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^fll!i«r«'s lirst firrt. 
 
 THE * 
 
 FIEST BOOK 
 
 OP 
 
 ARITHMETIC. 
 
 BY 
 
 DANA P. COLBURN, 
 
 PRINCIPAL OF THE EHODS ISLAND STATE NORMAL SCHOOIi, AITD 
 AUTHOR OF "ARITHMETIC AND ITS APPLICATIONS." 
 
 PHILADELPHIA: 
 } H. COWPERTHWAIT & CO. 
 ] BOSTON: SHEPARD, CLARK & BROWN. 
 1860. 
 
as 
 
 (>irj(f 
 
 Entered, according to Act of Congress, in the year 1856, by 
 
 DANA P. COLBURN, 
 
 in th« Clerk's Office of the District Court of the United States for the 
 
 District of Rhode Island. 
 
 STEREOTYPED BY J. rAGAIT. 
 
 (ii) 
 
PEEFACE. 
 
 The Docimal System of Numbers is one of the 
 most perfect things of man's invention. It is so 
 simple, that a child can understand it, yet so 
 comprehensive that it includes all possible num- 
 bers, represents them all by ten simple characters 
 and a point, and bases all numerical operations 
 on the combinations of the first ten, or primitive 
 numbers. 
 
 These primitive combinations can easily be 
 determined. 1 can be added to each number 
 from 1 to 10 ; so can each of the first ten num- 
 bers, and on these will depend all possible com- 
 binations in addition. For since 3 + 2 = 5, we 
 have 30 + 20 = 50, 300 + 200 = 500, &c., 23 + 2 
 == 25, 193 + 2 = 195, &c. 
 
 But subtraction is so closely connected with 
 addition, that, as far as the primitive numbers 
 are concerned, a knowledge of one implies a 
 knowledge of the other. What, for instance, are 
 2 + 3 = 5, 2 from 5 = 3, 3 from 5 = 2, 5 = 3 
 more than 2, &c., but different forms of expressing 
 the idea that 5 is made up of 2 and 3 ? 
 
 . __- 
 
PREFACE. 
 
 It is certainly more philosophical to present 
 these various forms in such connexion as to 
 show their mutual dependence, and thus secure 
 thoroughness from the outset, than to present 
 one form through a long series of lessons, and 
 then another form through another long series 
 of lessons, as though they have no connexion with 
 each other. The invariable experience of teachers 
 who have given both methods a fair trial is, that 
 elementary addition and subtraction can thus be 
 taught together, with greater ease than either can 
 be taught by itself. 
 
 All that has been said of addition and subtmc- 
 tion applies with equal force to multiplication and 
 division. Hence, the varied combinations of the 
 primitive numbers ought to be mastered by the 
 pupil before he attempts those of the derived 
 numbers; and when the latter are introduced, it 
 should be in such a way as to show their depen- 
 dence on the former. 
 
 Another reason for this careful elementary in- 
 struction is found in the fact that a child will rea- 
 dily understand and solve a problem involving 
 small numbers, when a similar one, involving 
 large numbers, will be entirely beyond his com- 
 prehension. There can be no doubt, then, that 
 his attention should be confined to small num- 
 bers, till his mathematical powers are so far 
 developed as to enable him to use large numbers 
 understandingly. 
 
r- 
 
 PR EFAGE. 
 
 The lessons of a First Book of Arithmetic 
 should be based on such principles. 
 
 They should be so arranged as to illustrate in 
 an easy and familiar manner the nature and uses 
 of numbers and of numerical operations, to call into 
 exercise and discipline the mental powers, form 
 accurate habits of thought and investigation, 
 impart a just self-reliance, develop a power of 
 following closely the most rigid reasoning pro- 
 cesses, and lay a sure foundation for future pro- 
 gress in mathematical studies. 
 
 They ought to be simple in their beginnings, 
 gradual in their developments, interesting in their 
 problems, varied in their exercises, and so con- 
 nected, that each shall follow naturally from those 
 that go before, and prepare the way for those that 
 come after. 
 
 Moreover, they should embody such a variety 
 and extent of exercises as to include all the essen- 
 tial principles of Arithmetic, and thus prepare the 
 way for any advanced treatise, and even give 
 those who have no further opportunity for study 
 in school^ such discipline as will enable them to 
 meet the demands of real life. 
 
 The author has endeavoured to prepare this 
 treatise in accordance with these views. In its 
 preparation he has drawn freely from the '' First 
 Steps in I^umbers," and the "Decimal System of 
 Numbers," both issued some years since, — the 
 the latter of which w^as written by himself, and 
 
PREFACE. 
 
 the former conjointly with Mr. George A. Wal- 
 ton, now of Lawrence, Mass. 
 
 Wlmtever may be its merits or defects, it is the 
 result of much careful thought and study, of con- 
 siderable experience as a Teacher, and of an 
 honest eftbrt to arrange such a course of lessons 
 as shall aid in developing the youthful mind, and 
 in forming correct habits of study. 
 
 DAN-A P. COLBUEN. 
 
 Providence, July, 1856. 
 

 COLBURN'S FIRST PART, 
 
 LESSOir I. 
 
 One boy. 1 boy. / ^tm. 
 
 One girl. 1 girl. / a^u. 
 
 One ball. 1 ball. / /a/f. 
 
 One. 1. /. I. 
 
 1. ITow many pictures are there on this page? 
 
 2. How many boys do you see in the picture ? 
 
 3. How many girls do you see in the picture ? 
 
 4. How many balls do you see in the picture ? 
 
 5. How many thumbs have you on your right hand ? 
 
 6. How many thumbs have you on your left hand ? 
 _ __ 
 
8 colburn's first part. 
 
 Note to the Teacher. — The pictures are not designed to take 
 the place of exercises -with visible objects, but rather as addi- 
 tional illustrations of the numbers introduced. With young 
 classes, the Teacher should give such easy lessons as the following, 
 making use of the most familiar objects as counters. 
 
 Even though the pupils have used numbers somewhat, such 
 lessons will make them better acquainted with their nature, and 
 will thus ensure a more rapid advancement. 
 
 Oral Lessox. — Teacher, taking a book, asks: "What have I 
 in my hand?" Ans. — *'A book." ** How many books?" Ans. — 
 '< One book." " How many pencils do I show you ?" Ans. — *' One 
 pencil." "How many chairs do I point at?" A?is. — "One 
 chair." " How many desks ?" Ans. — " One desk." 
 
 Tell the class to point to one boy, to one girl, to one windoV, 
 &c., &c. 
 
 The mark 1 (making it on the board) means one, as, 1 dog, 1 
 
 book. When writing, we make it thus : — ^ CiOG^j / t'OOri, 
 
 Note. — An oral lesson like the following may precede Lesson II. 
 
 Oral Lesson. — " How many pencils have I in my right hand?" 
 Ans. — " One pencil." " How many in my left hand?" Ans. — 
 "One pencil." " How many in both ?" Ans. — " Two pencils." 
 " How many pieces of chalk have I in my right hand?" Ans. — 
 "One." "In my left?" ^wj. — "One." "In both?" Ans.— 
 " Two." " How many fingers do I hold up ?" Ans. — " Two." 
 
 " One pen and one pen are, how many pens ?" " If I lay down 
 one pen, how many shall I have left ?" Lay down one pen, and 
 show the remaining one. " How many more must I get to have 
 two ?" Taking two pens in the right hand and one in the left, 
 ask : " How many pens have I in my right hand ?" " How many 
 in my left hand?" " How many more in my right hand than in 
 my left?" "How many less in my left hand than in my right 
 hand?" "If I should pass one from my right hand to my left 
 hand, how many would there be in my right hand?" "How 
 many would there be in my left?" 
 
 Ask such questions as these, illustrating each by familiar 
 objects, and continuing the exercise, till the numbers two, and 
 one, and their relations to each other, are perfectly understood. 
 
LESSON SECOND. 
 
 LESSON II. 
 
 Two boys. 2 boys. S ^ayd 
 
 Two soap-bubbles. 2 soap-bubbles. S i^ooA.'^OMed. 
 
 Two. 2. S. II. 
 
 The mark 2, or 2 j is called i]iQ figure two. 
 
 How many boys do you see in the picture ? 
 How many soap-bubbles do you see in the picture ? 
 
 A. To THE Teacher. — The following questions should first 
 be asked by substituting concrete in place of the abstract num- 
 bers. Thus : " How many apples are 1 apple and 1 apple ? 1 
 pear and how many pears are two pears?" &c. The pupils 
 should be taught to make such changes for themselves. The work 
 is not, however, mastered till the abstract numbers and operations 
 are mastered. 
 
 1. How many are 1 and 1 ? 
 
 2. 1 and how many are 2 ? 
 
 3. 1 from 2 leaves how many ? 
 
 4. How many must be taken from 2 to leave 1 ? 
 
 5. How many more are 2 than 1 ? 
 
10 COLB urn's first PART. 
 
 B. 1. George blew 1 soap-bubble, and Joseph 
 blew 1. How many did both blow? 
 
 2. There were 2 soap-bubbles in the air, but 1 of 
 them burst. How many remained ? 
 
 3. Sarah has 2 dolls, and Mary has 1. How 
 many more has Sarah than Mary? 
 
 4. A Story about James. — James was a little 
 boy who lived in the country, and studied the First 
 Book of Arithmetic. On his way to school one day, 
 he found 2 apples. At recess, he gave 1 of them to 
 his Teacher, and ate 1 ; but just before recess was 
 over, he received a present of 1 from a schoolmate. 
 After school, he found 1 under a tree, and gave 1 
 to a little boy whom he met. When he reached 
 home, he roasted all he had left. How many did he 
 roast ? 
 
 LESSON III. 
 
 Three rabbits. 3 rabbits. 3 ^a^'^cl 
 
 Three. 3. J. III. 
 
 The mark 3, or 3 ^ is called the figure three. 
 
LESSON THIRD. 11 
 
 To THE Teacher. — Oral lessons, like those in Lessons I. and II., 
 should be continued in this and the subsequent lessons. They 
 will, better than any lesson from the book, and better than any 
 mere description, lead the pupil to comprehend the nature of 
 numbers, and numerical operations. 
 
 A. 1. How many are 2 and 1 ? 
 
 2. How many are 1 and 2? 
 
 3. How many are 1 and 1 and 1 ? 
 
 4. 2 and how many are 3 ? 
 
 5. 1 and how many are 3 ? 
 
 6. 2 from 3 leaves how many? 
 
 7. 1 from 3 leaves how many? 
 
 8. How many more are 3 than 2 ? 
 
 9. How many more are 3 than 1? 
 
 B. 1. Edward had 2 tame rabbits, and his cou- 
 sin gave him 1 more. How many had he then ? 
 
 Solution. — If Edward had 2 tame rabbits, and his cousin gave him 
 1 more, he would then have 2 rabbits and 1 rabbit, which are 3 
 rabbits. 
 
