 
 
 Multiply 
 
 
 Multiply 
 
 27. 
 
 21346 by 31452. 
 
 31. 
 
 10786 by 31672. 
 
 28. 
 
 47309 by 45233. 
 
 32. 
 
 47396 by 73462. 
 
 29. 
 
 25737 by 63252. 
 
 33. 
 
 76448 by 54173. 
 
 30. 
 
 43629 by 28516. 
 
 34. 
 
 28354 by 31867. 
 
 51. CLASS IV.— When one or both terms con- 
 tain ciphers. 
 
 1. Multiply 5721 by 3006- also, 37000 by 2400. 
 
 OPERATION. 
 
 5721 
 
 3006 
 
 OPERATION. 
 37000 
 
 2400 
 
 34326 
 17163 
 
 17197326 
 
 148 
 74 
 
 88800000 
 
 Note. — In the first example, pass over the naughts, placing the right- 
 hand figure of the product by 3 directly under the 3. In the second 
 problem, we multiply by the significant figures, and then annex the 
 naughts to the product. 
 
 
 Multiply 
 
 
 Multiply 
 
 2. 
 
 3678 by 204. 
 
 11. 
 
 4500 by 2800. 
 
 3. 
 
 4107 by 307. 
 
 12. 
 
 67000 by 450. 
 
 4. 
 
 4178 by 1005. 
 
 13. 
 
 96000 by 2800. 
 
 5. 
 
 8675 by 3007. 
 
 14. 
 
 87000 by 4800. 
 
 6. 
 
 7276 by 6008. 
 
 15. 
 
 73500 by 32000. 
 
 7. 
 
 4136 by 2305. 
 
 16. 
 
 86700 by 47200. 
 
 8. 
 
 8449 by 3046. 
 
 17. 
 
 32800 by 346000. 
 
 9. 
 
 4592 by 5607. 
 
 18. 
 
 70900 by 407100. 
 
 to 
 
 8124 by 4801. 
 
 19. 
 
 85900 by 1030600 
 
 rO. Multiply four thousand six hundred and ten by 
 seven thousand and fort}-. 
 
60 .MULTIPLICATION. 
 
 EXAMPLES IN MULTIPLICATION. 
 
 1. If one orange cost 8 cents, what will 7 oranges cost 
 at the same rate ? 
 
 OPERATION. 
 
 Solution. — If one orange cost 8 cents, 7 8 
 
 oranges will cost 7 times 8 cents, which are 56 7 
 
 cents. 5 6 Ans . 
 
 2. If one pig cost 7 dollars, what will 6 pigs cost at 
 the same rate ? 
 
 3. If a yard of muslin cost 37 cents, what will 8 yards 
 cost at the same rate ? 
 
 4. If a boy writes 36 words in a day, how many will 
 he write in 13 days ? 
 
 5. What must I pay for 15 cows, if I pay 28 dollars 
 for each cow ? 
 
 6. If Henry takes 42 steps in a minute, how many 
 steps will he take in 15 minutes ? 
 
 7. If a car runs 25 miles in an hour, how far will it 
 run in 12 hours ? 
 
 . 8. If a boy learns 14 new words each clay, how many 
 will he learn in 11 days ? 
 
 9. Mary has 14 rose-bushes in her garden, and on 
 each bush there are 26 roses ; how many roses on all ? 
 
 10. How much must I pay for 16 pounds of tea, at 
 the rate of 78 cents a pound? 
 
 11. What are nine loads of hay worth, at the rate of 
 23 dollars a load ? 
 
 12. If one cord of wood is worth six dollars, how 
 much are 18 cords of wood worth ? 
 
 13. How many marbles will 7 boys have, if each boy 
 has 12 marbles ? 
 
 14. At the rate of 45 miles a day, how far will a per- 
 son travel in 23 days ? 
 
 15 If Henry can count 65 m a minute, how taany 
 can he count in 26 minutes ? 
 
MULTIPLICATION. 61 
 
 PRACTICAL EXAMPLES 
 in Multiplication. 
 
 1. What cost 24 horses at 245 dollars each ? 
 
 Solution. — If one "horse cost 245 dollars, 24 horses cost 24 times 
 245 dollars, which by multiplying we find to be 5880 dollars. 
 
 2. If a boat sails 246 miles in one day, how far will it 
 sail in 26 days? 
 
 3. If in one book there are 364 pages, how many 
 pages in 18 such books ? 
 
 4. In one barrel of flour there are 196 pounds ; how 
 many pounds in 25 barrels of flour ? 
 
 5. How much will 42 horses cost, at the rate of 150 
 dollars apiece ? 
 
 6. If an acre of land is worth 218 dollars, how much 
 will 76 acres cost ? 
 
 7. If in an orchard there are 32 rows of trees with 
 46 trees in a row, how many trees in all ? 
 
 8. A man bought 326 horses and 36 times as many 
 sheep ; how many sheep did he buy ? 
 
 9. What cost 125 yards of cloth at the rate of 325 
 cents a yard ? 
 
 10. How much will 236 bushels of wheat cost, at 175 
 cents a bushel? 
 
 11. There are 1760 yards in one mile; how many 
 yards in 12 miles ? 
 
 12. There are 5280 feet in a mile; how many feet in 
 18 miles? 
 
 13. There are 660 feet in one furlong ; how many feet 
 in 26 furlongs? 
 
 14. There are 5760 grains in one pound Troy ; how 
 many grains in 137 pounds ? 
 
 15. There are 256 drams in an ounce ; how many 
 drams in 420 ounces ? 
 
 16. There are 198 inches in one rod; how manv 
 inches in 76 rods ? 
 
(52 MULTIPLICATION. 
 
 17. There are 1728 pins in a great gross ; how many 
 pins in 256 great gross ? 
 
 18. There are 231 cubic inches in a wine gallon ; how 
 many inches in 48 wine gallons ? 
 
 19. There are 281 cubic inches in a beer gallon; how 
 many cubic inches in 345 beer gallons ? 
 
 20. There are 4840 square yards in one acre ; how 
 many square yards in 365 acres ? 
 
 21. There are 5280 feet in a mile ; how many feet in 
 156 miles ? 
 
 22. There are 63360 inches in a mile; how many 
 inches in 640 miles ? 
 
 23. There are 5280 feet in a mile; how many feet in 
 the diameter of the earth, if it is 7912 miles ? 
 
 PRACTICAL PROBLEMS 
 in Multiplication. 
 
 1. How much cost 75 barrels of flour, at 7 dollars a 
 barrel ? 
 
 OPERATION. 
 75 
 
 Solution. — If 1 barrel cost 7 dollars, 75 barrels - 
 
 will cost 75 times 7 dollars, which are 525 dollars. — - 
 
 525 Ans. 
 
 Note. — In practice, we multiply the 75 by 7, since it is more con- 
 venient to use the smaller number as the multiplier. 
 
 2. How much will 436 bushels of potatoes cost, at 48 
 cents a bushel ? 
 
 3. How much will 847 bushels of corn cost, at 56 cents 
 a bushel ? 
 
 4. How much will 936 yards of muslin cost, at 37 cents 
 a yard ? 
 
 5. A drover bought 4896 pigs, at 9 dollars each ; what 
 did they cost ? 
 
 6. How much will 3686 grammars cost, at 54 cents 
 apiece ? 
 
 7. At 6 cents a quart, what will 3678 quarts of milk 
 cost ? 9876 quarts ? 
 
MULTIPLICATION. 63 
 
 8. There are 60 seconds in one minute ; how many 
 seconds in 6725 minutes? In 9360 minutes? 
 
 9. If 27 men do a piece of work in 48 days, how long 
 will it take one man to do it? 
 
 10. If 145 men do a piece of work in 246 days, how 
 long at this rate would it take one man ? 
 
 11. If 29 men build a fence in 276 days, how long 
 would it take one man to do it? 
 
 12. If 200 acres of corn can be hoed by 157 boys in 
 19 days, how long would it take one boy ? 
 
 13. If 35S men cut 700 cords of wood in 179 days, 
 how Ionic would it take one man to do it ? 
 
 14. There are 16536 letters in a book ; how many 
 letters in 496 of the same books ? 
 
 15. If sound moves 1120 feet in one second, how far 
 ^vill it move in 9872 seconds? 
 
 16. If the earth moves in its orbit 1640000 miles in a 
 day, how many miles does it move in 305 days ? 
 
 17. The moon is 240000 miles from the earth, and the 
 sun about 396 times as far ; how far is the sun from the 
 earth ? 
 
 PRACTICAL PROBLEMS 
 in Addition, Subtraction, and Multiplication. 
 
 (Quite young pupils may omit these until review.) 
 
 1. A has 245 acres of land, and B has 3 times as much ; 
 how many acres has B ? how many acres have both ? 
 
 2. One farmer has 476 hens, and another farmer has 
 5 times as many, minus 392 hens j how many has the 
 second farmer ? 
 
 3. One ship sailed 1248 miles, and another sailed 8 
 times as far, lacking 697 miles ; how many miles did the 
 second ship sail ? 
 
 4. A has 485 dollars, and B has 692 dollars; how 
 
64 MULTIPLICATION. 
 
 much money have they both ? how much has C, if he 
 has 7 times as much as both ? 
 
 5. Mr. Shank has 6450 bushels of corn, and Mr. Frantz 
 has 16 times as much, minus 24986 bushels ; how many 
 bushels has Mr. Frantz ? 
 
 6. A has 456 dollars, B has 759 dollars, and C has 25 
 times as much as both, minus 8965 dollars ; how many 
 dollars have A and B ? how many has C ? 
 
 7. A man sold 24 cows at 35 dollars each, and 17 
 horses at 275 dollars each ; what did he receive for his 
 cows ? for his horses ? for all ? 
 
 8. A man sold his house for 4560 dollars, and 148 
 acres of land at 245 dollars an acre ; how much did he 
 receive for his house and land ? 
 
 9. A bought 126 pigs at 8 dollars each, and B bought 
 
 97 sheep at 12 dollars each; which cost the most, and 
 how much ? 
 
 10. B bought a house for 2960 dollars, and gave for it 
 
 98 cows at 24 dollars each, and the rest in money ; how 
 much money did he pay ? 
 
 11. One army contains 4575 men, and another 36 
 times as many, lacking 1936 men ; how many men in 
 the second army ? 
 
 12. Mr. Peters has 2461 gallons of coal oil, Mr. Martin 
 has 1146 gallons, and Mr. Benson has 147 times as much 
 as both ; how much has Mr. Benson ? 
 
 13. A farmer sold 129 cows at 37 dollars each, and re- 
 ceived in payment 2000 dollars ; how much yet remains 
 due ? 
 
 14. B sold 76 hens at 73 cents each, 96 turkeys at 324 
 cents each, and received in payment 24000 cents; how 
 much remains due ? 
 
 15. A's barn cost 2485 dollars, his house cost 3 times 
 as much, and his farm cost as much as both; what was 
 the cost of the house ? what was the cost of the farm ? 
 
 16. A drover bought 36 horses at 145 dollars a head, 
 
DIVISION. 65 
 
 and 96 cows at 28 dollars a head j which cost the most, 
 and how much? 
 
 17. A's book contains 248 pages, with 2850 letters on 
 a page, and B's contains 325 pages, with 3465 letters on a 
 page ; how many letters in A's book ? how many in B's ? 
 
 18. A man has 75 bags of apples, each bag containing 
 2 bushels j how much will he receive for them, at 125 
 cents a bushel ? 
 
 19. A farmer sold 25 firkins of butter, each firkin con- 
 taining 126 pounds, and received for each pound 37 
 cents ; how much did he receive for it all ? 
 
 DIVISION. 
 
 52 Division is the process of finding how many 
 times one number is contained in another. 
 
 53. The Dividend is the number which contains the 
 
 other. 
 
 54. The Divisor is the number contained in the 
 
 dividend. 
 
 55. The Quotient is the number which shows how 
 many times the dividend contains the divisor. 
 
 56. The sign of Division is -t-, and is read divided 
 by. It shows that the number on the left is to be divided 
 by the one on the right. 
 
 57. There are two methods of performing division 
 called Short Division and Long Division. 
 
 . Note to Teachers. — If the pupils .are not familiar with the ele- 
 mentary quotients, let them turn to page 21 and learn them. 
 
 SHORT DIVISION. 
 
 58. Sliort Division is the method of dividing wher 
 
 the partial dividends are not written. 
 
 6* 
 
06 SHORT DIVISION. 
 
 Case I. 
 59. When tlie divisor is one figure. 
 
 1. How many times is 2 contained in 6 ? 
 
 Solution 1. — We write the 6, draw a line be- operation. 
 neath and a curve to the left, and place the 2 2)6 
 
 to the left of the curve. Two is contained in 6, 3 
 
 three times, since 3 times 2 are 6. We write 
 the quotient 3 beneath the dividend. 
 
 Solution 2. — Two is contained in 6 three times, with no remainder. 
 
 (2-) 
 
 2)8 
 
 (3.) 
 
 3)6 
 
 2)4 
 
 (5.) 
 3)9 
 
 (6.) 
 4)8 
 
 (7.) 
 
 2)10 
 
 (8.) 
 
 2)12 
 
 (9.) 
 3)12 
 
 (10.) 
 2)14 
 
 (11.) 
 3)18 
 
 (12.) 
 2)20 
 
 (13.) 
 
 2)22 
 
 (14.) 
 
 2)24 
 
 (15.) 
 3)21 
 
 (16.) 
 
 3)27 
 
 (17.) 
 3)30 
 
 (18.) 
 3)36 
 
 (19.) 
 3)33 
 
 (20.) 
 4)16 
 
 (21.) 
 4)24 
 
 (22.) 
 4)28 
 
 (23.) 
 4)20 
 
 (24.) 
 4)36 
 
 (25.) 
 
 4)48 
 
 (26.) (27.) (28.) (29.) (30.) (31.) 
 
 5)15 5)25 5)35 5)20 5)40 5)55 
 
 (32.) (33.) (34.) (35.) (36.) (37.) 
 
 5)60 6)12 6)24 6)36 6)48 6)60 
 
 (38.) (39.) (40.) (41.) (42.) (43.) 
 
 7)21 7)35 7)49 7)63 7)77 7)84 
 
 (44.) (45.) (46.) (47.) (48.) (49.) 
 
 8)16 8)64 8)56 8)40 8)72 8)96 
 
 (50.) (51.) (52.) (53.) (54.) (55.) 
 
 9)27 9)45 9)63 9)81 9)108 9)99 
 
SHORT DIVISION. 67 
 
 Case I. 
 
 60. When the divisor is one figure and there arp 
 no remainders. 
 
 1. Divide 46 by 2. 
 
 Solution 1.— 2 is contained in 4 tens 2 tens operation. 
 
 timeo. 2 is contained in 6 units 3 units times ; 2)46 
 
 hence the quotient is 23. 23 
 
 Solution 2. — 2 is contained in 4, 2 times ; 2 is contained in 6, 
 3 times. 
 
 (2.) (3.) (4.) (5.) (6.) (7.) 
 
 2)42 2)48 2)26 2)64 2)84 2)86 
 
 (8.) (9.) (10.) (11.) (12.) (13.) 
 
 2)82 3)36 3)69 3)96 3)90 3)39 
 
 (14.) (15.) (16.) (17.) (18.) (19.) 
 
 4)48 4)44 4)88 4)40 4)80 5)50 
 
 (20.) 
 2)428 
 
 (21.) 
 2)228 
 
 (22.) 
 2)848 
 
 (23.) 
 2)408 
 
 (24.) 
 3)369 
 
 (25.) 
 3)693 
 
 (26.) 
 3)906 
 
 (27.) 
 3)609 
 
 (28.) 
 3)930 
 
 (29.) 
 4)480 
 
 (30.) 
 
 4)804 
 
 (31.) 
 4)408 
 
 (32.) 
 3)669 
 
 (33.) 
 4)488 
 
 (34.) 
 
 2)880 
 
 (35.) 
 
 2)804 
 
 (36.) 
 3)906 
 
 (37.) 
 
 2)886 
 
 Case III. 
 
 (38.) 
 2)468 
 
 (39.) 
 3)603 
 
 61. When the divisor is one 
 
 figure i 
 
 and there are 
 
 remainders. 
 
 
 
 
 
 1. Divide 7 
 
 by 3. 
 
 
 
 
(58 SHORT DIVISION. 
 
 Solution 1. — Three is contained in 7, 2 times, operation. 
 which we write under the 7 ; and since 2 times 3 (7 
 
 3 are 6, and 7 is 1 more than 6, hence 3 is 2 «f- 1 
 
 contained in 7, 2 times, and 1 remaining, which 
 we write after the 2 with the sign -j- before it. 
 
 Solution 2. — 3 is contained in 7, 2 times. 2 times 3 are 6, 6 from 
 7 leaves 1 ; hence the quotient is 2, with a remainder of 1. 
 
 (2.) 
 
 (3.) 
 
 (4-) 
 
 (5.) 
 
 (6.) 
 
 (7-) 
 
 2)9 
 
 2)11 
 
 2)19 
 
 2)21 
 
 2)13 
 
 "2)25 
 
 (8.) (9.) (10.) (11.) (12.) (13.) 
 
 3)5 3)11 3)8 3)14 3)17 3)23 
 
 (14.) (15.) (16.) (17.) (18.) (19.) 
 
 4)11 4)17 4)22 4)37 4)43 4)27 
 
 (20.) (21.) (22.) (23.) (24.) (25.) 
 
 5)12 5)19 5)28 5)38 5)47 5)58 
 
 (26.) (27.) (28.) (29.) (30.) (31.) 
 
 6)15 6)21 6)35 6)51 6)65 6)71 
 
 (32.) (33.) (34.) (35.) (36.) (37.) 
 
 7)23 7)29 7)38 7)46 7)58 7)80 
 
 (38.) (39.) (40.) (41.) (42.) (43.) 
 
 8)23 8)19 8)28 8)36 8)47 8)93 
 
 (44.) (45.) (46.) (47.) (48.) (49.) 
 
 9)31 9)26 9)52 9)61 9)70 9)83 
 
 Case IV. 
 
 62. When the quotient contains several figures 
 and there are successive remainders. 
 
 1. Divide 536 by 2. 
 
SHORT DIVISION. 69 
 
 Solution 1. — 2 is contained in 5 hundreds 2 hun- operation 
 ireds times, with 1 hundred remaining; 1 hundred 2)536 
 
 equals 10 tens, which, with 3 tens, equal 13 tens; 2 o^g 
 
 is contained in 13 tens 6 tens times, and 1 ten re- 
 maining ; 1 ten equals 10 units, which, with 6 units, equal 16 units ; 
 2 is contained in 16 units 8 units times. Hence the quotient is 268. 
 
 Solution 2. — 2 is contained in 5, 2 times, and 1 remaining; 2 is 
 contained in 13, 6 times, and 1 remaining ; etc. 
 
 Rule — 1. Write the divisor at the left of the dividend; 
 begin at the left hand, and divide the number denoted by 
 each figure of the dividend by the divisor, and write the quo- 
 tient beneath. 
 
 2. If there is a remainder after any division, regard it 
 as prefixed to the next figure, and divide as before. If any 
 partial dividend is less than the divisor, prefix it to the next 
 figure, and write a cipher in the quotient. 
 
 63. Proof. — Multiply the quotient by the divisor, 
 and add the remainder, if any, to the product. 
 
 (2-) 
 2)456 
 
 (3.) 
 2)736 
 
 (4.) 
 
 2)548 
 
 (5.) 
 2)374 
 
 (6.) 
 2)538 
 
 (70 
 
 3)735 
 
 (8.) 
 3)816 
 
 (9.) 
 3)522 
 
 (10.) 
 3)414 
 
 (11.) 
 
 3)738 
 
 (12.) 
 3)567 
 
 (13.) 
 3)513 
 
 (14.) 
 3)645 
 
 (15.) 
 3)765 
 
 (16) 
 3)825 
 
 (17.) 
 4)512 
 
 (18.) 
 4)624 
 
 (19.) 
 
 4)732 
 
 (20.) 
 
 4)576 
 
 (21.) 
 4^824 
 
 (22.) 
 4)736 
 
 (23.) 
 4)816 
 
 (24.) 
 4)972 
 
 (25.) 
 
 4)608 
 
 (26) 
 4)436 
 
 (27.) 
 5)615 
 
 (28.) 
 
 5)735 
 
 (29.) 
 5)645 
 
 (30.) 
 
 5)785 
 
 (31.) 
 
 5)840 
 
 (32.) 
 
 5)815 
 
 (33.) 
 
 5j935 
 
 (34.) 
 
 5)780 
 
 (85.) 
 
 5)765 
 
 (36.) 
 
 5)980 
 
70 
 
 ► 
 
 ] 
 
 LONG DIVISION. 
 
 
 (37.) 
 
 (38.) 
 
 (39.) 
 
 (40.) 
 
 (41.) 
 
 6)834 
 
 6)738 
 
 6)654 
 
 6)774 
 
 6)864 
 
 (42.) 
 
 (43.) 
 
 (44.) 
 
 (45.) 
 
 (46.) 
 
 6)1476 
 
 6)3336 
 
 6)2514 
 
 6)3654 
 
 6)7236 
 
 (47.) 
 
 (48.) 
 
 (49.) 
 
 (50.) 
 
 (51.) 
 
 7)2569 
 
 7)4732 
 
 7)8456 
 
 7)9359 
 
 7)9870 
 
 Divide 
 
 
 Divide 
 
 
 52. 8256 
 
 by 8. 
 
 
 59. 
 
 72352 by 8. 
 
 
 53. 7656 
 
 by 8. 
 
 
 60. 
 
 23769 by 9. 
 
 
 54. 9576 
 
 by 8. 
 
 
 61. 
 
 73145 by 5. 
 
 
 55. 9874 
 
 by 9. 
 
 
 62. 
 
 5882597 by 
 
 7. 
 
 56. 9756 
 
 by 9. 
 
 
 63. 
 
 1101032 by 
 
 8. 
 
 57. 9387 
 
 by 9. 
 
 
 64. 
 
 21820708 by 
 
 4. 
 
 58. 92565 by 9. 
 
 
 65. 
 
 6328476 by 
 
 9. 
 
 LONG DIVISION. 
 64. Long Division is the method of dividing when 
 the partial dividends are written. 
 
 Case I. 
 
 65. When the divisor and quotient are each one 
 figure. 
 
 OPERATION. 
 
 2)7(3 
 6 
 
 1. Divide 7 bv 2. 
 
 Solution 1. — 2 is contained in 7 three times. We 
 place ihe 3 at the right in the quotient, and multiply 
 the divisor by it. 3 times 2 are 6, which we write 
 under the 7. We then draw a line beneath, and sub- 1 
 
 tract, and have 1 remaining. 
 
 Solution 2. — 2 is contained in 7, 3 times ; 3 times 2 are 6 ; 6 from 
 7 leaves 1 ; hence the quotient is 3, and 1 remaining. 
 
 (2) 
 
 (3.) 
 
 (*•) 
 
 (5.) 
 
 (6.; 
 
 (?■) 
 
 2)5( 
 
 2)9( 
 
 2)10( 
 
 2)12( 
 
 2)15( 
 
 2)18( 
 

 
 long : 
 
 DIVISION 
 
 
 71 
 
 (8.) 
 
 2)13( 
 
 (9-) 
 2)11( 
 
 (10.) 
 
 3)6( 
 
 (11.) 
 
 3)9( 
 
 (12.) 
 
 3)8( 
 
 (13.) 
 
 3)12( 
 
 (14.) 
 
 8)18( 
 
 (15.) 
 
 3)21( 
 
 (16.) 
 
 3)17( 
 
 (17.) 
 3)19( 
 
 (18.) 
 3)23( 
 
 (19.) 
 3)27( 
 
 (20.) 
 4)8( 
 
 (21.) 
 4)12( 
 
 (22.) 
 4)20( 
 
 (23.) 
 4)28( 
 
 (24) 
 4)30( 
 
 (25) 
 4)10( 
 
 (26.) 
 
 4)13( 
 
 (27.) 
 4)23( 
 
 (28.) 
 4)27( 
 
 (29.) 
 
 5)10( 
 
 (30.) 
 4)20( 
 
 (31.) 
 5)25( 
 
 (32.) 
 
 6)45( 
 
 (33.) 
 5)25( 
 
 (34.) 
 
 5)27( 
 
 (35.) 
 5)38( 
 
 (36.) 
 
 5)43( 
 
 (37.) 
 
 5)47( 
 
 (38.) 
 6)12( 
 
 (39.) 
 
 6)24( 
 
 (40.) 
 
 6)34( 
 
 (41.) 
 6)50( 
 
 (42.) 
 6)59( 
 
 (43.) 
 6j59( 
 
 (44.) 
 
 7)28( 
 
 (45.) 
 7)49( 
 
 (46.) 
 
 7)50( 
 
 (47.) 
 7)60( 
 
 (48.) 
 
 7)48( 
 
 (49.) 
 
 7)57( 
 
 (50.) 
 
 8)24( 
 
 (51.) 
 8)37( 
 
 (52.) 
 8)70( 
 
 (53.) 
 8)69( 
 
 (54.) 
 8)59( 
 
 (55.) 
 8)76( 
 
 (56.) 
 9)27( 
 
 (57.) 
 9)63( 
 
 (58.) 
 
 9)57( 
 
 (59.) 
 9)70( 
 
 (60.) 
 
 9)76( 
 
 (61.) 
 
 9)89( 
 
 Case II. 
 
 66. When the divisor is one figure and the qno< 
 tient is several figures. 
 
 1. Divide 867 by 3. 
 
 Solution 1. — 3 is contained in 8 hundreds 2 operation. 
 
 hundreds times. 2 hundreds times 3 equal 6 3)867(289 
 
 hundreds. 6 hundreds from 8 hundreds leave 2 6 
 
 hundreds. 2 hundreds ami 6 tens are 26 tens. oq 
 
 3 is contained in 26 tens 8 tens times. 8 tens 24 
 
 times 3 are 24 tens. 24 tens from 26 tens leave ~~^~ 
 
 2 tens. 2 tens and 7 units are 27 units. 3 is 27 
 contained in 27 units 9 times. 9 times 3 are 27. 
 Subtracting, nothing remains. Hence, the quotient is 287. 
 
72 
 
 LONG DIVISION. 
 
 Solution 2. — 3 is contained in 8, 2 times ; 2 times 3 are 6 ; 6 from 
 8 leaves 2. Bring down the 6, and we have 26. 3 is contained in 
 26, 8 times; 8 times 3 are 24 ; 24 from 26 leaves 2. Bring down the 
 7. and we have 27. 3 is contained in 27, 9 times ; 9 times 3 are 27, 
 etc. 
 
 EXAMPLES FOR PRACTICE. 
 
 (2.) 
 
 (3.) 
 
 (4.) 
 
 (5.) 
 
 (6.) 
 
 2)36(/ 2)58( 
 
 2)54( 
 
 2)92( 
 
 2)97( 
 
 (7-) 
 
 (8.) 
 
 (9.) 
 
 (10.) 
 
 (11.) 
 
 3)576( 
 
 3)465( 
 
 3)723( 
 
 3)873( 
 
 3)675( 
 
 (12.) 
 
 (13.) 
 
 (14.) 
 
 (15.) 
 
 (16.) 
 
 4)852( 
 
 4)764( 
 
 4)932( 
 
 4)576( 
 
 4)748( 
 
 (17.) 
 
 (18.) 
 
 (19.) 
 
 (20.) 
 
 (21.) 
 
 5)735( 
 
 5)850( 
 
 5)975( 
 
 5)745( 
 
 5)835( 
 
 (22.) 
 
 (23.) 
 
 (24.) 
 
 (25.) 
 
 (26.) 
 
 6)732( 
 
 6)846( 
 
 6)924( 
 
 6)972( 
 
 6)834( 
 
 (27.) 
 
 (28.) 
 
 (29.) 
 
 (30.) 
 
 (31.) 
 
 7)784( 
 
 7)798( 
 
 7)833( 
 
 7)966( 
 
 7)959( 
 
 (32.) 
 
 (33.) 
 
 (34.) 
 
 (35.) 
 
 (36.) 
 
 8)896( 
 
 8)936(' 
 
 8)9440 
 
 8)976( ! n 
 
 8)992( 
 
 
 Divide 
 
 
 
 Divide 
 
 
 37. 
 
 37596 by 2. 
 
 
 46. 
 
 46542 by 3. 
 
 
 38. 
 
 57672 by 3. 
 
 
 47. 
 
 785641 by 6. 
 
 
 39. 
 
 78908 by 4. 
 
 
 48. 
 
 218030 by 8. 
 
 
 40. 
 
 93546 by 6. 
 
 
 49. 
 
 51600 by 4. 
 
 
 41. 
 
 73455 by 5. 
 
 
 50. 
 
 84507 by 7. 
 
 
 42. 
 
 75448 by 8. 
 
 
 51. 
 
 61243 by 2. 
 
 
 43. 
 
 45794 by 7. 
 
 
 52. 
 
 47065 by 5. 
 
 
 44. 
 
 36783 by 9. 
 
 
 53. 
 
 31696 by 6. 
 
 
 45. 
 
 487652 by 7. 
 
 
 54. 
 
 20040 by 9. 
 
 
LONG DIVISION. 73 
 
 Case III. 
 67. Wlien the divisor is two or more figures. 
 
 1. Divide 442 by 13. 
 
 Solution. — 13 is contained in 44 tens 3 tens operation. 
 
 times:, 3 tens times 13 equal 39 tens; 39 tens 13)442(34 
 from 44 tens leave 5 tens, and bringing down 39 
 
 the 2 units we have 52 units. 13 is contained in 52 
 
 52 units 4 times. 4 times 13 are 52 ; subtract- 52 
 
 ing, nothing remains. Hence the quotient is 34. 
 
 Note. — With young pupils, abbreviate the explanation, as in the 
 previous solutions. 
 
 Eule. — 1. Divide the number expressed by the least num- 
 ber of figures on the left that will contain the divisor, and 
 place the quotient on the right. 
 
 2. Multiply the divisor by this quotient ; write the product 
 under the partial dividend, and subtract, and to the re- 
 mainder annex the next figure of the dividend. 
 
 3. Divide as before until all the figures of the dividend 
 have been brought down ant used. 
 
 4. If any partial dividend will not contain the divisor, 
 place a cipher in the quotient, annex the next figure of the 
 dividend, and proceed as before. 
 
 6S. Proof. — Multiply the quotient by the divisor, 
 and add the remainder, if any, to the product. 
 
 Notes. — 1. The pupils will notice that there are four operations: 
 1st, Divide, 2d, Multiply, 3d, Subtract, 4th, Bring down. 
 
 2. If when we multiply the product is greater than the partial 
 dividends, the quotient figure is too large, and must be diminished. 
 
 3. When a remainder is equal to or greater than the divisor, the 
 quotient figure is too small, and must be increased. 
 
 4. A final remainder may be set off by itself, or it may be written 
 over the divisor and annexed to the quotient. 
 
74 
 
 LONG DIVISION. 
 
 EXAMPLES 
 
 Divide 
 
 2. 364 by 11. 
 
 3. 780 by 12. 
 
 4. 312 by 13. 
 
 5. 322 by 14. 
 
 6. 570 by 15. 
 
 7. 752 by 16. 
 
 8. 425 by 17. 
 
 9. 594 by 18. 
 
 10. 608 by 19. 
 
 11. 945 by 21. 
 
 12. 2760 by 22. 
 
 13. 2852 by 23. 
 
 14. 3168 by 24. 
 
 15. 5575 by 25. 
 
 16. 6396 by 26. 
 
 17. 6777 by 27. 
 
 18. 10136 by 28. 
 
 19. 11948 by 29. 
 
 20. 19778 by 31. 
 
 21. 16864 by 32. 
 
 22. 10725 by 33. 
 
 23. 20808 by 34. 
 
 24. 7875 by 35. 
 
 25. 20616 by 36. 
 
 26. 41602 by 37. 
 
 27. 39790 by 38. 
 
 28. 48725 by 39. 
 
 29. 67314 by 41. 
 
 30. 82307 by 42. 
 
 31. 57256 by 43. 
 
 32. 49378 by 44 
 
 33. 98716 by 45. 
 
 34. 60904 by 46. 
 
 35. 76704 by 47. 
 
 FOR PRACTICE. 
 
 Divide 
 
 36. 62377 by 49. 
 
 37. 84309 by 57. 
 
 38. 92736 by 83. 
 
 39. 41875 by 123, 
 
 40. 106750 by 50. 
 
 41. 120054 by 51, 
 
 42. 116532 by 52. 
 
 43. 133242 by 53. 
 
 44. 126414 by 54. 
 
 45. 132715 by 55. 
 
 46. 146552 by 56. 
 
 47. 154926 by 57, 
 
 48. 126614 by 58. 
 
 49. 191219 by 59. 
 
 50. 234606 by 61. 
 
 51. 259284 by 62. 
 
 52. 274176 by 63. 
 
 53. 301952 by 64. 
 
 54. 234455 by 65. 
 
 55. 186846 by 66. 
 
 56. 423867 by 69. 
 
 57. 478608 by 78. 
 
 58. C25902 by 87. 
 
 59. 811332 by 372. 
 
 60. 1234560 by 247, 
 
 61. 3456780 by 356. 
 
 62. 1646301 by 381. 
 
 63. 1985175 by 425. 
 
 64. 1787160 by 562. 
 
 65. 2100315 by 581. 
 
 66. 1019806 by 893. 
 
 67. 74818S8 by 1021. 
 
 68. 5226412 by 2036. 
 
 69. 23456789 by 465- 
 
LONG DIVISION. 
 
 75 
 
 OPERATION. 
 
 5,00)76,54 
 
 15-154 
 or 15i§A 
 
 Case IY. 
 69. When ciphers are on the right of the divisor. 
 
 1. Divide 7654 by 500. 
 
 Solution". — We find how many times 5 hun- 
 dreds is contained in 76 hundreds by dividing 76 
 by 5. It is contained 15 times, with a remain- 
 der of 1 hundred, which, with 54, equals 154. 
 
 Eule. — 1. Cut off the ciphers at the right of the divisor, 
 and as many places from the right of the dividend. 
 
 2. Divide the remaining part of the dividend by the 
 remaining part of the divisor ; prefix the remainder to the 
 figures cut off, for the true remainder. 
 
 Note. — When the divisor, with the ciphers cut off, is greater than 
 12, we will of course divide by long division. 
 
 Divide 
 
 2. 189 by 50. 
 
 3. 487 by 60. 
 
 4. 985 by 80. 
 
 5. 1837 by 400. 
 
 6. 2572 by 1100. 
 
 7. 4783 by 1200. 
 
 8. 8725 by 1300. 
 
 9. 4687 by 1400. 
 10. 9876 by 1500. 
 
 Divide 
 
 11. 18732 by 1600. 
 
 12. 28732 by 1700 
 
 13. 19873 by 1900. 
 
 14. 25307 by 2100. 
 
 15. 40302 by 2500. 
 
 16. 87316 by 3400. 
 
 17. 92913 by 4600. 
 
 18. 31200 by 5100. 
 
 19. 8732000 by 12300. 
 
 PRACTICAL PROBLEMS. 
 Case I. 
 TO. To divide a number by an equal part. 
 
 1. At 5 dollars each, how many sheep can you buy 
 
 for 675 dollars ? 
 
 Solution. — If 5 dollars will buy one sheep, 
 675 dollars will buy as many sheep as 5 is con- 
 tained times in 675, which are 135. Hence, 
 you can buy 135 sheep. 
 
 OPERATION. 
 
 5)675 
 
 135 Ans 
 
76 PRACTICAL PROBLEMS. 
 
 2. At 12 dollars each, how many pigs can you buy 
 for 3780 dollars ? Ana. 315. 
 
 3. At 6 cents apiece, how many oranges can you buy 
 for 354 cents ? Ans. 59. 
 
 4. At 11 cents a quart, how many quarts of cherries 
 can you buy for 1243 cents? Ans. 113. 
 
 5. In one pound there are 12 ounces; how many 
 pounds in 1728 ounces ? 
 
 6. In one minute there are 60 seconds ; how many 
 minutes in 12900 seconds? 
 
 7. How many cows can you buy for 2952 dollars, at 
 the rate of 24 dollars each ? 
 
 8. How many pounds of butter will 8100 cents buy, 
 at the rate of 25 cents a pound ? 
 
 9. There are 16 ounces in one pound; how many 
 pounds in 5472 ounces ? 
 
 10. In one bushel there are 32 quarts ; how many 
 bushels are there in 16182 quarts ? 
 
 11. How many acres of land at 56 dollars an acre can 
 you buy for 12152 dollars? 
 
 12. How long will it take a vessel to sail 6460 miles, 
 at the rate of 68 miles a day ? 
 
 13. The diameter of the earth is nearly 8000 miles ; 
 how long will it take a person to walk the distance, at 
 the rate of 48 miles a day ? 
 
 14. The circumference of the earth is nearly 25000 
 miles ; how long will it take a person to walk it, at the 
 rate of 50 miles a day ? 
 
 15. The distance to the moon is 240,000 miles ; how 
 long would it take a balloon to reach it, moving at the 
 rate of 75 miles an hour ? 
 
 16. The sun is 95,000,000 miles from the earth ; how 
 long would it require a cannon-ball to reach it, moving 
 at the rate of 48 miles a minute ? 
 
PRACTICAL PROBLEMS. 77 
 
 Case II. 
 
 Tl. To divide a number into equal parts. 
 
 1. A man divides 387 dollars equally among 9 boys; 
 how many dollars does each receive ? 
 
 Solution. — Each boy will receive as many operation. 
 dollars as 9 is contained times in 387, which 9)387 
 
 are 43 dollars. 43 ^ns. 
 
 2. A lady divides 4860 dollars equally among 12 girls; 
 how many dollars will each receive ? Ans. 405. 
 
 3. A man earns 2639 dollars in 13 weeks ; how much 
 does he earn in one week ? Ans. 203. 
 
 4. A man travels 1728 miles in 36 days ; how far does 
 he travel each day ? 
 
 5. There are 25 pounds in a quarter; how many 
 pounds are there in 34450 quarters ? 
 
 6. There are 6468 cubic inches in 28 gallons; how 
 many cubic inches in one gallon ? 
 
 7. Sound moves 37060 feet in 34 seconds ; how far will 
 it move in 48 seconds ? 
 
 8. There are 2583 gallons in 41 hogsheads; how 
 many gallons in one hogshead? 
 
 9. If a road 57 miles long cost 7695 dollars, how 
 much did it cost a mile ? 
 
 10. A man gave 1725 dollars for cows worth 25 dollars 
 each ; how many cows did he buy? 
 
 11. How many bushels of oats at 56 cents a bushel 
 can be bought for 13272 cents ? 
 
 12. A man gave 1905 dollars for saddles worth 15 
 dollars each ; how many did he buy ? 
 
 13. A farmer sold 24 horses for 5640 dollars; how 
 much did he receive apiece for them ? 
 
 14. A farmer sold a lot of horses for 7685 dollars; 
 how many did he sell, if he received 145 dollars each ? 
 
 15. How many mules can you buy for 8832 dollars, 
 
 at the rate of 184 dollars each ? 
 
 7* 
 
78 PRACTICAL PROBLEMS IN ADDITION. 
 
 MISCELLANEOUS PROBLEMS. 
 
 (Quite young pupils will omit these until review.) 
 
 PRACTICAL PROBLEMS IN ADDITION. 
 
 1. A man left 850 dollars to his daughter, and 9-45 
 dollars to each of his two sons ; how much did he leave 
 his two sons ? how much did he leave all ? 
 
 2. "Washington was born in the year 1732, Jefferson 
 11 years after, and Hamilton 15 years after Jefferson; 
 when was Jefferson born ? when was Hamilton born ? 
 
 3. A farmer owns three farms ; the first is worth 6560 
 dollars, the second 385 dollars more, and the third 1387 
 dollars more than the second j what is the value of the 
 second farm ? of the third farm ? of all ? 
 
 4. A has 7586 cents, B has 596 more than A, and C 
 has as many as A and B together; how many has B? 
 how many has C ? how many have all ? 
 
 5. B walked 876 miles, C walked 285 miles more than 
 B, and D walked 985 miles more than C ; how far did 
 C walk ? how far did D walk ? how far did they 
 together walk ? 
 
 6. A man gave to his wife 4675 dollars, to his son 7582 
 dollars, to his daughter 3594 dollars, and had 8575 dol- 
 lars left ; what was his fortune ? 
 
 7. A owns a farm worth 3750 dollars, a wood-lot worth 
 856 dollars more, and a store worth 987 dollars more 
 than both ; what was the value of the wood-lot ? of the 
 store ? of all three ? 
 
 8. A man had two sons and three daughters ; he gave 
 each son 5896 dollars, and each daughter 4385 dollars; 
 how much did he give to his sons ? to his daughters ? 
 to ail ? 
 
 9. A butcher sold to one man 876 pounds of meat, to 
 another man 587 pounds more, and to another, 395 
 
PRACTICAL PROBLEMS. 79 
 
 pounds more than both j how much did he sell to the 
 second man ? to the third man ? how much to all ? 
 
 10. A raised 345(3 bushels of wheat, which was 2475 
 bushels less than B raised, and D raised 3489 bushels 
 more than both ; how much did B raise ? how much 
 did D raise ? how much did all raise ? 
 
 11. A bought some land for 8759 dollars, a house for 
 3768 dollars, and sold them so as to gain 1389 dollars; 
 for what did he sell them ? 
 
 12. A man bought two lots for 3750 dollars each ; and 
 iD selling them he gained 278 dollars on the first, and 
 389 dollars on the second; how much did he gain on 
 both ? how much did he receive for both ? 
 
 13. A has 757 acres of land, B has 285 acres more than 
 A, and C has as many as A and B both ; how many 
 acres has B ? how many has C ? how many have all ? 
 
 14. William lends his brother 3785 dollars, his sister 
 4261 dollars, and a friend 485 dollars more than his 
 sister, and has 5858 dollars remaining; how much did 
 he lend his friend, and what was his whole fortune ? 
 
 PRACTICAL PROBLEMS 
 in Addition and Subtraction. 
 
 1. Find the sum of six hundred and five and 18 hun- 
 dred and ninety-seven. 
 
 2. Subtract one thousand and nine from four thousand 
 and seven. 
 
 3. Subtract 7567 + 896 from 4875 + 4736 -j- 2539. 
 
 4. A had 472 hens, and bought 589 hens, and then 
 sold 985 ; how many had he then ? 
 
 5. A farmer had 397 pigs, and bought 85 pigs, and 
 then sold 182 pigs ; how many had he then ? 
 
 6. A drover sold his cows for 257-") dollars, and his 
 sheep for 976 dollars, and gained 594 dollars ; how much 
 did he pay for them ? 
 
 7. A man having 1600 acres of land sold 546 acres 
 
80 PRACTICAL PROBLEMS. 
 
 to B, and 289 acres more to C than to B ; how much 
 did he sell to C ? how much to both ? how much re- 
 mained ? 
 
 8. Mr. Peters, having 4300 bushels of wheat, sold 1480 
 bushels, and then bought 1856 bushels more than he 
 sold ; how many bushels had he then ? 
 
 9. Henry had 756 dollars, and his mother gave him 
 enough to make his money 1200 dollars ; how much did 
 his mother give him ? 
 
 10. Sarah bought 575 pins ; her mother gave her 289, 
 and her sister gave her enough to make her number 
 1000 j how many did she receive from her sister ? 
 
 11. Mary's father left her 596 acres of land; she sold 
 484 acres, and then bought 396 acres ; how many acres 
 had she then ? 
 
 12. A sold 4760 bushels of grain, then sold 1780 
 bushels, and then had 1875 bushels ; how many bushels 
 had he at first ? 
 
 13. B sold 7560 bushels of rye, then bought 2580 
 bushels, and then had 5680 bushels ; how much did he sell 
 more than he bought ? how many bushels had he at first ? 
 
 14. William had 456 dollars ; his father gave him 2528 
 dollars, he then lost 1869 dollars, and gave away 286 
 dollars ; how many dollars had he then ? 
 
 15. Three men bought a farm for 20000 dollars ; the 
 first paid 7580 dollars, the second paid 6765 dollars, and 
 the third the remainder ; how much did the first and 
 second pay ? how much did the third pay ? 
 
 16. A man deposited 8000 dollars in the bank; he 
 drew out at one time 2575 dollars, at another 3467 dol- 
 lars, at another 1576 dollars ; how much remained in 
 the bank ? 
 
 17. Mr. Bowman, whose property was 35000 dollars, 
 willed 9650 dollars to each of his two sons, 8750 to his 
 daughter, and the remainder to his wife; how much did 
 the children receive? how much did his wife receive ? 
 
PRACTICAL PROBLEMS.- 81 
 
 PRACTICAL PROBLEMS 
 in Addition, Subtraction, and Multiplication. 
 
 1. What is the value of 467 X 672 — 31675 ? 
 
 2. What is the value of 672 X 36 plus 216 X 42 ? 
 
 3. A man sold his house for 27 times 98 dollars, plus 
 397 dollars ; how much did he receive for it ? 
 
 4. A sold 24 horses for 168 dollars each, and 63 cows 
 for 34 dollars each ; what did he receive for the horses ? 
 for the cows ? for all ? 
 
 5. I bought 76 oxen at 68 dollars each, and 327 sheep 
 at 12 dollars each ; how much did the oxen cost ? how 
 much the sheep ? how much did all cost ? 
 
 6. My barn cost 2318 dollars, my house 3 times as 
 much, and my farm as much as both ; what was the 
 cost of the house ? the cost of the farm ? 
 
 7. A man bought 235 cows at 24 dollars each, and 
 sold them for 32 dollars, each ; how much did he gain 
 by the transaction ? 
 
 8. A drover bought 78 horses at 164 dollars each, 
 215 oxen at 59 dollars each, and sold them all for 30000 
 dollars ; how much did he gain ? 
 
 9. How much must I pay for 7 building-lots, at 2348 
 dollars each, 5 houses, at 4250 dollars each, and 6 boats, 
 at 3980 dollars each ? 
 
 10. I bought 78 sheep at 7 dollars a head, and sold 
 them so as to gain 267 dollars ; how much did I receive 
 for them ? 
 
 11. I bought 185 acres of land at 95 dollars an acre, 
 and in selling it I lost 2486 dollars ; how much did I 
 receive for it ? 
 
 12. A speculator bought a farm of 327 acres at 79 
 dollars an acre, and sold it at 95 dollars an acre ; how 
 much did he gain ? 
 
 13. A man sold his oil stock for 14000 dollars, and 
 then bought a farm containing 93 acres, at 125 dollars 
 
82 PROBLEMS. 
 
 an acre ; how much money has he left after paying 
 for it ? 
 
 14. A clerk receives a salary of 75 dollars a month ; 
 he spends 18 dollars a month for "board, and 9 dollars 
 for other expenses ; how much can he save in 1 month ? 
 in 12 months ? 
 
 15. A farmer having 3420 dollars bought 35 cows at 
 24 dollars a head, and 36 oxen at 54 dollars a head; 
 how much has he left, after paying for them? 
 
 16. Thomas travels 24 miles a day, and Walton travels 
 52 miles a day ; how much farther does Walton travel 
 in 72 days than Thomas ? 
 
 17. A man bought 336 bushels of potatoes at 65 cents 
 a bushel, and 3 times as many bushels of apples at 98 
 cents a bushel ; what was the entire cost ? 
 
 18. A's barn cost 1980 dollars, his house 2150 dollars 
 more than the barn, and his farm cost 14 times as much 
 as the barn and house together; what was the cost of 
 the farm ? 
 
 PRACTICAL PROBLEMS 
 in Addition, Subtraction, Multiplication, and Division. 
 
 1. Divide 42624 by 36 and add 3146 to the quotient. 
 
 2. Divide 73305 by 45 and subtract the quotient from 
 3702. 
 
 3. Subtract 3125 from 5213, divide the remainder by 
 9, and add the quotient to 1745. 
 
 4. A man having 18000 dollars leaves his wife 4800 
 and divides the remainder equally among 6 children ; 
 what does each receive? 
 
 5. A farm of 24 acres was bought for 4056 dollars and 
 sold at a gain of 3168 dollars : for what was it sold per 
 acre ? 
 
 6. If 27 men share 11286 dollars equally, how much 
 would each have ? how much would A have, if he had 
 four times as much as each, plus 1245 dollars? 
 
PROBLEMS. 83 
 
 7. If 29 men earn 7946 cents in a day, and 25 boys 
 earn 5450 cents in a day, how much more does one man 
 earn in a day than one boy ? 
 
 8. A horse and 18 oxen are worth 1001 dollars; now, 
 if the horse is worth 245 dollars, what is the value of 
 the oxen ? of each ox ? Ans. 42 dollars. 
 
 9. The value of 3 horses and 15 cows is V 55 dollars; 
 if the value of each horse is 225 dollars, what is the 
 ralue of each cow ? - Ans. 32 dollars. 
 
 10. If you divide G0466 by 49, by what number must 
 I multiply the quotient to produce 9872 ? Ans. 8. 
 
 11. The income of a man who " struck oil" is 400 
 dollars per day ; how many teachers would this employ 
 at a salary of 730 dollars a year ? 
 
 12. I bought 326 barrels of flour for 2608 dollars, pad 
 46 dollars for transportation, and sold it at a gain cf 
 280 dollars ; what did I receive a barrel ? 
 
 Ans. 9 dollars. 
 
 13. I sold a farm containing 190 acres for 65 dollars 
 an acre, and bought with the proceeds another farm at 
 95 dollars an acre ; how many acres in the latter farm ? 
 
 14. A drover bought 234 cows at 25 dollars each, and 
 sold 95 of them at cost each ; how much must he re- 
 ceive a head for the remainder, to gain 973 dollars? 
 
 Ans. 32. 
 
 15. If the President of the United States expends 52 
 dollars daily, how much can he save in a year of 365 
 «iays, out of his salary of 25000 dollars? 
 
 16. If the Vice-President expends 35 dollars daily, 
 how much can he save at the end of the year, if he has 
 an income of 6450 dollars, besides his salary of 8000 
 dollars a year ? 
 
 17. If the Secretary of State expends 16 dollars a 
 day, how much can he save in a year, his salary being 
 8000 dollars a year and his private income 28 dollars a 
 week ? 
 
84 UNITED STATES MONEY. 
 
 SECTION IV. 
 
 UNITED STATES MONEY. 
 
 72. United States Money is the money of the United 
 
 States. 
 
 TABLE. 
 
 10 mills (m.) equal .... 1 cent, c. 
 
 10 cents " 1 dime, d. 
 
 10 dimes " 1 dollar, $. 
 
 10 dollars " 1 eagle, E. 
 
 Coins are pieces of metal, stamped by the authority 
 of the government, to be used as money. 
 
 GOLD COINS. SILVER COINS. 
 
 Eagle, value $10 
 
 Double-eagle, •' 20 
 
 Half -eagle, " 5 
 
 Quarter-eagle, " 2\ 
 
 Dollar, " 1 
 
 Three-dollar, " 3 
 
 Dollar, value $1 
 
 Half-dollar, " 50 c. 
 
 Quarter-dollar, " 25 c. 
 
 Dime, " 10 c. 
 
 Half-dime, " 5 c. 
 
 Three-cent, " 5 c. 
 
 NICKEL. BRONZE. 
 
 Three-cent, value 3 c. 
 
 Five-cent, " 5 c. 
 
 Two-cent, value 2 c. 
 
 Cent, " ic. 
 
 NUMERATION AND NOTATION. 
 
 73. The doZtar is indicated by the symbol $. The 
 eagle and dollar are read as a number of dollars : thus, 
 3 eagles and 5 dollars are read, 35 dollars. 
 
 74. The dime is one-tenth of a dollar, and is written 
 to the right of the dollar and separated from it by a 
 point, called a separatrix; thus, $3.4 represents 3 dol- 
 lars and 4 dimes, ta ' ! . 
 
 75. The cent is 1 tenth of a dime, or 1 hundrbdth of a 
 dollar. It is written two places to the right of dollars; 
 thus, $4.58 represents 4 dollars, 5 dimes, and 8 cents. 
 
UNITED STATES MONEY. 85 
 
 76. Dimes and cents are usually read as so many 
 cents ; thus, $7.45 is read, 7 dollars and 45 cents. 
 
 77. The mill is 1 tenth of a cent, and is written one 
 place to the right of cents • thus, $5,475 is read, 5 dollars, 
 47 cents, and 5 mills. 
 
 PRACTICAL PROBLEMS. 
 
 EXAMPLES IN NUMERATION. 
 
 1. Write and read $24.75. 
 
 Solution. — The pupil will write this upon the elate or black- 
 board, and say: This is read, 24 dollars, 7 dimes, and 5 cents; or, 
 24 dollars and 75 cents. 
 
 The pupil will write and read the following : 
 
 2. $14.25 
 
 3. $24.67 
 
 4. $19.84 
 
 5. $28,574 
 
 6. $48.50 
 
 7. $50.06 
 
 8. $48,408 
 
 9. $96,004 
 
 10 $105 076 
 
 11. $976,705 
 
 12. $350,035 
 
 13. $847,008 
 
 EXAMPLES IN NOTATION. 
 
 1. "Write six dollars and twenty-five cents. 
 
 2. Write twenty-five dollars and thirty-six cents. 
 
 3. Write eight dollars, forty-five cents, and six mills. 
 
 4. Write twenty dollars, seventy-five cents, and two 
 mills. 
 
 5. Write six eagles, seven dollars, and eighty-four 
 cents. 
 
 6. Write four dollars, six dimes, and seven cents. 
 
 7. Write 25 dollars, five cents, and eight mills. 
 
 REDUCTION OF UNITED STATES MONEY. 
 
 78. Reduction consists in changing the denomina~ 
 tion without changing the value. From the table we 
 derive the following principles : 
 
 79. To reduce cents to mills, we multiply the cents by 10, 
 or annex one cipher. 
 
66 UNITED STATES MONEY. 
 
 80. To reduce dollars to cents, we annex two ciphers. 
 
 81. To reduce dollars to mills, we annex three ciphers. 
 
 82. To reduce a number of dollars and cents to cents, 
 we remove the decimal point ; thus, $5.24 = 524 cents. 
 
 Case I. 
 To reduce to lower terms. 
 
 1. Eeduce 6 dollars to cents. 
 
 Solution. — In 1 dollar there are 100 cents ; operation. 
 hence, in 6 dollars there are 6 times 100 cents, $6 = 000 cents- 
 or 600 cents ; or we annex two ciphers. 
 
 2. Eeduce $18 to cents. 
 
 3. Eeduce $24 to cents. 
 
 4. Eeduce $385 to cents. 
 
 5. Eeduce $27 to mills. 
 
 6. Eeduce 85 cents to mills. 
 
 7. Eeduce $5.47 to cents. 
 
 8. Eeduce $27.05 to cents. < 
 
 9. Change $9 607 to mills. 
 
 Case II. 
 To reduce to higlier terms. 
 
 S3. From the table we have the following principles : 
 
 1. To reduce cents to dollars, place the point two places 
 from the right. 
 
 2. To reduce mills to dollars, place the point three 
 places from the right. 
 
 1. Eeduce 2347 cents to dollars. 
 
 Solution. — There are 100 cents in 1 dollar, operation. 
 and in 2347 cents there are as many dollars as 2347 -f- 100 
 100 is contained times in 2347, which are =$23.47 
 
 $23.47 ; or we place the point two places from 
 the right. 
 
 2. Eeduce 845 cents to dollars. Ans. $8.45. 
 
 3. Eeduce 2835 cents to dollars. Ans. $28.35. 
 
UNITED STATES MONEY. 87 
 
 4. Reduce 4G785 cents to dollars. 
 
 5. Eeduce 7895 mills to dollars. 
 
 6. Eeduce 27005 mills to dollars. 
 
 7. Eeduce 4800 cents to dollars. 
 
 8. Eeduce 9600 mills to dollars. 
 
 ADDITION OF UNITED STATES MONEY. 
 84. Addition of United States Money is performed 
 as in simple numbers, according to the following 
 
 Eule. — 1. Write dollars under dollars, cents under 
 cents, etc. 
 
 2. Add as in simple numbers, and place the separatrix 
 between dollars and cents. 
 
 1. Find the sum of 824.30, 890.58, and 875.42. 
 
 Solution. — We -write dollars under dollars operation. 
 
 and cents under cents, and commence at the $24.86 
 
 right to add. 2 and 8 are 10, and 6 are 16 cents ; 96.58 
 
 which equals 6 cents and 1 dime; we -write the 75.42 
 
 6 cents under the column of cents, and add the QiQf on 
 1 dime to the next column, etc. 
 
 2. Find the sum of 648.50, 839.46, 824.G7, and 881.09. 
 
 3. Add 823.84, 897.30, 852.75, and 898.27. 
 
 4. Add 873.75, 848.50, 839.87, and 875.48. 
 
 5. Add 840.375, 897.283, 872.475, and 88.390. 
 0. Add 8150.90, 8284.070, 89.27, and 885.735. 
 
 7. A man bought a cow for 824.75, a horse for 8150.50, 
 a wagon for 8287.75, and a carriage for 8375.87 ; how 
 much did he pay for all ? 
 
 8. A merchant bought flour for 857.35, some calicc 
 for 896.87, some cloth for 884.50, some boots for 852.87, 
 and some muslin for 875.75; what did they all cost? 
 
 9. A tailor sold a coat for $34.75, a vest for 88.50, 
 a cloak for 852.25, a pair of pants for 89.75, and some 
 other things for 828.45; what did he receive for all? 
 
 10. I bought a table for 818.25, a looking-glass for 
 
S8 UNITED STATES MONEY. 
 
 $25.75, a bedstead for $36.50, a bureau for $46.25 ; wnat 
 did they all cost ? 
 
 11. A owes $624.30, B owes $467.56, C owes $359.45, 
 D owes $95.12, E owes $43.84, F owes $27.75, G 
 owes $968.47, H owes $7.75 ; required the sum of 
 their debts. 
 
 SUBTRACTION OF UNITED STATES MONEY. 
 
 85. Subtraction of United States 3Ioney is per- 
 formed as in subtraction of simple numbers, according 
 to the following 
 
 Eule. — 1. Write dollars under dollars, cents under 
 cents, etc. 
 
 2. Subtract as in simple numbers, and place the separatrix 
 between dollars and cents. 
 
 1. Subtract $21.48 from $46.73. 
 
 Solution. — We cannot subtract 8 cents from operation. 
 6 cents, hence we add 10 cents to 8 cents, $46.73 
 
 making 13 cents; 8 cents from 13 cents leave 27.48 
 
 5 cents. Now, since we added 10 cents, or 1 $19.25 
 
 dime, to the minuend, we must add 1 dime to 
 the 4 dimes, making 5 dimes : 5 dimes from 7 dimes leave 2 dimes, etc. 
 
 (2.) (3.) (4.) (5.) 
 
 $78.25 $57.52 $96.43 $75.75 
 
 13.16 23.28 28.14 23.28 
 
 6. From $129.39 take $48.91. 
 
 7. Find the difference between $234.16 and $471.24. 
 
 8. A man bought a horse for $234.50, and sold it for 
 $228.25 ; what did he lose ? 
 
 9. A merchant bought cloth for $96.75, and sold it for 
 6110.29 ; what did he gain ? 
 
 10. A bought a farm for $3640.25, and sold it for 
 64000 ? what was the gain ? 
 
 11. My house cost $3480.75, and I sold it for $4000.50; 
 what did I gain ? 
 
UNITED STATES MONEY. 89 
 
 12. My horse cost $240.50, and my carriage cost 
 $386.25 ; I sold them for $680.50 ; what did I gain ? 
 
 13. A merchant bought cloth for $325.50, muslin for 
 $436.75, and flour for 8625.80; he sold them all for $1300; 
 how much did he lose ? 
 
 14. I paid $4637.25 for a farm, paid $3675.25 foi 
 building a house, and $2896.87 for building a barn ; I 
 sold my property for $13000 ; how much did I gain ? 
 
 15. I paid $246.75 for a horse, $325.45 for a mule, 
 $42.25 for an ox, $37.50 for a cow; I sold them all for 
 $603.50 j what was the loss ? 
 
 MULTIPLICATION OF UNITED STATES MONEY. 
 
 86. Multiplication of United States Money is per- 
 formed, like multiplication of simple numbers, according 
 to the following: 
 
 o 
 
 Rule. — Multiply as in simple numbers, and place the 
 separatrix between dollars and cents. 
 
 1. Multiply $36.25 by 3. 
 
 Solution. — Three times 5 cents are 15 cents, operation. 
 
 which equal 1 dime and 5 cents; we write the $36.25 
 
 5 cents, and reserve the 1 dime to add to the next 3 
 
 product. 3 times 2 dimes are 6 dimes, and 6 £108 75 
 dimes plus 1 dime are 7 dimes, etc. 
 
 Multiply 
 
 2. $26.14 by 4. 
 
 3. $37.27 by 5. 
 
 4. $48.96 by 7. 
 
 5. 837.52 by 8. 
 
 6. $79.35 by 9. 
 
 Multiply 
 
 7. $48.25 by 12. 
 
 8. $72.27 by 13. 
 
 9. $85.58 by 15. 
 
 10. $92.83 by 32. 
 
 11. $75.32 by 46. 
 
 12. If one yard of cloth cost $3.25, what cost 5 yards? 
 
 13. What will 12 horses cost at the rate of $150.75 
 a piece ? 
 
 14. A man bought 27 oxen at the rate of $36.25 each , 
 what did they cost ? 
 
 8* 
 
90 UNITED STATES MONEY. 
 
 15. A farmer sold 325 bushels of wheat at $1 25 a 
 bushel ; how much did he receive for it ? 
 
 16. A miller sold 472 barrels of flour at $7.87 a barrel; 
 how much did he receive for it ? 
 
 17. A man bought 47 cows for $24.30 each, and sold 
 them for $28.10 each ; what was the gain ? 
 
 18. A drover bought 247 horses for $130.75 each, and 
 gold them for $180.30 each; what did he gain? 
 
 19. A farmer bought 327 acres of land at $76.25 an 
 acre, and sold it at $92.50 an acre; what did he gain? 
 
 DIVISION OF UNITED STATES MONEY. 
 
 87. Division of United States Money is performed 
 like division of simple numbers. 
 
 Case I. 
 
 88. To divide a number irato equal parts. 
 
 Rule. — Divide as in simple numbers, and -place the sepa- 
 ratrix between dollars and cents. 
 
 1. Divide $7.32 in 3 equal parts, or find 1 third of it. 
 
 Solution. — 1 third of 7 dollars is 2 dollars, operation. 
 and 1 dollar remaining; 1 dollar equals 10 dimes, 3)$7.32 
 
 which, added to 3 dimes, equal 13 dimes. 1 $2.44 Ans. 
 
 third of 13 dimes equals 4 dimes, and 1 dime 
 remaining, etc. 
 
 2. Divide $9.24 into 4 equal parts. 
 
 3. Divide $7.25 into 5 equal parts. 
 
 4. Divide $17.22 into 6 equal parts. 
 
 5. If 7 pigs cost $36.75, what will one pig cost ? 
 
 6.- If 8 cows cost $172.80, what will one cow cost? 
 
 7. If 3 oxen cost $325.20, what will 5 oxen cost? 
 
 8. If 7 hens cost $3.15, what will 12 hens cost? 
 
 9. What cost 15 sheep, if 4 sheep cost $29.24? 
 
 10. What cost 25 pounds of butter, if 7 pounds cost 
 62.38 ? 
 
UNITED STATES MONEY. 91 
 
 11. What cost 34 acres of land, if 12 acres cost $5.04? 
 
 12. What cost 28 cows, if 35 cows cost 987 dollars? 
 
 13. What cost 75 oxen, if 38 oxen cost 1615 dollars? 
 
 14. What cost 234 hens, if 75 hens cost $25.50 ? 
 
 Case II. 
 
 89. To divide one sum of money by another, 
 
 Eule. — Reduce both sums to the same denomination, and 
 divide as in simple numbers. 
 
 1. Divide $736 by $4. 
 
 OPERATION. 
 
 Solution. — Dividing as in simple numbers, 4)736 
 
 we have 184. ~Jj^ Ans# 
 
 2. Divide $9600 by $16. 
 
 3. Divide 728 cents by 4 cents. 
 
 4. Divide 3625 cents by 5 cents. 
 
 5. Divide $26325 by 81 dollars. 
 
 6. At 24 dollars each, how many cows can you buy 
 for 1344 dollars? 
 
 7. At 42 dollars each, how many oxen can be bought 
 for $3276 ? 
 
 8. At $3.25 apiece, how many pigs can you buy for 
 $120.25? 
 
 9. A earned $3.75 a day ; how many days did he work 
 to earn $78.75 ? 
 
 10. A drover paid $6972 for horses, at $145.25 apiece; 
 how many did he buy ? 
 
 11. How many cords of wood can you buy for $312, 
 at $3.25 a cord ? 
 
 12. William earned $3.25 a day, and paid 75 cents for 
 board; in how many days would he save S912.50? 
 
 13. A merchant received $853.25 for a case of silk, 
 including $1.25 cost of box. How many pieces of silk 
 were in the case, if he received $53.25 apiece ? 
 
92 
 
 BILLS AND ACCOUNTS. 
 
 BILLS AND ACCOUNTS. 
 