 Solution 2d. — The 2 rabbits which he had, and the 1 rabbit which 
 his cousin gave him, would make 2 rabbits and 1 rabbit, which are 3 
 rabbits. 
 
 Note. — Such reasoning processes as the foregoing are of great 
 value ; for they teach children how to trace the connection be- 
 tween the problems and the numerical operations, and thus how 
 to reason; and they prepare the way for the solution of more 
 complicated problems. A little attention to them now, will save 
 much labor both to teacher and pupil in the higher departments 
 of Arithmetic. 
 
 2. A cross dog afterwards killed 2 of Edward's 
 rabbits. How many had he left ? 
 
 3. Emma had 3 rabbits, 1 of them was black, and 
 the rest were white. How many were white ? 
 
12 colburn's first part. 
 
 4. A Story about Carrie. — Carrie was a bright- 
 eyed little girl who lived in a village. One day she 
 cut out 2 paper dolls, and the next day she cut out 
 1 more. She then gave 1 to her playmate, Martha, 
 who came to see her, and 1 to Maria. She after- 
 wards cut out 2 more, but through carelessness, let 
 1 fall into the fire, when her mother cut out 1 very 
 nice one, and gave it to her. How many had she 
 then ? 
 
 To THE Teacher. — Make additional problems, and encourage 
 the pupils to do it for themselves, arranging them somewhat in 
 the form of stories, to increase their interest. One problem pro- 
 posed by a pupil, and solved by a class, will be of more value in 
 an educational view than many proposed by a teacher or author. 
 
 LESSON IV. 
 
 Four reapers. 4 reapers. A lea^ieid. 
 
 Four. 4. A. IV. 
 
 The mark 4, or Aj is called t\\(i figure four. 
 
 Note. — It should be made a part of the lesson for the pupil to 
 write out the exercises in abstract numbers. He will thus learn 
 
LESSON FOURTH. 13 
 
 to use figures and mathematical signs, and to write out arithme- 
 tical work neatly and correctly. 
 
 Explanation. — A cross made thus, -\-, is sometimes used in place 
 of "and" in such questions as, how many are 1 and 2? In like man- 
 ner, "2 -|- 2 are 4" mean the same as " 2 and 2 are 4." This sign is 
 also called ;;?m«, and sometimes the sign of addition. 
 
 A. 1. 3+1? 3. 2+2? 5. 1+1+2? 
 2. 1 + 3? 4. 1 + 2 + 1? 6. 2+1 + 1? 
 
 Explanation. — Read and perform the following questions, and 
 similar ones throughout the book, as though the words " how many" 
 were put in the place of the star. Thus, the question "2 + * = 4?'' 
 means the same as "2 and how many are 4?" 
 
 B. 1. 2 + *are4? 
 
 2. l + *are4? 
 
 3. 3 + -are4? 
 
 C. 1. 1 from 4? 
 2. 2 from 4 ? 
 
 D. How many more are — 
 L 4 than 3? 
 2. 4 than 1? 4. 3 than 1? 
 
 E. 1. George has 2 apples, and Rufus has 1. 
 How many have both ? How many more has George 
 than Rufus ? 
 
 2. Edward had 2 marbles, and his father gave 
 him a cent, with which he bought 2 more. How 
 many had he then ? 
 
 3. He afterwards lost 1, and gave away 2. How 
 many had he left ? 
 
 4. Jane had 1 picture-book, and on her birth-day 
 her father gave her 1 more ; her mother gave her 1, 
 and her uncle Henry sent her 1, which was very 
 pretty. How many had she then ? 
 
 4. 
 5. 
 6. 
 3. 
 4. 
 
 l + l + *are4? 
 l + 2 + *are4? 
 2+1 + * are 4? 
 3 from 4 ? 
 1 from 3 ? 
 
 3. 
 
 4 than 2? 
 
14 
 
 COLBURN S FIRST PART. 
 
 5. There were 3 robins on a cherry-tree, but 1 of 
 them flew away, and 2 others came to the tree. A 
 naughty boy throw a stone to knock down some 
 cherries, which so frightened the robins that 4 of 
 them flew away. How many robins were left on the 
 tree? 
 
 LESSON V. 
 
 Five toy-horses. 5 toy-horses. 5 ^^-no^^ed. 
 
 Five. 5. 5. V. 
 
 The mark 5, or 6 j is called the figure five. 
 
 A. 1. 4 + 1? 4. 2 + 3? 7. 2 + 1 + 2? 
 
 2. 1+4? 5. 3 + 1 + 1? 8. 1 + 1 + 3? 
 
 3. 3 + 2? 6. 1 + 2 + 2? 9. 2 + 2 + 1? 
 
 B. ExPLAXATiON. — Two parallel lines drawn thus, =, form what 
 is called the sign of eqxuility, which is often used in place of " are" in 
 such cases as "2+2 are 4," which would then be written " 2 + 2 = 4." 
 This may be read "2 and 2 are 4/' or "2 plus 2 are 4/* or "2 plus 2 
 equal 4." 
 
 1. 2+*=5? 4. 
 
 2. l + *=5? 6. 
 
 3. 2 + *=5? 6. 
 
 2+l+*=5? 
 2+2+*=5? 
 
LESSONFIFTH. 15 
 
 c. 
 
 1. 2 from 5? 
 
 3. 
 
 1 from 5 ? 
 
 
 2. 4 from 5 ? 
 
 4. 
 
 3 from 5 ? 
 
 D. 
 
 How many more 
 
 are — 
 
 - 
 
 
 1. 5 than 2? 
 
 3. 
 
 5 than 3 ? 
 
 
 2. 5 than 4? 
 
 4. 
 
 5 than 1 ? 
 
 E. 1. Edwin had 2 cents, but he afterwards found 
 3, and spent 4. How many had he left ? 
 
 2. Arthur had a half-dime, which, as you know, 
 is worth just 5 cents. He went to a store and bought 
 some nuts for 2 cents, and some candy for 1 cent, 
 giving in payment his half- dime. How many cents 
 ought he to receive back ? 
 
 3. Mr. French had 3 black horses, 2 white horses, 
 and 1 grey horse. He sold his grey horse, and 1 
 of his black ones. How many had he left ? 
 
 4. Near a village lived a poor w^oman named Lucy, 
 but everybody called her Aunt Lucy. In the sum- 
 mer she would pick blackberries to sell. One day 
 she picked 3 quarts in one pasture, and 1 in another, 
 when, meeting a gentleman from the village, she sold 
 him 2 quarts. She picked 3 quarts more, and started 
 to go home. On her way, she sold 2 quarts to one 
 man, and 1 quart to another. How many had she 
 left ? 
 
16 
 
 colburn's first part. 
 
 
 LESSON VI 
 
 
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 ^ 1 
 
 L 
 
 
 ^^ - ^!^?^^^^ 
 
 ^ 
 
 t. 
 
 
 r^^mMky^y^WwoBjK 
 
 
 j^^^i 
 
 
 jr^^ffiM^^Pg|jB|«fflB|B 
 
 
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 Six birds. 6 birds. 
 
 ^ vek(/<i- 
 
 
 Six. 6. 6. VL 
 
 
 
 The mark 6, or ^^ is called the figure six. 
 
 A. 
 
 1. 5 + 1? 4. 2+4? 6. 2 + 2 + 2? 
 
 
 2. 1 + 5? 5. 3 + 3? 7. 1 + 3 + 2? 
 
 
 3. 4 + 2? 
 
 8. 1 + 2+2? 
 
 B. 
 
 1. 2 + *=G? 4. 
 
 l+l+l+*=6? 
 
 
 2. 3 + *=6? 5. 
 
 l+2+*=6? 
 
 
 3. 4 + *=6? 6. 
 
 l+l + 2 + *6? 
 
 C. 
 
 1. 4 from 6 ? 4. 
 
 3 from 6 ? 
 
 
 2. 1 from 6? 6. 
 
 5 from 6 ? 
 
 
 3. 2 from 6? 6. 
 
 6 from 6 ? 
 
 D. 
 
 Explanation. — 6 less 2 means 6 diminished by 2, or made 
 
 smalle 
 
 r by 2, which is just the same 
 
 as " 2 from 6." Hence 6 less 
 
 2=4; 
 
 5 less 3 = 2, <fcc. 
 
 
 
 1. 6 less 2? 3. 6 less 3 ? 5. 4 less 2 ? 
 
 
 2. 5 less 3? 4. 5 less 1 ? 6. 6 less 4? 
 
 E. 
 
 1. There were 6 boys at play ; 2 of them were 
 
LESSON SEVENTH. 
 
 17 
 
 flying their kites, and the rest were rolling their 
 hoops. How many were rolling their hoops? 
 
 2. A pedler had 6 plaster birds on a tray. 2 of 
 them were painted yellow, 1 of them was painted red 
 and brown, and the rest were painted red and black. 
 How many were painted red and black ? 
 
 3. A hunter shot 1 partridge, 3 quails, and 2 
 pigeons: How many birds did he shoot in all ? 
 
 4. Julia picked 3 white roses, and 3 red ones. 
 How many did she pick in all ? She gave 2 red 
 roses and 1 white rose to her teacher, and 1 white 
 rose to her friend Lydia. How many had she ? 
 
 LESSON VII. 
 
 Seven hens. 7 hens. ^ nend* 
 Seven. 7. /. VII. 
 The mark 7, or 7^is called the fiijfure seven. 
 1. 6+1? 4. 2 + 5? 7. 3+2+2? 
 
 2. 
 3. 
 
 1 + 6? 
 5+2? 
 
 5. 
 6. 
 
 2* 
 
 3+4? 
 
 4 + 3? 
 
 9. 
 
 1 + 1 + 5? 
 
 2+1+4? 
 
18 
 
 COLBURN S FIRST PART. 
 
 B. 1. 
 
 2. 
 3. 
 
 2+*=7? 
 2+3+*=7? 
 
 C. 1. 
 
 2. 
 
 6 from 7 ? 3. 
 4 from 7 ? 4. 
 
 4. 2 + l + *-7? 
 
 5. l + 2 + l + *=7? 
 
 6. 2 + 2 + 2+*=7? 
 
 2 from 7? 5. 1 from 7? 
 
 3 from 7? 6. 5 from 7 ? 
 
 D. 1. 7 less 3 ? 3. 7 less 2 ? 5. 7 less 5? 
 2. 7 less 1? 4. 7 less 6? 6. 7 less 4? 
 
 E. 1. A farmer had seven grej liens. He sold 2 
 of them, and a fox killed 1 of them. How many did 
 he hnve left ? If he should afterwards sell 2 more, 
 and huy 4 small white hens, how many would he 
 then have? 
 
 2. Alfred had a half-dime and 4 €ents ; hut he 
 exchanged tlie half dime fur its value in cents. How 
 many cents did he then have ? He was so unfortu- 
 nate as to lose 3 of his cents, and he gave 3 more 
 for a three-cent piece. How many cents had he then 
 left? 
 
 3. 6 boys were at play together ; 1 of them got 
 hurt, and went home, and 2 were called a\^ay by 
 their friends ; but very soon 4 more boys cm me out 
 to play. These played together till all but 3 got 
 tired, and sat down to rest. How many sat down 
 to rest ? 
 
LESSON EIGHTH. 
 
 19 
 
 LESSON VIII. 
 
 Eight persons. 8 persons. 8 ^ZdontX. 
 
 Eight. 8. S. VIII. , 
 
 The mark 8, or 8 j is called \he figure eight. 
 
 A. 1. 7 + 1? 
 
 2. 1+7? 
 
 3. 6+2? 
 
 B. 1. 2+*=8? 
 
 2. 4+*=8? 
 
 3. 8+*=8? 
 
 C. 1. 7 from 8? 
 
 2. 2 from 8 ? 
 
 3. 4 from 8 ? 
 
 4. 2 + 6? 
 
 5. 5 + 3? 
 
 6. 3 + 5? 
 
 7. 4+4? 
 
 8. 2 + 2+2 + 2? 
 
 4. 3+2+*=8? 
 
 5. l + l+2+l + *==8? 
 
 6. • l + 2 + 3 + *=8? 
 
 4. (5 from 8? 
 
 5. 3 from 8 ? 
 
 6. 5 from 8 ? 
 
20 COLBURN*S FIRST PART. 
 
 D. llow many more are — 
 
 1. 8 than 5? ^ 8 than 3? 
 
 2. 8 than 1? 5. 8 than 2? 
 
 3. 8 than 6 ? 6. 8 than 4 ? 
 
 E. Explanation. — A single mark made like a dash, thus, — , is 
 often used in place of the -word " less." For instance : "8 — 3 = 6" 
 means the same as " 8 less 3 = 5." 
 
 1. 8—3? 3. 8—5? 6. 8—6? 
 
 2. 8—7? 4. 8—2? 6. 8—4? 
 
 F. 1. In a ferry-boat were 4 ferrymen, 2 ladies, 
 and 2 gentlemen. How many persons were in the 
 boat? 
 
 2. A farmer had 8 little pigs. He sold 2 to 
 one man, and 2 to another. How many had he left ? 
 
 3. Sarah's mother gave her 3 dresses for her doll, 
 her sister Susan gave her 2, and her aunt Mary gave 
 her enough to make up 8. How many did her aunt 
 Mary give her ? 
 
 4. Alfred found 3 chestnuts under one tree, 4 
 under another, and 1 under another. He soon after 
 ate 2, when, having the ill luck to fall, he lost 3. 
 He afterwards found 4 more, when, seeing a pretty 
 squirrel run into a hole in a tree, he put in 3 chest- 
 nuts for the squirrel to eat. How many chestnuts 
 had he left? 
 
LESSON NINTH. 
 
 21 
 
 LESSON IX. 
 
 ,il-s^'-,lir1:ri!«rtj 
 
 <j^/^-: 
 
 Nine ducklings. 9 ducklings, p cUi,cn/cna<i: 
 
 Nine, 9. p. IX. 
 
 The mark 9, or P ^ is called the figure nine. 
 
 A. 1. 
 
 2. 
 3. 
 4. 
 
 B. 1. 
 
 2. 
 3. 
 4. 
 
 C. 1. 
 
 2. 
 
 3. 
 
 8 + 1? 
 
 1 + 8? 
 
 7 + 2? 
 
 2 + 7? 
 
 5+*=9? 
 2+*=9? 
 4+*=9? 
 3+*=9? 
 
 8 from 9 ? 
 6 from 9 ? 
 
 3 from 9 ? 
 
 6+3? 
 
 3 + 6? 
 5+4? 
 
 4 + 5? 
 
 5. 
 6. 
 
 7. 
 8, 
 
 9. 
 10. 
 11. 
 12. 
 
 3+1+3? 
 2+2+5? 
 3+2+3? 
 1+2+4? 
 
 :9? 
 
 2 + 5 + 
 3+5+*=9? 
 3+3+*=9? 
 2+2+2+*=9? 
 
 4. 
 5. 
 6. 
 
 7 from 9 ? 
 4 from 9 ? 
 2 from 9 ? 
 
22 coLB urn's first part. 
 
 D. 1. 9—5? 4. 9—4—2? 
 
 2. (?— 7? 5. 9—2—3? 
 
 3. 9— 1>? 6. 9—5—2? 
 
 E. 1. If a boy should have 6 cents, and receive 
 a present of 2 more, how many would he have ? If 
 he should spend 3, and then have 4 given him, how 
 many would he have ? 
 
 2. Daniel had 3 baskets. The first was a red 
 one, which held 3 quarts ; the second was a blue one, 
 which held 4 quarts ; the third was a yellow one, 
 which held 2 quarts. How many quarts would all 
 hold? 
 
 3. One day he picked so many berries, that he 
 filled them all ; but he sold what there was in the 
 blue basket. How many quarts had he left ? 
 
 4. He emptied the contents of his red basket into 
 his blue basket. How many more quarts would it 
 take to fill it? 
 
 5. David had 3 marbles, and Austin had 4; but 
 David found 4, and Austin lost 2. Thej then agreed 
 to put what they had into a littie box. How many 
 marbles did they put into the box ? 
 
LESSON TENTH. 
 
 23 
 
 LESSON X. 
 
 Ten herrings. 10 herrings. ^0 nezuna^. 
 
 Ten. 10. ^0. X. ■ 
 
 The mark is called ihejlgure naught, or zero. 
 
 A. 1. 9 + 1? 
 
 2. 1+9? 
 
 3. 8 + 2? 
 
 4. 2 + 8? 
 
 5. 7 + 3? 
 3 + 7? 
 6+4? 
 
 6. 
 
 7. 
 
 1. 2+*=10? 
 
 2. 3+*==10? 
 
 3. 5 + *=10? 
 
 4. 4+*=10? 
 
 8. 4+6? 
 
 9. 5+5? 
 
 10. 4 + 3 + 2? 
 
 11. 2 + 3 + 5? 
 
 12. 1+2 + 2 + 2 + 2? 
 
 13. 2+2+2+2+2? 
 
 6. l+4+2+*=10? 
 
 6. 3+l + 2+*=10? 
 
 7. 2 + 4+3 + *=10? 
 
 8. 4 + l + 2 + *=10? 
 
24 colburn's first part. 
 
 (J. How many more are — 
 
 1. 10 than 6? 4. 7 than 4? 
 
 2. 8 than 3? 5. 10 than 7? 
 
 3. 10 than 5 ? 6. 10 than 8 ? 
 
 D. 1. 10—8? 5. 10—5+3? 
 
 2. 10—7? 6. 10—4—3 + 6? 
 
 3. 10—6? 7. 10—3 + 2—4? 
 6. 10—3? 8. 10—4—4+5? 
 
 E. 1. Ahunter shot 3Hrds from oneflock, 2fiom 
 another, and 5 from another. How many did he 
 shoot in all ? 
 
 2. Anna says she has 10 picture-books, of which 
 her mother gave her 3, her teacher gave her 1, her 
 aunt gave her 1, her uncle gave her 1, and her fa- 
 ther gave her the rest. How many did her father 
 give her ? 
 
 3. Edward had a half-dime, a three-cent piece, 
 and 2 cents. How many cents were they worth? 
 He bought an apple for 2 cents, an orange for 3 
 cents, some candy for 1 cent, and some raisins for 3 
 cents. How many cents had he left ? 
 
 4. Albert and Timothy went a-fishing one day. 
 Albert caught 3 perch, 2 pickerel, and 4 trout. 
 Timothy caught 2 perch, 3 pickerel, 3 trout, and 2 
 eels. How many more fish did Timothy catch than 
 Albert ? Albert gave his 4 trout in exchange for 
 Timothy's 2 perch and 3 pickerel. How many fish 
 had each boy then ? 
 
LESSON ELEVENTH. 
 
 25 
 
 LESSON XI. 
 
 Eleven arrows. 11 arrows. 
 Eleven. 11. //. XI. 
 
 /'/ aao-iefj^. 
 
 A. 
 
 1. 
 
 10 + 1? 
 
 8. 
 
 6+5? 
 
 
 2. 
 
 9+2? 
 
 9. 
 
 5+6? 
 
 
 3. 
 
 2+9? 
 
 10. 
 
 4+2+3? 
 
 
 4. 
 
 8 + 3? 
 
 11. 
 
 2+5+4? 
 
 
 5. 
 
 3+8? 
 
 12. 
 
 2+2+1+2+2+2? 
 
 
 6. 
 
 7+4? 
 
 13. 
 
 1+2+2+2+1+2? 
 
 
 7. 
 
 4+7? 
 
 14. 
 
 3+1+3+2+2? 
 
 B. 1. 5+*=ll? 
 
 2. 2 + *=ll? 
 
 3. 4 + *=ll? 
 
 C. 1. * from 11 = 5? 
 
 2. * from 11=6? 
 
 3. * from 11 = 7? 
 
 4. 2+2+5 + *=ll? 
 
 5. 3+4 + *=ll? 
 
 6. 3 + 3 + 3+*=ll? 
 
 4. 
 5. 
 6. 
 
 * from 11=4? 
 
 * from 11 = 8? 
 
 * from 11 = 3 ? 
 
26 COL burn's rmsT part. 
 
 D. 1. 11—4? 8 11—3? 
 
 2. 11—2? 9. 11—10? 
 
 3. 11—6? 10. 5+4 + 2-6? 
 
 4. 11—8? 11. 1 + 2+4 + 4—3? 
 
 5. 11—5? 12. 2 + 5 + 3—8? 
 
 6. 11—9? 13. 3 + 1 + 5—3? 
 
 7. 11—7? 14. 7 + 4—3—3? 
 
 E. 1. A person was shooting arrows at a target, 
 and I observed that when he had shot 3 arrows, and 
 placed another in his bow, there were 7 lying on the 
 ground. How many were there in all ? 
 
 2. William owned 3 arrows, George owned 2, and 
 Rufus 5. How many did they all own ? One after- 
 noon, as they were playing with their bows and 
 arrows, William lost 1 arrow, Rufus lost 1 and broke 
 1, and George found a very nice one, which some boy 
 had lost. How many arrows had the boys then ? 
 