 90. A Bill is a written statement of goods bought 
 and sold, the quantity, price, and entire cost. 
 
 91. An Account is a bill in which each of the parties 
 has received value of the other. 
 
 92. The party who owes is the debtor; the party 
 who is owed is the creditor. A bill is made out by the 
 following 
 
 Kule.— Multiply the cost of each article by the whole 
 number, and find the sum of the products. 
 
 92J. In an account, find the difference between the debit 
 and credit amounts. 
 
 Make out the following bills : 
 
 (1.) Millersville, May 8, 186%. 
 
 Mr. Harry Bowman, 
 
 Bought of HENR Y MARTIN, 
 
 
 8 
 
 yds. 
 
 of 
 
 muslin, 
 
 at 
 
 
 12 
 
 a 
 
 of 
 
 cloth, 
 
 a 
 
 
 15 
 
 a 
 
 of 
 
 silk, 
 
 a 
 
 27, 
 2.37, 
 l.t 
 
 Amount due, 
 
 (2-) 
 Theo. Miller, 
 
 Lancaster, April 6, 1864. 
 Bought of DANIEL MO ONE Y, 
 
 24- 
 37 
 
 ¥> 
 28 
 
 b*=*=s 
 
 o 
 
 ©=*=*& 
 
 pairs boots, at $5.25, 
 gaiters, " 3.75, 
 slippers, " 1.37, 
 rubbers, " 1.25, 
 
 a 
 
 u 
 
 Amount due, 
 
 Received Payment, 
 
 Theo. Miller. 
 
BILLS AND ACCOUNTS. 
 
 93 
 
 (3.) 
 John J. Brooks, 
 
 New York, Dec. 17, 1862. 
 Bought of CHARLES HOYT, 
 
 47 
 
 28 
 
 97 
 
 U6 
 
 €>SS6^ 
 
 bbls. of flour, at $7.35, 
 lbs. of beef, " 0.37, 
 yds. of cloth, " 2.75, 
 bu. of wheat, " 1.12, 
 
 Amount, 
 Received Payment, 
 
 Charles Hoyt. 
 
 (4-) 
 John Smith, Dr. 
 
 1S66. 
 
 
 Jan. 
 
 1 
 
 Feb. 
 
 5 
 
 Jan. 
 
 7 
 
 Feb. 
 
 2 
 
 To 75 lbs. of sugar, at $0.35, 
 " 4.7 yds. of cloth, " 3.25, 
 
 Cr. 
 
 By 75 bu. of corn, at $0.78, 
 " 83 bu. of apples, " 1.25, 
 
 Balance due, 
 
 $ 
 
 
 f 
 
 (5.) Philadelphia, April 1, 1860. 
 
 Mr. Henry Farnam, Dr. 
 
 To Edwin Lamborn. 
 
 I860. 
 
 
 Jan. 
 
 4. 
 
 Jan. 
 
 10. 
 
 Jan. 
 
 20. 
 
 I860. 
 
 
 Jan. 
 
 3. 
 
 Jan. 
 
 12. 
 
 Feb. 
 
 21^. 
 
 To 145 bu. wheat, at $1.25, 
 " 236 " rye, « 1.05, 
 " 176 " oats, " 0.65, 
 
 Cr. 
 By 45 yds. cloth, at $3.65, 
 " 72 " silk, " 2.12, 
 
 " £# " cassimere, " i.75, 
 
 Bed a nee due, 
 Received Payment, 
 
 Edwin Lamborn. 
 
94 COMMON FRACTIONS. 
 
 SECTION V. 
 COMMON FEACTIONS. 
 
 93. A Fraction is a number of equal parts of a 
 unit ; as one half, two thirds, etc. 
 
 94. A fraction is expressed by figures with a line 
 between ; thus, § expresses 2 thirds. 
 
 95. The number denoted by the figure below the 
 line is called the denominator ; it shows the number of 
 equal parts into which the unit is divided. 
 
 98. The number denoted by the figure above the line 
 is the numerator; it shows the number of equal parts 
 considered. 
 
 97. A Proper Fraction is one whose value is less 
 than a unit ; as §, f , | 5 etc. 
 
 98. An Improper Fraction is one whose value is 
 equal to or greater than a unit; as |, |, 2 B l , etc. 
 
 99. A Compound Fraction is a fraction of a fraction ; 
 as \ of |. 
 
 100. A Mixed Number consists of a whole number 
 and a fraction; as 2 J, 5'f, etc. 
 
 To Teachers. — Give pupils a clear idea of a fraction by dividing 
 some object, as an apple, by lines upon the blackboard, etc. For 
 Mental Exercises, see Primary Mental Arithmetic. 
 
 MENTAL EXERCISES. 
 
 1. What is one-half? 
 Ans. One-half of any thing is one of the two equal part,- of it 
 
 What is What is 
 
 2. One-third? 
 
 3. One-fourth ? 
 
 4. One-fifth? 
 
 5. One-sixth ? 
 
 6. One-seventh ? 
 
 7. One-eighth ? 
 
 8. One-tenth ? 
 
 9. One-twelfth? 
 
COMMON FRACTIONS. 
 
 95 
 
 1. What is two-thirds? 
 Ans. Two-thirds of any thing is two of the three equal parts of it. 
 
 What is What is 
 
 2. Two-fourths? 
 
 3. Three-fourths? 
 
 4. Two-fifths? 
 
 5. Three-fifths? 
 
 6. Four-fifths? 
 
 7. Two-sixths? 
 
 8. Three-sevenths? 
 
 9. Four-ninths ? 
 
 1. What is J of 6 ? 
 
 Ans. \ of 6 is 3, since 2 times 3 are 6. 
 
 2. 
 
 Find I of 8. 
 
 5. 
 
 Find | of 15. 
 
 o 
 O. 
 
 Find | of 12. 
 
 6. 
 
 Find | of 20. 
 
 4. 
 
 Find \ of 16. 
 
 7. 
 
 Find 4 of 30. 
 
 
 
 1. 
 
 2. 
 
 NUMERATION AND NOTATION. 
 Eead the following fractions. 
 
 2 3 . _6_ 
 °' 5 > ll' 
 
 A 11. 8 
 
 ^' T3 > 10"" 
 
 5 . 6 
 
 G> 7" 
 
 7 . 3 
 
 3 > 5 
 
 5. 
 6. 
 
 i f? . 
 
 •2 i > 
 
 7 
 To' 
 
 53- 115 
 
 Write the following: fractions. 
 
 1. Two-thirds. 
 
 2. Four-fifths. 
 
 3. Five-sevenths. 
 
 1. Analyze the fraction f. 
 
 4. Eight-tenths. 
 
 5. Seven-ninths. 
 
 6. Eleven-fifteenths. 
 
 Solution. — In the fraction f, the denominator, 4, shows that the 
 unit is divided into 4 equal parts, and the numerator, 3, shows that 
 3 of these parts are taken. 
 
 Analyze the following: 
 
 2. 
 
 3. 
 4. 
 
 ■2 . 
 
 3 > 
 
 5 . 
 
 e > 
 3 . 
 
 7 > 
 
 4 
 
 7" 
 
 K 4 . 3 
 °' B"> 77* 
 
 8. 
 
 9 . 7 
 
 15? 18"' 
 
 4 
 5" 
 
 6 7 . 8_ 
 U * 9 > 11' 
 
 9. 
 
 13. 16 
 2 1 ) 2i' 
 
 2 
 
 7 12. 8 
 1 ' 13 J 14' 
 
 10. 
 
 21.34 
 
 1 1" 
 
 
 "3 1 ' 4 4* 
 
 PRINCIPLES OF FRACTIONS. 
 
 lOO-LWe will now solve a number of problems, and 
 derive some of the principles of fractions. 
 1. Multiply the numerator of | by 2. 
 
06 
 
 COMMON FRACTIONS. 
 
 Solution. — Multiplying the numerator of | operation. 
 by 2, we have 6 fifths, which is 2 times as great |X2 == | 
 
 as 3 fifths. Hence the following 
 
 Principle I. — Multiplying the numerator of a fraction 
 by any number multiplies the fraction by that number. 
 
 Multiply the fraction 
 
 2. | by 5. Ans. L«. 
 
 3. | by 7. 
 
 4. }| by 8. 
 
 5. if by 11. 
 
 Multiply the fraction 
 
 6. \ § by 14. 
 
 7. || by 18. 
 
 8. if by 17. 
 
 9. Jf by 20. 
 
 OPERATION 
 9 — 2 
 
 4 
 ~5 
 
 1. Divide the numerator of J by 2. 
 
 Solution. — Dividing the numerator of f by 
 2, we have 2 fifths, which is 1 half of 4 fifths. 
 Hence the following 
 
 Principle II. — Dividing the numerator of a fraction by 
 any number divides the fraction by that number. 
 
 Divide the fraction 
 2. f by 3. Ans. f. 
 
 3. 
 4. 
 5. 
 
 | by 4. 
 J? by 5. 
 
 14 V 7. 
 
 Divide the fraction 
 6. if by 4. 
 7- if by 9. 
 
 8. iff by 12. 
 
 9. |Sf by 32. 
 
 OPERATION. 
 3 V 1 — 1 
 
 1. Multiply the denominator of f by ! 
 
 Solution. — Multiplying the denominator by 
 2, we have 3 eighths, which is one-half as much 
 as 3 fourths, since eighths are only half as large 
 as fourths. Hence the following 
 
 Principle III. — Multiplying the denominator of a frac- 
 tion by any number divides the fraction by that number. 
 
 
 Divide the fraction 
 
 
 Divide the fraction 
 
 2. 
 
 i by 4. 
 
 Ans. A. 
 
 1 - 
 
 7. 
 
 IfbyS 
 
 3. 
 
 H b >' 1- 
 
 Ans. i|. 
 
 8. 
 
 1 by 6. 
 
 4. 
 
 ! by 5. 
 
 
 9. 
 
 11 by 12. 
 
 5 
 
 H by 7. 
 
 
 10. 
 
 t 9 u by 11. 
 
 6. 
 
 1 by 3. 
 
 
 11. 
 
 if by 13. 
 
COMMON FRACTIONS. 97 
 
 1. Divide the denominator of f by 2. 
 
 Solution. — Dividing the denominator by 2, we have 3 halves, and 
 S halves is twice as great as 3 fourths, since halves are twice as large 
 as fourths. Hence the following 
 
 Principle IV. — Dividing the denominator of a fraction 
 by any number multiplies the fraction by ihat number. 
 
 Multiply, by dividing the denominator, 
 
 2. § by 2. 
 
 3. 13 by 6. 
 
 4. | by 3. 
 
 5. i| by 9. 
 
 6. | by 4. 
 
 7. a? by 7. 
 
 
 
 8. j% by 5. 
 
 9. f| bv 12. 
 
 10. j§ by 6. 
 
 11. <J by 19. 
 
 12. ii by 10. 
 
 13. T V ? by 36. 
 
 1. Multiply both numerator and denominator of § by 2. 
 
 Solution. — Multiplying both numerator and de- operation. 
 nominator by 2, we have £ ; and this equals f , since f X f ~ t 
 we both multiplied and divided § by 2, and hence 
 did not change its value. Hence we have the following 
 
 Principle V. — Multiplying both numerator and denomi- 
 nator of a fraction by the same number does not change the 
 value of the fraction. 
 
 2. Multiply both numerator and denominator of | by 
 3; | by 6; | by 5 ; § by 3 ; i» by 9 ; j| by 6. 
 
 1. Divide both numerator and denominator of | by 2. 
 
 Solution. — Dividing both numerator and denomi- operation. 
 nator by 2, we have f ; and this equals |, since we |-^|=| 
 both divided and multiplied | by 2, and hence did 
 not change its value. From this we have 
 
 Principle VI. — Dividing both numerator and denomina- 
 tor by the samenumber does not change the value of the fraction. 
 
 2. Divide both numerator and denominator of g by 2; 
 l%by4; J> 3 by 3/ ig by 2 ; Jjby4j IJbylO: |$byl$ 
 
 9 
 
98 
 
 REDUCTION OF FRACTIONS. 
 
 REDUCTION OF FRACTIONS. 
 
 101. Reduction of fractions is the process of changing 
 their form without changing their value. 
 
 Case I. 
 
 102. To reduce mixed numbers to fractions. 
 
 1. How many thirds in 4# ? operation. 
 
 Solution. — In 1 there are f, and in 4 there 4| 
 
 are 4 times f, which are ^ 2 , and ^ -f § equal 3 
 
 J 3 4 . From this solution we have the following 14 thirds = i- 4 . 
 
 Rule. — Multiply the tohole number by the denominator, 
 add the numerator, and write the denominator under the 
 result. 
 
 
 Reduce 
 
 to 
 
 improper 
 
 fractions 
 
 2. 
 
 
 
 Ans. 
 
 23 
 
 5 • 
 
 
 7. 181. 
 
 3. 
 
 1 4- 
 
 
 
 
 
 8. 21}. 
 
 4. 
 
 Q5 
 
 
 
 
 
 9. 19JA. 
 
 5. 
 
 7§. 
 
 
 
 
 
 10. 25 T V 
 
 6. 
 
 131. 
 
 
 
 
 
 11. 35}§. 
 
 
 
 
 
 Casi 
 
 s II. 
 
 Ans. 9 ,- 3 . 
 
 103. To reduce improper fractions to whole or 
 mixed numbers. 
 
 1. How many ones in \ 5 ? 
 
 Solution. — In one there are 4 fourths, and 
 in 15 fourths there are as many ones as 4 is 
 contained times in 15, which are 3-|. From 
 this solution we have the following 
 
 is 
 
 OPERATION 
 
 = 15 
 
 4 = 3|. 
 
 Rule. — Divide the numerator by the denominator, and 
 the quotient will be the whole or mixed number. 
 
 Reduce to whole or mixed numbers 
 
 7. 
 8. 
 
 2. 
 3. 
 4. 
 5. 
 6. 
 
 9 
 
 4"* 
 1 1 
 
 3 ' 
 1 9 
 
 5 * 
 32 
 
 4 
 
 ¥■ 
 
 Ans. 2\ 
 
 9. 
 10. 
 11. 
 
 47 
 
 e • 
 
 92 
 
 1 !* 
 
 2 5 
 
 i 2" 
 
 2 35 
 77 • 
 
 7 2_4 
 3 J ' 
 
 Ans. Y 
 
REDUCTION OF FRACTIONS. 99 
 
 Case III. 
 10 1. To reduce fractions to liiguer terms. 
 
 1. How many twelfths in |? 
 
 Solution. — Multiplying both numerator and operation. 
 
 denominator of a fraction by the same nuin- 3 3 X 3 __ _<?__ 
 
 ber does not change its value, Prin. V. ; hence, 4 ^ 3 
 
 multiplying both numerator and denominator by 
 
 3, and we have f = T 9 2 . From this solution we have the following 
 
 Eule. — Multiply both numerator and denominator b§ 
 any number which will give the required denominator. 
 Ecduce Reduce 
 
 2. f to 12ths. Ans. T 8 3 . 
 
 3. § to 30ths. Ans. §§. 
 
 4. I to 16ths. 
 
 5. T 9 to 20ths. 
 
 6. I to 27tks. 
 
 7. ]± to 36ths. 
 
 8. }| to GOths. 
 
 9. 1 to 81sts. 
 
 10. }| to 80ths. 
 
 11. if to 8-ltks. 
 
 Case IV. 
 
 105. To reduce a fraction to lower terms. 
 
 106. A fraction is reduced to lower terms when it is 
 reduced to one having a smaller numerator and denomi- 
 nator. 
 
 lot. A fraction is in its lowest terms when it cannot 
 be reduced to any lower terms. 
 
 1. Reduce T % to fourths. 
 
 Solution. — Dividing both numerator and de- operation. 
 nominator of a fraction by the same number 3)& = f 
 
 does not change its value, Prin. YI. ; hence, to 
 
 reduce T 9 T to fourths wc divide both numerator and denominator by 
 3, and we have |. From this solution we have the following 
 
 Rule. — 1. To reduce a fraction to lower terms, divide 
 both numerator and denominator by the same number or 
 numbers. 
 
 2. To reduce to lowest terms, divide in this way until 
 the fraction cannot be reduced any lower. 
 
100 
 
 COMMON DENOMINATOR. 
 
 Reduce to lowest terms Reduce to lowest terms 
 
 2. 
 
 3. 
 4. 
 5. 
 6. 
 
 f> 1 4 
 
 "S5 VI 1 * 
 
 8 
 i 2 J 
 
 1 
 I 2> 
 
 1 6 
 24? 
 
 2Bj 
 
 12 
 
 1 8* 
 
 2 5 
 30 
 
 18 
 2 7' 
 
 48 
 84* 
 
 Ans. § 
 
 7 
 
 Ans. f . 
 
 8 
 
 Ans. |. 
 
 9 
 
 
 10 
 
 
 11 
 
 24 
 40' 
 70 
 8 0' 
 
 4 5 
 
 5 0' 
 9 9 
 
 108' 
 
 96 
 
 27 
 
 Ans. ?. 
 
 
 
 84 
 
 108 
 I Z 0~ " 
 
 1 2 1 
 732' 
 1 44 
 
 10 4' 156' 
 
 108 
 1 
 
 Case V. 
 To reduce compound fractious to simple. 
 
 What are f of % ? 
 
 Solution. — \ of f 
 
 -X, since mul- 
 tiplying the denominator of a fraction 
 by 3 divides the fraction by 3 ; and if 
 of f = T %, f of f equals 2 times T 4 , 
 
 OPERATION. 
 
 2V4 
 3X0 3X5 T * 
 
 15' 
 
 which are -A- From this solution we have the following 
 
 R ULE . — Multiply the numerators together, and the de- 
 nominators together. 
 
 2. 
 
 3. 
 4. 
 5. 
 6. 
 
 What is 
 -2 of 2? 
 
 5 f 7? 
 
 6 U1 9 ' 
 
 4 f IS? 
 
 7 Ui 15" 
 
 £ Of A-i ? 
 9 Ui 1 2 * 
 
 * Of - 9 - ? 
 
 8 Ui 10* 
 
 Ans. 
 
 21 
 
 32" 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 
 What is 
 
 11 of A6? 
 
 12 Ui 33 • 
 
 3 f 5 f 7? 
 7 0f 
 
 9 Of 
 TO U1 
 
 li? 
 1 6 " 
 
 % Of 2 Of ' 7 ? 
 
 8 
 
 7 8 
 
 5 of - 3 Of i 4 - ? 
 3 Ui 7 ^ 3 tS ' 
 
 12. A had | of a ton of hay, and sold his neighbor J 
 of it ; how much did he sell ? 
 
 f of a ton of hay, and 
 
 OPERATION. 
 
 fXf = T 8 sAns. 
 
 Solution. — If A had f 
 sold his neighbor § of it, he sold his neighbor 
 I of I of a ton, which is T 8 3 of a ton. 
 
 13. A boy picked § of a bushel of strawberries, and 
 sold § of them ; how many did he sell ? Ans. {§, or §. 
 
 14. A man had § of a bushel of barley, and sold j of 
 it ; how much did he sell ? Ans. |. 
 
 15. A little girl had J of a melon, and gave her brother 
 4 of it ; how much did her brother receive ? Ans. £. 
 
 16. Says Jennie to Kate, My father owns f of f of f 
 of a ship j what part of the ship did he own ? Ans. 
 
 o 
 
 5' 
 
COMMON FRACTIONS. 101 
 
 COMMON DENOMINATOR. 
 109. Fractions have a Common Denominator when 
 they have the same number for a denominator. 
 1. Reduce | and i to a common denominator. 
 
 Solution. — Multiplying both numerator and operation. 
 
 denominator of f by 5, the denominator of -f, 3X*> _ 15 
 
 we have |4; and multiplying both numerator "4X5 "^ 
 
 and denominator of f by 4, the denominator of 4 4X4 16 
 
 f, we have ^§; and this makes the fractions have z 5X4 20 
 the same denominator; hence the following 
 
 Rule. — Multiply both numerator and denominator of 
 each fraction by the denominators of the other fractions. 
 
 Or, Multiply both numerator and denominator of each 
 fraction by any number that will make the denominators 
 ilike. 
 
 Reduce to a common denominator 
 
 2. 
 
 -§ and |. Ans. 
 
 1 
 
 1 5 5 
 
 1 2 
 1 5' 
 
 7. 
 
 12 anc l 13 
 1 3 allu 14' 
 
 3. 
 
 | and {. Ans. 
 
 24 
 
 30? 
 
 2 5 
 
 "3 0* 
 
 8. 
 
 | 3 j, and |. 
 
 4. 
 
 I and j. 
 
 
 
 9. 
 
 3 5 anc i 4 
 4 , 6 , aiiu 5 . 
 
 5. 
 
 | and f . 
 
 
 
 10. 
 
 4 § and 5 
 
 6. 
 
 9 ari d - 8 - 
 
 
 
 11. 
 
 1 - G and 1 
 
 65 7' a u 6* 
 
 ADDITION OF FRACTIONS. 
 110. Addition of Fractions is the process of finding 
 the sum of two or more fractions. 
 
 Case I. 
 To add when the denominators are alike. 
 
 1. What is the sum of I and i o ? 
 
 Solution. — 2 fifths plus 4 fifths equals 6 fifths, operation. 
 which equals li. f + f = f = H 
 
 2. What is the sum of § and § ? Ans. §. 
 
 3. What is the sum of § and J ? Ans. §, or 1J. 
 4 What is the sum of § and J ? Ans. H- 
 5. What is the sum of I and | ? Ans. \ 5 , or If. 
 
 9* 
 
102 
 
 COMMON FRACTIONS. 
 
 6. Mary had § of a dollar and Sarah had f of a dollar; 
 how much did they both have ? 
 
 7. Lucy gave me § of a peach, and Fanny gave me | 
 of a peach; how much did I receive? Ans. 1^. 
 
 8. George and Susie had each J of a pine-apple ; how 
 much had they together? Ans. If. 
 
 9. If I walk | of a mile and ride | of a mile, how far 
 
 do I go ? 
 
 Ans. 11 mile. 
 
 10. A had | of a dollar, B had J of a dollar, and C 
 had | of a dollar ; how much had they all ? 
 
 Case II. 
 To add when tlae denominators are unlike. 
 
 1. What is the sum of § and | ? 
 Solution. — We first reduce the fractions to a 
 
 common denominator: -| 
 
 8 • 3 
 12 ' ? 
 
 T 9 j ; 8 twelfths 
 
 plus 9 twelfths are 17 twelfths. Hence f + f - 
 
 12' 
 
 OPERATION. 
 
 2 _|_ 3 
 
 3 T 4 
 
 T2 + T% = TT 
 
 From this we have the following 
 
 Eule. — Reduce the fractions to a common denominator; 
 
 add the numerators, and place the sum over the common 
 
 denominator. 
 
 N 0TE . — Reduce each fraction to its lowest terms before reducing 
 to a common denominator, and also the result after addition. 
 
 Find the sum of 
 
 2. 
 
 3. 
 4. 
 
 5. 
 6. 
 
 7. 
 8. 
 
 and 
 
 2 
 
 | and \. 
 | and |. 
 | and § . 
 | and g. 
 % and T q 
 i and §. 
 
 Ans. 
 Ans. 
 
 16 
 1 5* 
 
 31 
 
 20' 
 
 Find the sum of 
 Ans. 
 
 Ans. 
 
 | and f . 
 9 anc * TO* 
 
 4 ar]C l 12 
 6 <UJU 14' 
 
 97 
 5 6' 
 
 31 
 
 3 0* 
 
 9. 
 10. 
 
 11. o 
 
 12. f| and Jf. 
 
 13. if and flf. 
 
 14. 1« and -}f . 
 
 15. i§ and U. 
 
 of a pie; how much 
 
 16. A has f of a pie, and B f 
 have they both ? 
 
 17. B having j of a ton of hay bought % of a ton; 
 Vow much had he then? 
 
COMMON FRACTIONS. 103 
 
 18 Henry owned § of a vessel, and bought \ of the 
 ressel ; how much did he then own ? 
 
 19. A had 7f dollars, and B gave him 8| dollars ; how 
 many had he then ? 
 
 Note.— Add the 7 and 8, and then add £ and f; 8 + 7 = 15: 
 
 ! + f=A + A = H = 1 A' 15+1 T 5 . = 10 T V Ans. 
 
 20. A had 25 r acres of land, and then bought 17| 
 acres ; how many had he then ? 
 
 21. B had 57| dollars, and C had 96| dollars; what 
 was the sum of their money ? 
 
 22. C sold 9G| yards of cloth, and then had 147 T 9 9 
 yards left ; how much had he at first ? 
 
 SUBTRACTION OF FRACTIONS. 
 
 111. Subtraction of Fractions is the process of find- 
 ing the difference between two fractions. 
 
 Case I. 
 To subtract when the denominators are alike. 
 
 1. Subtract | from §. 
 
 Solution.— 3 eighths subtracted from 7 eighths operation. 
 leave 4 eighths, and f reduced to its lowest f — 1 = |, or £ 
 terms equals \. 
 
 2. Subtract f from f . 
 
 3. Subtract \ from f. 
 
 4. Subtract | from f . 
 
 5. Subtract T 5 3 from }£. 
 
 6. Mary had g of an apple, and gave away | of an 
 apple ; what part of an apple had she left ? Ans. £. 
 
 7. Peter found i of a dollar, and spent § of a dollar ; 
 what part, of a dollar had he left? 
 
 8. If I buy \\ of a ton of hay and sell j 5 . 2 of a ton, 
 what part of a ton will I have left ? 
 
 i). Peter has T % of a dollar, John /„ of a dollar, and 
 
104 COMMON FRACTIONS. 
 
 Jacob T 4 o of a dollar ; how much more have Peter and 
 John than Jacob ? Ans. $^. 
 
 10. Mary has § of a dollar, Sarah i of a dollar, and 
 Jane § of a dollar ; how much more have Mary and 
 Sarah than Jane ? Ans. $1. 
 
 Case II. 
 To subtract when the denominators are unlike. 
 
 1. Subtract | from f. 
 
 Solution. — We will first reduce the fractions operation. 
 
 to a common denominator. f = xf and f =1& t — I 
 
 and 12 fifteenths minus 10 fifteenths is 2 fifteenths. if — \% 
 From this solution we have the following 
 
 2 
 T5 
 
 Eule. — Reduce the fractions to a common denominator, 
 subtract the numerators, and place the result over the com- 
 mon denominator. 
 
 Note. — Reduce each fraction, and also the difference, to its lowest 
 terms. 
 
 Subtract 
 
 2. I from f . Ans. T ^. 
 
 3. | from |. Ans. y 1 ^. 
 
 4. | from {. 
 
 5. | from §. 
 
 6. | from §. 
 
 Subtract 
 
 8. § from ig. Ans. 5 \. 
 
 9. £ from J. Ans. J g . 
 
 10. f from jf. 
 
 11. | from {I. 
 
 12. £ from T V 
 
 13. II from U. 
 
 " r . « from {I. 
 
 14. Mary had | of a dollar and gave away \ of a dol- 
 lar ; how much had she left ? 
 
 Solution. — If Mary had f of a dollar, and 
 gave away \ of a dollar, she had left the differ- operation. 
 
 ence between | of a dollar and £ of a dollar, f — \ — 
 
 which by reducing to a common denominator \% — ^ = ^ ff Ans. 
 and subtracting, we find to be -fa of a dollar 
 
 15. Willie gave Sallie § of a quart of peanuts, and 
 Sallie gave him back | of a quart ; what part of a quart 
 did Sallie keep ? 
 
COMMON FRACTIONS. 105 
 
 16 A has 2 of a pie; if he gives B | of a pie, how 
 much vi ill remain ? 
 
 17. B bought T 9 of a ton of hay, and sold C f of a 
 ton; how much did B retain? 
 
 18. C owned ] of a vessel, H bought J of this, and 
 then sold I of what he bought: how much did H keep? 
 
 19. The sum of two fractions is £{j, and one fraction 
 is li • what is the other fraction ? 
 
 20. If D had | of a certain sum of money, and then 
 earned f of the same sum, arid then spent J of the sum, 
 how much remained ? 
 
 21. The sum of three fractions is If, and the other 
 two fractions are -\ and \ ; what is the third fraction ? 
 
 22. A has | of a sum of money; he owes B § of that 
 sum and C 1 of the sum: how much will remain after 
 paying his debts ? Ans. /<j. 
 
 PRACTICAL PROBLEMS 
 in Addition and Subtraction of Fractions. 
 1. Subtract 4f from 74. 
 
 4 "* OPERATION. 
 
 Solution.— 1\ equals 6 -f £ -f | = 6£; 4| 7£ = 6f 
 
 Dtr 
 
 acted from 6| leaves 2|. 
 
 
 43 
 
 2f Ans. 
 
 
 Subtract 
 
 
 Subtract 
 
 2. 
 
 5| from 9J. 
 
 5. 
 
 I65 from 2( H 
 
 3. 
 
 7| from 10$. 
 
 6. 
 
 19f from 30§. 
 
 4. 
 
 8| from 13f. 
 
 7. 
 
 24| from 36|. 
 
 8. A had 17? dollars, and gave B 12| ; how much 
 remained ? 
 
 9. A has 6| dollars, and B has 7^ dollars; how much 
 have both ? 
 
 10. A read 4^ pages one day and 7 J another day ; 
 how many pages did he read in the two days? 
 
 11. Mary had 20 dollars; she gave her brother $9| 
 and her sister $l'l ; how much remained ? 
 
106 COMMON FRACTIONS. 
 
 12. William having $100 gave $26- to the poor and 
 spent S18| for clothing; how much remained? 
 
 13. My father gave me $3|, my mother gave me $5|, 
 and I then gave my sister $4| ; how much remained? 
 
 14. What is the sum of h of -J and | of \ ; and what, 
 also, is their difference ? 
 
 15. ] had $24, and gave \ of it to my sister and \ of 
 it to my brother ; how much remained ? 
 
 16. Sarah had $23, and gave \ of it to the poor and § 
 of it for a dress ; how much remained ? 
 
 17. Henry's father gave him $16|, and his mother 
 gave him $18£j he then spent f of it; how much re- 
 mained? 
 
 18. A man had $24f , and then earned $161, and then 
 spent A of it; how much remained? 
 
 19. Peter had $17-J, and then lost $11 -J, and then 
 earned $14 ; how much had he then ? 
 
 20. Harold had $|, then lost $|, and then earned $| ; 
 how much had he then ? 
 
 MULTIPLICATION OF FRACTIONS. 
 
 112. Multiplication of Fractions is the process oi 
 multiplying when one or both terms are fractions. 
 
 Case I. 
 
 113. To multiply a fraction by a whole number. 
 
 1. Multiply | by 4. 
 
 Solution.— 4 times § equal \°, according to operation. 
 
 Prin. I. Or, 4 times f equal f, since di- f X 4 = \° 
 viding the denominator multiplies the frac- or, | X 4 — I 
 tion, according 4« Prin. IV. From this we 
 have the following 
 
 Rule— To multiply a fraction by an integer, multiply 
 the numerator or divide the denominator by the integer-. 
 
COMMON FRACTIONS. 
 
 107 
 
 
 Multiply 
 
 
 
 Multiply 
 
 
 2. 
 
 A by 5. 
 
 Ans. J. 
 
 6. 
 
 13 by 9. 
 
 Ans. 8J 
 
 3. 
 
 Uby4. 
 
 Ans. y. 
 
 7. 
 
 i? by 7. 
 
 Ans. 6? 
 
 4. 
 
 I^by7. 
 
 
 8. 
 
 P by 12. 
 
 
 5. 
 
 i! by 3. 
 
 
 9. 
 
 mby36. 
 
 
 10. A has }J of a ton of hay, and B has 3 times as 
 nuch; how much have both? 
 
 Case II. 
 114. To multiply a whole number by a fraction. 
 
 1. Multiply 8 by f ; also by 4§. 
 
 Solution 1. — 8 multiplied by ^ equals % of 8, 
 or f, and 8 multiplied by § equals 2 times f, 
 
 16 
 
 or * 
 
 Solution 2. — We multiply 8 by 2 and divide 
 by 8, and have 51; then multiply by 4 and add 
 the product 32 to 51, making 371. Hence, in a 
 mixed number we multiply first by the fraction, 
 and then by the integer. From these solutions 
 we have the following 
 
 OPERATION. 
 
 8X2 
 
 OPERATION. 
 8 
 
 ^3 
 
 3)16 
 
 32 
 3Ti 
 
 Rule. — Multiply the whole number by the numerator of 
 the fraction, and divide the product by the denominator. 
 
 Multiply 
 
 2. 16 by |. 
 
 3. 18 by §. 
 
 4. 12 by l. 
 
 5. 20 by 2 T V 
 
 6. 35 by 45. 
 
 Ans. 12. 
 Ans. 15. 
 
 Multiply 
 
 7. 45 by §. 
 
 8. 43 by g. 
 
 9. 28 by 5}. 
 
 10. 76 by 4|. 
 