 3. One beautiful afternoon in June, Emma and 
 Hannah went out to gather wild flow^ers, and make 
 boquets. Emma made 4, and Hannah made 5, when 
 they put the rest of their flowers together, and made 
 2 very pretty boquets. They put them all in a bas- 
 ket, and went home. They gave 3 to Hannah's 
 mother, and 1 to her sister ; and they gave 3 to 
 Emma's aunt, and 2 to her teacher ; after which, 
 Emma took 1, and Hannah took the rest. How 
 many did Hannah take? 
 
LESSON TWELFTH. 
 
 27 
 
 LESSON ZII. 
 
 Twelve eggs. 12 eggs. /J* 
 Twelve. 12. /J*. XII. 
 
 1. 
 
 2. 
 3. 
 4. 
 
 5. 
 6. 
 
 7. 
 
 10 + 2? 
 
 2 + 10? 
 9+3? 
 
 3 + 9? 
 
 8+4? 
 
 4 + 8? 
 7+5? 
 
 B. 1. 4+*=12? 
 
 2. 5 + *=12? 
 
 3. 3 + *=12? 
 
 0. 1. 6 from 12 ? 
 
 2. 9 from 12? 
 
 3. 2 from 12? 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 14. 
 
 5+7? 
 6 + 6? 
 
 3+3+3+3? 
 3+2+3+2? 
 
 1+4+3+3? 
 2+5+2+3? 
 4+2+2+4? 
 
 4. 3+4+3 + *==12? 
 
 5. 2 + l + 4 + * + 12? 
 
 6. 2 + l+2+*=12? 
 
 4. 8 from 12? 
 
 5. 10 from 12? 
 
 6. 7 from 12 ? 
 
28 colburn's first part. 
 
 D. 1. 12—4? 6. 12—9? 
 
 2. 12—3? 7. 12—3—4? 
 
 3. 12—7? 8. 2 + 7 + 3—6? 
 
 4. 12—8? 9. 12—5—2? 
 
 5. 12—10? 10. 1 + 8 + 2—4? 
 
 E. 1. Alfred found a hen's nest, with a large 
 number of eggs in it. He took out 3, and then took 
 out 4 more, when he found that there were 5 left in 
 the nest. How many were there in the nest at first ? 
 When he was putting the eggs back, he carelessly 
 broke 2. How many were then left ? 
 
 2. Alice had 11 little chickens, but 2 of them 
 died, 2 of them got lost, and a rat killed 3. How 
 many then remained ? 
 
 3. Benjamin earned a half-dime, and found a 
 three-cent piece and 4 cents. He bought a top for 
 8 cents, after which he received a present of 8 cents. 
 How much money had he then ? 
 
 4. Annie had 7 pictures, and Emma had 5. Annie 
 gave away 3 pictures, and Emma received a present 
 of 2 ; after which Emma lost 3, and Annie found 1. 
 How many pictures had each girl then ? How many 
 had both ? 
 

 
 LESSON 
 
 IHIRTEENTH. 29 
 
 
 
 LESSON XIII 
 
 
 « 
 
 /f"^*^^/^^^^ 
 ^W^ '^^^#*I 
 
 
 ^ 
 
 
 g 
 
 
 
 ||KMyfev^ 
 
 
 1 
 
 
 
 11^^ 
 
 
 f 
 
 Chirteen sheep. 
 
 18 sheep. /J* ^/^^. 
 
 
 Thirteen. 13. 
 
 /3. 
 
 XIII. 
 
 A. 
 
 1. 
 
 10 + 3? 
 
 7. 
 
 7+6? 
 
 
 2. 
 
 3 + 10? 
 
 8. 
 
 6+7? 
 
 
 3. 
 
 9+4? 
 
 9. 
 
 2+3+4+3? 
 
 
 4. 
 
 4+9? 
 
 10. 
 
 1+4+2+6? 
 
 
 5. 
 
 8 + 5? 
 
 11. 
 
 4+3+3+4 
 
 
 6. 
 
 5 + 8? 
 
 12. 
 
 •1 + 5 + 3+4? 
 
 B. 
 
 1. 
 
 3+*=13? 
 
 4. 
 
 3 + 4 + *=13? 
 
 
 2. 
 
 5 + *=13? 
 
 5. 
 
 4 + 2 + 4+*=13? 
 
 
 3. 
 
 6 + *=13? 
 
 6. 
 
 2 + 3+4 + *=13? 
 
 C. 
 
 1. 
 
 9 from 13 ? 
 
 4. 
 
 6 from 13? 
 
 
 2. 
 
 7 from 13 ? 
 
 6. 
 
 5 from 13 ? 
 
 
 3. 
 
 4 from 13 ? 
 
 6. 
 
 8 from 13 ? 
 
 
 3* 
 
 
 
 
H: 
 
 80 colburn's first part. 
 
 D. 1. 13—*=7? 4. 13— *=9? 
 
 2. 13— *=4? 5. leS— *=6? 
 
 3. 13— *=8? 6. 13— *=5? 
 
 E. 1. Mr. Green owns 13 sheep, and Mr. Allen 
 owns 7. How many more does Mr. Green own than 
 Mr. Allen ? If Mr. Green should sell 4 sheep to 
 Mr. Allen, how many would each have ? 
 
 2. A farmer had 3 sheep in one pasture, 5 in 
 another, and 4 in another ; but at night he drove 
 them all into one pen. How many were there in 
 the pen ? The next day he drove 6 of them into 
 one pasture, 2 into another, and the rest into an- 
 other. How many did he drive into the last pasture ? 
 
 3. A little boy had 13 marbles. He lost 4, and 
 gave away 3, when, finding it was school-time, he 
 put the rest into a box. When he came from 
 school, he found his little brother Erastus had been 
 playing with the box, and had lost 3 of the mar- 
 bles. How many were left in the box ? His father 
 afterwards gave him 2 cents, with which he pur- 
 chased 5 marbles. How many marbles had he 
 then ? 
 

 
 LESSON 
 
 FOURTEENTH. 31 
 
 
 
 LESSON 
 
 XIV. 
 
 
 i 
 
 
 
 
 I 
 
 ^^t^^saw^^^^^^^C^ 
 
 
 ^^p^^s 
 
 
 
 ^^^^^m 
 
 
 ^j^^^K 
 
 
 
 
 
 S^ 
 
 
 Fourteen barrels 
 
 3. 14 barrels. /^ ^atte/d. 
 
 
 Fourteen. 14. 
 
 /A. 
 
 XIV. 
 
 A. 
 
 1. 
 
 10+4? 
 
 6. 
 
 6 + 8? 
 
 
 2. 
 
 4 + 10? 
 
 7. 
 
 7 + 7? 
 
 
 3. 
 
 9 + 5? 
 
 8. 
 
 2+1+7+4? 
 
 
 4. 
 
 5+9? 
 
 9. 
 
 4+2+8? 
 
 
 5. 
 
 8r6? 
 
 10. 
 
 3+4+2+4? 
 
 B. 
 
 1. 
 
 7 + *=14? 
 
 4. 
 
 3+4 + *=14? 
 
 
 2. 
 
 5 + *=14? 
 
 6. 
 
 4+6 + *=14? 
 
 
 3. 
 
 6 + *=14? 
 
 6. 
 
 2 + 4+3 + *^14? 
 
 C, 
 
 1. 
 
 9 from 14 ? 
 
 4. 
 
 8 from 14? 
 
 
 2. 
 
 6 from 14 ? 
 
 5. 
 
 7 from 14? ,,,- 
 
 i 
 
 3. 
 
 4 from 14 ? 
 
 6. 
 
 4 from 14?' '^ 
 
32 colburn's first part. 
 
 D. 1. 14—4—3? 4. 4 + 5 + 5—6 + 3? 
 
 2. 14—6 + 3? 5. 8 + 2 + 3—5—3? 
 
 3. 14—7+5? 6. 3 + 2+4 + 5—6—3? 
 
 E. 1. A teamster had a load of 14 barrels. He 
 unloaded 6 at the store of Shaw & Co., 3 at a railroad 
 depot, and the rest at the store of Saunders k 
 Brown. How many did he unload at the last place ? 
 Not long after, he had a load of 13 barrels, and he 
 unloaded 4 of them at one place, and 3 at another, 
 after which he took on his truck 8 barrels more. 
 How many had he then on his load ? 
 
 2. Susan has 6 books without pictures, and 7 books 
 with pictures. How many books has she ? 
 
 3. Austin has 14 books. Waldo had 6. His mo- 
 ther gave him 3, and his father gave him enough to 
 make as many as Austin. How many did his father 
 give him ? 
 
 4. One day, Henry went out to look for chest- 
 nuts. He found 6 under one tree, 3 under another, 
 and 5 under another. After eating 8 of them, and 
 finding 6 more, he went home, carrying his chest- 
 nuts with him. He gave 3 of them to his father, 3 to 
 his mother, 4 to his little sister Lucy, and the rest to 
 his brother Francis. How many did he give to 
 Francis ? 
 
LESSON FIFTEENTH. 
 
 33 
 
 LESSON XV. 
 
 Fifteen apples. 15 apples. ^5 a/?A^a 
 Fifteen. 15. <f5. XV. 
 
 A. 1. 10 + 5? 
 2. 5+10? 
 
 B. 1. 8 from 15 ? 
 
 2. 6 from 15 ? 
 
 3. 10 from 15? 
 
 C. 1. 15-7 + 3? 
 
 2. 15-6+4? 
 
 3. 1,5-9-5? 
 
 D. 1. 15-*=7? 
 
 2. 15-*=6? 
 
 3. 15-*=-9? 
 
 3. 9 + 6? 
 
 4. 6+9? 
 
 5. 8+7? 
 
 6. 7 + 8? 
 
 4. 7 from 15? 
 
 5. 5 from 15? 
 
 6. 9 from 15? 
 
 4. 8 + 2+5-7? 
 
 5. 4+3+8-6? 
 
 6. 2 + 7 + 5-8 + 3? 
 
 4. 15-*=5? 
 
 5. 15-*==8? 
 
 6. 14-*=10? 
 
 C 
 
34 colburn's fibst part. 
 
 E. 1. Mary found 3 apples under one tree, 4 apples 
 under another, and 8 under another. How many 
 did she find in all ? As she was bringing them to 
 the house, she stopped to play with her kitten, and 
 accidentally dropped most of them, as you see in the 
 picture. On picking them up, she found that 6 of 
 them were bruised a little, and 1 of them, which the 
 kitten played with, was bruised very badly. The rest 
 were not bruised at all. How many were not bruised 
 at all? 
 
 2. Julia made 4 squares of blue patch-work, 3 
 squares of brown, and 8 squares of red. How many 
 did she make in all ? 
 
 3. Hattie hemmed 15 handkerchiefs, 5 of them 
 were for her sister Lydia, 2 were for her brother 
 Cyrus, 3 for her father, and the rest for her mother. 
 How many did she hem for her mother? 
 
 4. Augusta received 4 merit-marks on Monday, 3 
 on Tuesday, 1 on Wednesday, 2 on Thursday, and 5 
 on Friday, and on Saturday school was not in ses- 
 sion. How many merit-marks did she receive through 
 the week? How many more than Emeline, who 
 obtained but 9 during the week ? 
 
LESSON SIXTEENTH. 
 
 35 
 
 LESSON XVI. 
 
 Sixteen tents. 16 tents. ^6 lent^. 
 Sixteen. 16. ^6. XVI. 
 
 A. 1. 10 + 6? 
 
 2. 6 + 10? 
 
 3. 9 + 7? 
 
 B. 1. 8+*=16? 
 
 2. 6 + *=16? 
 
 3. 2 + 5 + *=16? 
 
 C. 1. 16-6? 
 
 2. 16-9? 
 
 3. 16-7? 
 
 4. 7+9? 
 
 4. 3+4+2 + *=16? 
 
 5. 4+2+4 + *=16? 
 6 l+3+2+*=16? 
 
 4 IG— *=-7? 
 
 5 16-*=9? 
 
 6. 16— *=8'^ 
 
30 colburn's first part. 
 
 D. 1. 14=4+*? 6. 16=104-*? 
 
 2. 16 = 6f*? 7. 13=10+*? 
 
 3. 13=3+*? 8. 14=10 + *? 
 
 4. 12=2 + *? 9. 12=10 + *? 
 
 5. 11=1 + *? 10. 11=10 + *? 
 
 E. 1. 4 + 2? 6. 3 + 13? 
 
 2. 14 + 2? 7. 1+4? 
 
 3. 4 + 12? 8. 11+4? 
 
 4. 3 + 3? 9. 1 + 14? 
 
 5. 13 + 3? 
 
 F. 1. On a certain muster-field, tliere were 8 
 tents in one row, and 8 in another. How many were 
 there in both rows ? 
 
 2. Albert was asked how many chestnuts he had, 
 to which he replied, " If I should give my father 3, 
 my mother 4, my little sister Anna 4, and my bro- 
 ther George 3, T should have but 2 left." How many 
 chestnuts had he ? 
 
 3. Lucy read 6 pages of history in the morning, 
 and 10 in the afternoon, but when questioned about 
 it, she found that she had forgotten all but 4 pages. 
 How many had she forgotten ? 
 
 4. I had 3 dollars, and received 6 dollars of one 
 man, 3 of another, and 2 of another, when I paid 
 away 8 dollars, after which I received 4 dollars. 
 How many had I then ? 
 
LESSON SEVENTEENTH, 
 
 37 
 
 LESSON XYII. 
 
 Seventeen birds. 17 birds. ^7 ^irc/d-. 
 Seventeen. 17. //. XVII. 
 
 A. 1. 10+7? 
 
 2. 7+10? 
 
 3. 8+9? 
 
 B. 1. 4+5+*=17? 
 
 2. 3+7 + *=17? 
 
 4. 9 + 8? 
 
 5. 3 + 5+9? 
 
 6. 7+2+8? 
 
 3. 4 + 3+*=17? 
 
 4. 4 + 4 + *=17? 
 
 C. 1. * from 17=8 ? 3. * from 17=7 ? 
 2. * from 17=10 ? 4. =f: from 17=9 ? 
 
 D. 1. 3 + 1 + 3? 
 
 2. 13+1 + 3? 
 
 3. 3 + 11 + 3? 
 
 4. 2 + 2 + 2? 
 
 5. 12 + 2 + 2? 
 
 6. 2 + 12 + 2? 
 
38 COLBURN*S FIRST PART. 
 
 E. 1. 4-f*-7? 7. 3 from 7? 
 
 2. 4 + *=17? 8. 3 from 17? 
 
 3. 14 + *=17? 9. 13 from 17? 
 
 4. 3 + *=6? 10. 1 from 6? 
 
 5. 3 + *-16? 11. 1 from 16? 
 
 6. 134->K=16? 12. 11 from 16? 
 
 F. 1. Jane gave 6 cents to a beggar-woman, Lucy 
 gave 3, Sarah gave 5, and Abbj gave 3. How many 
 (lid all give her ? 
 
 2. The beggar-woman spent 10 cents for bread, 
 after which Julia gave her 4 cents, Nancy gave her 
 3 cents, and Susan gave her 2 cents. How many 
 cents had she then ? 
 
 3. A gardener picked 8 roses from one bush, 7 
 from another, and 2 from another. How many did 
 he pick in all ? He put 4 of the roses in one boquet, 
 5 in another, and 3 in another, and the rest in 
 another. How many did he put in the last boquet ? 
 
 4. Samuel bought a quart of molasses for 10 
 cents, and then had 6 cents left. How many cents 
 had he at first ? 
 
 5. He made his molasses into candy, 9 sticks of 
 which he sold for 7 cents, and the remaining 8 sticks 
 he sold for 6 cents. How many sticks did he sell ? 
 How many cents did he receive for it ? How many 
 more cents did he receive for his candy than he paid 
 for his molasses ? 
 
LESSON E I G H T K E N T U . 
 
 39 
 
 LESSON XVIII. 
 
 Eighteen books. 18 books, /o M<m<S: 
 Eighteen. 18. /<?. XVIII. 
 
 A. 1. 10 + 8? 
 
 2. 8 + 9? 
 
 B. 1. 8 + *=18? 
 
 2. 9+*=18? 
 
 3. 10+*=18? 
 
 C. 1. * from 7=3? 
 
 2. * from 17=3? 
 
 3. * from 17=13? 
 
 3, 9+9? 
 
 4. 2 + 2 + *=14? 
 
 5. 12 + 2+*=14? 
 
 6. 2+12 + *=14? 
 
 4. * from 8=4? 
 
 5. * from 18=4 ? 
 
 6. * from 18 = 14? 
 
40 COL burn's first part. 
 
 D. 1. 8—2—2—2? 5. 9 + 6 + 3—4? 
 
 2. 18—2—2—2? 6. 9 + 3+4—2? 
 
 3. 18—12—2—2? 7. 10 + 3 + 4—3? 
 
 4. i7_3_3 + 4? 8. 18—4—4 + 3 + 2? 
 
 E. 1. One " Fourth of July," Robert's father 
 gave him a dime, his mother gave him a half-dime, 
 and his uncle gave him a three-cent piece ; but he 
 exchanged them all for their value in cents. How 
 many cents did he receive for them ? 
 
 2. He paid 8 cents for a bunch of crackers, and 
 4 cents for torpedoes, and the rest of his money to 
 see some animals, which were exhibited in a tent. 
 How many cents did he pay to see the animals ? 
 
 3. Mr. Gay owns a garden, a pasture, a wood-lot, 
 and an orchard. His garden contains 2 acres, his 
 pasture 6 acres, his wood-lot 4 acres, and his orchard 
 enough to make up 18 acres. How many acres does 
 his orchard contain ? If he should sell his wood-lot 
 and orchard, how many acres would he have left ? 
 
 4. I had 17 dollars this morning, but I have since 
 bought a hat for 4 dollars, and a pair of boots for 6 
 dollars. I have also received 11 dollars, which a 
 friend owed me, and paid a debt of 7 dollars. How 
 much money have I now ? 
 

 LESSON NINETEENTH. 41 
 
 
 LESSON XIX. 
 
 
 -z^^-r-^'T?^ 
 
 %^-^-T r-„ 
 
 
 ^^^^Hipp 
 
 Wk^m 
 
 
 ^^^R3 
 
 ^^p 
 
 Nineteen wild geese. 19 wild 
 
 geese. ^PwuUaffede. 
 
 Nineteen. 19. /^. XIX. 
 
 A. 
 
 1. 10 + 9? 5. 
 
 14 + 2 + *=19? 
 
 
 2. 9 + 10? 6. 
 
 4+2 + *=19? 
 
 
 3. 10 + *=19? 7. 
 
 19 3 3 ? 
 
 
 4. 9 + *-19? ■ 8. 
 
 19 3 13? 
 
 B. 
 
 Ho;y many more are — 
 
 
 
 1. 9 than 2 ? 4. 
 
 8 than 3 ? 
 
 
 2. 19 than 2? 5. 
 
 18 than 3 ? 
 
 
 3 19 than 12 ? 6. 
 
 18 than 13 ? 
 
 C. 
 
 1. 4 + 3+4+5? 6. 
 
 4 + 6 + *=19? 
 
 
 2. 8 + 3+*=19? 7. 
 
 9 + 3 + 5— *=8? 
 
 
 3. 5 + 7+4 8? 8. 
 
 10+4 + 4— *=11? 
 
 
 4. 8+6 + 6—7? 9. 
 
 3+3+3+3+3+3? 
 
 
 5. 6 + 4+4+4? 10. 
 
 1+3+3+3+3+3? 
 
42 colburn's first part. 
 
 D. 1. One day, Edward and Susan saw a flock of 
 wild geese, which contained just 19. Some sports- 
 men shot 4 of them, which so frightened the rest, 
 that 6 of them flew towards the east, and the re- 
 mainder towards the west. Another party of hunt- 
 ers seeing those which were flying towards the west, 
 shot 2 of them, when the rest flew towards the east, 
 and joined that part of the flock which had first 
 flown in that direction. How large a flock was there 
 then ? 
 
 2. Frank has money enough to buy a pencil for 
 3 cents, a pen for 6 cents, some ink for 4 cents, and 
 an inkstand for 5 cents. How much money has he ? 
 
 3. Frank's sister has money enough to buy a pen, 
 and inkstand like Frank's, and 5 cents worth of 
 paper. How many cents has she ? 
 
 4. Walter and Reuben had each 12 cents. But 
 Walter earned 5 cents by doing an errand, and 
 Reuben spent 5 cents for confectionery. How many 
 cents had each of the boys then ? How many more 
 had Walter than Reuben ? 
 
LKSSON TWENTIETH. 
 
 43 
 
 LESSON XX. 
 
 Twenty soldiers. 20 soldiers. SO Mu/ieu 
 Twenty. 20. 20. XX. 
 
 A. 1. 10+10? 
 2. 10 + *=20? 
 
 3. 20=* tens? 
 
 4. 20—10 ? 
 
 B. 1. 3+*=.10? 4. 1+*=:10? 
 
 2. 3 + *=20? 5. l-l-*=20? 
 
 3. 13+*=20? 6. ll+*=20? 
 
 C. 1. 3 + 5+7+4? 4. 2+1+4+9 + 4? 
 
 2. 2+9+4 + 6? 5. 4 + 9+2 + 2 + 2? 
 
 3. 6 + 7 + 3 + 5? 6. 2 + 4+4+4+4? 
 
44 colburn's first part. 
 
 7. 2 + 2 + 2 + 2 + 24-2 + 2 + 2 + 2 + 2? 
 