 11. 85 by 8 T V 
 
 Ans. 40. 
 Ans. 35| 
 
 12. A has 18 tons of hay, and B has 4| times as much, 
 plus 31 tons; how much has B? 
 
 Case III. 
 115. To multiply a fraction by a fraction. 
 
 1. Multiply I by j. 
 
108 
 
 DIVISION OF FRACTIONS. 
 
 | multiplied by \ equals \ of 
 
 III. ; and 
 | multiplied by f equals 3 times -fa or fa. 
 
 Solution. — 
 
 which is fa, according to Prin. 
 
 OPERATION. 
 
 3 V 3 
 
 "5 A £ — 
 
 3 X 3 _ 9 
 
 — tv 
 
 5X4 
 
 Eule. — Multiply the numerators together for the numera- 
 tor, and the denominators together for the denominator of 
 the product. 
 
 Note. — Reduce the result to its lowest terms. 
 What is the product of 
 
 2. 
 3. 
 4. 
 5. 
 6. 
 
 I by |? 
 
 I by 
 
 ft? 
 
 6 * 
 
 J> ? 
 
 1 4 * 
 
 15? 
 
 1 6 
 
 4? 
 
 t 9 o by° 
 iiby 
 
 M*y 
 
 Ans. 
 Ans. 
 
 3 
 
 TO* 
 
 7 
 TO"' 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 
 I? by 
 
 by 
 
 2 5 
 2 9 
 
 V 
 
 28 ? 
 25 ' 
 17 ? 
 3 5 ' 
 
 Ans 
 
 Ans. 
 
 Ans. : 
 
 is 
 
 24* 
 
 1 6 
 TO- 
 SS 
 103* 
 
 1 by | of |? 
 
 I of 
 
 5 U J 2 8' 
 
 12- A has | of a ton of hay, and B has f as much 
 plus 2| tons ; how much has B ? 
 
 DIVISION OF FRACTIONS. 
 
 116. Division of Fractions is the process of dividing 
 when one or both terms are fractional. 
 
 OPERATION. 
 
 8 ^_4 
 
 9 • * 
 
 Case I. 
 117. To divide when the dividend is a fraction, 
 
 1. Divide | by 4. 
 
 Solution. — | divided by 4 equals f , according to 
 Prin. I. When the numerator will not contain- 
 Ihe divisor, we multiply the denominator, accord- 
 ing to Prin. III. 
 
 Rule. — Divide the numerator, or multiply the denomt 
 nator, by the divisor. 
 
 Divide 
 
 t 9 o by 3. Ans. 
 
 by 4. Ans. 
 
 by 6. 
 
 2. 
 3. 
 4. 
 
 5. _9 T by 4. 
 
 6. | by a 
 
 _8_ 
 1 1 
 12 
 1 3 
 
 3 
 
 TO* 
 _2_ 
 
 i r 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 
 Divide 
 
 H by 7. 
 if by 5 
 
 V 8 by 8. 
 
 31 by 9. 
 5| by 12. 
 
 Ans. 
 
 Ans. 
 
 12 
 91' 
 i 6 
 
 5 5' 
 
MISCELLANEOUS EXAMPLES. ■ 109 
 
 12. A gave 3A dollars to 6 little girls ; how much did 
 each receive ? 
 
 Case II. 
 118. To divide when the divisor is a fraction. 
 
 1. Divide f by -1. 
 
 Solution. — | divided by 1 equals f, hence -J operation. 
 divided by A equals 5 times f , and -| divided by | -r- f = 
 £ equal £ ot 5 times f, or J times f, which |Xf = xf 
 equal ^|. Hence, we see the divisor becomes 
 inverted, and we have the following 
 
 Rule — Invert the divisor, and multiply the dividend by 
 the resulting fraction. 
 
 Divide Divide 
 
 2. 
 
 ffcyf 
 
 Ans. |. 
 
 8. 
 
 14 bv -9- 
 
 Ans. 1J. 
 
 o 
 O. 
 
 i by I 
 
 Ans. f . 
 
 9. 
 
 2.1 h V 14 
 
 3 2 U J lS' 
 
 Ans. f J. 
 
 4 
 
 I by §. 
 
 Ans. |i. 
 
 10. 
 
 2 |) V 18 
 2 1 U J 3 5* 
 
 Ans. Iff. 
 
 5. 
 
 t 9 o *y !• 
 
 
 11. 
 
 16 V)V 8 
 
 Ans. 2 T 8 T . 
 
 6. 
 
 H by {. 
 
 
 12. 
 
 15 u J 3 5* 
 
 
 7. 
 
 I? by fj. 
 
 
 13. 
 
 3 2 Kv 4 8 
 3 6 U J 5 0' 
 
 
 14. How many yards of cloth at | of a dollar a yard 
 can you buy for 4^ dollars ? 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. Reduce 32| to an improper fraction. 
 
 2. Reduce 47 T fi T to an improper fraction. 
 
 3. Reduce 4 T ° 4 8 to a mixed number. 
 
 4. Reduce ^f 8 to a mixed number. 
 
 5. Reduce |J| to its lowest terms. 
 
 6. Reduce y 4 ^ to its lowest terms. 
 
 7. Reduce f of -f of § to a simple fraction. 
 
 8. Reduce | of || of -^ 8 to a simple fraction. 
 
 9. Reduce §, |, and J to a common denominator. 
 
 10. Reduce |, |, and -j Q to a common denominator 
 
 11. Find the sum of J, J, i, and J. 
 12- Find the sum of f, j, <, and {. 
 
110 • ANALYSIS. 
 
 13. Subtract § of | from § of |. 
 
 14. Subtract ? of f. from the sum of ? and |. 
 
 4 o 4 
 
 15. Multiply the sum of ^ and ] by | plus i. 
 1G. Multiply I -J- | by the sum of § and |. 
 
 17. Divide I -{- 1 by the sum of \ and f . 
 
 18. Divide the sum of § and J- by | minus §. 
 
 19. What cost 24 apples at | of a cent each? 
 
 20. What cost 45 oranges at 2§ cents apiece ? 
 
 21. How much cost 16 j yards of cloth at $6 a yard ? 
 
 22. What cost 16- yards of muslin at 12 J cents a yard? 
 
 23. If one yard of cloth cost $9, how many yards can 
 you buy for $48 ? Ans. 5 J yds. 
 
 24. How many yards of muslin at 16J cents a yard 
 can you buy for 208^ cents? Ans 12^ yds. 
 
 AEITHMETICAL ANALYSIS. 
 
 119. Analysis is the process of solving problems by 
 a comparison of their elements. In comparing, we 
 reason to the unit and from the unit, the unit being the 
 basis of the reasoning process. 
 
 Case I. 
 
 120. To pass irons, one integer to another. 
 
 1. If 5 cows cost $80, what will 7 cows cost at the 
 same rate ? 
 
 OPERATION. 
 
 Solution. — If 5 cows cost $80, one cow costs | 5)80 
 of $80, which is $16, and 7 cows will cost 7 times 16 
 
 $16, which are $112. _7 
 
 112 Ans. 
 
 2. If 6 hens cost 186 cents, what will 9 hens cost at 
 the same rate? 
 
 3. If 5 pigs cost $35, what will 11 pigs cost at the 
 same rate ? 
 
ANALYSIS. 
 
 Ill 
 
 4. If 8 horses cost $1200, what will 12 horses cost at 
 the same rate? 
 
 5. If 7 yards of cloth cost 842, what will 25 yards 
 cost at the same rate ? 
 
 6. How much must I pay for 3G cows, at the rate of 
 7 cows for 196 dollars ? 
 
 7. What will 17 books cost, at the rate of 8 books for- 
 810.80? 
 
 8. A man bought 72 ducks at the rate of $21 for 7; 
 
 what did they cost? 
 
 9. If a man can walk 324 miles in 9 days, how far 
 can he walk in 69 days? 
 
 10. In 26 years there are 9490 days ; how many days 
 are there in 75 years ? 
 
 11. In 5 square miles there are 3200 acres ; how many 
 acres in 64 square miles ? 
 
 12. If a car run 2736 miles in 18 days, how far will 
 it run in 54 days ? 
 
 Case II. 
 121. To pass from a fraction to an integer. 
 
 1. If | of an acre of land cost $96, what will one acre 
 cost? 
 
 OPERATION. 
 
 Solution. — If § of an acre cost $96, I of an § — $96 
 
 acre cost A of $96, or $48, and if A of an acre cost i == $48 
 
 $48, | of an acre, or one acre, will cost 3 times f = $144 Ans. 
 $48, or $144. 
 
 2. If | of a sum of money is $72, required the sum. 
 
 3. If | of the cost of a cow is $25, required the cost 
 of the cow. 
 
 4. What cost 2 boxes of raisins, if -? of a box cost 6 
 dollars ? 
 
 5. What is the distance from Lancaster to Philadel- 
 phia, if | of the distance is 51 miles? 
 
 6 If the cost of | of an acre of land is $120, what 
 will 4 acres cost at the same rate? 
 
112 COMMON FRACTIONS. 
 
 7. If f of a farm cost $7200, what will the whole 
 farm cost at that rate ? 
 
 8. How much will 7 loads of hay weigh, if J of a 
 load weighs 840 pounds ? 
 
 9. What will 17 horses cost me, if f of the price of 
 a horse is 93 dollars ? 
 
 10. A merchant bought 236 barrels of flour at the 
 rate of $8 for § of a barrel ; how much did they cost 
 him? 
 
 Case III. 
 122. To pass from a nnit or fraction to a fraction, 
 
 1. If one barrel of flour coHs $12, what will j of a 
 barrel cost ? 
 
 Solution. — If one barrel of flour costs $12, operation. 
 1 fourth of a barrel will cost \ of $12, or $3, 4)12 
 
 and | of a barrel will cost 3 times $3, or $9. 3 
 
 3 
 
 9 Ans. 
 
 2. If one acre of land is worth $125, what is 4 of an 
 acre worth ? 
 
 3. A paid $1650 for a pleasure-boat ; how much would 
 he have paid if he had given § as much ? 
 
 4. If | of a barrel of flour cost $8, what will f of a 
 barrel cost ? 
 
 5. If there are 40 pounds in | of a bushel of clover- 
 seed, how many pounds are there in § of a bushel ? 
 
 6. If there are 50 pounds in | of a bushel of wheat, 
 how many pounds are there in j] of a bushel ? 
 
 7. If there are 49 pounds in J of a bushel of rye, 
 how many pounds are there in | of a bushel? 
 
 8. If there are 147 pounds in | of a barrel of flour, 
 how many pounds are there in | of a barrel ? 
 
 9. If there are 154 cubic inches in f of a gallon, how 
 many cubic inches in ^ of a gallon ? 
 
COMMON FRACTIONS. 113 
 
 10. /f there are 1536 cubic inches in § of a cubic foot, 
 how many cubic inches in ji of a cubic foot? 
 
 Case IT. 
 123. Given a fractional part and the remainder, 
 to find the whole. 
 
 1. A man spent | of his money, and then had $24 re* 
 maining ; how much money had he at first ? 
 
 Solution. — If he spent § of his money, operation. 
 
 there remained § of his money minus f of f — f = § == $24 
 his money, which is § of his money, which ^ = $12 
 
 is S24. If | of his money is §24, I of his $ = $60 Ans. 
 
 money is | of $24, which is $12, and § of 
 his money is 5 times $12, or $00. 
 
 2. A man spent f of his money, and then had $30 re- 
 maining; how much had he at first? 
 
 3. William sold I of his hens, and then had 60 remain- 
 ing; how many had he at first ? 
 
 4. Henry sold § of his bank-stock, and the remainder 
 was worth $550 j how much had he at first ? 
 
 5. After giving \ of his income to the poor, Samuel 
 had $960 remaining; what was his income ? 
 
 6. A pole stands J in the mud and | in the water, and 
 12 feet in the air; required the length of the pole. 
 
 7. One-fourth of a drove of animals are cows, I are 
 pigs, and the remainder, 132, are sheep; how many 
 animals in the drove? 
 
 8. Two-fifths of my money is in bank, | in govern- 
 ment bonds, and $480 in cash; what was my money ? 
 
 9. A sold J of his land to B, and | to C, and then had 
 90 acres remaining; how much had he at first? 
 
 10. A man walked § of the distance from Lancaster to 
 Philadelphia one day, | of the distance the next day, 
 and the remaining distance, 22 miles, the third day; 
 how far did he walk each day ? Ans. 20 ; 28 ; 22. 
 
 10* 
 
114 DECIMAL FRACTIONS. 
 
 SECTION VI. 
 
 DECIMAL FRACTIONS. 
 
 124. A Decimal Fraction is a number of the deci- 
 mal divisions of a unit; that is, a number of tenths, hun- 
 dredths, etc. 
 
 125. A decimal fraction is usually expressed by 
 placing a point before the numerator and omitting the 
 denominator. Thus, .5 represents T 5 D ; .05 represents 
 
 TOO' e ^ c - 
 
 126. The point is called the decimal point, or separa- 
 trix. The decimal fraction thus expressed is called a 
 decimal. 
 
 127. This method of expressing decimal fractions is 
 but an extension of the method of notation for integers. 
 This method, as applied to integers and fractions, is ex- 
 hibited in the following 
 
 o 
 
 NOTATION AND NUMERATION TABLE. 
 
 DO 
 
 a 
 a 
 
 CO 
 
 m o 
 
 o *t 
 
 
 
 CO ,-j CO 
 
 Xl 2 o> o 5 © 
 6666666 6. 6666666 
 
 
 
 
 05 
 
 M 
 
 
 
 
 
 
 ^ 
 
 C3 
 CO 
 
 3 
 
 
 
 
 
 • 
 
 -3 
 
 
 ^q 
 
 02 
 
 -2 
 
 03 
 0) 
 
 CO 
 
 co 
 
 a 
 
 CO 
 
 3 
 O 
 
 O 
 
 I 
 
 S- 
 
 H3 
 
 CO 
 
 o 
 
 o 
 3 
 
 0) 
 
 a 
 
 5 
 — | 
 
 1 
 
 g 
 
 .— 
 
 0) 
 
 Eh 
 
 i-M 
 
 l-i-C 
 
 H 
 
 H 
 
 
 <s 
 
 H 
 
 
 EXAMPLES IN NUMERATION. ■ 
 
 1. Read the decimal .36. 
 
 Solution. — This expresses 3 tenths and 6 hundredths, or, since 
 3 tenths equals 30 hundredths, and 30 hundredths plus 6 hundredths 
 equal 36 hundredths, it may also be read 36 hundredths. 
 
DECIMAL FRACTIONS. 
 
 115 
 
 128. Hence there are two methods of reading deci- 
 mals, which are expressed by the following rules: — 
 
 Eule 1. — Commencing at tenths, read each figure in 
 order toward the right, giving it Us proper denomination. 
 
 Rule 2. — Read the decimal as a whole number, and 
 give it the denomination of the last figure on the right; 
 numerating toward the point to determine the numerator, 
 and from the point to determine the denominator. 
 Read the following decimals: — 
 
 2. 
 
 .45. 
 
 6. .046. 
 
 10. 2.0123. 
 
 3. 
 
 .83. 
 
 7. .007. 
 
 11. 4.2057. 
 
 4. 
 
 .126. 
 
 8. .3216. 
 
 12. 13.0205. 
 
 5. 
 
 .324. 
 
 9. .1357. 
 
 13. 27.0027. 
 
 examples in notation. 
 1. Express 25 hundredths in the form of a decimal. 
 
 Solution.— 25 hundredths equals 2 tenths and 5 hundredths, and 
 this is expressed by writing a decimal point before 25, thus, .25 
 Hence the following * 
 
 Rule. — Write the decimal as we would a whole number, 
 placing the decimal point so as to give each figure its proper 
 place, using ciphers after the decimal point if necessary. 
 
 Express the following in the decimal form. 
 
 2. Thirty-four hundredths. 
 
 3. Seventy-five hundredths. 
 
 4. Two-tenths and six-hundredths. 
 
 5. Twenty-five thousandths. 
 
 6. Four-tenths and 7-thousandths. 
 
 7. Seven-tenths and 8-thousandths. 
 
 8. Five hundred and 25-thousandths. 
 
 9. Three-tenths and 7 ten-thousandths. 
 
 10. Four hundredths and 96 millionths. Ans. .0400096. 
 
 PRINCIPLES OF DECIMAL NOTATION. 
 129. We now present the following principles of 
 Decimals, which the pupils will illustrate. 
 
116 DECIMAL FRACTIONS. 
 
 1. Changing the decimal point one place toward the right 
 multiplies by 10; two places, by 100, etc. 
 
 2. Changing the decimal point one place toward the left 
 divides by 10 ; two places, by 100, etc. 
 
 3. Placing a cipher between the decimal point and a deci- 
 mal divides the decimal by ten. 
 
 REDUCTION OF DECIMALS. 
 
 130. The Reduction of Decimals is the process of 
 changing their form without changing their value. 
 
 There are two cases : — 
 
 1. To reduce decimals to common fractions. 
 
 2. To reduce common fractions to decimals. 
 
 Case I. 
 
 131. To reduce a decimal to a common fraction. 
 
 1. Eeduce .45 to a common fraction. 
 
 * 
 
 Solution. — .45 expressed in the form of a operation. 
 
 common fraction, is T 4 5 o> which, reduced to its .45 = T \ 5 5 
 
 lowest terms, equals ^V Hence we have the = -fa Ans. 
 following 
 
 R ULE . — Write the denominator under the decimal, omiU 
 ting the decimal point, and reduce the common fraction to 
 its lowest terms. 
 
 Eeduce the following decimals to common fractions. 
 
 2. 
 
 .35. 
 
 Ans. -Jq. 
 
 6. 
 
 9.75. 
 
 Ans. 
 
 » 4 « 
 
 3. 
 
 .48. 
 
 Ans. i$. 
 
 7. 
 
 .725. 
 
 Ans. 
 
 29 
 4 0' 
 
 4. 
 
 .125. 
 
 Ans. |. 
 
 8. 
 
 .075. 
 
 Ans. 
 
 3 
 
 40' 
 
 0. 
 
 .625. 
 
 Ans. g. 
 
 9. 
 
 .0125. 
 
 Ans. 
 
 1 
 HO* 
 
 Case II. 
 132. To reduce a common fraction to a decimal. 
 
 1. Eeduce f to a decimal. 
 
DECIMAL FRACTIONS. 117 
 
 Solution.— | equals \ of 3. 3 equals 30 operation. 
 
 tenths, and \ of 30 tenths is 7 tenths and 2 f = \ of 3 = 
 
 tenths remaining. 2 tenths equals 20 hun- 4 )3.00 
 dredths, and \ of 20 hundredths is 5 hun- .75 
 
 dredths ; hence f = .75. From this we have 
 the following 
 
 Eule. — 1. Annex ciphers to the numerator and divide by 
 the denominator. 
 
 2. Point off as many places in the quotient as there are 
 ciphers annexed. 
 
 Reduce the following common fractions to decimals. 
 
 2. 4- Ans. .25 
 
 4 
 
 7. T V Ans. .4375. 
 
 1 D 
 
 3. |. Ans. .125. 
 
 4 5 
 
 5* 
 
 5. 
 
 6. -V Ans. .3125. 
 
 8. T 9 S . Ans. .5625. 
 
 1 o 
 Q 1 1 
 
 10 ±4 
 
 11 P 
 ±x ' 64* 
 
 OPERATION 
 
 7.5 
 
 18.25 
 
 21.36 
 
 47.45 
 
 ADDITION OF DECIMALS. 
 
 133. Addition of Decimals is the process of finding 
 the sum of two or more decimals. 
 
 1. What is the sum of 7.5, 18.25, 21.36 and 47.45? 
 
 Solution. — "We write the numbers so that 
 figures of the same order shall stand in the 
 same column, and commence at the right to 
 add. 5 hundredths, plus 6 hundredths, plus 5 
 hundredths, equal 16 hundredths, which equal 
 1 tenth and 6 hundredths; we write the 6 hun- 94.56 
 
 dredths, and add the 1 tenth to the next sum. 
 
 4 tenths, plus 3 tenths, plus 2 tenths, plus 5 tenths, are 14 tenths, 
 and the 1 tenth added are 15 tenths, which equals 1 unit and 5 
 tenths; we write the 5 tenths, and add the 1 unit to the sum of 
 the units, etc. 
 
 Rule. — 1. Write the numbers so that units of the same 
 order shall stand in the same column. 
 
 2. Add, as in whole numbers, placing the decimal point 
 in its proper place in the sum. 
 
118 DECIMAL FRACTIONS. 
 
 2. Find the sum of 12.05, 33.24, 47.62, 96.47. 
 
 3. Find the sum of 76.24, 89.45, 36.40, 85.75. 
 
 4. Find the sum of 79.76, 85.08, 95.42, 237.675. 
 
 5. Add 18.79, 147.072, 856.709, 185.8761, 397.05784. 
 
 6. Add 59.874, 435.095, 672.328, 976.309, 8467.500843. 
 
 7. Add together 9 and 7 tenths, 41 and 8 hundredths, 
 75 and 54 hundredths, 128 and 187 thousandths. 
 
 Ans. 254.507. 
 
 8. Add together 76 and 49 hundredths, 127 and 49 
 thousandths, 496 and 167 thousandths, 985 and 98 ten- 
 thousandths, and 99 and 99 hundred-thousandths. 
 
 SUBTRACTION OF DECIMALS. 
 134. Subtraction of Decimals is the process of find- 
 ins: the difference between two decimals. 
 
 1. From 67.35 take 42.63. 
 
 Solution. — We write the numbers so yiat operation 
 figures of the same order stand in the same 67.85 
 
 column, and begin at the right to subtract. 3 42.63 
 
 hundredths from 5 hundredths leave 2 hun- 24.72 
 
 dredths ; 6 tenths we cannot subtract from 3 
 tenths ; we therefore take 1 unit from the 7 units, which with 3 tenths 
 equal 13 tenths ; then 6 tenths from 13 tenths leave 7 tenths, etc. 
 
 Rule. — 1. Write the smaller number under the greater, 
 so that figures of the same order stand in the same column. 
 
 2. Subtract as in simple numbers, and place the decimal 
 point in its proper place in the difference. 
 
 2. From 63.72 take 25.81. 
 
 3. From 96.32 take 73.15. 
 
 4. From 123.16 take 75.84. 
 
 5. From 247.125 take 167.183. 
 
 6. From 1 and 1 tenth take 1 tenth and 1 thousandth. 
 
 7. From 2 and 2 hundredths take 2 tenths and 2 
 thousandths. 
 
 8. From 3 tenths take 3 ten-thousandths. 
 
 9. From 7 take 7 tenths and 707 millionths. 
 
DECIMAL FRACTIONS. 119 
 
 MULTIPLICATION OF DECIMALS. 
 
 135. Multiplication of Decimals is the process of 
 multiplying when one or both terms are decimals. 
 
 1. Multiply 7.23 by .46. 
 
 Solution 1. — Multiplying as in whole num- operation. 
 
 bers, we have 33258; now, if the multiplicand 7.23 
 
 alone were hundredths, the product would be one- .46 
 
 hundredth of this, or 332.58 ; but since the mul- 4303 
 
 tiplier is also hundredths, the product is one-hun- 2892 
 
 dredth of 332.58, which, by moving the decimal 3.8258 
 point two places to the left, becomes 3.3258. 
 
 Solution 2.-7.23 X -46 = HfX^ = -^ =to!ooX -258 
 
 10000 
 
 = 3.3258. From either of these solutions we derive the following 
 
 Rule. — Multiply as in loliole numbers, and point off as 
 many decimal places in the product as there are in both mul- 
 tiplier and multiplicand, prefixing ciphers when necessary. 
 
 2. Multiply 15.17 by .IS. 
 
 3. Multiply 26.18 by .25. 
 
 4. Multiply 53.46 by .35. 
 
 5. Multiply 67.38 by 1.26. 
 
 6. Multiply 138.25 by 2.47. 
 
 7. Multiply 466.72 by 5.29. 
 
 8. Multiply 407.03 by 7.35. 
 
 9. Multiply 620.75 by 12.36. 
 
 10. Multiply 725.82 by 23.08. 
 
 11. Multiply .00723 by .0317. 
 
 12. Multiply 1.0309 by .00321. 
 
 DIVISION OF DECIMALS. 
 
 136. Division of Decimals is the process of dividing 
 when one or both terms are decimals. 
 
 1. Divide 7.8315 by 2.27. 
 
120 DECIMAL FRACTIONS. 
 
 Solution. — If we divide as in whole num- operation. 
 
 bers, we obtain a quotient of 345 ; now, since 2.27)7.8315(3.45 Ana 
 the dividend is the product of the divisor and c 81 
 
 quotient, the number of decimal places in the 1 021 
 
 dividend must equal the number in the divisor 908 
 
 and quotient: hence, the number of decimal 1135 
 
 -i -i or 
 
 places in the quotient must equal the number Xl0 ° 
 
 of decimal places in the dividend diminished 
 
 by the number in the divisor ; hence, there should he four minus two, 
 
 or tioo decimal places in the quotient, therefore the quotient is 3.45. 
 
 Q n 7 QQ1 K . 9 97 78315 _i_ 227 78 315 V W ■ 
 
 C50LLTI0N Z. — i.OoLO — £—t — TO ±($0 TOO A Hz7 — 
 
 78315 = T ^ x 7 f I?- 5 = jh X 3 ^5 = 3 - 45 - From either of these 
 
 100^227 
 
 solutions we derive the following 
 
 Rule. — Divide as in whole numbers, and point off as 
 many decimal places in the quotient as the number of deci- 
 mal places in the dividend exceeds the number in the divisor. 
 
 Note 1. — When there are not as many decimal places in the divi- 
 dend as in the divisor, annex ciphers to make the number of place? 
 equal. 
 
 2. When the number of figures in the quotient is less than tho 
 excess of the decimal places in the dividend over those in the divisor, 
 ciphers must be prefixed to the quotient. 
 
 2. Divide 14.1372 by 4.5. Ans. 3.1416. 
 
 3. Divide 196.1875 by 10.75. Ans. 18.25. 
 
 4. Divide 25.1328 by 8. Ans. 3.1416. 
 
 5. Divide 65.9736 by 3.1416. 
 
 6. Divide 2450.448 by .5236. 
 
 7. Divide 2748.9 by .7854. 
 
 8. Divide 127.328 by .07958. 
 
 9. Divide 15.90435 by 20.25. 
 10. Divide 352.0625 by 32.75. 
 
 PRACTICAL PROBLEMS. 
 
 1. What cost 43.45 acres of land at $38.5 an acre? 
 
 Ans. $1672.825. 
 
 2. "What cost 57.75 tons of hay at $12.25 a ton ? 
 
 Ans. $707.4375. 
 
DECIMAL FRACTIONS. 121 
 
 3. If 31.25 yards of muslin cost $7.8125; how much 
 is that a yard ? Ans. $0.25. 
 
 4. A man sold 35.25 pounds of butter for $5,875 ; how 
 much is that a pound? Ans. $0,166+. 
 
 5. There are 7.92 inches in a link ; how many inches 
 in 990 links ? Ans. 7840.8 in. 
 
 6. There are 31.5 gallons in a barrel ; how many 
 barrels in 2756.25 gallons ? Ans. 87.5 barrels. 
 
 . 7. If 14.5 yards of cloth cost $68,875, how much is 
 that a yard ? Ans. $4.75. 
 
 8. If a man walk 112-1184 miles in 9.16 days, how 
 many miles does he walk each day ? Ans. 12.24 miles. 
 
 9. How many yards of cloth at $4.28 a yard, can a 
 person buy for $44.9828 ? Ans. $10.51. 
 
 10. What is the value of 54.6 multiplied by 80.5, and 
 the product divided by 2 ? Ans. 2197.65. 
 
 11. The circumference of a water-wheel is 64 feet, and 
 the diameter equals this divided by 3.1416; required 
 the diameter ? Ans. 20.3718 feet. 
 
 12. If 25.5 yards of cloth cost 195.375, how much will 
 45.25 yards cost ? Ans. $346,696+. 
 
 13. If an imperial gallon contains 277.274 cubic inches, 
 how many cubic inches in 328.55 gallons? 
 
 Ans. 91098.3727. 
 
 14. A gallon of distilled water weighs 8.33888 pounds; 
 how many gallons in 1000 pounds of such water ? 
 
 Ans. 119.92+. 
 
 15. A cubic inch of water weighs 252.458 grains; how 
 many cubic inches in 157786.25 grains ? 
 
 Ans. 625cu. in. 
 
 16. A drew 41.25 barrels, of 31.5 gallons each, from a 
 cistern containing 2000 gallons; how much remained? 
 
 Ans. 700.625. 
 
 17. A bought 7S.25 acres of land at $128.5 an acre, 
 and sold it fur $97^1.25; what was the loss on each 
 acre? Ans. $3.50. 
 
 11 
 
122 PROBLEMS. 
 
 REVIEW OF FUNDAMENTAL RULES. 
 
 HISTORICAL, GEOGRAPHICAL, ETC. PROBLEMS. 
 Suggestion. — The teacher should explain the nature of the facts 
 presented, and require the pupils to remernher some of the mora 
 important numbers and dates. 
 
 PROBLEMS 
 
 on Battles of the Revolution. 
 
 1. At the battle of Lexington, the Americans lost 90 
 men, the British 190 more ; how many did the British 
 Jose? 
 
 2. At the battle of Bunker Hill, the Americans had 
 1500 men, the British 1500 more; how many had the 
 British ? 
 
 3. In this battle the Americans lost 450 men, the 
 British 604 more; how many did the British lose? 
 
 4. At the battle of Long Island, the British lost 367 
 men, the Americans 1233 more; how many did the 
 Americans lose ? 
 
 5. At the battle of Trenton, the British lost 45 in 
 killed and wounded, and 1000 prisoners; what was 
 their loss ? 
 
 6. In the battle of Brandywine, the British lost 800 
 men, and the Americans 450 more ; how many did the 
 Americans lose ? 
 
 7. In the battle of Germantown, the British lost 600 
 men, and the Americans lost 600 more ; how many did 
 the Americans lose ? 
 
 8. At the battle of Bennington, the Americans lost 
 about 100, and the British 600 more; required the 
 British loss. 
 
 9. At the battle of Monmouth, the Americans lost 70 
 ii? killed, and the British 230 more; required the British 
 loss. 
 
 10. In taking Stony Point, Gen. Wayne lost 15 killed 
 and 83 wounded, and the British lost 500 more in killed, 
 wounded, and prisoners ; required the British loss. 
 
PROBLEMS. 123 
 
 11. At the battle of Sander's Creek, the British lost 
 325, and the Americans 675 more; what was the Ame- 
 rican loss? 
 
 12. At the battle of Kind's Mountain, the Americans 
 lost 20 men, and the British 280 more; how many did 
 the British lose ? 
 
 13. At the battle of 'Guilford, the Americans lost 400 
 men, the British lost 100 more; required the British 
 loss. 
 
 14. At Hobkirk's Hill, the British loss was about 258 
 men, and the Americans 8 more; how many did the 
 latter lose? 
 
 15. At Xinety-Six, the Americans lost 51 in killed 
 and Avounded ; the British lost 1 more than this in 
 killed, and 283 more in wounded; required the British 
 loss. 
 
 1G. At the battle of Eutaw Springs, the Americans 
 lost 555, and the British 138 more; required the British 
 loss. 
 
 17. At Yorktown, Washington had 11,000 Americans 
 and 5000 French, and the British had 2000 more than 
 the French ; what was the force on each side ? 
 
 18. At Yorktown, the Americans lost about 75 killed, 
 and 225 wounded ; the British lost 156 killed, 170 more 
 than this wounded, and 70 missing ; what w T as the loss 
 on each side, not including prisoners ? 
 
 PROBLEMS IN AMERICAN HISTORY. 
 
 1. America was discovered by Columbus in 1492, and 
 Jamestown was settled in 1607; w T hat was the differ- 
 ence of time? 
 
 2. Plymouth was settled in 1620 ; how long was that 
 after America was discovered, and how long after the 
 settlement of Jamestown? 
 
 3. The battle of Lexington was fought in 1775; how 
 long was that after the settlement of Plymouth? 
 
 4. The .Declaration of Independence was made in 
 
124 PROBLEMS. 
 
 1776 ; how long was that after the settlement of James, 
 town ? 
 
 5. The surrender of Burgoyne took place in 1777 j 
 how long was that after the discovery of America? 
 
 6. The Inauguration of Washington took place iL 
 1789; how long was that after the battle of Bunker 
 Hill, in 1775 ? 
 
 7. The battle of New Orleans took place in 1815; 
 how long was that after the inauguration of Washing- 
 ton ? 
 
 8. The frigate Constitution captured the British fri- 
 gate Guerriere in 1812; how long was it after the De- 
 claration of Independence? 
 