 8. 1 + 2 + 2 + 2 + 2 + 2 + 2 + 2+2 + 2? 
 
 9. 3 + 3 + 3-13 + 3 + 3? 
 
 10. 1 + 3 + 3 + 3 + 3 + 3 + 3? 
 
 11. 2+3 + 3 + 3 + 3 + 3 + 3? 
 
 12. 20—2—2—2—2—2—2—2—2 + 2 + 2 ? 
 
 13. 19-2—2-2—2—2—2-2—2-2 ? 
 
 14. 20—3—3—3—3—3—3 ? 
 
 15. 19—3—3—3—3—3—3 ? 
 
 16. 18—3—3—3—3—3—3 ? 
 
 D. 1. Little Willie had 20 toy-soldiers. Replaced 
 3 in front for officers, and then arranged the rest in 
 3 rows, placing 7 in the first row, 6 in the second, 
 and the rest in the third. How many did he place 
 in the third row? 
 
 2. Arthur was telling his mother about the boys 
 who went to his school. He said that 4 of them had 
 neither hoops nor kites, that 6 had kites only, that 
 7 had hoops only, and that 3, including himself, had 
 both hoops and kites, and this comprised all the 
 boys in the school. How many boys were there in 
 the school? 
 
 3. Isaac and Francis were playing ball with Au- 
 gustus and Reuben. Isaac batted the ball 11 times, 
 and Francis batted the ball 9 times. Augustus batted 
 it 10 times, and Reuben 7 times. How many times 
 did Isaac and Francis bat it ? How many times did 
 Augustus and Reuben bat it ? Isaac caught the ball 
 9 times, and Francis caught it 7 times. Augustus 
 caught it 9 times, and Reuben caught it 10 times. 
 
LESSON TWENTIETH. 45 
 
 How many more times did Augustus and Reuben 
 catch it than Isaac and Francis ? 
 
 4. Three idle boys, Thomas, Joseph, and Samuel, 
 were disputing about their examples. Samuel said 
 he performed 6 examples on Monday, 3 on Tuesday, 
 and 5 on Wednesday. Joseph said he performed 3 
 on Monday, 5 on Tuesday, and 6 on Wednesday. 
 Thomas said he performed 6 on Monday, 3 on Tues- 
 day, and 5 on Wednesday. Each thought he had 
 performed more than either of the others ; so they 
 quarreled about it. N6w, can you tell who had done 
 the most ? William, who was an industrious boy, 
 performed 20 examples on Monday. How many 
 more than Samuel did he perform on that day ? 
 How many more than Joseph ? How many more 
 than Thomas ? How many more than each of the 
 others performed in three days ? 
 
 5. Susan and Mary had each 9 oranges. Susan 
 gave 5 of hers to Mary, and Mary ate 2. They then 
 put what they had left together, intending to keep 
 them until the next week; but before that time 4 
 of them had spoiled. They then so divided the good 
 ones among them, that Mary had 6. How many 
 had Susan ? 
 
 G. Mr. Wheelock had 6 dollars, but he has since 
 received 9 dollars, spent 10 dollars for broad-cloth, 
 received an old debt of 2 dollars, found 2 dollars, 
 received 7 dollars for work, paid 8 dollars for a bar- 
 rel of flour, 2 dollars for a barrel of apples, and lost 
 6 dollars. How many dollars has he now ? 
 
46 colburn's first part. 
 
 LESSON XXI. 
 
 2 tens = twenty, and is wi'itten 20, or SO, 
 
 3 tens = thirty, and is written 30, or 30. 
 
 4 tens = forty, and is written 40, or AO, 
 
 5 tens = fifty, and is written 50, or SO, 
 
 6 tens=sixty, and is written 60, or OO, 
 
 7 tens= seventy, and is written 70, or /^O 
 
 8 tens = eighty, and is written 80, or oO, 
 
 9 tens^ninety, and is written 90, or ^0, 
 
 10 tens = one hundred, and is Avritten 100, or /OO. 
 
 E. How many tens^re there — 
 
 1. In 50? 4. In 70? 7. In 90? 
 
 2. In 80? 6. In 20? 8. In 40? 
 
 3. In 30? 6. In 60? 9. In 100? 
 
 C. "What number is equal to each of the following : 
 
 1. 3 tens? 4. 4 tens? 7. 6 tens? 
 
 2. 9 tens ? 6. 8 tens ? 8. 2 tens ? 
 
 3. 10 tens ? 6. 7 tens ? 9. 5 tens ? 
 
 D. How will you write each of the following num- 
 bers in fio-ures ? 
 
LESSON TWENTY-FIRST. 47 
 
 1. Forty? 4. Ninety? 7. Thirty? 
 
 2. Eighty? 5. Seventy? 8. Sixty? 
 
 3. Twenty? 6. Fifty? 9. One hundred ? 
 
 E. 1. 4 tens + 5 tens ? Then 40 + 60 ? 
 
 2. 4 tens + 4 tens ? Then 40 + 40 ? 
 
 3. 7 tens + 3 tens ? Then 70 + 30 ? 
 
 4. 6 tens -|- * tens = 9 tens ? Then 60 + * =: 90 ? 
 6. 4 tens -f * tens = 10 tens ? Then 40 + * = 100 ? 
 
 6. 2 tens + * tens = 6 tens ? Then 20 + ^ = 60 ? 
 
 7. 8 tens — 5 tens ? Then 80 -— 50 ? 
 
 8. 4 tens — 3 tens ? Then 40 — 30 ? 
 
 9. 10 tens — 3 tens ? Then 100 — 30 ? 
 
 F. 1. 404-30+20? 7. 30+30+40—50? 
 
 2. 50+20+30? 8. 20+20+20+20+20—70? 
 
 3. 20+30+30? 9. 100—20—20—20—20+50? 
 
 4. 20+40+20? 10. 90—00—20+^=60? 
 6. 40+20+30? 11. 20+30+40—^ = 70? 
 
 6. 30+30+40? 12. 100— 30— 30— 20+)t = 80? 
 
 G. 1. A man gave 20 cents for Harpers' Magazine, 20 for Put- 
 nam's, and 50 for the North American Review. How many cents 
 did he pay for all ? 
 
 2. A provision-dealer had 20 bushels of potatoes. He after- 
 wards bought 20 bushels of one man, 30 of another, and 30 of 
 another, and then sold 40 bushels to one man, and 30 to another. 
 How many bushels had he left ? 
 
 8. A farmer, who owned 90 sheep, kept 40 in one pasture, 30 
 in another, and the rest in another. How many did he keep in 
 the last pasture ? He drove 10 sheep from the second pasture 
 
48 colbuiin's first part. 
 
 into the first, and 20 from the second into the third. How many 
 were then in each pasture ? 
 
 4. Sarah found 20 walnuts under one tree, and 30 under an- 
 othef, while Lydia found 30 under one tree, and 20 under another. 
 How many did each find ? How many did both find ? 
 
 5. Sarah gave 20 nuts to Jane, and Lydia gave her 10. How 
 many did both give her ? How many had each left ? How many 
 had the three girls ? 
 
 6. George has 20 cents, Williams has 20, and Edward has 20 
 more than George and William together. How many has Edward? 
 How many have all the boys? They agreed to put their money 
 together and purchase some articles with it. They bought some 
 paper for 20 cents, some pencils for 20 cents, and some pens for 
 10 cents, and a pretty story-book for the rest of their money. 
 How much did the story-book cost ? 
 
 LESSON XXII. 
 
 A. A unit is a single thing, or one. 
 
 2 tens -|- 1 unit = twenty-one = 21. 
 2 tens -\- 2 units = twenty-two = 22. 
 
 2 tens -|- 9 units = twenty-nine = 29. 
 
 3 tens -f- 1 unit = thirty-one = 31. 
 
 8 tens -|- 7 units = eighty-seven = 87. 
 
 9 tens -f- 9 units = ninety-nine = 99. 
 
 B. Count from twenty to one hundred^ thus : — 
 Twenty-one, twenty-two, twenty-three, &c., &c. 
 
 What is the value of — 
 
 1. 20-J-8? 3. 30-1-7? 5. 60 -f 9? 
 
 2. 40 -I- 6 ? 4. 90 -I- 3 ? 6. 20 -}- 5 ? 
 
LESSON TWENTY-SECOND, 49 
 
 C. How many tens and units are there — 
 
 1. In 64? 4. In 28? 7. Li23? 
 
 2. In 87? 6. In 60? ^ 8. In 19 ? 
 
 3. In 73? 6. In 07? 9. In 91 ? 
 
 Explanation. — The figure vrhich represents the units of a number 
 is called the units' figure, and that which represents the tens is 
 called the tens* figure. 
 
 Point out the tens' figure, and also the units' figure, in the 
 numbers written under letter C. 
 
 The position, or place occupied by the unit's figure, is called the 
 units* place, and that occupied by the tens* figure is called the tens' 
 place. The tens' place is always just at the left of the units. 
 
 A period, called the decimal point, is often used to aid in determin- 
 ing the place of figures ; the first place at the left of the point being 
 the units' place, the second at the left being the tens' place, and the 
 third the hundreds'. When the point is not written, it is understood 
 to belong at the right of the given number, thus making the right- 
 hand figure the units' figure. 
 
 E. Write each of the following numbers : — 
 
 1. Seventy-nine. 3. Fifty-seven. 5. Eighty-six. 
 
 2. Twenty-four. 4. Sixty -nine. 6. One hundred. 
 
 F. 1. 4 + 2 + 2? 9. 3+4+4f = 9? 
 
 2. 24 + 2 + 2? 10. 3 + 4+^ = 29? 
 
 3. 84 + 2 + 2? 11. 3+24+^ = 29? 
 
 4. 4 + 72 + 2 ? 12. 3 + 74 + * = 79 ? 
 6. 6 + 3 + 2? 13. 2 + 4 + ^- = 10? 
 
 6. 65 + 3 + 2^? 14. 2 + 4 + * = 40? 
 
 7. 95 + 3 + 2? 15. 2 + 4 + ^ = 70? 
 
 8. 85 + 3 + 2? 16. 2 + 4 + ^ = 100? 
 
 G. 1. 2 + 5 + 3 + 3 + 2 + 5+3 + 5+2+3? 
 2. 2 + 7 + 1 + 2 + 7 + 1 + 2 + 7 + 1+2? 
 
 6 . — p = 
 
' 1 
 
 GO 
 
 COLB urn's first PART. 
 
 3. 
 
 3 + 3-f4-f34-34-4-f3-|-3-f4 + 3? 
 
 4. 
 
 53 + 3_[-4-f3 + 3 + 4-f3-f3-f44.3? 
 
 6. 
 
 24 4-2 + 4 + 1 + 7 + 2 -I. 4 + 6 -f. 2 + 5? 
 