 9. Commodore Perry won his great naval victory in 
 1813 ; how long was that after the battle of Lexington? 
 
 10. General Jackson won his great victory at New 
 Orleans in 1815 ; how long is it from then till the pre- 
 sent ? 
 
 11.. General Packenham had 12000 men, and General 
 Jackson 6000 ; how many more had the British J /;han 
 the Americans ? 
 
 12. The British lost 1700 in killed and wounded, the 
 Americans 13 men; what was the difference? 
 
 13. War with Mexico commenced in 1846 ; how long 
 was that after the battle of Lexington ? 
 
 14. At the battle of Palo Alto, General Taylor had 
 2300 men and the Mexicans 6000 ; what was the differ- 
 ence? 
 
 15. The battle of Buena Yista was fought in 1847 ; 
 how long was this after the battle of Bunker Hill ? 
 
 16. At this battle General Taylor had 4759 men, while 
 Santa Anna had 20000 men ; required the difference of 
 the forces. 
 
 17. General Scott took the Mexican capital in 1847; 
 how long is it from that time to the present ? 
 
PROBLEMS. 125 
 
 PROBLEMS ON THE AREA OF STATES. 
 NEW ENGLAND STATES. 
 
 1. The area of Maine is 30000 square miles, and of 
 jjSew Hampshire 9280 square miles; how mueh larger 
 is the former? 
 
 2. Vermont contains 9056 square miles, and Massa- 
 chusetts 7800 square miles; how much larger is the 
 former than the latter? 
 
 3. Ehode Island contains 1306 square miles, and Con- 
 necticut 4674 square miles; how much larger is Maine 
 than both of these? 
 
 4. Which is larger, and how much, Maine or all the 
 rest of the .New England States ? Which is larger, and 
 how much, New Hampshire and Vermont together, or 
 Massachusetts and Connecticut together ? 
 
 MIDDLE STATES. 
 
 5. New York contains 47000 square miles, and New 
 Jersey 8320 square miles; how much larger is the former? 
 
 6. Pennsylvania contains 46000 square miles, and 
 Delaware 2120 square miles; how much larger is Penn. 
 sylvania ? 
 
 7. Maryland contains 9356 square miles; how much 
 larger is Maryland than New Jersey ? 
 
 8. How much larger is Pennsylvania than New Jer- 
 sey, Delaware, and Maryland all together? 
 
 9. How much larger are the Middle States than the 
 New England States? 
 
 WESTERN STATES. 
 
 10. Ohio contains 39964 square miles, and Indiana 
 33809 square miles; how much larger is the former 
 than the latter ? 
 
 11. Michigan contains 56243 square miles, and Illinois 
 55405 square miles; how much larger is the formed 
 than the latter ? n * 
 
126 PROBLEMS. 
 
 12. How much larger are Ohio and Michigan than 
 Indiana and Illinois? 
 
 13. Wisconsin contains 53924 square miles, and Iowa 
 55045 square miles; how much larger is the latter than 
 the former ? 
 
 14. Missouri contains 67380 square miles, and Ken- 
 tucky 37680 square miles; how much larger is Mis- 
 souri ? 
 
 15. Which would make the larger State, Wisconsin 
 and Iowa, or Missouri and Kentucky? 
 
 16. California contains 189000 square miles, and Ore- 
 gon 95000 square miles; which is the larger, and how 
 much ? 
 
 17. How much larger are the Western States than 
 the New England and Middle States together ? 
 
 SOUTHERN STATES. 
 
 18. "Virginia contains 41352 square miles, and West 
 Virginia 20000 square miles; how much larger is Vir- 
 ginia than West Virginia ? 
 
 19. North Carolina contains 45000 square miles, and 
 South Carolina 24500 square miles; how much larger 
 is North Carolina than South Carolina ? 
 
 20. Georgia contains 58000 square miles, and Louisiana 
 46431 square miles ; how much larger is G-eorgia than 
 Louisiana? 
 
 21. Alabama contains 50722 square miles, and Missis- 
 sippi 47156 square miles ; how much larger is Alabama 
 than Mississippi ? 
 
 22. Arkansas contains 52198 square miles, and Tennes- 
 Bee 45600 ; which is the larger, and how much ? 
 
 23. Florida contains 59628 square miles, and Texas 
 237321 square miles ; how much larger is Texas than 
 Florida ? 
 
 24. Which is larger, and how much, Texas, or all the 
 other States taken together ? 
 
BUSINESS PROBLEMS. 127 
 
 BUSINESS PROBLEMS. 
 
 Suggestion. — Pupils will put these in the form of accounts, as on 
 
 page 93. 
 
 1. A merchant sold a farmer 125 yards of calico, at 
 18 cents a yard, 150 yards of drilling, at 15 cents a yard, 
 and bought of the farmer 225 bushels of oats, at 40 cents 
 a bushel, and 90 bushels of rye, at $1.25 a bushel ; which 
 owes the other, and how much? 
 
 2. A mechanic sold a farmer a wagon for $56.50, two 
 plows, at $7.50 each, and 6 wheel-barrows, at $5.25 each; 
 and bought of the farmer 50 bushels of potatoes, at 75 
 cents a bushel, and 75 bushels of wheat, at 85 cents a 
 bushel ; which owes the other, and how much ? 
 
 3. A farmer sold a merchant 4 cows, at $28.50 each, a 
 yoke of oxen for $95, and 7 sheep, at $6.25 each; and 
 took in payment 40 yards of carpet, at $2.25 a yard, 35 
 yards of cloth, at $3.25 a yard, and 58 yards of muslin, 
 at 15 cents a yard; how much remains due? 
 
 4. A farmer bought of a mechanic, 2 wagons, at $76 
 each, 4 drags, at $6.50 each, 3 harrows, at $12.25 each; 
 and sold him 45 bushels of apples, at 55 cents a bushel, 
 3 barrels of cider, at $5.25 a barrel, 28 bushels of corn, 
 at 42 cents a bushel, and 3 cows, at $28.75 each; which 
 owes the other, and how much ? 
 
 5. A mechanic bought of a merchant 
 
 28 pounds of sugar, at 18cts. a pound, 
 
 36 pounds of rice, at 17cts. a pound, 
 45 yards of muslin, at 18cts. a yard, 
 28 yards of cloth, at $5.25 a yard, 
 
 37 barrels of flour, at $7.25 a barrel ; 
 And sold him 
 
 4 wagons, at $75 each, 
 
 6 wagon-racks, at $13.50 each, 
 
 2 mowing-machines, at $157 each, 
 
 3 ox-yokes, at $6.75 each ; 
 Which owes the other, and how much? 
 
128 DENOMINATE NUMBERS. 
 
 SECTION" VII. 
 
 DENOMINATE NUMBERS. 
 
 137. A Concrete Number is one which refers to 
 some particular unit, as 2 books, 3 pounds, etc. 
 
 138. Concrete numbers are of two kinds ; those in 
 which the unit is natural, and those in which it is arti- 
 ficial. 
 
 139. A Denominate Number is a concrete number 
 in which the unit is artificial, as 3 pounds, 4 yards, 5 
 minutes, etc. 
 
 14 0. Reduction is the process of changing a number 
 from one denomination to another without changing its 
 value. 
 
 141. Reduction Descending is the process of re- 
 ducing from a higher to a lower denomination. 
 
 142. Reduction Ascending is the process of re- 
 ducing from a lower to a higher denomination. 
 
 ENGLISH MONEY. 
 
 143. English, or Sterling Money, is the money of 
 England. 
 
 TABLE. 
 
 4 farthings (far., or qr.) equal 1 penny, . . . d. 
 
 12 pence " 1 shilling, . . . s. 
 
 20 shillings " 1 pound,* . . . £. 
 
 21 shillings " 1 guinea. 
 
 to' 
 
 S b * s"^- 1 ^"" s* 
 
 Mental Exercise;* — Repeat the table of English Money. 
 
 How many far. in 2d. ? in 3d. ? in Gd. ? in 8d. ? 
 
 How many pence in I2far. ? in lGfar. ? in 20far. ? in 28far. ?, 
 
 How many pence in 2s. ? in 3s. ? in 5s. ? in 6s. ? 
 
 How many far. in Is. ? in 2s. ? in 3s. ? in 5s. ? 
 
 * The £ coined in gold is called a sovereign. Its value is $4.84. A five- 
 shilling piece in silver is called a crown. A two-and-a-half-shilling pieo* 
 in silver is called a half-Town. 
 
DENOMINATE NUMBERS. 129 
 
 REDUCTION DESCENDING. 
 1. How many farthings in 8 pence and 3 farthings? 
 
 8 
 4 
 
 Solution. — In one penny there are 4 operation. 
 
 farthings, hence 4 times the number of d. far. 
 
 pence equal the number of farthings; 4 
 times 8 are 32, and 32far. plus the 3 far. 
 equal 35 farthings. 32 
 
 3 
 
 3 ofar. Ans. 
 
 2. How many farthings in 14d. 2far.? Ans. 58far. 
 
 3. How many pence in 15s. 9d. ? Ans. 189d. 
 
 4. How many shillings in £23 10s.? Ans. 470s. 
 
 5. How many farthings in 23s. lOd. 3far. ? 
 
 6. How many pence in £32 19s. 3d. ? 
 
 REDUCTION ASCENDING. 
 
 1. How many shillings, pence, and farthings, in 1487 
 farthings ? 
 
 Solution. — There are 4 farthings in one operation. 
 
 penny, hence in 1487far. there are as 4) 1487 
 
 many pence as 4 is contained times in 12)371-3far. 
 
 1487, which are 371 pence, and 3far. 30-lld. 
 
 remaining. There are 12 pence in one 
 
 shilling, and in 371 pence there are as many shillings as 12 is con- 
 tained times in 371, which are 30 shillings, and 11 pence remaining. 
 Hence, 1485far. equal 30s., lid., 3far. 
 
 2. How many shillings, pence, and farthings in 989 
 farthings ? Ans. 20s. 7d. lfar. 
 
 3. Reduce 2676 farthings to shillings and pence. 
 
 Ans. 55s. 9d. 
 
 4. How many pounds in 3178 farthings ? 
 
 Ans. £3 6s. 2d. ?far, 
 
 5. How many pounds in 9761 pence? 
 
 6. How many guineas in 17654 farthings ? 
 
loO 
 
 DENOMINATE NUMBERS. 
 
 TROY WEIGHT. 
 144. Troy Weight is used in weighing gold, silver, 
 jewels, etc. 
 
 TABLE. 
 
 24 grains (gi\) . equal 1 pennyweight, . . pwt. 
 20 pennyweights . " 1 ounce, . . . . oz. 
 12 ounces . . . . " 1 pound, .... lb. 
 
 Mental Exercises. — How many oz. in 21b. ? in 31b. ? in 51b. ? 
 How many \\. in 3Goz. ? in 48oz. ? in 60oz. ? in lpwt. ? in 2pwt. ? 
 How many pwt. in 2oz. ? in 3oz. ? in 4oz. ? in 48grs. ? in 72grs. ? 
 
 1. How many grains in 
 15oz. 16pwt. 13grs.? 
 oz. pwt. 
 15 16 
 
 20 
 
 grs. 
 
 13 
 
 300 
 '16 
 
 316 pwt. 
 24 
 
 2. How many penny- 
 weights in 74597grs.? 
 
 grs. 
 24) 7597 
 
 2|0)31|6 — 13grs. 
 15 — 16pwt. 
 Ans. 15oz. 16pwt. 13grs. 
 
 Note. — In dividing by large 
 numbers, like 24, we, of course, 
 divide by Long Division. We in- 
 dicated the result above. 
 
 1264 
 632 
 
 7584 
 13 
 
 7597grs. Ans. 
 
 How many 
 
 3. Grains in 6pwt. 12grs. ? 
 
 4. Pounds, oz., and pwts. in 963pwts. ? 
 
 Ans. 48oz. 3pwt 
 
 5. Grains in 3oz. llpwt. 14gr. ? 
 
 6. Oz., pwt., and grs. in 5170grs. ? 
 
 7. Pounds, oz. etc., in 15786grs. ? 
 
 Ans. 156grs. 
 
DENOMINATE NUMBERS. 
 
 131 
 
 APOTHECARIES WEIGHT.. 
 
 145. Apothecaries Weight is used in mixing medi- 
 cines. Medicines are bought and sold by Avoirdupois 
 Weight. 
 
 TABLE. 
 
 20 grains (gr.) 
 3 scruples 
 8 drams . . 
 
 12 ounces . 
 
 equal 1 scruple, . . . . 9 . 
 
 1 dram, .... 3. 
 
 1 ounce, . . . . 3. 
 
 1 pound, .... lb. 
 
 a 
 it 
 
 u 
 
 Mental Exercises. — 1. How many grs. in 2 scruples? in 3 scru- 
 ples ? in 4 scruples. 
 
 2. How many scruples in 40grs. ? in GOgrs. ? in 80grs. ? inl60grs.? 
 
 3. How many scruples in 2 drams? in 4 drams? in 2 ounces? in 
 4 ounces ? 
 
 4. How many drams in 2 ounces? inlft)? in!25?in365? in!20grs.? 
 
 1. How many grains in 
 153 2B 12gr. ? ' 
 
 OrERATION. 
 
 15 2 12 
 3 
 
 47 
 
 20 
 
 952gr. Ans. 
 
 Note. — "When convenient, add 
 in the numbers as we multiply. 
 
 How many 
 
 1. Drams in 71b. 53 ? Ans. 7123. 
 
 2. Pounds and ounces in 2305 ? Ans - 19tb - 11 5- 
 
 3. Scruples in 191b. 85 53 2£ ? Ans. 5681B- 
 
 4. Pounds, etc., in 92375o;r.? Ans. 161b. 85 I3 15gr- 
 
 2. How many drams in 
 952 grains? 
 
 OPERATION. 
 
 2(0)952 
 
 3)47^ 12gr. 
 
 15 — 29 
 Ans. 153 29 12gr 
 
? 
 
 132 
 
 DENOMINATE NUMBERS. 
 
 AVOIRDUPOIS WEIGHT. 
 
 146. Avoirdupois Weight is used for weighing every 
 thing except jewels, precious metals, etc. 
 
 
 
 TABLE. 
 
 16 drams (dr.) . equal 1 ounce, oz. 
 
 16 ounces . " 1 pound, lb. 
 
 25 pounds . . u 1 quarter, qr. 
 
 4 quarters . . " 1 hundred-weight, cwt. 
 20 hundred-weight " 1 ton, T. 
 
 Mental Exercises. — -Ask mental questions upon this and the 
 following tables, similar to those suggested under the previous 
 tables. 
 
 
 
 N 
 
 )w many drams in 
 
 2. How many poundf 
 
 oz. 9dr ? 
 
 in 3289 drams?" 
 
 OPERATION. 
 
 OPERATION. 
 
 lb. oz. dr. 
 
 drams. 
 
 12 13 9 
 
 16)3289 
 
 16 
 
 16) 205 — 9dr. 
 
 72 
 
 12 — 13oz. 
 
 12 
 
 Ans. 121b. 13oz. 9dr. 
 
 13 
 
 
 205 oz., etc. 
 
 
 How many 
 3 Drams in 171b. 12oz. lldr. ? 
 
 4. Ounces in 3qr 151b. 13oz. ? 
 
 5. Pounds in 5cwt. 3qr. 161b. ? 
 
 6. Pounds in 15675 drams ? 
 
 7. Quarters in 27392 ounces ? 
 
 8. Drams in 20cwt. 3qr. 141b. lloz. 9dr. ? 
 9 Hundred-weight in 17896754 drams ? 
 
DENOMINATE NUMBERS. 13S 
 
 WINE MEASURE. 
 
 147. Wine Measure is used for measuring nearly 
 all kinds of liquids. 
 
 TABLE. 
 
 4 gills (gi.) .... equal 1 pint, pt. 
 
 2 pints " 1 quart, qt. 
 
 4 quarts " 1 gallon, gal. 
 
 Note. — The wine gallon contains 231 cubic inches, while the beer gallon, 
 used in measuring beer, and sometimes milk, contains 281 cu. in. In the 
 old tables were given 3U gals. = 1 barrel; 63 gals. = 1 hogshead: 2 
 hogsheads = 1 pipe ; 2 pipes = 1 tun. These are not measures, however, 
 but vessels of variable capacity. 
 
 How many 
 
 1. Pints in 4gal. 3qt. lpt.? 
 
 2. Gallons in 976 pints ? 
 
 3. Gills in 17gal. 2qt. 3gi. ? 
 
 4. Gallons in 1763 gills ? 
 
 DRY MEASURE. 
 
 148. Dry Measure is used in measuring dry sub- 
 stances, as grain, fruit, salt, coal, etc. 
 
 TABLE. 
 
 2 pints (pt.) . . . equal 1 quart, qt. 
 8 quarts .... "1 peck, pk. 
 4 pecks "1 bushel, bu. 
 
 How many 
 
 1. Pints in 3pk. 6qt. lpt. of berries? 
 
 2. Bushels in 314 quarts of clover seed? 
 
 3. Bushels in 3157 pints of cranberries ? 
 
 4. What cost .4 bushels of berries at 2 cents a pint f 
 
134 DENOMINATE NUMBERS. 
 
 APOTHECARIES' FLUID MEASURE. 
 
 149. Apothecaries' Fluid Measure is used for mea- 
 suring liquids in preparing medical prescriptions. 
 
 TABLE. 
 
 60 minims (Tfp) . equal 1 nuidrachm, f^. 
 
 8 fluidrachms . " 1 fluidounce, fg. 
 16 fluidounces . " 1 pint, ' O. 
 
 8 pints .... "1 gallon, Cong. 
 
 ^" 0TE . — o is the initial of octans, the Latin for one-eighth, the pint being 
 | of a gallon. Cong, is the abbreviation of congiarium, the Latin fol 
 gailon. 
 
 How many 
 
 1. Minims in 20. 5f^ ? 
 
 2. Pints in 8000 minims ? 
 
 3. Fluidounces in BCong. 70. 5f J ? 
 
 4. Gallons in 78561 minims ? 
 
 MEASURE OF LENGTH. 
 
 150. Measure of Length, or Long Measure, is used 
 for measuring length, breadth, height, distances, etc. 
 
 1. A Line is that which has length 
 
 Without breadth or thickness. . / D 
 
 2. An Angle is the opening between j / 
 
 two lines which diverge from a common A g B 
 
 point. Thus, ACD and DCB are angles. 
 
 3. A Right Angle is formed by one line perpendicular to 
 another, as ACE or ECB. 
 
 TABLE. 
 
 12 inches (in.) . 
 
 3 feet . 
 5 A yards 
 40 rods . 
 
 8 furlongs 
 
 equal 1 foot, 
 
 ft. 
 
 " 1 yard, 
 
 yd. 
 
 " 1 rod, 
 
 rd. 
 
 " 1 furlong, 
 
 fur. 
 
 " 1 mile, 
 
 mi. 
 
 . ■ Note.— Cloth Measure is not now used. Cloth, muslin, etc. are bought 
 by the yard, half-yard, eighth, etc. 
 
DENOMINATE NUMBERS. 
 
 135 
 
 1. How many feet in 
 I2rd. 3yd. 2ft. ? 
 
 OPERATION. 
 
 rd. yd. ft. 
 
 12 3 2 
 _5| 
 
 63 
 6 
 
 69yd. 
 3 
 
 209ft. Ans. 
 
 Note. — We multiply by 5, and add 
 to the product the 3 yds., and then 
 multiplying by £, we have 69 yd. 
 
 2. How many rods in 
 209 ft, ? 
 
 OPERATION, 
 feet. 
 3)209 
 
 5*)69ft, 
 
 2 2 
 
 11)138 
 
 12— 6halves=3yi 
 Ans. 12rd. 3yd. 2ft. 
 
 Note. — To divide by b\, we re- 
 duce both to halves, then the re- 
 mainder is halves, which we reduce 
 to wholes, by dividing by 2. 
 
 Ans. 281ft. 
 
 Ans. 635in. 
 
 Ans. 87yds. etc. 
 
 How many 
 
 3. Feet in 16rd. 5yd. 2ft. ? 
 
 4. Inches in 17yd. 1ft. llin. ? 
 
 5. Yards in 3146 inches? 
 
 6. Eods in 6547 inches ? 
 
 7. Feet in 7fur. 32rd. 4yd. ? 
 
 8. Furlongs in 4389 feet? 
 
 9. Miles in 19280 feet? 
 
 10. Inches in 2m. 6fur. 4rd. 8in. ? 
 
 11. How many inches from New York to Philadelphia, 
 if the distance is 96 miles ? 
 
 SURFACE OR SQUARE MEASURE. 
 
 151. Surface or Square Measure is used in mea- 
 suring surfaces, as land, boards, etc. 
 
 1. A Surface is that which has length and breadth -without 
 thickness. 
 
 2. A Square is a surface which has four equal 
 sides and four right angles, as in the margin. 
 
136 
 
 DENOMINATE NUMBERS. 
 
 3. A Rectangle is a surface which has four 
 sides and four right angles. A slate, a door, 
 the sides of a room, etc., are examples of 
 rectangles. 
 
 4. The Area of a surface is expressed by the number of times 
 it contains a small square as aunit of measure. 
 
 5. The area of a square or rectangle is equal to the length multi' 
 plied by the breadth. For, in the rectangle above, the whole 
 number of little squares is equal to the number in each row 
 multiplied by the number of rows : that is, 4 X 3 which equals 
 12, which is the same as the number of units in length multiplied 
 by the number in breadth. 
 
 TABLE 
 
 144 square inches (sq. in.) equal 1 square foot, 
 
 9 square feet . . 
 30 \ square yards . 
 40 perches . . . 
 
 4 roods . . i . 
 640 acres . . . . 
 
 1. How many square 
 feet in 28P. 18sq. yd. 5 
 sq. ft. ? 
 
 OPERATION. 
 P. sq. yd. sq. ft. 
 28 18 5 
 
 30* 
 
 865sq. yd. 
 9 
 
 7790sq. ft. 
 
 Note. — We multiplied by 30, 
 added in the 18 sq. yds., and then 
 multiplied by i and took the sum. 
 
 sq. ft. 
 1 square yard, sq. yd. 
 1 perch, or sq. rod, P. 
 1 rood, R. 
 
 1 acre, A. 
 
 1 square mile, sq. mi. 
 
 2. How many perches 
 are there in 7790sq. ft. ? 
 
 OPERATION 
 sq. ft. 
 
 9)7790 
 
 30^)865 — 5sq. ft. 
 4 4 
 
 121)3460 
 
 28 — 72 fourths, 
 or 18sq. yd. 
 Ans. 27P. 18sq. yd. 5sq. ft. 
 
 Note. — To divide by 30J we re- 
 duce both divisor and dividend to 
 4ths, and then divide ; the remain- 
 der ie 72 fourths, or 18 sq. yd. 
 
DENOMINATE NUMBERS. 
 
 137 
 
 How many 
 
 3. Square inches in 2sq. yd. 3sq. ft. ? 
 
 4. Square feet in 2R. 13P. 16sq. yd. ? 
 
 5. Perches in 8765 square feet ? 
 
 6. Acres in 1997 perches ? Ans. 12A. IE. 37P. 
 
 7. Perches in 29A. 3R. 19P.? Ans. 4779P. 
 
 8. Acres in 89763 square yards ? 
 
 Ans. 18A. 2E. 7P., etc. 
 
 3 feet long. 
 
 CUBIC OK SOLID MEASURE. 
 
 152. Cubic or Solid Measure is used in measuring 
 things which have length, breadth, and thickness. 
 
 1. A Volume is that which has 
 length, breadth, and thickness. A 
 volume is also called a solid. 
 
 2. A Cube is a volume bounded 
 by six equal squares. A Rect- 
 angular Volume is one bounded 
 by rectangles. Cellars, boxes, rooms, 
 etc., are examples of rectangular 
 volumes. 
 
 3. The Contents of a volume are expressed by the number of 
 times it contains a cube as a unit of measure. 
 
 The contents of a Cube or Rectangular Solid are equal to the pro- 
 duct of the length, breadth, and height. 
 
 Fur in the volume above, the number of cubic units on the base 
 is equal to the length multiplied by the breadth, and the whole 
 number of cubic units equals the number on the base multiplied 
 by the number of layers ; hence the whole number equals 3 X 
 3X3 = 27. 
 
 4. A cord of wood is a pile 8 
 feet long, 4 feet wide, and 4 feet 
 high. A cord foot is a part of 
 this pile, 1 foot long; it equals 
 10 cubic feet. 
 
 12* 
 
 1 cord. 
 
138 
 
 DENOMINATE NITOIBERS. 
 
 TABLE. 
 
 1728 cubic inches (cu. in.) equal 1 cubic foot, 
 27 cubic feet . . . 
 16 cubic feet . . . 
 8 cord feet, or "| 
 128 cubic feet, j 
 40 feet of round timber, or 
 50 feet of square timber 
 
 u 
 
 u 
 
 ii 
 
 , or I 
 
 cu. ft. 
 1 cubic yard* cu. yd 
 1 cord foot, cd. ft. 
 
 1 cord of wood, Cd. 
 1 ton, tn. 
 
 How many 
 
 1. Cu. in. in 7cu. ft. 96cu. in. ? 
 
 2. Cu. in. in 12cu. yd. 25cu. ft.? 
 
 3. Cu. ft. in 8469 cubic inches? 
 
 4. Cu. yd. in 60463 cubic inches? 
 
 5. Cords in 8192 cubic feet? 
 
 MEASURE OF TIME. 
 153. Time is the measure of duration. 
 
 60 seconds (sec.) 
 60 minutes 
 24 hours . 
 7 days 
 4 weeks . 
 52 weeks . 
 365 days 
 100 years . 
 
 TABLE. 
 
 equal 1 minute, min 
 
 1 hour, h. 
 
 1 day, da. 
 
 1 week, wk. 
 
 1 month, mo. 
 
 1 year, yr. 
 1 common year, yr. 
 
 1 century, cen. 
 
 How many 
 
 1. Minutes in 1 day ? 
 
 2. Seconds in 1 day ? 
 
 3. Hours in 1 year? 
 
 4. Minutes in 1 year? 
 
 5. Hours in 56780 seconds ? 
 
 6. Days in 600000 seconds ? 
 
 Ans. 1440. 
 
 Ans. 86400. 
 
 Ans. 8760. 
 
 Ans. 525600. 
 
 Ans. 15K etc. 
 
 Ans. 
 
DENOMINATE NUMBERS. 
 
 139 
 
 CIRCULAR MEASURE. 
 
 154. Circular Measure is used to measure angles 
 and directions, latitude and longitude, etc. 
 
 1. A Circle is a figure bounded by a curve 
 line, every point of which is equally distant 
 from a point within, called the centre. 
 
 2. The Circumference is the bounding 
 line ; any part of the circumference, as BC, is 
 an arc ; AB is the diameter, and OC the radius. 
 
 3. For the purpose of measuring angles, the circumference is 
 divided into 3G0 equal parts, called degrees; each degree into 60 
 equal parts, called minutes; each minute into 60 equal parts, called 
 seconds. 
 
 4. Any angle at the centre, as COB, is measured by the arc 
 BC included between its sides. A right angle is measured by 
 90 degrees ; half a right angle, by 45 degrees, etc. 
 
 TABLE. 
 
 60 seconds (") . . 
 
 60 minutes . . . 
 
 30 degrees . . . 
 
 12 signs, or 360°, . 
 
 How many 
 
 1. Seconds in 24' 32"? 
 
 2. Seconds in 23° 24' 15"? 
 Minutes in 1472"? 
 
 <( 
 
 a 
 
 u 
 
 r 
 o 
 
 o 
 O. 
 
 equal 1 minute, 
 
 1 degree, . . 
 
 1 sign . . . . S. 
 
 1 circumference, C. 
 
 Ans. 1472". 
 Ans. 84255". 
 Ans. 24' 32". 
 Ans. 23° 24' 15". 
 
 4. Degrees in 84255" ? 
 
 5. What is the difference between the number of 
 minutes in a day and the number of minutes in a cir. 
 en inference ? 
 
 6. "What is the difference between the number of 
 seconds in a day and the number of seconds in a cir- 
 cumference ? 
 
 7. If you study 6 hours a day, for 5 days in a week, 
 how many minutes will you study in a week ? 
 
liO 
 
 DENOMINATE NUMBERS. 
 
 MISCELLANEOUS TABLES. 
 155. Counting. 
 
 12 units 
 12 dozen 
 12 gross 
 20 units 
 
 < 
 
 24 sheets 
 
 20 quires 
 
 2 reams 
 
 5 reams 
 
 equal 1 dozen. 
 1 gross. 
 1 great gross. 
 1 score. 
 
 a 
 
 i 
 it 
 
 156. Paper. 
 
 . . . equal 1 quire. 
 1 ream. 
 1 bundle. 
 1 bale. 
 
 u 
 a 
 
 << 
 
 157. Weight, Capacity, Length, etc. 
 
 56 pounds of rye or corn equal L bushel. 
 
 60 pounds of wheat or clover seed 
 
 60 pounds of beans or potatoes 
 100 pounds of fish 
 
 196 pounds of flour " 
 
 220 pounds of shad or salmon " 
 
 200 pounds of other fish " 
 
 200 pounds of beef or pork " 
 
 14 pounds (by English law) " 
 
 8 bushels of wheat " 
 4 inches " 
 
 9 inches " 
 22 inches (in Scripture) " 
 
 A knot or nautical mile is 6086.7 feet. 
 
 A surveyor's chain of 100 links is 4 rods long. 
 
 1. How many units in a gross? 
 
 2. How many pins in a great gross? 
 
 3. How many sheets in a ream ? In a bundle? 
 
 4. How many sheets in a bale ? In 12 bales ? 
 
 5. How many bushels of rye will weigh as much aa 
 14 bushels of wheat ? Ans. 15bu. 
 
 6. How many bushels of beans will weigh as much as 
 30 bushels of corn ? Ans. 28bu. 
 
 7. If Dr. "Windship lifts 3000 pounds, how many bar« 
 rels of beef can he lift ? Ans. 15 barrels. 
 
 1 bushel. 
 
 1 bushel. 
 
 1 quintal. 
 
 1 barrel. 
 
 1 barrel. 
 
 1 barrel. 
 
 1 barrel. 
 
 1 stone. 
 
 1 quarter. 
 
 1 hand. 
 
 1 span. 
 
 1 cubit. 
 
DENOMINATE NUMBERS. 14] 
 
 MISCELLANEOUS PROBLEMS. 
 
 1. Reduce 907 pence to pounds. 
 
 2. Reduce 184U pence to pounds. 
 
 3. Reduce 24S0 farthings to shillings. 
 
 4. How many pounds in 8000 grains Troy ? 
 
 5. How many pounds in 10000 ounces Avoirdupois? 
 
 6. Reduce £12 9s. Gd. to pence. 
 
 7. Reduce 81b. 7oz. lopwt. to pennyweights. 
 
 8. Reduce 121b. 14oz. 15dr. to drams. 
 
 9. How many seconds in 24 hours, or one day ? 
 
 10. How many pounds in 16cwt. 3qr. 131b. ? 
 
 11. How many tons in 9876 pounds? 
 
 12. How many tons in 165762 ounces? 
 
 13. Change 63 29 12gr. to grains. 
 
 14. Reduce 9^ 4 3 19 lOgr. to grains. 
 
 15. How many pounds in 5876^ ? 
 
 16. How many pounds in 765429 ? 
 
 17. Reduce 3m. 7fur. 4rd. 2yd. to yards. 
 
 18. Reduce 47692 feet to miles. 
 
 19. Reduce 1234560 inches to miles. 
 
 20. Adam died at the age of 930 years ; how many 
 seconds was this? 
 
 21. Methuselah died at the age of 969 years; how 
 many seconds old was this ? 
 
 22. If the pulse beat 75 times a minute, how often 
 does it beat in a day? Ans. 108000 times. 
 
 23. How long will it take to count a million, at the 
 rate of a hundred a minute, working 12hrs. a day? 
 
 Ans. 13d. lOh. 40m. 
 
 24. If the distance around the earth is 25000 miles, 
 how long will it take to walk the distance, walking 4 
 miles an hour? Ans. 260d. lOh. 
 
 25. If £1 equals 84.84, what is the value of £5 in 
 United States Money? 
 
142 DENOMINATE NUMBERS. 
 
 26. If £1 equals $4.84, required the value of £7 15s 
 in the money of the United States. Ans. $37.51. 
 
 27. If £2 equals $9.68, what is the value of $37.51 in 
 English Money? Ans £7 15s. 
 
 28. If 12 of Henry's peaches fill a quart measure, how 
 many will there be in a bushel? Ans. 384. 
 
 29. How many square rods in a rectangular field 32 
 rods long and 12 rods wide? Ans. 384sq. rd. 
 
 30. How many square feet in a board 18 feet long and 
 2| feet wide? Ans. 45sq. ft. 
 
 31. How many cubic feet in a block of stone 6 feet 
 lonjj, 3 feet wide, and 2 feet thick? Ans. 36cu. ft. 
 
 32. Required the value of a rectangular lot 36 rods 
 long and 20 rods wide, at $3 a square rod. Ans. $2160., 
 
 33. How many cords in a pile of wood 48 feet long, 
 4 feet wide, and 4 feet high ? Ans. 6 cords. 
 
 34. How many cords in a pile of wood 16 feet long, 
 8 feet high, and 4 feet wide? Ans. 4 cords. 
 
 35. What must I pay for a pile of wood 24 feet long, 
 12 feet high, and 4 feet wide, at $1.50 a cord? 
 
 Ans. $13.50. 
 
 36. How much time is wasted by taking an hour's 
 nap each afternoon, for 24 years of 365 days each ? 
 
 Ans. 1 year. 
 
 ADDITION OF DENOMINATE NUMBERS. 
 158, Addition of Denominate Numbers is the pro- 
 cess of finding the sum of two or more denominate 
 numbers of the same kind of quantity. 
 
 1. Find the sum of £8 7s. 5d.; £9 8s. 6d. ; £7 14s. 9d 
 Solution. — We write the numbers so that operation. 
 units of the same kind shall stand in the same £ s. d. 
 
 column, and begin at the right to add. 9d plus 8 7 5 
 
 6d. plus 5d. equal 20d., which, by reduction, we 9 8 6 
 
 find equal Is. and 8d. We write the 8d. under 7 14 9 
 
 the pence column, and reserve the Is. to add to 25 10 8 
 the column of shillings. Is. plus 14s. plus 8s. 
 
DENOMINATE NUMBERS. 143 
 
 plus 7s. equal 30s., which, by reduction, we find equal £1 and 10s. 
 We write the 10s. in shillings column, and add the £1 to the column 
 of pounds, etc. Hence the following 
 
 Rule. — 1. Write the numbers so that units of the same 
 name stand in the same column, and commence at the right 
 to add. 
 
 2. Add as in simple numbers, reduce by division the sum 
 of each column to the next higher denomination, write the 
 remainder under the column added, and add the quotient to 
 the next column. 
 
 3. Proceed in the same manner icith all the columns to 
 the last, under which write the entire sum. 
 
 Proof. — The same as in simple numbers. 
 
 (2.) 
 
 
 (3.) 
 
 
 
 (4-) 
 
 £ s. d. 
 
 £ 
 
 s. d. 
 
 
 £ 
 
 s. d. 
 
 24 12 6 
 
 25 
 
 16 8 
 
 
 123 
 
 14 6 
 
 25 13 9 
 
 17 
 
 13 9 
 
 
 137 
 
 18 10 
 
 17 18 10 
 
 14 
 
 17 11 
 
 
 246 
 
 19 11 
 
 (5.) 
 
 
 (6.) 
 
 
 
 (70 
 
 lb. oz. pwt. 
 
 lb. 
 
 oz. pwt. 
 
 
 lb. 
 
 oz. pwt. 
 
 17 9 16 
 
 18 
 
 9 16 
 
 
 92 
 
 7 12 
 
 25 6 12 
 
 36 
 
 8 21 
 
 
 71 
 
 3 17 
 
 72 11 13 
 
 29 
 
 7 23 
 
 
 28 
 
 9 10 
 
 57 10 19 
 
 42 
 
 11 17 
 
 
 36 
 
 11 18 
 
 (8) 
 
 
 (9.) 
 
 
 
 (10.) 
 
 cwt. qr. lb. oz. 
 
 qr. 
 
 lb. oz. 
 
 dr. 
 
 rd. 
 
 yd. ft. in 
 
 20 3 12 11 
 
 12 
 
 16 12 
 
 11 
 
 17 
 
 4 2 6 
 
 16 2 16 12 
 
 13 
 
 23 9 
 
 10 
 
 21 
 
 2 17 
 
 17 22 20 
 
 14 
 
 24 14 
 
 15 
 
 23 
 
 3 8 
 
 19 1 18 19 
 
 15 
 
 16 15 
 
 8 
 
 25 
 
 5 2 9 
 
144 DENOMINATE NUMBERS. 
 
 (11.) (12.) (13.) 
 
 ft). 
 
 
 5 
 
 9 
 
 gr- 
 
 gal. 
 
 qt. 
 
 pt. 
 
 L. 
 
 mi. 
 
 fur. 
 
 rd. 
 
 28 
 
 11 
 
 7 
 
 2 
 
 16 
 
 36 
 
 2 
 
 1 
 
 16 
 
 2 
 
 7 
 
 30 
 
 19 
 
 9 
 
 5 
 
 1 
 
 23 
 
 42 
 
 1 
 
 1 
 
 14 
 
 2 
 
 7 
 
 32 
 
 27 
 
 8 
 
 3 
 
 2 
 
 17 
 
 25 
 
 3 
 
 
 
 28 
 
 1 
 
 6 
 
 28 
 
 24 
 
 7 
 
 2 
 
 1 
 
 18 
 
 28 
 
 3 
 
 1 
 
 34 
 
 
 
 5 
 
 37 
 
 SUBTRACTION OF DENOMINATE NUMBERS. 
 
 159. Subtraction of Denominate Numbers is the 
 
 process of finding the difference between two compound 
 numbers of the same kind of measure. 
 
 1. From lOoz. 12pwt. 20gr. take 7oz. 15pwt. 16gr. 
 
 Solution. — We write the subtrahend under operation. 
 the minuend, writing units of the same name in oz. pwt. gr. 
 the same column, and commence at the lowest 19 12 20 
 denomination to subtract. 16gr. subtracted 7 15 16 
 
 from 20gr. leave 4grs., which we write under 2 17 4 
 
 the grains. 15pwt. from 12pwt. we cannot 
 take; we will therefore take loz. from the lOoz., leaving 9oz. ; 'ioz. 
 equal 20pwt., which added to 12pwt. equal 32pwt. ; 15pwt. sub- 
 tracted from 32pwt. equal 17pwt., which we write under pwts. 7oz. 
 from 9oz. (or since it will give the same result, we may add loz. to 
 7oz. and say, 8oz. from lOoz.) leave 2oz. Hence the following 
 
 Rule. — 1. Write the subtrahend under the minuend, with 
 units of the same denomination in the same column. 
 
 2. Commence at the lowest denomination, and subtract 
 each number in. the subtrahend from the corresponding 
 number in the minuend. 
 
 3. If the number in the subtrahend exceeds the number in 
 the minuend, add to the latter as many units of that deno- 
 mination as make one of the next higher, and then subtract ; 
 add also one to the next number in the subtrahend before 
 subtracting. 
 
 4. Proceed in the same manner with each denomination 
 to the Last. 
 
 Proof. — The same as in simple numbers. 
 
DENOMINATE NUMBERS. 145 
 
 (2-) 
 £ s. d. far. 
 
 (3.) 
 £ s. 
 
 d. far. 
 
 lb. 
 
 (4.) 
 oz. pwt. gr 
 
 143 11 10 2 
 
 930 17 
 
 7 3 
 
 16 
 
 10 16 18 
 
 115 14 6 3 
 
 246 19 
 
 8 1 
 
 13 
 
 11 17 15 
 
 27 17 3 3 
 
 
 
 2 
 
 10 19 3 
 
 (5.) 
 lb. oz. pwt. gr. 
 
 125 8 14 20 
 
 (6.) 
 cwt. qr. lb. 
 112 3 17 
 
 oz. 
 12 
 
 T. 
 236 
 
 cwt. qr. lb. oz 
 13 2 18 12 
 
 96 9 10 23 
 
 37 1 10 
 
 13 
 
 127 
 
 11 4 22 10 
 
 (8.) 
 hhd. gal. qt. pt. 
 
 128 27 1 
 
 (9.) 
 yr. mo. wk 
 216 10 2 
 
 . da. h. 
 
 5 16 
 
 (10.) 
 
 sq.yd. sq.ft. sq.in 
 
 226 20 120 
 
 106 30 2 1 
 
 123 10 3 
 
 2 20 
 
 S. 
 
 L34 25 130 
 
 (11.) 
 A. II. P. 
 
 (12.) 
 L. mi. fur. 
 
 rd. 
 
 (13.) 
 
 o / // 
 
 426 1 30 
 
 16 2 7 
 
 30 
 
 25 
 
 20 30 40 
 
 207 3 35 
 
 14 2 7 
 
 32 
 
 20 
 
 30 40 50 
 
 14. A farmer had 200bu. of wheat, and sold 28bu.2pk. 
 5qt. lpt. to one man, and as much more to another ; how 
 much remained ? Ans. 142bu. 2pk. 5qt. 
 
 15. A miner having 1121b. of gold sent his mother 
 171b. lOoz. 15pwt. 20gr. and 31b. 16pwt. less to his 
 father ; how much did he retain ? 
 
 Ans. 791b. 3oz. 4pwt. 8gr. 
 
 16. Subtract 16dol. 57cts. 5* mills from $25 20cts. ~\ 
 mills, and add 2 eagles and 25 £ dimes to the result. 
 
 Ans. 31dol. 20cts. 6| mills. 
 IT. Add 961b. 9oz. lOpwt. 23gr. to 1251b. 8oz. 14pwt 
 20grs. and subtract the sum from the sum of 1021b. 
 iloz. 16pwt. and 2561b. 9oz. 19pwt. 
 
 13 
 
146 DENOMINATE NUMBERS. 
 
 MULTIPLICATION OF DENOMINATE NUMBERS 
 
 16©. Multiplication of Denominate Numbers is the 
 
 process of multiplying a denominate number by an ab- 
 stract number. 
 
 1. Multiply £12 lis. 7d. by 8. 
 
 Solution. — 8 times 7d. are 56d., which by operation. 
 
 reduction we find is equal to 4s. and 8d. We £ s. d. 
 
 write the 8 pence under the pence, and re- - 12 11 7 
 
 serve the 4s. to add to the next product. 8 8 
 
 times lis. are 88s., which added to the 4s. ^(X) 12 8 
 equal 92s., which we find by reduction equal 
 
 £4 and 12s. 8 times £12 are £96, which added to £4 equal £100, 
 Hence the following 
 
 Rule. — Write the multiplier under the lowest denomina- 
 tion of the multiplicand, multiply as in simple number^ 
 reducing as in addition. 
 
 Proof. — The same as in simple numbers. 
 
 EXAMPLES FOR PRACTICE. 
 
 (2.) (3.) (4.) 
 
 cwt. qr. lb. oz. lb. oz. pwt. gr. M. da. h. min. pec. 
 
 18 3 21 9 16 8 15 17 50 10 20 30 40 
 
 5 3 7 
 
 (5.) (6.) (7.) 
 
 £ s. d. far. hhd. gal. qt. pt. lb 3 3 J} gr. 
 
 13 12 9 2 21 35 3 1 12 8 7 2 20 
 
 8 9 11 
 
 8. Multiply 12L. 2mi. 5fur. 32rd. by 5, by 6, by 7, by 8 
 
 9. Multiply 23ch. 18bu. 2pk. 7qt. lpt. by 4, by 5, by 
 9, by 10. 
 
 10. A farmer sold 5 loads of hay, each containing 
 15cwt. 3qr. 151b. ; how much did he sell ? 
 
 Ans. 79cwt. 2qr. 
 
DENOMINATE NUMBERS. 147 
 
 11. Multiply 13yr. lOmo. 3wk. 5da. by 5, and that 
 product by 3. Ans. 208yr. "mo. 3wk. 5da. 
 
 12. If a man walk 17mi. 7 fur. 20rd. in each of 21 
 days, how far will he walk in all? 
 
 Ans. 3T6mi. 5fur. 20rd. 
 
 13. If a farmer raise 60bu. 3pk. 6qt. lpt. of grain on 
 one acre, how much can he raise at the same rate on 
 48 acres? * Ans. 2925bu. 3pk. 
 
 14. A owned 1000A. of land ; he sold B 96A. 3K. 
 30P., and 4 times as much to C ; how much remained ? 
 
 Ans. 515A. 1H. 10P. 
 
 DIVISION OF DENOMINATE NUMBERS. 
 
 161. Division of Denominate Numbers is the pro- 
 ;ss of dividing when one or bot 
 number. There are two cases. 
 
 cess of dividing when one or both terms is a denominate 
 
 Case I. 
 
 162. To divide a denominate number into equal 
 parts. 
 
 1. Divide £103 7s. 6d. into 5 equal parts; that is, take 
 l of it. 
 
 Solution. — \ of £103 is £20 and £3 re- operation. 
 
 maining. £3 equal GOs., which added to 7s. £ s. d. 
 
 equal 67s. ; i of 67s. is 13s. and 2s. remain- 5)103 7 6 
 
 ing. 2s. equal 24d., which added to 6d. 20 13 6 
 equal 30d. ; \ of 30d. is 6d. Hence the 
 following 
 
 *o 
 
 Rule. — 1. Begin at the highest denomination, and divide 
 as in simple numbers. 
 
 2. If there is a remainder, reduce it to the next loioer de-. 
 nomination, add to it the number of that denomination 
 and divide as before, and thus continue to the last. 
 
148 DENOMINATE NUMBERS. 
 
 
 
 EXAMPLES 
 
 FOR 
 
 PRACTICE. 
 
 
 
 (2.) 
 
 
 (3.) 
 
 
 
 (4.) 
 
 
 £ 
 
 s. d. 
 
 lb. 
 
 oz. 
 
 pwt. 
 
 gr. 
 
 T. 
 
 cwt. qr. 
 
 A. 
 
 4)61 
 
 18 4 
 
 9 7 
 
 6)76 
 
 10 
 
 14 
 
 12 
 
 7)112 
 
 ! 16 2 
 
 16 
 
 15 
 
 
 
 
 
 16 
 
 2 1 
 
 13 
 
 
 (5.) 
 
 
 
 
 
 (6.) 
 
 
 cwt. 
 
 qrs. lb. 
 
 oz. 
 
 dr. 
 
 
 hhd. gal. 
 
 qt. pt. 
 
 g 1 - 
 
 8)125 
 
 3 19 
 (7.) 
 
 12 
 
 8 
 
 
 9)1' 
 
 08 42 
 
 2 1 
 
 2 
 
 
 
 (8.) 
 
 
 min. 
 
 fur. rd. 
 
 yd. 
 
 ft. 
 
 
 
 A. 
 
 R. P. sq.yd. 
 
 11)120 
 
 7 33 
 
 3 
 
 2 
 
 
 
 5)112 
 
 3 24 
 
 (10.) 
 
 24 
 
 
 (9.). 
 
 
 
 • 
 
 bu. 
 
 pk. qt 
 
 . P t. 
 
 
 
 
 lb. 
 
 oz. pwt. 
 
 g r - 
 
 9)1137 
 
 3 4 
 
 1 
 
 
 
 
 8)37 
 
 10 17 
 
 16 
 
 11. A miner divides 371b. lOoz. 17pwt. 16gr. of gold 
 among 8 sisters ; how much does each receive ? 
 
 Ans. 41b. 8oz. 17pwt. 5gr. 
 
 12. A man walked 376mi. 6fur. 36rd. in 22 days; what 
 was the average distance each day ? 
 
 Ans. 17 mi. lfur. l T 7 7 rd. 
 
 13. If 26 casks contain 21hhd. llgal. 2qt. lpL, what 
 is the capacity of each cask? Ans. 51gal. lqt. Upt. 
 
 Case II. 
 
 163. To divide a denominate number by a simi- 
 lar denominate number. 
 
 1. Divide £26 6s. 2d. by £4 15s. 8d. 
 
 Solution.— £26 6s. 2d. we find by operation. 
 
 reduction equal 6314 pence ; £4 15s. £26 6s. 2d . = 6314d. 
 
 8d. equals 1148 pence; and dividing £415s. 8d. = 1148d. 
 
 6314d. by 1148d. we obtain a quotient 1148)6314f5 1 Ans. 
 
 of 5£. From this solution we have 6314 
 the following 
 
 Bule.— Reduce both dividend and divisor to the lowest 
 
DENOMINATE NUMBERS. 149 
 
 denomination mentioned in either, and then divide as in 
 simple numbers. 
 
 Remark. — The division may also be made before the reduction to 
 lower denomination ; and this will be shorter when there is no re- 
 mainder. 
 
 2. Divide £48 7s. 4d. by £6 lid. Ans. 8. 
 
 3. Divide 69bu. 3pk. 6qt. by 6bu. 3pk. 6qt. 
 
 Ans. 10^. 
 
 4. Divide 80bu. 2pk. 4qt. by 13bu. 3pk. 5qt. 
 
 .ci.ns. Ojt-jj. 
 
 5. Divide 6971b. 7oz. 5dr. by 601b. lOoz. 6dr. 
 
 Ans. 11A. 
 
 6. A man travelled 3mi. 6fur. 36rd. 4yd. in one hour; 
 in what time will he travel 247mi. 2fur. 30rd. 3yd. ? 
 
 Ans. 64 hrs. 
 
 7. A drove of cattle ate 6T. 15cwt. 3qr. 121b. of hay 
 in a week ; how long will 33T. 19cwt. lqr. 101b. last 
 them ? Ans. 5 weeks. 
 
 PROBLEMS IN TIME. 
 
 1. 'Washington was born Feb. 22d, 1732, and died 
 Dec. 14th, 1799; what was his age? 
 
 Solution. — We write the number of the 
 year, month, and day of both periods, and 
 subtract the one from the other, as is shown 
 in the margin. 
 
 67 9 22 
 
 2. John Adams was born the 19th of October, 1735, 
 and died the 4th of July, 1826 ; required his age. 
 
 3. Thomas Jefferson was born xVpril 2d, 1743, and 
 died July 4th, 1826; what was his age? 
 
 4. James Madison was born March 16th, 1751. and 
 died June 28th, 1836 ; required his age. 
 
 5. James Monroe was born April 28th, 1758, and died 
 July 4th, 1831 ; required his age. 
 
 13* 
 
 OPERATION. 
 
 y r - 
 
 mo. 
 
 da. 
 
 1799 
 
 12 
 
 14 
 
 1732 
 
 2 
 
 22 
 
150 DENOMINATE NUMBERS. 
 
 6. John Quincy Adams was born July 11th, 1767, and 
 died Feb. 23d, 1848 ; what was his age? 
 
 7. Andrew Jackson was born March 15th, 1767. and 
 died June 8th, 1845; required his age. 
 
 8. Martin Van Buren was born Dec. 5th, 1782, and 
 died July 24th, 1862 ; required his age. 
 
 9. William Henry Harrison was born Feb. 9th, 1773, 
 and died April 4th, 1841 ; required his age. 
 
 10. James K. Polk was born Nov. 2d, 1795, and died 
 June 15th, 1849 ; required his age. 
 
 11. General Zachary Taylor was born Nov. 24th, 1784, 
 and died July 9th, 1850 ; required his age. 
 
 12. How Ions has a note to run which is dated Dec. 
 30th, 1862, and made payable Jan. 16th, 1864? 
 
 Ans. lyr. 16da. 
 
 13. The Revolution was commenced the 19th of April, 
 1775, and terminated January 20th, 1783 ; how long did 
 it continue? Ans. 7yr. 9mo. Ida. 
 
 PROBLEMS IN LATITUDE AND LONGITUDE. 
 
 1. The latitude of Boston is 42° 21' 23" north ; that 
 of Charleston, 32° 46' 33" north ; what is the difference 
 of latitude ? 
 
 2. The latitude of New York is 40° 24' 40" K; that 
 of New Orleans, 29° 57' 30" N. ; what is the difference 
 of latitude ? 
 
 3. The latitude of Philadelphia is 39° 56' 39" ; that 
 of Savannah is 32° 4' 56"; what is the difference of 
 latitude ? 
 
 4. The latitude of Baltimore is 39° 17' 23"; that of 
 St. Louis is 38° 37' 28"; what is the difference of 
 latitude ? 
 
 5. The latitude of the Cape of Good Hope is 30° 55' 
 15" S.; that of Cape Horn, 55° 58' 30"; what is the 
 difference of latitude ? 
 
INTRODUCTION 
 
 TO THE 
 
 METRICAL SYSTEM OF WEIGHTS AND MEASURES. 
 
 The old system of weights and measures in our country is irregular^ 
 difficult to learn, and inconvenient to apply. The same is true with 
 She old systems of all nations. Originating by chance, rather than 
 science, they lacked the simplicity of law, and were, therefore, irregu- 
 lar and chaotic. 
 
 In 1795, France adopted a system of weights and measures called 
 the Metric System, based upon the decimal method of notation ; all 
 the divisions and multiples being by 10. It was regarded as so 
 great an improvement upon the old methods that it has since been 
 adopted by Spain, Belgium, and Portugal, to the exclusion of all 
 other weights and measures, and is in partial use in Holland, Italy, 
 Germany, and Austria, and also in many parts of Spanish America. 
 
 In 1864, the British Parliament passed an act permitting its use 
 throughout the empire whenever parties should agree to use it. In 
 18G6, Congress authorized its use in the United States, and provided 
 for its introduction into the post-offices for the weighing of letters 
 and papers. 
 
 To facilitate its adoption, a convenient standard of comparison was 
 furnished, by making the new five-cent piece five grams in weight 
 and one fiftieth of a meter, or two centimeters, in diameter. This 
 system will, without doubt, in a few years be in general use in this 
 country. 
 
 The advantages of the Metric System are numerous and important. 
 
 1. It is easily learned ; a school-boy can learn it in a single after- 
 noon. 
 
 2. It is easily applied , all the operations being the same as in 
 simple numbers. 
 
 3. It does away with addition, subtraction, multiplication, division, 
 and reduction of compound numbers and fractions. 
 
 4. It will facilitate commerce, giving the nations a universal 
 
 system of weights and measures. 
 
 151 
 
152 
 
 THE METRIC SYSTEM. 
 
 THE METEIC SYSTEM. 
 
 164. The Metric System of Weights and Measures is 
 based upon the decimal system of notation. 
 
 k«3. In this system we first establish the unit of 
 each measure, and then multiply and divide it by 10. 
 
 166. Names. — We first name the unit of any measure, 
 and then derive the other denominations by prefixing 
 words to the unit flame. 
 
 167. The higher denominations are expressed by pre- 
 fixing to the name of the unit, 
 
 Deka, 
 
 10 
 
 Hecto, 
 
 100 
 
 Kilo, 
 
 1000 
 
 Myria. 
 
 10,000 
 
 168. The lower denominations are expressed by pre- 
 fixing to the name of the unit, 
 
 I>eci, 
 
 l^ 
 
 10 
 
 Centi, 
 
 l 
 
 loo 
 
 Milli. 
 
 l 
 
 1000 
 
 169. Units. — The following are the different units, 
 with their English pronunciation : — 
 
 Measure. 
 
 Unit. 
 
 Pronunciation. 
 
 Measure. 
 
 Unit. 
 
 Pronunciatio 
 
 Length, 
 
 Meter, 
 
 (meter.) 
 
 Capacity, 
 
 Liter, 
 
 (leeter.) 
 
 Surface, 
 
 Are, 
 
 (air.) 
 
 Weight, 
 
 Gram, 
 
 (gram.) 
 
 Volume, 
 
 Stcre, 
 
 (stair.) 
 
 Value, 
 
 Dollar* 
 
 
 MEASURE OF LENGTH. 
 170. The Meter is the unit of length. It is the ten- 
 millionth part of the distance from the equator to the 
 poles, and equals 39.37 inches, or 3.28 feet. 
 
 TABLE. 
 
 10 millimeters (m.m.) equal 1 centimeter, 
 
 10 centimeters 
 10 decimeters 
 10 meters 
 
 10 dekameters 
 10 hectometers 
 10 kilometers 
 
 u 
 
 a 
 
 a 
 
 a 
 
 a 
 
 n 
 
 1 decimeter, 
 1 meter, 
 
 1 dekanieter, 
 1 hectometer, 
 1 kilometer, 
 
 cm. 
 
 d.m. 
 
 M. 
 
 D.M. 
 
 H.M. 
 
 K.M. 
 
 1 myriameter, M.M. 
 
THE METRIC SYSTEM. 153 
 
 Notes. — 1. The meter is very nearly 3 feet, 3 inches, and 3 eighths of 
 an inch in length, which may be easily remembered as the rule of 
 three threes. 
 
 2. Cloth, etc. are measured by the meter; very small distances, by 
 the millimeter; great distances, by the kilometer. 
 
 3. The new 5-cent piece is -fa of a meter in diameter: hence its 
 diameter is ^ of a decimeter, or 2 centimeters. 
 
 4. A decimeter is about 4 inches ; a kilometer, about 200 rods, or 
 | of a mile ; a millimeter, about ^ of an inch. The inch is about 2i 
 centimeters ; the foot, 3 decimeters ; the rod, 5 meters ; the mile, 
 1600 meters, or 16 hectometers. 
 
 QUESTIONS. 
 
 1. How many centimeters in a meter? 
 
 2. How many millimeters in a meter? 
 
 3. How many decimeters in a dekameter? 
 
 4. How many meters in a hectometer ? 
 
 5. How many meters in a kilometer ? 
 
 MEASURES OF SURFACE. 
 
 171. The Are is the unit of surface used to measure 
 land. The are is a square dekameter. It equals 119.6 sq. 
 yd., or 0.0247 acre. 
 
 
 TABLE. 
 
 
 10 milliares (m. 
 
 ,a.) equal 1 centiare, 
 
 c.a. 
 
 10 centiares 
 
 u 
 
 1 deciare, 
 
 d.a. 
 
 10 declares 
 
 a 
 
 1 are, 
 
 A. 
 
 10 ares 
 
 tt 
 
 1 dekare, 
 
 D.A. 
 
 10 dekares 
 
 u 
 
 1 hectare, 
 
 H.A. 
 
 10 hectares 
 
 it 
 
 1 kilare, 
 
 K.A. 
 
 10 kilares 
 
 u 
 
 1 myriare, 
 
 M.A. 
 
 Notes. — 1. The are, centiare, and hectare are the denominations 
 principally used, as these are exact squares. The centiare is a 
 square whose side is 1 meter ; the hectare is a square whose side is 
 100 meters. 
 
 The are = 100 square meters. The centiare = 1 square meter. 
 The hectare = 10,000 square meters. 
 
 2. The deciare is not a square, it is merely the tenth of an are; 
 the dekare is not a square, it is merely 10 ares. 
 
154 THE METRIC SYSTEM. 
 
 3. A hectare equals very nearly 21 acres ; a centiare equals nearly 
 1 \ sq. yd. An acre is very nearly 40 ares. 
 
 MEASURES OF OTHER SURFACES. 
 
 172. All surfaces besides land are measured by the 
 square meter, square decimeter, etc. The measures are 
 shown by the following table : — 
 
 TABLE. 
 
 2 
 
 2 
 
 ■ 
 
 2 
 
 100 sq. millimeters (m.m. 2 ) = l sq. centimeter, cm. 
 100 sq. centimeters =1 sq. decimeter, d.m. 
 
 100 sq. decimeters =1 sq. meter, M. 
 
 Note. — The measures higher than these are not generally usod. 
 
 QUESTIONS. 
 
 1. How many centiares in an are? 
 
 2. How many ares in a hectare ? 
 
 3. How many square meters in an are? 
 
 4. How many square decimeters in an are? 
 
 5. How many ares in 640 square meters ? 
 
 MEASURES OF VOLUME. 
 173. The Stere is the unit of volume. It is a cubic 
 meter, and equals 35.3166 cubic feet, or 1.308 cu. yd. 
 
 TABLE. 
 
 10 millisteres (m.S.) equal 1 centistere, c.S. 
 
 10 centisteres " 1 decistere, d.s. 
 
 10 decisteres " 1 stere, S. 
 
 10 steres " 1 dekastere, D.S. 
 
 10 dekasteres " 1 hectostere, H.S. 
 
 10 hectosteres " 1 kilostere, K.S. 
 
 10 kilosteres " 1 myriastere, M.S. 
 
 Note.— 1. Wood is measured by this measure. The stere, ieci- 
 &tere, and dekastere are principally used. 3.6 steres, or 36 deci- 
 steres, very nearly equal the common cord. 
 
THE METRIC SYSTEM. 155 
 
 MEASURES OF OTHER VOLUMES. 
 
 174, Other solid bodies are usually measured by the 
 cubic meter and its divisions. The measures are shown 
 by the following table. 
 
 TABLE. 
 
 1000 cubic millimeters(m.m. 3 )=l cubic centimeter.c.m.* 
 1000 cubic centimeters =1 cubic decimeter, d.m. 3 
 
 1000 cubic decimeters =1 cubic meter, M. 3 
 
 N 0TE . — The higher denominations are not generally used. 
 
 QUESTIONS. 
 
 1. How many centisteres in a stere ? 
 
 2. How many decisteres in a dekastere ? 
 
 3. How many dekasteres in a kilostere ? 
 
 4. How many cubic meters in a hectostere? 
 
 MEASURES OF CAPACITY. 
 
 175. The Liter is the unit of capacity. It equals a 
 cubic decimeter; that is, a cubic vessel whose size is one- 
 tenth of a meter. 
 
 This measure is used for measuring liquids and dry 
 substances. The liter is a cylinder, and holds 2.1135 
 pints wine measure, or 1.816 pints dry measure. 
 
 TABLE. 
 
 10 milliliters (m. 
 
 ,1.) equal 1 centiliter, 
 
 C.I. 
 
 10 centiliters 
 
 u 
 
 1 deciliter, 
 
 d.l. 
 
 10 deciliters 
 
 u 
 
 1 liter. 
 
 L. 
 
 10 liters 
 
 it 
 
 1 dekaliter, 
 
 D.L. 
 
 10 dekaliters 
 
 a 
 
 1 hectoliter, 
 
 H.L. 
 
 10 hectoliters 
 
 « 
 
 1 kiloliter, 
 
 K.L. 
 
 10 kiloliters 
 
 u 
 
 1 myrialiter. 
 
 M.L. 
 
 Notes. — 1. The liter is principally used in measuring liquids, and 
 the hectoliter in measuring grains, etc. 
 
 2. The liter equals nearly 1J 5 liquid quarts, or T 9 j of a dry quart. 
 or nearly -^ of a bushel measure. 
 
156 THE METRIC SYSTEM. 
 
 3. The hectoliter is about 2| bushels, or | of a barrel. 4 liters are 
 a little more than a gallon; 35 liters, very nearly a bushel. 
 
 QUESTIONS. 
 
 1. How many liters in a hectoliter? 
 
 2. How many liters in a kiloliter? 
 
 3. How many deciliters in a dekaliter ? 
 
 4. How many liters in a cubic meter? Ans. 1000. 
 
 5. How many liters in a stere ? Ans. 1000. 
 
 MEASURE OF WEIGHT. 
 
 176. The Gram is the unit of weight. It is the weight 
 of a cubic centimeter of distilled water at the tempera- 
 ture of melting ice. The gram equals 15.432 troy grains. 
 
 
 TABLE. 
 
 
 10 milligrams (m 
 
 .g.) equal 1 centigram, 
 
 e.g. 
 
 10 centigrams 
 
 a 
 
 1 decigram, 
 
 d.g. 
 
 10 decigrams 
 
 a 
 
 1 gram, 
 
 G. 
 
 10 grams 
 
 u 
 
 1 dekagram, 
 
 D.G. 
 
 10 dekagrams 
 
 u 
 
 1 hectogram, 
 
 H.G. 
 
 10 hectograms 
 
 a 
 
 1 kilogram, 
 
 K.G.,orK. 
 
 10 kilograms 
 
 to 
 
 1 myriagram, 
 
 M.G. 
 
 Notes. — 1. The gram is used in weighing letters, in mixing and 
 compounding medicines, and in weighing all very light articles. 
 The new 5-cent coin (dated 1866) weighs 5 grams. 
 
 2. The kilogram is the ordinary unit of weight, and is generally 
 abbreviated into kilo. It equals about 2^ pounds avoirdupois. 
 Meat, sugar, etc. are bought and sold by the kilogram. 
 
 3. In weighing heavy articles, two other weights, the quintal 
 (100 kilograms) and the tonneau (1000 kilograms), are used. The 
 tonneau is between our short ton and long ton. 
 
 4. The avoirdupois ounce is about 28 grams; the pound is a little 
 less than J a kilo. 
 
 QUESTIONS. 
 
 1. How many grams in a kilogram? 
 
 2. How many milligrams in a gram ? 
 
 3. How many decigrams in *•> kilogram? 
 
 4 How many hectograms in a myriagram ? 
 
THE METRIC SYSTEM. 157 
 
 NUMERATION AND NOTATION. 
 
 177. In the Metric System the decimal point is 
 placed between the unit and its division, tne whole 
 quantity being regarded as an integer and a decimal. 
 Thus, 3 dekagrams, 5 grams, 6 decigrams, and 8 centi- 
 grams, are written thus : 35.68 grams. 
 
 178. The initials of the denomination may be placed 
 either before or after the quantity ; thus, 27 grams may 
 be written G.27, or, 27 G. 
 
 EXERCISES IN NUMERATION. 
 