 6. 
 
 35 + 3 + 2+3 + 7 + 1 + 9 + 2 + 4 + 2? 
 
 7. 
 
 42 + 8 + 5 + 3 + 2 + 4 + 6 + 2+7+1? 
 
 8. 
 
 67 + 3+1 + 3 + 5 + 1 + 2 + 3 + 4+1? 
 
 9. 
 
 14+6 + 4+6+3 + 7 + 5+5 + 4+4? 
 
 10. 
 
 31 + 2 + 4+3 + 5 + 5+8 + 2 + 3 + 4? 
 
 H. 1. 
 
 9 — 3 — 4? 4. 10 — 2 — 5? 
 
 2. 
 
 29 — 3 — 4 ? 5. 40 — 2 — 5 ? 
 
 3. 
 
 99 — 3 — 4? 6. 50 — 2 — 45? 
 
 I. 1. 
 
 70 — 4 — 3 — 3 — 4 — 3 — 3—4? 
 
 2. 
 
 100 — 3 — 5 — 2 — 3 — 5 — 2 — 3? 
 
 3. 
 
 80 — 2 — 4 — 4 — 2 — 4 — 4 — 2? « 
 
 4. 
 
 40 — 8 — 2 — 5 — 5 — 2 — 4 — 3? 
 
 5. 
 
 70 — 6 — 4 — 2 — 8 — 3 — 5 — 2? 
 
 6. 
 
 100 — 3 — 4 — 3 — 10 — 5 — 5 — 3? 
 
 7. 
 
 25 + 3 + 2 + 8 — 5 — 3 — 6 — 4? 
 
 8. 
 
 83 + 7 + 2+8 — 3 — 7—5 — 5 
 
 9. 
 
 63 + 4 + 2+1 + 2 + 4 — 6 — 5? 
 
 10. 
 
 90 — 8 — 2—3 — 7—9+6+3? 
 
 J. 1. 
 
 One day, George was reckoning up his money. He said his 
 
 father gave him 25 cents at one time, and 5 at another ; that his | j 
 
 mother 
 
 gave him 3 cents at one time, and 7 at another ; and that 
 
 his uncle Rufus gave him 6 cents at one time, and 3 cents at an- 1 1 
 
 . other. 
 
 How many cents had he ? 
 
 2 Samuel think3 that the sum of23 + 4 + 3 + G + 4 + 4 + 5, is | 
 
LESSON TWENTY-SECOND. 51 
 
 49, while William thinks it is 48, and Lydia thinks it is 47. How 
 much do you think it is ? 
 
 3. Sarah had 50 cents ; she spent 10 cents for ribbon, 4 cents 
 for sewing-silk, and 5 cents for needles. How many cents had 
 she left ? 
 
 4. As Eliza was picking up shells on the sea-shore, she found 
 15 in one place, 5 in another, 3 in another, 7 in another, and 
 enough to make up 40 in another. How many did she find in the 
 last place ? 
 
 5. Erastus and Edwin played *' odd or even," beginning their 
 game with 20 grains of corn a-piece. The first time Erastus won 
 3 grains from Edwin, the second time he won 3 grains, and the 
 third time he won 4. How many had each boy then ? 
 
 6. A party of hunters, on counting their game, found that they 
 had shot 23 pigeons, 7 partridges, 20 quails, 3 woodcocks, and 3 
 snipes. How many birds had they shot in all ? 
 
 1: Laura found 44 blackberries in one place, 6 in another, 2 in 
 another, 8 in another, and 9 in another, when, feeling tired, she 
 sat down to rest. She ate 9, and put 4 in a hole for a squirrel to 
 eat, and threw C to some birds. She then found 20, and started 
 for home, but on her way she unfortunately lost 8. How many 
 had she to carry home ? 
 
 8. Mr. Day went out to pay some debts that he owed, and to 
 collect some money that was due him, taking with him 30 dollars. 
 He paid 7 dollars to a shoemaker, 3 dollars to a laborer, and 5 
 dollars to a hatter. He then received 5 dollars from Mr. Baker, 
 30 dollars from Mr Smith, and 8 dollars from Mr. Sumner, after 
 which he paid Mr. Gay 6 dollars for groceries, and 2 dollars for 
 cloth, and Dr. Fogg 7 dollars for services as a physician. How 
 much money had he left ? 
 
52 
 
 C L B U R X 
 
 S FIRST PART. 
 
 
 LESSO]^ XXIII. 
 
 A. 1. 
 
 9+8? 
 
 6 
 
 6+89! 
 
 2. 
 
 19 + 8? 
 
 7. 
 
 4+7? 
 
 3. 
 
 49+8? 
 
 8. 
 
 14+7? 
 
 4. 
 
 6+9? 
 
 9 
 
 4+37? 
 
 6. 
 
 36+9? 
 
 10. 
 
 4 + 67? 
 
 B. 1. 
 
 6 + 8+9? 
 
 7. 
 
 6 + -x- = 14? 
 
 2. 
 
 16 + 8 + 9? 
 
 a 
 
 6 + * = 34 ? 
 
 3. 
 
 66 + 8 + 9 ? 
 
 9. 
 
 46 + * = 54? 
 
 4. 
 
 9 + 7 + 8? 
 
 10 
 
 3 + ^ = 11? 
 
 5. 
 
 89+7+8? 
 
 11. 
 
 13 + ^ = 21? 
 
 6. 
 
 9+77 + 8? 
 
 12. 
 
 3+*== 61? 
 
 C. 1. 
 
 13 — 8? 
 
 6. 
 
 94 — 6? 
 
 2. 
 
 23 — 8? 
 
 7. 
 
 11—5? 
 
 3. 
 
 93 — 8? 
 
 8. 
 
 81—5? 
 
 4. 
 
 14 — 6? 
 
 9. 
 
 41 — 5? 
 
 6. 
 
 44 — 6? 
 
 10. 
 
 91 — 5? 
 
 To THE Teacher. — Vary and extend the preceding exercises till 
 the scholars appreciate the connexion between 9 + 8, 15 + 8, 29 -t- 8, <fcc., 
 12 + 9, 23 + 9, 83 + 9, &c., and understand fully that as 9^4- 8 = 17, or 
 7 more than 10, so 49 + 8 = 7 more than 50, or 57 ; that as 13 — 9 = 4, 
 go 23 — 9 = 14, 83 — 9 = 74, &c. The great objects to be aimed at are 
 accuracy and promptness, the latter being scarcely less important than 
 the former. 
 
r 
 
 1 
 
 LESSON T W E N T y - X II I 11 D . 
 
 1' 
 
 58 
 
 D. 1. 48 + 4? 
 
 12. 
 
 73+9 + 6 + 9 
 
 9 
 
 2. 37 + 6? 
 
 13. 
 
 28+5+8+7 
 
 + 4? 
 
 3. 29 + 7? 
 
 14. 
 
 49+7 + 6+8 
 
 ? 
 
 4. 53 + 8? 
 
 15. 
 
 67+8 + 4 + 9^ 
 
 
 6. 74+7? 
 
 16. 
 
 67+8+4+6 
 
 ? 
 
 6. 23+9+5 + 8? 
 
 17. 
 
 57 + 6—2 — 8: 
 
 
 7. 2 
 
 7 + 6 + 8+5? 
 
 18. 
 
 67+8 — 9 — 6? 
 
 
 8.- 34+8+9—7? 
 
 19. 
 
 33+8+4+9+7+6+5+3 : 
 
 9. 26+3+5—6? 
 
 20. 
 
 56+6+9+3+8+5+4+6 ? 
 
 10. 2 
 
 5+8+4-9 ? 
 
 21. 
 
 45+8+9+7+4+9+6+5 ? 
 
 11. 29+7+9+4 ? 
 
 22. 
 
 38+6+7+7+5+8+9+8 ? 
 
 E. Find the sum of the following columns : — j 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 2 
 
 9 
 
 3 
 
 3 
 
 7 
 
 7 
 
 5 
 
 6 
 
 9 
 
 9 
 
 A 
 
 9 
 
 6 
 
 6 
 
 A 
 
 B 
 
 7 
 
 A 
 
 2 
 
 3 
 
 5 
 
 6 
 
 5 
 
 7 
 
 § 
 
 9 
 
 9 
 
 B 
 
 5 
 
 6 
 
 3 
 
 3 
 
 7 
 
 A 
 
 2 
 
 3 
 
 9 
 
 2 
 
 A 
 
 6 
 
 6 
 
 5 
 
 3 
 
 S 
 
 A 
 
 7 
 
 9 
 
 6 
 
 3 
 
 3 
 
 9 
 
 A 
 
 7 
 
 9 
 
 — 
 
 — 
 
 — 
 
 — 
 
 — 
 
 6* 
 
54 colburn's first part. 
 
 To THE Teacher. — It will be a valuable exercise for the pupil to 
 count by twos, threes, fours, &c., i. e., to call the results obtained by 
 successive additions of the same number to itself, or some other num- 
 ber, till all possible combinations are exhausted. 
 
 Thus, in adding threes, we shall exhaust the varieties of combination 
 by beginning thus ; three, six, nine, twelve, &c. ; two, five, eight, 
 ELEVEN, &c. ; ONE, FOUR, SEVEN, TEN, <fcc. A similar course may be 
 taken in subtraction. 
 
 F. 1. Henry bad 42 cents. He earned 9 cents hy doing errands, 
 5 cents by holding a gentleman's horse, and 8 cents by delivering 
 a letter, after which he spent 7 cents. How many cents had he left ? 
 
 2. A newsboy bought 8 copies of the Boston Post, 9 of the 
 Atlas, 10 of the Traveller, 7 of the Bee, 10 of the Journal, and 1 
 of the Transcript. How many papers did he buy in all ? He sold 
 all but 7 of them. How many did he sell ? 
 
 3. A trader bought 8 yards of cloth, for which he paid 13 dol- 
 lars ; 6 yards for which he paid 9 dollars, 9 yards for which he 
 paid 8 dollars, and 6 yards for which he paid 8 dollars. How 
 many yards of cloth did he buy in all? How many dollars did 
 he pay for it ? 
 
 4. A man bought a horse for G3 dollars, and was obliged to sell 
 him for 8 dollars less than he cost him. For how much did he 
 sell him ? 
 
 5. A farmer who had 33 bushels of corn, sold 6 bushels for 5 
 dollars. IIow many bushels had he left? 
 
 6. Mr. Adams owned 40 acres of land, and bought enough to 
 make up 54 acres. How many did he buy ? 
 
 7. Sarah had 32 roses. She gave 9 to one of her companions, 
 8 to another, and, when she had given some to another, she had 
 7 left. How many did she give to the last? 
 
 8. A trader bought a lot of grain for 54 dollars, and it cost him 
 3 dollars more to have it carried to his store. For how much 
 must he sell it to gain 9 dollars ? 
 