 L Eead M. 28.35. 
 
 Solution. — This is read 28 and 35 hundredths meters; or it may 
 be read 2 dekameters, 8 meters, 3 decimeters, and 5 centimeters. 
 
 Read the following. 
 
 2. M. 15.37. 
 
 3. M.46.75. 
 
 4. A.57.34. 
 
 5. A. 75.25. 
 
 6. S. 134.09. 
 
 7. S. 325.125. 
 
 8. L.57.45. 
 
 9. L.68.25. 
 
 10. G.72.325. 
 
 11. G.416.318. 
 
 12. G. 207.305. 
 
 13. M.3056.705. 
 
 EXERCISES IN NOTATION. 
 
 1. "Write 7 meters and 5 centimeters. 
 
 Solution. — We write the 7 meters with a 
 decimal point to the right, and then, since there operation. 
 
 are no decimeters, we write a naught in the M.7.05 
 
 tenths place, and then write the 5 centimeters 
 in the place of centimeters. 
 
 2. Write 8 meters and 6 centimeters. Ans. M.8.06. 
 
 3. Write 12 meters and 56 decimeters Ans. M. 12.56. 
 
 4. Write 25 meters and 8 millimeters. Ans. M. 25.008. 
 
 5. Write 13 ares, 3 declares, 5 centiares. 
 
 Ans. A. 13.35. 
 
 6. Write 24 ares, 5 centiares, 7 milliares. 
 
 Ans. A.24.057. 
 
158 THE METRIC SYSTEM. 
 
 7. Write 7 dekares, 4 declares, 5 centiares. 
 
 Ans. A. 70.4. 
 
 8. "Write 6 hectares, 8 declares, 2 milliares. 
 
 Ans. A. 600.802. 
 
 9. Write 25 steres, 6 decisteres, 5 centisteres. 
 
 Ans. S. 25.65. 
 
 10. Write 8 hectosteres, 7 steres, 4 centisteres. 
 
 Ans. S. 807.04. 
 
 11. Write 53 liters, 8 deciliters, 5 milliliters. 
 
 Ans. L.53.805. 
 
 12. Write 17 grams, 5 decigrams, 6 centigrams, 4 milli- 
 grams. Ans. G. 17. 564. 
 
 13. Write 42 kilograms, 8 dekagrams, 3 decigrams. 
 
 * Ans. K.42.0803. 
 
 14. Write 27 kilograms, 9 grams, 5 decigrams. 
 
 Ans. K. 27.0095. 
 
 15. Write 8 myriagrams, 4 kilograms, 3 dekagrams, 
 5 grams, 6 decigrams. Ans. K. 84.0356. 
 
 REDUCTION. 
 
 179. Reduction in the Metric System is very simple, 
 since the numbers are expressed in the decimal system. 
 
 ISO. Since ten of any denomination equal one of 
 the next higher, we can reduce from one denomination 
 to another by simply changing the position of the deci- 
 mal point. 
 
 Case I. 
 181. To reduce a number to lower denominations. 
 
 1. Reduce 25.75 dekameters to meters. 
 
 Solution. — Since 10 meters equal 1 deka- 
 
 metcr, ten times the number of dekameters oteratiox. 
 
 equal the number of meters; ten times 23.75 D.M.25.75 = 
 
 equal 257.5, which changes the decimal point one M. 257.5 
 place to the right. Hence the following rule: 
 
 .Rule. — Remove the decimal point as many -places to the 
 
THE METRIC SYSTEM. 159 
 
 right as there are tenths required to make the lower denomi- 
 nations. 
 
 Reduce 
 
 2. 47.125 hectometers to meters. Ana. M. 4712. 5. 
 
 3. 35.25 dekagrams to grams. Ans. G.352.5. 
 
 4. 46.75 dekaliters to liters. Ans. L.467.5. 
 
 5. 7.0375 kilograms to grams. Ans. G. 7037. 5. 
 
 6. 9.5063 hectares to ares. Ans. A. 950. 63. 
 
 7. 5.6304 hectometers to meters. Ans. M. 563.04. 
 
 8. 53.025 steres to centisteres. Ans. C.S. 5302.5. 
 
 9. 365.24 square meters to square decimeters. 
 
 Ans. 36524d.m 2 . 
 
 10. 432.15 square dekameters to square meters. 
 
 Ans. M 2 . 43215. 
 
 11. 356.25 ares to square meters. Ans. M\ 35625. 
 
 12. 56 cubic meters to cubic decimeters. 
 
 Ans. 56000c. d 3 . 
 
 Case II. 
 
 182. To reduce a number to Mglaer denomi- 
 nations. 
 
 1 Reduce 257.5 meters to dekameters. 
 
 Solution. — Since 1 dekameter equals 10 
 meters, 1 tenth of the number of meters equals operation. 
 
 the nuinber of dekameters; 1 tenth of 257.5 is M.257.5 = 
 25.75, which changes the decimal point one D.M.25.75 
 place to ine left. Hence we have the following 
 rule : 
 
 Rule.— Remove the decimal 'point as many places toward 
 the left as there are tens required to make the higher denomi- 
 nation. 
 
 Reduce' 
 
 2. 4712.5 meters to hectometers. Ans. 47.125H.M. 
 
 3. 352.5 grams to dekagrams. Ans. 35.25D.G. 
 
 4. 467.5 liters to dekaliters. Ans. 4('>7">D.L. 
 
 5. 7037.5 grams to kilograms. Ans. 7.0375K.G. 
 
IGO THE METRIC SYSTEM. 
 
 6. 950.63 ares to centiares. Ans. 9. 5063c. a. 
 
 7. 563.04 meters to centimeters. Ans. 5.6304c. m. 
 
 8. 5302.5 steres to centisteres. Ans. 53.025c. s. 
 
 9. 36524 d.m 2 . to square meters. Ans. 365.24 M 2 . 
 
 10. 43215 M\ to square dekameters. Ans.432.15D.IVl 2 . 
 
 11. 35625 M\ to ares. Ans. 356.25 ares. 
 
 12. 56000 c.d 3 . to cubic meters. 
 
 Ans. 56 cubic meters. 
 
 ADDITION IN THE METRIC SYSTEM. 
 183. The Metric System being founded upon the deci- 
 mal system, addition is performed the same as in simple 
 numbers. 
 
 EXAMPLES. 
 
 1. Find the sum of 35.25 meters, 76.45 meters, 89.28 
 meters, and 36.46 meters. 
 
 2. Find the sum of 75.45 grams, 84.67 grams, 45.84 
 grams, and 97.52 grams. 
 
 3. Find the sum of 125.65 ares, 223.87 ares, 97.423 
 ares, and 867.055 ares. 
 
 4. Find the sum of 72.125 liters, 99.72 liters, 125.406 
 liters, and 237.125 liters. 
 
 5. A man bought at one time 26.25 hectoliters of 
 grain, at another time 38.50 hectoliters, at another, 46.75 
 hectoliters, and at another, 86.25 hectoliters; how much 
 did he buy in all ? 
 
 6. Mr. Behmer bought at one time 26.25 kilos of 
 meat, at another time 38.75 kilos, at another, 29.50 
 kilos, and at another, 46.75 kilos j how much did he buy 
 in all ? 
 
 7. A man owns three farms ; the first contains 916.25 
 ares, the second 829.50 ares, the third 765.75 ares, the 
 fourth 1227.75 ares ; how many ares does he own in all ? 
 
 8. A merchant sold 6.25 liters of molasses to one man, 
 12.75 liters to another, 8.50 liters to another and 25.25 
 
THE METRIC SYSTEM. 161 
 
 Titers to another ; how many liters of molasses did he 
 sell to the four men ? 
 
 9. A postmaster mailed five letters this morning ; the 
 first weighed 2.25 grams, the second 5.50 grams, the 
 third 4.85 grams, the fourth 6.65 grams, and the fifth 
 7.54 grams ; what did they all weigh ? 
 
 SUBTRACTION IN THE METRIC SYSTEM. 
 184. The Metric System being based upon the deci- 
 mal system, subtraction is performed the same as in 
 simple numbers. 
 
 EXAMPLES. 
 
 1. Subtract 35.75 meters from 53.20 meters. 
 
 2. Subtract 96.73 grams from 104.34 grams. 
 
 3. Subtract 378.25 ares from 523.40 ares. 
 
 4. A man bought 72.125 liters of wine, and sold 36.375 
 liters ; how much did he retain ? 
 
 5. If I own 906.25 ares of land and sell 376.42 ares, 
 how much will remain ? 
 
 6. If I buy 82 hectoliters of grain, and sell 962 liters 
 of it ; how much will remain ? 
 
 7. If my father owns 1064 ares of land, and sells 565 
 square meters, how much will remain ? 
 
 8. From a barrel containing 150 liters of wine, I 
 drew out 47.25 liters, and there leaked out 25.37 liters j 
 how much remained in ? 
 
 9. A man had 127.45 hectares of land ; he sold 75 ares 
 to one man, and 148 ares to another ; how many ares of 
 land remained ? 
 
 10. A merchant had a rope one kilometer in length ; 
 he sold 25 meters to one man, 34 meters to another, and 
 128 meters to another ; how many meters of the rope 
 remained ? 
 
 11. A man bought 167.5 kilos of sugar, and sold 83.25 
 kilos j how much sugar remained ? 
 
 14* 
 
162 THE METRIC SYSTEM. 
 
 MULTIPLICATION IN THE METRIC SYSTEM. 
 
 185. The Metric System being based upon the deci- 
 mal system of notation, multiplication is performed as 
 in simple numbers. 
 
 EXAMPLES. 
 
 1. Multiply 28.25 grams by 5. 
 
 2. Multiply 35.76 meters by 8. 
 
 3. If a new five-cent piece weighs 5 grams, what will 
 25 of such pieces weigh ? 
 
 4. If the diameter of the five-cent piece is 2 centi- 
 meters, what is the length of a row of 50 of them? 
 
 5. If one decistere of wood is worth 36 cents, what 
 are 25.5 decisteres worth ? 
 
 6. How much will 28.25 liters of cider cost at 15 cents 
 a liter ? 
 
 7. Henry bought 28 f hectoliters of grain at $4.25 a 
 liter; what did he pay for it? 
 
 8. Mr. Baker bought 56f hectoliters of grain at $3.50 
 a liter, and sold it at $3.25 a liter; what did he lose ? 
 
 9. A man travelled at the rate of 128.5 kilometers in 
 a day ; how far did he travel in 24 days ? 
 
 10. A laborer can disc 4.25 cubic meters of ditch in a 
 day ; how much, at that rate, can he dig in 5£ days ? 
 
 DIVISION IN THE METRIC SYSTEM. 
 
 186. The Metric System being based upon the deci- 
 mal system of notation, division is performed as in 
 simple numbers. 
 
 EXAMPLES. 
 
 1. Divide 17.25 grams by 5. 
 
 2. Divide 29.25 meters by 15. 
 
 3. If 25 meters of cloth cost $86.25, what will one 
 meter cost ? 
 
 4. If 27 hectares of land cost $7701.75, what will one 
 hectare cost ? 
 
THE METRIC SYSTEM. 163 
 
 5. How much will a gram of jewels cost, if 12 grams 
 And 5 decigrams cost $81.25 ? 
 
 6. How many liters of wine can you buy for $38.12 £, 
 at the rate of $1.25 a liter? Ans. 30.5 liters. 
 
 7. If 5 dekasteres of wood cost $12.75, what must I 
 pay for 8 hectosteres, 6 decisteres of wood ? 
 
 8. What cost one meter of cloth, if 36 meters, 4 deci- 
 meters, and 5 centimeters cost $169.49 ? Ans. $4.65. 
 
 9. I paid $194.18 for sugar, at the rate of 56 cents per 
 kilogram ; how many kilograms did I buy ? 
 
 10. If 48.625 meters of cloth cost $183.75, for what 
 ought I to sell 9.725 meters so as neither to gain nor 
 lose any thing ? Ans. $36.75. 
 
 MISCELLANEOUS PROBLEMS. 
 
 1. The new 5-cent piece weighs 5 grams ; how much 
 will 50 of them weigh ? 
 
 2. The diameter of the new 5-cent piece is 2 centi- 
 meters ; how long will a row of 50 of them be ? 
 
 Ans. 1 meter. 
 
 3. A meter of cloth costs 52 cents ; what will a hecto- 
 meter cost at the same rate ? 
 
 4. One decistere of wood is worth 32 cents ; what is 
 a stere of wood worth ? 
 
 5. If a liter of wine weighs 872 grams, what will a 
 hectoliter of wine weio-h ? 
 
 6. If a letter weighs 2.5 grams, how many such 
 letters will it take to weigh a kilogram ? 
 
 7. A hectoliter of corn cost $51; at this rate, what 
 will a liter of corn cost ? 
 
 8. Mary bought 10.5 meters of silk for a dress, at $4.50 
 a meter; what did it cost her? 
 
 9. A grocer bought 186 kilograms of sugar at 18 } 
 eents a kilo ; what did it cost ? 
 
 10. I bought 5£ kilos of beef at 37£ cents a kilo ; how 
 much did it cost me ? 
 
164 THE METRIC SYSTEM. 
 
 11. A man sold 36 hectares of land at $4.25 a hectare ; 
 what did he receive for it ? 
 
 12. What must I pay for 324 liters of coal oil, if it 
 cost me 15? cents a liter? 
 
 13. If a kilogram weighs about 2\ pounds, how much 
 will a myriagram weigh ? 
 
 14. If a tonneau equals 1000 kilos, how many myria- 
 grams in 3 tonneaux? Ans. 300 M.G. 
 
 15. If a kilogram of coffee cost 96 cents, what will 
 a tonneau of coffee cost ? 
 
 16. If a tonneau equals 1000 kilos, how many ton- 
 neaux in 16700 kilos? Ans. 16.70 tonneaux. 
 
 17. A "quintal is 1 tenth of a tonneau ; how many 
 kilograms are there in a quintal ? 
 
 18. If a kilogram of pork costs 37£ cents, what will a 
 quintal of pork cost ? 
 
 19. If a man walks 6.5 kilometers in an hour, how 
 far will he walk in 12 hours ? 
 
 20. If an are of land is worth $26.50, what are 7 
 hectares of land worth ? 
 
 21. I bought 1600 ares of land at $15 £ an are, and 
 sold it at $175 a hectare j how much did I gain ? 
 
 22. I bought 15 kilograms of drugs at $27.50 a kilo, 
 and retailed them at the rate of 5 cents a gram ; what 
 did I gain ? 
 
 23. A cask of cider lost by leakage 144L. in 6 hours; 
 how much leaked out in an hour ? 
 
 24. If 5 steres of wood cost $7.50, what must I pay for 
 7.5 steres of the same kind of wood ? 
 
 25. If the height of a pole is 54.57GM., how long will 
 it take a worm to climb to its top at the rate of 12 M. a 
 
 day? 
 
 26. If a kilogram of sugar costs 18:] cents, how many 
 kilograms can be bought for 75 dollars? 
 
 27. A man bought a lot of land containing 45 hectares, 
 and sold from it 150 square meters; how much remained? 
 
THE METRIC SYSTEM. 165 
 
 28. If 6.5 meters of cloth cost £9.25, what will be the 
 cost of 32.5 meters at the same rate ? 
 
 29. A block of marble 2.5 meters long, .75 meters 
 wide, and .5 meters thick cost $10 ; what would a cubic 
 meter of marble cost at the same rate ? 
 
 EEDUCTION 
 
 FROM ONE SYSTEM TO THE OTHER. 
 
 187. Until the Metric System has gone into general 
 use, it will be necessary to reduce from one system to 
 the other ; hence the following exercises are presented. 
 
 MEASURES OF LENGTH. 
 
 188. A 3Ieter equals 39.37in., or 3.28ft. ; hence, to re- 
 duce from one measure of length to another, we have 
 the following rules : — 
 
 Eule I. — Multiply the number of meters by 39.37, and 
 it will give the number of inches; or by 3.28, and it will give 
 the number of feet. 
 
 Eule II. — Divide the number of inches by 39.37. or the 
 number of feet by 3.28, and it will give the number of meters. 
 
 1. How many inches in 16 meters ? 
 
 2. How many feet in 2.5 meters ? 
 
 3. How many rods in 25 meters ? 
 
 4. How many miles in 12.5 kilometers ? 
 
 5. How many meters in 9.20 feet ? 
 
 6. How many meters in 629.92 inches? 
 
 MEASURES OF SURFACE. 
 
 189. An Are equals 119.6sq. yd., or 0.00247 acre; 
 hence we may readily reduce from one measure of sup 
 face to the other. 
 
 1. How many sq. yd. in 25 ares ? 
 
 2. How many ares in 2990sq. yd. ? 
 
 3. How many acres in 360 ares? 
 
166 THE METRIC SYSTEM. 
 
 4. How many ares in 8.892 acres ? 
 
 5. How many ares in 20A. 2E. ? 
 
 6. How many hectares in 20 A.2E. ? 
 
 MEASURES OF VOLUME. 
 
 190. A Steve equals 35.3166 cubic feet, or 1.308 cu yd., 
 or .2759 cord ; hence we may readily reduce from one 
 measure to the other. 
 
 1. How many cu. ft. in 25 steres ? 
 
 2. How many cu. yd. in 3.5 steres ? 
 
 3. How many steres in 8829.15 cu. ft. ? 
 
 4. How many steres in 4.578 cu. yd. ? 
 
 5. How many cords in 12.5 steres ? 
 
 6. How many steres in 34.4875 cords? 
 
 MEASURES OF CAPACITY. 
 
 191. A Liter equals 1.0567 quarts liquid measure, or 
 .908 quarts dry measure; hence we can readily reduce 
 from one measure to the other. 
 
 1. How many liquid quarts in 23 liters? 
 
 2. How many dry quarts in 50 liters ? 
 
 3. How many gallons in 23 liters ? 
 
 4. How many bushels in 5 hectoliters ? 
 
 5. How many liters in 24.3041 liquid quarts ? 
 
 6. How many liters in 45.40 dry quarts? 
 
 MEASURES OF WEIGHT. 
 
 192. The Gram equals 15.432 Troy grains; the Mo- 
 gram equals 2.2046 lbs. avoirdupois; hence we can readily 
 reduce from one system to the other. 
 
 1. How many grains in 4.25 grams? 
 
 2. How many pounds in 8.5 kilograms? 
 
 3. How many grams in 65.586 grains? 
 
 4. How many kilograms in 18.7391 pounds? 
 
 5. How many pounds in 16 quintals? 
 
 6. How many pounds in 24 tonneaux ? 
 
THE METRIC SYSTEM. 
 
 167 
 
 To assist pupils in becoming familiar with the value 
 of the different denominations of the Metric System, we 
 present the following Tables. 
 
 193. Tables showing the Kelation of the De- 
 nominations of the Metric System to the Common 
 System. 
 
 Measures of Length. 
 
 Names. 
 
 Value in Meters. 
 
 Value in the Common System. 
 
 Millimeter, 
 
 .001 
 
 .0394 inch. 
 
 Centimeter, 
 
 .01 
 
 .3937 inch. 
 
 Decimeter, 
 
 .1 
 
 3.937 inches. 
 
 Meter, 
 
 1. 
 
 39.37 inches. 
 
 Dekameter, 
 
 10 
 
 393.7 inches. 
 
 Hectometer, 
 
 100 
 
 328^ feet. 
 
 Kilometer, 
 
 1000 
 
 .62137 mile. 
 
 Myriameter, 
 
 10000 
 
 6.2137 miles. 
 
 Measures of Surface. 
 
 Names. 
 
 Value in M 2 . 
 
 ■i 
 Value in the Common System. 
 
 Centare, 
 Are, 
 Hectare, 
 
 1 
 
 100 
 
 10000 
 
 1550 sq. in. 
 119.6 sq. yd. 
 2.471 acres. 
 
 Measures of Capacity. 
 
 Names. 
 
 Value 
 in Liters. 
 
 Dry Measure. 
 
 Wine Measure. 
 
 Milliliter, 
 
 .001 
 
 .061 cu. in. 
 
 .27 fluid dr'm. 
 
 Ten til iter, 
 
 .01 
 
 .6102 cu. in. 
 
 .338 fluid oz. 
 
 Deciliter, 
 
 .1 
 
 6.1022 cu. in. 
 
 .845 pill. 
 
 Liter, 
 
 1 
 
 .908 qt. 
 
 1.0567 qt. 
 
 Dekaliter, 
 
 10 
 
 9. OS qt. 
 
 2 6417 gal. 
 
 Hectoliter, 
 
 100 
 
 2.8375 bu. 
 
 26.417 gal. 
 
 Kiloliter, or St ere, 
 
 1000 
 
 1.308 cu. yd. 
 
 264.17 gal. 
 
168 
 
 THE METRIC SYSTEM; 
 
 Weights. 
 
 Names. 
 
 Value 
 in Grams. 
 
 Common Weights. 
 
 1 
 Quantity of Water. 
 
 Milligram, 
 
 .001 
 
 .0154 gr. Tr. 
 
 1 m. m 3 . 
 
 Centigram, 
 
 .01 
 
 .1543 gr. " 
 
 10 m. m 3 . 
 
 Decigram, 
 
 •.1 
 
 1.543 gr. " 
 
 T \ c. m 3 . 
 
 Gram, 
 
 1 
 
 15.432 gr. " 
 
 1 c. m 3 . 
 
 Dekagram, 
 
 10 
 
 .3527 oz. av. 
 
 10 c. m 3 . 
 
 Hectogram, 
 
 100 
 
 3.5274 oz. " 
 
 1 deciliter. 
 
 Kilogram, 
 
 1000 
 
 2.2046 lb. " 
 
 1 liter. 
 
 Myriagram, 
 
 10,000 
 
 22.046 lb. " 
 
 10 liters. 
 
 Quintal, 
 
 100,0U0 
 
 220.46 lb. » 
 
 1 hectoliter. 
 
 Tonneau, 
 
 1,000,000 
 
 2204.6 lb. 
 
 1 M 3 , or 1 K. L. 
 
 Common System compared with the Metric. 
 
 1 inch 
 
 = .0254 meter. 
 
 1 gallon = 
 
 3.786 liters. 
 
 1 foot 
 
 = .3048 meter. 
 
 1 bushel = 
 
 .3524 hectoliter. 
 
 1 yard 
 
 == .9144 meter. 
 
 1 cu. feet == 
 
 .2832 hectoliter. 
 
 1 mile 
 
 = 1.6094 kilometer. 
 
 1 cu. yd. = 
 
 .7646 stere. 
 
 1 sq. ft. 
 
 = .0929 sq. meter. 
 
 1 cord = 
 
 3.625 steres. 
 
 1 sq. yd. 
 
 = .8362 sq. meter. 
 
 1 grain = 
 
 .0648 gram. 
 
 1 sq. rd. 
 
 = .2529 are 
 
 1 Av. oz. = 
 
 .0283 kilogram. 
 
 1 acre 
 
 = .4047 hectare. 
 
 1 lb. Troy= 
 
 .373 kilogram. 
 
 1 sq. mile 
 
 = .259 hectare. 
 
 1 lb. Av. = 
 
 .4536 kilogram. 
 
 1 quart 
 
 == .9465 liter. 
 
 1 ton = 
 
 .9071 tonneau. 
 
PERCENTAGE. 169 
 
 SECTION VIII. 
 
 PEBCENTAGE. 
 
 194. Percentage is a process of computation in which 
 the basis of comparison is a hundred. 
 
 195. The term per cent, means by or on a hundred; 
 thus, 5 per cent, of any thing means 5 of a hundred of it. 
 
 196. Hence, 1 per cent, of a number is y i^ of it ; 2 
 per cent, is t §q of it ; 5 per cent, is T ^ of it, etc. It is 
 also evident that 100 per cent, of a number is the whole 
 
 01 it. 
 
 197. The sign of Percentage is %, and is read per 
 cent. The per cent, is also indicated by a common frac- 
 tion or a decimal ; thus, 5% = T ^ = .05. 
 
 19S. In percentage there are four quantities con- 
 sidered : 
 
 1. The Base, or number on which percentage is 
 estimated. 
 
 2. The Pate, or number denoting how many of a 
 aaudred. 
 
 3. The Percentage, denoting how many of the basis'. 
 
 4. The Amount or Difference of the basis and per- 
 
 centage. 
 
 
 
 
 
 
 
 RATE EXPRESSED : 
 
 BY A 
 
 FRACTION. 
 
 
 
 2 per cent. 
 
 equals 
 
 .02, 
 
 or Too' or 
 
 1 
 50" 
 
 
 4 per cent. 
 
 a 
 
 .04, 
 
 a 4 a 
 TTJO' 
 
 1 
 25"' 
 
 
 5 per cent. 
 
 u 
 
 .05, 
 
 a 5 " 
 l oO' 
 
 1 
 20' 
 
 
 20 per cent. 
 
 u 
 
 .20, 
 
 a 2 a 
 TOO' 
 
 1 
 5' 
 
 
 25 per cent. 
 
 u 
 
 .25, 
 
 U 2 5 it 
 
 i oO' 
 
 1 
 4~* 
 
 
 125 per cent. 
 
 it 
 
 1.2&, 
 
 U 125 U 
 TOO' 
 
 5 
 4* 
 
 A per cent, is .00.^, or .005 ; i per cent, is .001, or .002; 
 12£ per cent, is .12^, or .125, etc. 
 
 15 
 

 
 
 PERCENTAGE. 
 
 
 
 
 EXERCISES. 
 
 
 
 Express 
 
 decimally 
 
 Express in a 
 
 common fraction 
 
 1. 
 
 6%. 
 
 Ans. .06. 
 
 6. 6%. 
 
 Ans. 3 V 
 
 2. 
 
 12%. 
 
 Ans. .12. 
 
 7. 12%. 
 
 Ans. 2 V 
 
 3. 
 
 163%. 
 
 Ans. .16?. 
 
 9 
 
 8. 50%. 
 
 Ans. } 2 . 
 
 4. 
 
 24%. 
 
 Ans. .24. 
 
 9. 121%. 
 
 Ans. I. 
 
 5. 
 
 33|%. 
 Express 
 
 Ans. .33|. 
 as a per cent. 
 
 10. 16g%. 
 
 Ans. J. 
 
 1. 
 
 l 
 
 4* 
 
 Ans. 25 % . 
 
 4 ! 
 
 Ans. 12* %, 
 
 o 
 
 Li, 
 
 1 
 
 Ans. 20%. 
 
 5. J. 
 
 Ans. 16f%, 
 
 3. 
 
 1 
 
 2* 
 
 Ans. 50%. 
 
 6. 1. 
 
 Ans. 66f%. 
 
 Case I. 
 
 199. Given the base and rate, to find the per- 
 centage. 
 
 1. What is 5 per cent, of $250? 
 
 OPERATION. 
 
 SoLtiYiON.— 5 per cent, of $250 is T fo of $250, $250 
 
 or .05 times $250, which, by multiplying, we find .05 
 
 to be $12.50. Hence the following «->2 50 
 
 Eule. — Multiply the base by the rate, expressed deci* 
 molly. 
 
 2. What 
 
 3. What 
 
 4. What 
 
 5. What 
 
 What 
 What 
 What 
 What 
 What 
 "What 
 What 
 What 
 
 6 
 7 
 8 
 9 
 10 
 11 
 
 19 
 
 13 
 
 is 5 per cent, of $280 ? 
 is 6 percent, of $190? 
 is 7 per cent, of $240 ? 
 is 8 per cent, of 125yds. ? 
 is 9 per cent, of 3641bs.? 
 is 10 per cent, of 982ft. ? 
 is 12 per cent, of 831in.? 
 is 12^ per cent, of 320oz.? 
 is 16- per cent, of 630yds. ? 
 is 35 per cent, of 1286 miles ? 
 is 40 per cent, of 2467 pounds? 
 is 75 pei cent, of 3182 perches * 
 
 Ans. $14. 
 Ane. $11.40. 
 Ans. $16.80. 
 Ans 10yds. 
 
PERCENTAGE. 171 
 
 14. A man bought a cow for $30, and sold her at a 
 gain of 25% ; what did he gain? Ans. $7h 
 
 15. A man bought a horse for $150, and sold him at 
 a gain of 30% ; how much did he gain ? Ans. 845. 
 
 16. A lady bought 360 acres of land, and sold 12i% 
 of it ; how much did she retain ? Ans. 315 acres. 
 
 17. A man bought a horse for $4250, and sold it at a 
 gain of 5% ; how much did he receive for it ? 
 
 18. My salary is $1500 a year ; I spend 25% of it the 
 first half year, and 35% the second half; how much do 
 I save in a year ? Ans. $600. 
 
 Case IT. 
 
 200. Given the percentage and rate, to find the 
 base. 
 
 1. 60 is 5 per cent, of what number? 
 
 Solution.— If 5% of a number is 60, 1 % operation. 
 
 of the number is \ of 60, or 12, and 100% 5% =60 
 
 of the number, which is the whole number, 100% -— 1200 Ans. 
 
 is 100 times 12, or 1200. or, 
 
 Since this is equivalent to multiplying 60-f-.05 = 1200 
 by 100 and dividing by 5, and this last is 
 equivalent to dividing by .05, we have the following 
 
 Eule. — Divide the percentage by the rate expressed 
 decimally. 
 
 Note. — For young pupils the analysis will be simpler than the solution 
 by the rule. 
 
 2. 45 is 20 per cent, of what number? Ans. 225. 
 
 3. 75 is 25 per cent, of what number? Ans. 300. 
 
 4. 96 is 20 per cent, of what number? 
 
 5. 230 is 5 per cent, of what number? 
 
 6. 1121b. is 40 per cent, of what weight ? 
 
 7. 456 acres are 30 per cent, of how many? 
 
 8. 237 cows are 25 per cent, of how many ? 
 
 Ans. 948. 
 * 9. 157yds. are 12^ per cent, of how many ? 
 
 Ans. 1256, 
 13* 
 
Y12 PERCENTAGE. 
 
 10. 644 pigs are 35 per cent, of how many ? 
 
 11. $78.18 is 33J per cent, of how much? 
 
 Ans. $234.54. 
 
 12. A man spends $500 a year, which is 25% of his 
 salary ; what is his salary ? Ans. $2000. 
 
 13. A has 280 acres of land, and this is 35% of what 
 B has ; how much has B ? 
 
 14. A sold 36 pigs, which is 8 % of what he now has ; 
 how many had he before the sale ? 
 
 15. A boy found $15, which is 30% of what he then 
 had ; how much had he at first ? Ans. $35. 
 
 16. A man had $13681.60 in a bank, and drew out 
 35 % of it ; how much did he draw out ? 
 
 Ans. $4788.56. 
 
 Case III. 
 
 201. Given the base and percentage, to find the 
 rate. 
 
 1. 25 is what per cent, of 125 ? 
 
 OPERATION. 
 
 Solution.— 125 is 100 per cent, of itself, 125 = 100% 
 
 and 25, which is ffc of 125, is ffe of 100 25 = T 2 ^ X 100% 
 
 per cent., or \ of 100 per cent,, which is =$X 1( > % = 20 % 
 20 per cent. Hence the following 
 
 Eule. — Take such a part of 100 as the percentage is of 
 the base; or, multiply the percentage by 100 and divide by 
 the base. 
 
 2. 75 is what per cent, of 300 ? Ans. 25%. 
 
 3. 90 is what per cent, of 360 ? Ans. 25%. 
 
 4. 45 is what per cent, of 225 ? Ans. 20%. 
 
 5. 72 is what per cent, of 216 ? Ans. 33£%. 
 
 6. 96 is what per cent, of 128 ? 
 
 7. 48 is what per cent, of 120 ? 
 
 8. 112 is what per cent, of 896 ? 
 
 9. A man had $960, and lost $240 ; what per cent, did 
 h°> lose ? 
 
PERCENTAGE. 173 
 
 10. B lost 825, and then had $125 ; what per cent, of 
 his money did he lose ? 
 
 11. C sold 50 cows, which was 25 per cent, of the re- 
 mainder; how many had he at first? 
 
 12. D gave his sister $180, and had $960 left; his 
 money is now what per cent, of what he had at first ? 
 
 SIMPLE INTEREST. 
 
 202. Interest is money charged for the use of money. 
 It is estimated at a certain rate per cent, per annum. 
 
 203. The Principal is the sum on which interest is 
 
 reckoned. 
 
 204. The Rate of Interest is the interest on 100 for 
 
 one year. 
 
 205. The Time is how long the money is on interest. 
 
 206. The Amount is the sum of the principal and 
 
 interest. 
 
 207. Simple Interest is interest upon the principal 
 only. Compound Interest is interest upon the prin- 
 cipal and interest. 
 
 208. Legal Interest is the rate established by law. 
 Usury is a rate greater than the legal rate. The taking 
 of usury is prohibited by law. 
 
 209. The quantities considered in Simple Interest 
 are the Principal, Bate, Time, Interest, and Amount 
 There are four cases. 
 
 Case I. 
 
 210. Given the principal, rate per cent., and 
 time, to lind the interest or amount. 
 
 First Method. 
 1. What is the interest of $2400, for 6yr. 7mo. 15da.. 
 at7%? lft . 
 
174 PERCENTAGE. 
 
 Solution. — By reduction we find that 6yr. deration 
 
 Tmo. 15da. equal C|yr. If the interest of $1 $2400 
 
 for lyr. is 7ct., the interest of $2400 for lyr. is .07 
 
 2400 times 7ct., which is $168, and for 6§yr. it 168.00 
 
 is 6| times $168, which by multiplying we find g| 
 
 is $1113. Hence the following $1113 00 Ans 
 
 Eule. — Multiply the principal by the rate per cent. t 
 expressed decimally, and that product by the time expressed 
 in years. 
 
 Kequired the interest 
 
 2. Of $180 for 3yr. 6rao. at 7%. Ans. $44.10. 
 
 3. Of $470 for 7yr. 8mo. at 6%. Ans. $216.20. 
 
 4. Of $172 for 5yr. 9mo. at 5%. Ans. $49.45. 
 5 Of $480 for 5yr. lOmo. at 12%. 
 
 8. Of $1080 for 3yr. 7mo. 6da. at 5%. 
 
 7. Of $1260 for 2yr. 2mo. 12da. at 5%. 
 
 8. Of $1000 for 3yr. 8mo. 12da. at 10%. 
 
 Second Method. 
 
 211. The second method, called the " Six per cent, 
 method," is perhaps the method most generally used by 
 business men. 
 
 1. What is the interest of $240 for 2yr. 8mo. 12da. 
 at 6% f 
 
 Solution. — 2yr. 8mo. equal 32mo. The in- operation. 
 
 terest of $1 for 12mo. is 6cts., and for lmo. it 2yr. 8mo. = 32mo. 
 is T \ of 6cts., or Jet, and for 32mo. it is 32X£ = $ - 16 
 
 32><^ = 16cts. Also, since the interest on $1 12X1- - 002 
 
 for lmo., or 30cla., is Jet. or 5 mills, for Ida. it $0,162 
 
 is 3 L of 5 mills, or l of a mill, and for 12da. 240 
 
 it is 12 X \ = 2 mills : uence tne interest on $38~88 
 
 #1 for 32mo. and 12da. is 16cts. plus 2 mills, 
 
 or $0,162. If the interest on $1 is $0,162, on $240 ft is 240 times 
 $0,162 which equal $38.88. From this we have the following 
 
PERCENTAGE. 175 
 
 Rule — 1, Call one-half of the number of months tents, 
 and one-sixth of the number of days mills, and their sum 
 will be the interest of one dollar, for the given time, at 
 6 per cent. 
 
 2. Multiply this by the principal, and the product will be 
 the entire interest at 6 per cent. For any other rate, take 
 as many sixths of it as that rate is of six. 
 
 Notb.— 1. For 1% add J, for Sf add J, for 9% add %, for 5% subtract 
 J, for 4 % subtract J, etc. 
 
 2. When the time is brief, the rule of business men is as follows : " Multi- 
 ply dollars by days, and divide by 60." 
 
 Required the interest 
 
 1. Of $360 for 5yr. 6mo. 12da. at 6%. 
 
 Ans. 8119.52. 
 
 2. Of $480 for 3yr. 8mo. 18da. at 6%. 
 
 Ans. $107.04. 
 
 3. Of 8256 for 7yr. 4mo. 24da, at 6%. 
 
 Ans. 8113.66. 
 
 4. Of 848.25 for 3yr. 6mo. 6da. at 6%. 
 
 Ans. 810.18. 
 
 5. Of 850.50 for 6yr. lOmo. 18da. at 7%. 
 
 Ans. 824.33. 
 
 6. Of 828.25 for 5yr. 7mo. 24da. at 5%. 
 
 7. "What is the amount of 8360 for 2yrs. and 6mo. at 
 6 per cent, ? Ans. 8414. 
 
 8. What is the amount of $250 for Syr. 8mo. 18da. 
 at 6 per cent. ? Ans. 8305.75. 
 
 9. What is the amount of $620 for 5yr. lOmo. 24da. 
 at 7 per cent. ? Ans. 8876.06. 
 
 10. Mary's father put out 8500 on interest, at 10 % at 
 her birth ; how much will she be worth when she is 21 
 years of age ? 
 
 11. Required the difference between the interest and 
 the amount of $624 for 3yr. 8mo. 15da., at 5 per cent. 
 
 For other exercises under this rule solve the problems in the pre- 
 vious and following cases. 
 
176 
 
 PERCENTAGE. 
 
 Third Method. 
 
 212. A metfhocl of computing interest by taking 
 aliquot parts. 
 
 1. What is the interest of $2400 for 6yr. 7mo. 15da. 
 
 at 7%? 
 
 Solution. — We find 
 the interest for 1 yr. and 
 then for 6yr., and then 
 proceed thus: 7mo. = 
 Gmo. plus lmo., and 
 since 6mo. equal \ of a 
 year, the interest for 
 6mo. is \ of $1G8, which 
 is $84 ; and since lmo. 
 is i of 6mo., the interest 
 
 
 OPERATION. 
 
 
 $2400 
 
 
 .07 
 
 
 168.00 = Int. for lyr. 
 
 
 
 6 
 
 
 
 1008.00 = Int for6yr. 
 
 6mo. 
 
 =$yr- 
 
 84.00 = Int. for 6mo. 
 
 lmo. 
 
 = - of lyr. 
 
 14.00 = Int. for lmo. 
 
 15da. 
 
 -Juio. 
 
 7.00 = Int. for 15da. 
 1113.00, Ans. 
 
 for lmo. is \ of $84, 
 
 which is $14 ; and since 
 
 15 days is £ of a month, the interest for 15 days is | of $14, which 
 
 is $7 ; and the whole interest is the sum of these, which we find to 
 
 be $1113. Hence the following 
 
 Eule. — 1. Find the interest for the number of years, as 
 by the first method. 
 
 2. Find the interest for the number of months by taking 
 convenient fractional parts of one year's interest. 
 
 8. Find the interest for the number of days by taking 
 fractional parts of one month'' s interest. 
 
 Required the interest 
 
 2. Of $780 for 4yr. 8mo. at 6%. Ans. $218.40. 
 
 3. Of $960 for 7yr. 9mo. at 7%. Ans. $520.80. 
 
 4. Of $1260 for 3yr. 6mo. 15da. at 8 %. Ans. $357. 
 
 5. Of $2480 for 5yr. 5mo. lOda. at 5%. 
 
 Ans. $675.11. 
 
 Note. — Let the pupil solve by this method the problems under the first 
 u.nd second methods. 
 
STOCKS AND DIVIDENDS. 177 
 
 STOCKS AND DIVIDENDS. 
 
 213. The Stoclc of a company represents the money 
 mvcsted in its business by the stockholders or owners. 
 
 Stock is divided into equal parts, called shares, 
 
 214. A Dividend is a sum to be paid to the stock- 
 holders out of the gains of the company. It is divided 
 according to the par value of stock held by them. 
 
 215. The Par Value of stock is its nominal value as 
 fixed by the charter, or articles of agreement, of the 
 company. It is usually $50 or $100 per share; although 
 other sums are often agreed upon. 
 
 216. The Heal Value of stock is what it will sell for. 
 
 217. Premium is how much its real value exceeds its 
 par value. 
 
 21S. Discount is how much its real value is less than 
 its par value. 
 
 Case I. 
 
 219. Given the stock and rate of dividend, to 
 find tlie dividend. 
 
 1. A owns 50 shares of bank-stock, at $100 each; 
 the bank declares a dividend of 6 % ; required A's divi- 
 dend. 
 
 OPERATION. 
 
 Solution. — If one share is worth SI 00, 50 50 
 
 shares are worth 50 times $100, or $5000. Hence 100 
 
 A r s stock is worth $5000. His dividend is .06 5000 
 
 times $5000, or $300. .06 
 
 $ 30()! 00 
 
 Eule. — Multiply the par value of the number of shares 
 by the rate of dividend. 
 
 2. A man owns 56 shares of stock, at $50 per shar<? 
 
178 STOCKS AND DIVIDENDS. 
 
 par value ; the company declares a 5% dividend; what 
 is his dividend ? Ans. $140. 
 
 3. A lady has 175 shares of stock, at $10 par per share ; 
 the company declares 8J% dividend; what is her divi- 
 dend ? Ans. $148.75. 
 
 4. I hold 250 shares of mining stock, at $20 par a 
 share; the company has divided h\°J \ what is my 
 dividend ? Ans. $275. 
 
 Case II. 
 
 220. Given the par value and rate of premium 
 or discount, to find the premium, discount, or real 
 value. 
 
 1. A bought 25 shares of stock ($50), at 4 % premium ; 
 required the premium and the real value. 
 
 OPERATION. 
 
 Solution. — The par value of 25 $50 X 25 = $1250 = par value, 
 shares at $50 a share is 25 times .04 
 
 $50, or $1250. The premium is .04 50.00 = premium, 
 
 times $1250, or $50, and the pre- 1250. 
 
 mium added to the par value equals $1300 = real value. 
 
 $1300, the real value. 
 
 Eule. — Multiply the par value by the rate, for amount of 
 premium or discount. Add or subtract this from par value, 
 for real value. 
 
 2. I bought 75 shares of gas stock ($20), at 3$f pre- 
 mium ; required the premium and the cost. 
 
 Ans. $52.50; $1552.50. 
 
 3. I sold 120 shares of stock ($15), at 5% discount; 
 required the discount, and the amount received for it. 
 
 Ans. $90 ; $1710. 
 
 4. Sold 34 shares of E. E. stock ($50), at 2%% dis- 
 count; what was received for it ? Ans. $1657.50. 
 
 5. Bought 12 shares bank stock ($50), at 12£% pre- 
 mium, and afterwards sold the same at $60 per share ; 
 did I gain or lose, and how much ? Ans. Grained $45. 
 
STOCKS AND DIVIDENDS. 179 
 
 UNITED STATES BONDS. 
 
 221. United States Bonds are printed U.S. promises 
 to pay the holder a certain sum of money, with interest 
 payable at stated periods in the interval. 
 
 222. U.S. Bonds may be of various kinds : — of these, 
 Seven- Thirties, Five- Twenties, Ten-Forties, and Twenty- 
 years-Sixes, are at present the principal. These may 
 also be either Coujyon or Registered bonds. 
 
 223. Seven- Thirties are so called because they draw 
 an annual interest of 7 T 3 per cent., or a daily interest 
 of two cents on every $100. The interest for less than 
 6 months is always calculated by days. They are indi- 
 cated thus : 7-30's. 
 
 224. Five- Twenties are so called because they are 
 payable at the option of the Government any time 
 between 5 years and 20 years after they are issued. 
 They are marked 5-20's. 
 
 225. Ten-Forties are so called because they are pay- 
 able at the option of the Government any time between 
 10 years and 40 years after they are issued. They are 
 marked 10-40's. 
 
 226. Seven-Thirties as now issued pay 7 T 3 % interest 
 in legal currency. Five-Twenties pay 6% interest in gold ; 
 and Ten-Forties, 5% interest in gold. Interest on each 
 is due semi-annually. 
 
 227. Coupon Bonds have attached to them "cou- 
 pons," or certificates for each six months' interest 
 accruing, payable as they become due. These coupons 
 may be cut off, and are payable to the bearer of them. 
 
 228. Registered Bonds have no coupons attached, 
 and the interest is payable only to the bond-holder 
 whose name is registered in the Treasury Department, 
 or to his order. 
 
 229. United States bonds usually sell at a premium 
 over their face value, and are quoted or priced at their 
 current value per S100. 
 
180 STOCKS AND DIVIDENDS. 
 
 Case I. 
 
 230. To find the interest due on a United States 
 Bond. 
 
 1. What is the semi-annual interest of a $500 7-30 
 bond? 
 
 OPERATION. 
 
 Solution. — If on $1 the interest for a year is 500 
 
 7 T 3 Q cents, on §500 the interest is 500 times 7 T 3 ^, .073 
 
 which is $36.50. If the interest for 1 year is 2)36 500 
 
 $36.50, for £ year it is * of $36.50, or $18.25. " ' 
 
 2. What is the semi-annual interest on a $500 5-20 
 bond worth in currency, gold being at a premium of 
 
 30%? 
 
 OPERATION. 
 
 Solution. — If the interest for 1 year is 6%, 500 
 
 for J year it will be 3%. The interest on 03 
 
 $500 at 2>f is $15.00. Since gold is at a pre- 15 qq 
 
 mium, we must calculate the premium on $15 oq 
 
 and add it to the $15. 30% of $15 is $4.50, 4 ,- 0Q - 
 
 which added to $15 equals $19.50, or worth 1( - 
 
 of the interest in currency. — 
 
 $19.50 
 
 Eule. — 1. Find the interest on the face of the bond for 
 the given time and rate. 
 
 2. In gold-bearing bonds, add the premium on the interest 
 to the interest. 
 
 3. I have $4000 7-30 bonds with the half-year's interest 
 due ; how much is due on them ? Ans. $146. 
 
 4. My father has a $5000 5-20 bond with a half-year's 
 interest due ; how much is the interest worth in cur- 
 rency, gold at 35% premium? Ans. $202.50. 
 
 5. I hold two $5000 5-20 bonds with the half-year's 
 interest due ; how much will I get in " greenbacks," 
 gold at 25% premium? Ans. $375. 
 
 6. Miss Smith has $4000 7-30's and $4000 5-20's; for 
 which does she get the most interest in currency in a 
 year, — gold at 35 % premium ? Ans $32 on the 5-20's. 
 
STOCKS AND DIVIDENDS. 181 
 
 Case II. 
 
 231. To find the cost of bonds when they are at 
 a premium. 
 
 1. What must I pay for a $500 7-30 bond, when 7-30'm 
 are at a premium of 5%, or sell at 105 ? 
 
 OPERATION. 
 
 Solution.— Five per cent, of $500 is $25, 500 
 
 which added to $500 equals $525. Hence I .05 
 
 must pay $525 for a $500 bond at 5$, premium. 25.00 
 
 Or, if a $100 bond is worth $105, my $500 bond "ivv^Tno 
 
 is worth five times as much, or $525. „,.„. 
 
 ' or $10o 
 
 5 
 
 $525 
 
 Rule. — 1. Find the premium, and add it to the face of 
 the bond. 
 
 Eule. — 2. Multiply the price per $100 by the number of 
 hundreds bought or sold. 
 
 Note. — If there is interest due on the bond, it must be added. If 
 a coupon not due is cut off, deduct the unaccrued interest thus 
 retained. 
 
 2. What must I pay for $2000 7-30's, when they com- 
 mand a premium of 4£%, or sell at 104} ? Ans. $2090. 
 
 3. How much will a $1000 7-30 bond cost, when there 
 is 93 days' interest due and the premium is 4^ % ? 
 
 Ans. $1066.10. 
 
 4. When 7-30's are at a premium of 5£%, how much 
 must I pay for $3500 bonds with 124 days' interest due 
 on them ? Ans. $3779.30. 
 
 5. I bought $3000 5-20's with 2 months' interest due, 
 the bond being at a premium of 12% ; what did it cost 
 me? Ans. $3390. 
 
 Note. — No premium for gold is allowed on less than 6 months' 
 interest. 
 
 16 
 
182 MISCELLANEOUS PROBLEMS. 
 
 MISCELLANEOUS PROBLEMS. 
 
 1. A has $4685, B has $1245 more than A, and C has 
 as much as A and B both ; how many dollars has C ? 
 
 2. C has 438 acres of land, D has 179 acres less, and 
 E has 48 less than C and D together ; how many acres 
 have D and C ? 
 
 3. Henry can walk 30 miles a day, and William can 
 walk 37 miles ; how much farther can William walk in 
 45 days than Henry? 
 
 4. Two men start from the same point and walk in 
 opposite directions, one traveling 25, the other 32, miles 
 an hour ; how far apart will they be in 148 hours ? 
 
 5. If a boy can earn $28 a month, and a man $47 a 
 month, how much will 6 boys and 9 men earn in a 
 month ? 
 
 6. If a clerk earns $150 a month, and spends $48, how 
 much can he save in 12 months, or a year? 
 
 7. A merchant gave $8.25 a barrel for 96 barrels of 
 flour, and sold it for $1000 ; what was the gain ? 
 
 8. How many bushels of apples can you buy, at $2£ 
 a bushel, for 56 barrels of flour, at $7.50 a barrel ? 
 
 Ans. 168. 
 
 9. A farmer has 137 hens; now, if he should lay out 
 $625 for hens, at the rate of 25 cents apiece, how many 
 would he then have? Ans. 2637. 
 
 10. If a steamboat goes 14 miles an hour, and a rail- 
 road train 32 miles an hour, how far will the steamboat 
 go while the train goes 672 miles ? Ans. 294 miles. 
 
 11. Mary and Susan had each 1420 cents; after Mary 
 had given Susan 360 and Susan had given Mary 480, 
 bow many had each ? Ans. Mary, 1540 ; Susan, 1300. 
 
 12. Six men and 8 boys earned a sum of money, and 
 divided it so that each man had $75 and each boy $63 ; 
 bow much did each earn ? 
 
 13. A had 369 acres of land, then bought 720 acres, 
 
MISCELLANEOUS PROBLEMS. 183 
 
 and then divided the whole into 9 equal farms, and sold 
 6 of them ; how many acres remain ? 
 
 14. A boy earns $2.50 a day, and pays 75 cents a day 
 for his board j how much can he thus save in a week ? 
 
 Ans. $9.75. 
 
 15. What number must I add to the product of 126 
 and 72, to make 10000? 
 
 1G. A lady went to the city with 600 eggs, and sold 
 them at 15 cents a dozen ; what did she receive for 
 them ? 
 
 17. The sum of two numbers is 7809, and one of the 
 numbers is 3725 ; required the other number, and their 
 difference. 
 
 18. The sum of three numbers is 4082; the first num- 
 ber is 1028, the second 2372 ; required the third number. 
 
 19. The difference between two numbers is 709, and 
 one-half of the smaller number equals 482 ; what is the 
 larger number ? 
 
 20. A merchant bought 26 bales of cloth, each bale 
 containing 32 pieces, and each piece 24 yards; how 
 many yards did he buy ? 
 
 21. If a boat sail 378 miles in 9 days, how far can it 
 sail in 15 days at the same rate ? Ans. 630 miles. 
 
 22. If 28 men can build a lot of wall in 18 days, how 
 many men can build the same wall in 21 days ? 
 
 Ans. 24 men. 
 
 23. A drover bought 365 cows, at $25 a head, and 
 758 sheep, at $3.50 a head ; what was the cost of all ? 
 
 24. A merchant bought 96 barrels of flour for S960 ; 
 he sold 58 barrels at $8 a barrel, and the remainder at 
 $12 a barrel ; what was the loss ? 
 
 25. What is the sum of J, j, J, and J ? 
 
 26. Subtract the sum of ~ and f from the sum of ^ 
 and J. 
 
 27. Multiply f by }, and add the result to the product 
 of 4 and j-?-. 
 
184 MISCELLANEOUS PROBLEMS. 
 
 28. Multiply § by |, and divide the result by the pro- 
 ct of | and ]~. 
 
 29. Find the difference between 1 off of 3 and i of js 
 
 off 
 
 30. If J of a barrel of flour costs $5.60, what will 4 
 barrels cost at the same rate ? Ans. $33.60 
 
 31. If | of a ton of coal is worth $7.50, what will 12 
 tons cost at the same rate? Ans. $120. 
 
 32. A man sold I of his land for $9750 ; what was it 
 all worth at the same rate ? Ans. $16250. 
 
 33. J\!ary lost \ of her money, and then had $960 re- 
 maining: how much had she before the loss ? 
 
 34. If § of a ton of coal is worth $4.50, what is % of a 
 ton worth at the same rate ? Ans. $9.37^. 
 
 35. If | of a lot of land is worth $234, what is | of 
 the same lot worth ? Ans. $260. 
 
 36. When rye is worth | of a dollar a bushel, how 
 many bushels can be bought for J of a dollar ? Ans. J. 
 
 37. If one yard of cloth is worth 2} dollars, how many 
 yards can be bought for $5| ? Ans. 2 T 3 Qyds. 
 
 38. A man owned g of a farm, and sold -J of his share ; 
 what part of the whole farm remained ? Ans. T V 
 
 39. Plow many times will 14 gallons fill a vessel that 
 holds 2 1 gallons ? Ans. 6 times. 
 
 40. A father divided 510 acres of land equally among 
 his sons, giving them 63| acres apiece; how many sons 
 had he ? 
 
 41. A merchant bought 6| cords of wood, at $6§ a 
 cord ; and paid for it with corn, at -| of a dollar a bushel; 
 how many bushels did it take ? Ans. 55bu. 
 
 42. If the sum of two fractions is |, and one of them 
 is ~, what is the other? Ans. |-J. 
 
 43. If the difference of two fractions is T 3 y, and tl*. 
 smaller is T \, what is the other fraction ? Ans. |J. 
 
 44. What fraction multiplied by 2| will give a pru 
 duct of 1J? Ans.fi. 
 
MISCELLANEOUS PROBLEMS. 18l 
 
 45. A pole stands J in the mud, | in the water, and 
 20 feet out of the water; what is the length of the pole 1 
 
 Ans. 48 feet. 
 
 46. Reduce £12 13s. lOd. to pence. 
 
 47. Reduce £17 17s. 9d. 3far. to farthings. 
 
 48. Reduce 281b. lOoz. 16pwt. 22grs. to grains. 
 
 49. Reduce 52876 farthings to pounds. 
 
 50. Reduce 89726 grains, Troy, to pounds. 
 
 51. Reduce 89726 grains, Apothecaries', to pounds. 
 
 52. Reduce 31207 drams to higher denominations. 
 
 53. How many drams in 1 ton ? 
 
 54. How many inches in 1 mile ? 
 
 55. How many square inches in 1 acre? 
 
 56. How many square feet in 1 square mile ? 
 
 57. How many cubic inches in 1 cord of wood ? 
 
 58. How many pounds of medicine would a physician 
 use in one year, if he averaged 10 prescriptions a day, 
 of 10 grains each? Ans. 61b 43 1^. 
 
 59. A merchant bought 6cwt. 321b. of sugar at 8£ 
 cents a pound ; what did it cost ? 
 
 60. How often will a wheel 18ft. 4in. in circumference 
 revolve in going 50 miles? Ans. 14400 times. 
 
 61. How many square rods in a rectangular field 50 
 rods long and 30 rods wide ? Ans. 1500sq. rd. 
 
 62. How many acres in a rectangular field 50 rods 
 long and 30 rods wide? Ans. 9A. 1R. 20P. 
 
 63. How many rods of fence will enclose the field in 
 the preceding problem ? Ans. 160 rods. 
 
 64. In an orchard, i of the trees bear apples, \ bear 
 pears, and the remainder, which is 33, bear peaches; 
 how many trees are there in the orchard ? 
 
 Ans. 60 trees. 
 
 65. How many yards of carpeting 1 yard in width 
 will carpet a floor 16 foot long and 9 feet wide ? 
 
 Ans. 16 yards. 
 
 16* 
 
IStf MISCELLANEOUS PROBLEMS. 
 
 66. How many cubic feet in a pile of wood 25 feet 
 long, 12 feet high, and 4 feet wide ? 
 
 67. How many cords in a pile of wood 64 feet long. 
 16 feet hio-h, and 4 feet wide ? Ans. 32 cords. 
 
 68. If a load of wood be 16 feet long and 6 feet wide, 
 how high must it bo to make a cord ? Ans. ljffc. 
 
 69. If my bedroom is 12 feet long, 10 feet wide, and 
 8 feet high, and I breathe 12 cubic feet of air in a minute, 
 how lone: will it take to breathe as much air as the 
 room holds? Ans. 80min. 
 
 70. America was discovered by Columbus, Oct. 11, 
 1492 ; how long from that time until Aug. 8, 1865 ? 
 
 71. The Kevolution commenced April 19, 1775, and 
 peace was declared January 20, 1783 ; how long did the 
 war continue ? 
 
 72. A boy lost £ of his kite-string, and then added 20 
 feet, and then found the string was f of its original 
 length ; what was the length at first ? Ans. 240ft. 
 
 73. Mary spent 20 per cent, of $500 for a watch, and 
 20 per cent, of the remainder for a chain; how much 
 then remained ? Ans. $320. 
 
 74. A man had 250 acres of land, and sold 25% to A, 
 and 20% to B; how much remained? 
 
 Ans. 137£ acres. 
 
 75. A man bought a horse for $250, and sold it at a 
 gain of 20% ; what did he receive for it? Ans. $300. 
 
 76. A house was bought for $1280, and sold for $1600; 
 what was the gain per cent. ? Ans. 25%. 
 
 77. Eequired the interest of $2400 for 6yr. lOmo. 
 24da., at 6 per cent. Ans. $993.60. 
 
 78. Eequired the amount of $960 for 2yr. 6mo. 12da., 
 at 7 per cent. Ans. $1130.24. 
 
 79. I would now like each little boy and girl, who has 
 gone through the book, to tell me how many months old, 
 then how many weeks old, and then how many days old, 
 he or she is. 
 
 THE END. 
 
PUBLICATIONS OF SOWER, POTTS &. CO., PHILADELPHIA. 
 
 PRICE 
 
 Brooks's Normal Geometry and Trigonom. $1.25 
 
 By the aid of Brooks's Geometry the principles of this beautiful science can be easily acquired in 
 one term. It is so condensed that the amount of matter is reduced one-half, and yet the chain 
 of logic is preserved intact and nothing- essential is omitted. The subject is made interesting 
 and practical by the introduction of Theorems for original demonstration, Practical Problems, 
 Mensuration, etc., in their appropriate places. The success of the work is very remarkable. 
 
 PRICE 
 
 Brooks's Normal Algebra • » • « $1.25 
 
 The many novelties, scientific arrangement, clear and concise definitions and principles, and 
 masterly treatment contained izi this quite new work, make it extremely popular. Each topic 
 is so clearly and fully developed that the next follows easily and naturally. Young pupils can 
 handle it, and should take it up before studying Higher Arithmetic. LiketheGeometry.it 
 can be readily mastered in one term. Jt^only needs introduction to make it indispensable. 
 
 
 PRICE 
 
 12mo 
 
 .... $1.60 
 
 18mo 
 
 .... .80 
 
 Peterson's Familiar Science. 
 Peterson's Familiar Science. 
 
 This popular application of science to every-day results is universally liked, and has an immense 
 circulation. No school should be without it. Inexperienced teachers have no difficulty in 
 teaching it. 
 
 PRICE 
 
 Roberts's History of the United States . 75 cts. 
 
 Short, compact and interesting, this History is admirably arranged to fix facts in the memory. 
 These only are dealt with, leaving causes for more mature minds. It ends with the close of 
 the late war. 
 
 PRICE 
 
 Sheppard's Text-Book of the Constitution $1.25 
 Sheppard's First Book of the Constitution .75 
 
 The ablest jurists and professors in the country, of all political denominations, have given these 
 works their most unqualified approval. Every young voter should be master of their contents. 
 
 PRICE 
 
 Bouvier's Astronomy $3.25 
 
 Bouvier's Astronomy. Abridged 2.25 
 
 No educational work published in this country has received so many recommendations from the 
 ablest Astronomers and Professors in Europe and America as these have had. They lead the 
 pupil from the elements to a thorough knowledge of the science. Being in the form of ques- 
 tions and answers, the subject is easily taught. The works are profusely and elegantly 
 illustrated. 
 
 PRICE 
 
 Hillside's Geology 94 cts. 
 
 This is a familiar and easy compend of Geology for schools and families in the form of questions 
 and answers. It is handsomely illustrated. No scholar can be considered well educated with- 
 out a knowledge of the noble and elevating sciences of Geology and Astronomy. 
 
ID OJOU4 
 
 PUBLICATIONS OF SOWER, POTTS & CO.. PHILADELPHIA. 
 
 Fairbanks's Bookkeeping . 
 
 PRICE 
 
 §4.00 
 
 Bookkeeping is thorou;,. 'y taught h '-e aid 01 lis comprehens. -eludes complete 
 
 sets of books in every hi of S* , from the country ritaile. ..^ui. bank, prepared 
 
 by practical book pcrs in v. .ent Shipping. Importing, Commission, Jobbing and Retail 
 
 Houses, Banks, Bain is I ' pensabte o every teacher, be .keeper, lawyer and 
 business-man as a wor*. 
 
 Jarvis's Chim" 
 
 PRICE 
 
 90 cts. 
 
 A music-book for schools compiled by one of the most accomplished musicians of the country. It 
 is full of choice pieces by eminent composers. 
 
 £ f- -I)- PRICE 
 
 Pel 
 
 1. 1 
 
 2. I 
 
 3. B 
 
 4. IN 
 
 5 
 
 6 
 
 Pelt 
 Pelt 
 
 9f\ a 
 
 t. IX 
 
 >. tA 
 
 5. M 
 
 Ub 
 
 This be TH£ UNIVERSITY OF CALIFORNIA LIBRARY 
 
 seems t 
 of Phys 
 
 Politica „ — r~j- 
 
 Notwithstanding the numerous outline maps that have been published since Pelton's series origin- 
 ated this, the true method of teaching Geography, the popularity of these elegant maps is 
 undiminished. 
 
 4®=- Sample copies sent to Teachers and School Officers for examination upon re- 
 ceipt of one-half of above prices. 
 
 Introduction Supplies furnished upon most liberal terms. 
 Catalogues and C'roulars sent fr. , judication. 
 
 Correspond^) -cb .ol Cafnloc;. . dicited. 
 
 For books ' atiou addr~* 
 
 SOW. TS & CO., 
 
 PU. 5 AND BOOKSELLERS, 
 
 530 Market St.. Philadelphia. 
 
c » ; 
 
 THE NORMAL SERIES 
 
 OF 
 
 ritfcmttitt m& IJ^tamita, 
 
 BY 
 
 EDWARD BROOKS, A. M. 
 
 PROFESSOR OF MATHEMATICS IN THE PEKNA. STATE NORMAL SCHOOL 
 
 AT MILLERSVILLE, PA. 
 
 ♦♦..♦♦ • 
 
 BROOKS'S NORMAL PRIMARY ARITHMETIC. 
 
 An Introduction to Mental Arithmetic for Beginners. 
 
 BROOKS'S NORMAL MENTAL ARITHMETIC. 
 
 A thorough and complete course by Analysis and Induction. 
 
 BROOKS'S METHODS OF TEACHING MENTAL ARITHMETIC, AND 
 KEY TO NORMAL MENTAL ARITHMETIC. 
 
 Containing many suggestions and methods for Arithmetical contrac- 
 tions, and a collection of Problems of an interesting and amusing 
 character for Class Exercise. 
 
 BROOKS'S NORMAL ELEMENTARY WRITTEN ARITHMETIC. 
 
 An Introduction to Written Arithmetic, containing the rules and 
 problems most used in every-day business. 
 
 BROOKS'S NORMAL WRITTEN ARITHMETIC, 
 
 By Analysis and Synthesis: for Common Schools, Normal Schools, 
 High Schools, Academies, <fcc. 
 
 BROOKS'S METHODS OF TEACHING WRITTEN ARITHMETIC, 
 
 And Key to Normal "Written Arithmetic. Containing, in addition to 
 Solutions to Problems, many valuable suggestions to Teachers. 
 
 BROOKS'S NORMAL GEOMETRY AND TRIGONOMETRY. 
 
 Just Published. 
 
 Designed as an Elementary Work for Popular Use. Containing many 
 new and simplified demonstrations. 
 
 OFFICE OF CONTROLLERS OF PUBLIC SCHOOLS. 
 
 Piiila., March 31, 1864. 
 Resolved: That Brooks's Mental Arithmetic, Primary Arithmetic and Key, be used in the Public 
 Schools of this District. JAS. D. CAMPBELL, Secy. 
 
 "His Mental Arithmetics I consider the best published." 
 
 JOS. W. WILSON. Prof, of Pract. Math., 
 Phila. Central High School. 
 
 " To those who wish to carry out the Normal System of teaching I can recommend no better work." 
 
 HENRY B. PIERCE, Prof, of Math., 
 State Normal School, Trenton, AT. J. 
 
 "I have no hesitation in saying that Prof. Brooks's Series of Arithmetics is the best now in u-e. 
 They are more progressive than any 1 have seen. I have had better success in teaching Arithmetic 
 with Brooks's Analysis than any other. My conclusions are drawn from a thorough examination and 
 careful comparison of his books with others." J. MORROW, Teacher Peebles District, 
 
 Pittsburg, Pa. 
 
 PUBLISHED BY 
 
 SOWER, POTTS 8x (DO. 
 530 Market St. and 523 Minor St., Philada.