LESSON TWENTY-FOURTH. 65 
 
 9. Mr. Edwards sold a colt for 57 dollars, a sheep for 8 dollars, 
 a calf for 5 dollars, and a cow for 30 dollars, and in part payment 
 received a horse worth 93 dollars. How much still remained due ? 
 
 10. Mr. Boy den and Mr. Manchester each bought a yoke of 
 oxen. Mr. Boyden gave in paj^ment for his oxen, a cart worth 
 47 dollars, 1 ten-dollar bill, 1 five-dollar bill, 1 three-dollar bill, 7 
 one-dollar bills, and 9 silver dollars. Mr. Manchester gave in 
 payment for his oxen, a cow worth 30 dollars, a double eagle 
 worth 20 dollars, an eagle worth 10 dollars, a half-eagle worth 5 
 dollars, an ox-yoke worth 10 dollars, and 9 dollars worth of hay. 
 Which paid the most for his oxen, and how much the most ? 
 
 LESSON XXIV. 
 
 A. 1. 30+50? 7. 40 + 20-1-20? 
 
 2. 32 + 50? 8. 43 + 20 + 20? 
 
 3. 39 + 50? 9, 40 + 28 + 20? 
 
 4. 20 -^ 60? 10. 30 + 20 + 40 ? 
 
 5. 25 + 60? 11. 38 + 20 + 40? 
 
 6. 20 + 68? 12. 30+20 + 47? 
 
 B. 1. How many are 24 -f 67 ? 
 
 SoLUTioJf.--24 and 60 are 84, and 7 are 91. 
 
 2. 63 + 29? 11. 27 + 58+12? 
 
 3. 26+55? 12. 33 + 47 + 16? 
 
 4. 37+48? 13. 24 + 29+47? 
 
 5. 24 + 37? 14. 24 + 29+37? 
 
 6. 73 + 19? 15. 27 + 27+27? 
 
 7. 28+53? 16. 11 + 46 + 25? 
 
 8. 37 + 67? 17. 34 + 26 + 27? 
 
 9. 29 + 29? 18. 16 + 17+19? 
 
56 colburn's first part. 
 
 -\] 
 
 C. 1. 80—20? 7. 90 — 80? 
 
 2. 86 — 20? 8. 97 — 80? 
 
 3. 60 — 30? 9. 80 — 40? 
 
 4. 67 — 30? 10. 86 — 40? 
 
 5. 70 — 40? 11. 60 — 30? 
 
 6. 77 — 40? 12. 53 — 30 
 
 D. 1. 68-26? 
 
 Solution. — 68 minus 20 are 48, minus 6 are 42. 
 
 2. 43 — 17? 7. 81 — 23? 
 
 3. 92 — 67? 8. 52 — 27? 
 
 4. 83 — 48? 9. 48 — 29? 
 
 5. 61 — 23? 10. 97 — 58? 
 
 6. 56 — 19? 
 
 E. 1. 63 + 37 — 82? 4. 64 + 36 — 48? 
 
 2. 48 + 35 — 27? 5. 25 + 39—42? 
 
 3. 24 + 67 — 19? 6. 27 + 64 — 18? 
 
 F. 1. Joseph bought a "First Book of Arithmetic" for 25 
 cents, and a slate for 13 cents. How much did he pay for both ? 
 
 2. Martha's mother gave her 75 cents with which to purchase 
 school-books and paper. She bought a Primary Geography for 
 37 cents, a Spelling Book for 17 cents, and spent the rest of her 
 money for paper. How much did she spend for paper ? 
 
 3. A farmer sold a horse for 93 dollars, which was 26 dollars 
 more than he gave for him. How much did he give for him ? 
 
 4. A horse dealer bought a horse for 54 dollars, and after pay- 
 ing 17 dollars for keeping him, he sold him for 96 dollars. How 
 much did he gain by the transaction ? 
 
LESSON TWENTY-FOURTH. 57 
 
 5. A man bought a sleigh for 21 dollars. He paid 9 dollars for 
 painting and repairing it, and then gave it and 18 dollars in mo- 
 ney for another sleigh. How much did the second sleigh cost 
 him? 
 
 6. From a cask containing 64 gallons of oil, 18 gallons were 
 drawn out at one time, and 25 at another, after which 17 gallons 
 were put in. How many gallons were then in the cask ? 
 
 7. There were 18 sheep in one flock, 27 in another, and 39 in 
 another ; but at night they were all put into the fold. How many 
 were there in the fold ? The next day, 23 were driven to one 
 pasture, 26 to another, and the rest to another. How many were 
 driven to the last pasture ? 
 
 8. Ralph shot 27 pigeons, 15 partridges, 14 woodcocks, and as 
 many quails as there were partridges and woodcocks together. 
 How many quails did he shoot? How many birds in all? 
 
 9. Mr. Thompson owes 13 dollars to Mr. Baker, 9 dollars more 
 to Mr. Ellis than to Mr. Baker, and as much to Mr. French as he 
 owes to Mr. Thompson and Mr. Ellis together. How much does 
 he owe to each, and how much to all ? 
 
 10. Mr. Talbot bought a large lot of apples. His son George asked 
 how much they cost him, to which he replied: " I paid 17 dollars 
 in silver, 25 in gold, and 13 dollars more in bank bills than in 
 silver and gold together. Now, if yon will tell me what they cost, 
 I will give you the difference between their cost and 100 dollars." 
 George answered correctly. What was his answer, and how much 
 money did his father give him ? 
 
58 colbukn's first part. 
 
 LESSON XXV. 
 
 A. The numbers above one hundred are counted 
 thus : — ' 
 
 One hundred one, one hundred two, one hundred three, Jfc, to one 
 hundred ninety-eight, one hundred ninety-nine, two hundred, two hun* 
 dred one, two hundred two, two hundred three, ^'C, to ten hundred, 
 which is generally called one thousand, ten hundred one, or one 
 thousand one, ten hundred ttvo, or one thousand two, &c., to elei^en 
 hundred one, or one thousand one hundred one, &c., to nineteen hun- 
 dred ninety-nine, or one thousand nine hundred ninety-nine, twenty 
 hundred, or two tltousand. 
 
 Ten hundred, or one thousand is written 1000. Twenty hun- 
 dred, or two thousand, is written 2000. Thirty hundred, or three 
 thousand, is written 3000. Ninety hundred, or nine thousand, is 
 written 9000. 
 
 The following exercises suggest the manner of reading and 
 writing numbers above one hundred : — 
 
 100 -f 2 = 102. 1000 -f 10 == 1010. 
 
 400 -f 9 = 409. 1100 -f. 3 = 1103. 
 
 100 -f 10 = 110 1000 4- 28 = 1028. 
 
 100 + 11 = 111. 1100 4- 11 = 1111. 
 
 300 4-12 = 312. 11004- 17=1117. 
 
 600 4- 20 = 620. 1000 4- 117 = 1117. 
 
 100 4- 29 = 129. 3200 4- 20 = 3220. 
 
 1000 4- 1 = 1001. 4200 4- 34 = 4234. 
 
 1000 4- 4 == 1004. 4000 4- 234 = 4234. 
 
LESSON TWENTY-SIXTH. 69 
 
 B. Eead the following numbers : — 
 
 /. A27 5. 5^ A p. //i'cf 
 
 2. ^6B 6. s^oy ^0 6oo6 
 
 3. S60 7. ^5S6 //. S22A 
 
 A. 630 8. 37M ^^' ^^'^7 
 
 C. Write each of the following in figures : — 
 
 1. Three hundred twenty-seven. 
 
 2. Eight hundred four. 
 
 3. Seventeen hundred twenty-eight. 
 
 4. Forty-six hundred thirty-six. 
 
 5. Four thousand six hundred thirty-six. 
 
 6. Twenty-six hundred six. 
 
 To THE Teacher. — For other exercises in Notation and Numera- 
 tion, see Arithmetic and its Applications. 
 
 LESSON XXVI. 
 
 Addition. 
 
 A- All such questions as " How many are 6 -|- ^ + ^ ?" 
 <<4^8-|-9?" &c., are questions in Addition. We are required 
 in the first to add 6, 9, and 7 together ; and in the second, to add 
 4, 8, and 9. It is obvious that in each, we are required to find a 
 number equal in value to all the given numbers. Thus, in the 
 first question, we are required to find a number equal in value to 
 6-f 9-f 7. 
 
 Addition is a process by which we find a number equal in value to 
 several given numbers. 
 
60 colburn's first part. 
 
 The number tlms found is called the sum or amount of the 
 I given numbers. Thus the sum of C, 9, and 7 is 22, for 
 6-1-9 + 7=- 22. 
 
 B. When writing large numbers for addition, we place them in 
 a column, so that the figures of the same denomination shall come 
 under each other, i. e., so that units shall come under units, tens 
 under tens, &c. We then begin at the right hand, and add the 
 columns separately, as in the following examples : — 
 
 1. What is the sum of 723 + 896 + 589 -j- 967 ? 
 
 Solution. — Writing the numbers as opposite, we first '/^ ^ 9 
 
 add the units' column, 7 + 9-f-6-J-3==25 units == 2 tens ' 
 
 and 5 units. Writing 5 units, and adding 2 tens to the 
 tens' column, we have 2-|-6-|-8 + 9-|-2 = 27 tens = 2 
 hundreds, and 7 tens. Writing 7 tens, and adding 2 hun- 
 dreds to the hundreds' column, we have 2 -f- 9 -j- 5 -f" 
 8 4-7 = 31 hundreds, which, being the sum of the last 
 column, we write. The answer, therefore, is 3175. 
 
 3/75 
 
 To test the correctness of the work, examine it carefully to see 
 if any error can be detected. Or, add the numbers again, begin- 
 ning at the top of the column. 
 
 To THE Teacheu. — The design of thi^ work renders it impracticable 
 to give further illustrations here, but the Teacher can readily supply 
 them if they are needed by the class. (See Arithmetic and its 
 Applications, Sect. IV.) 
 
 C. Add the following : — 
 
 12 3 4 6 
 
 SAP A/7 SS7 Sp6 BA3 
 
 37 s 2S6 6/3 /7B 2AB 
 
 6 AS A3p ASB S5p 537 
 
 376 67P 85 A A3B AS>p 
 
LESSON TWENTY-SEVENTH. 61 
 
 6. 4254-487 + 569 + 837+694? 
 
 7. 854 + 308 + 560 + 716 + 593 ? 
 
 8. 672 + 481 + 326 + 425 + 519 ? 
 
 9. 243 + 495 + 826 + 324 + 476 ? 
 10. 627+ 756 + 434+ 874+999 ? 
 
 LESSON XXVII. 
 
 Subtraction. 
 
 A. Such questions as " 4 from 9?" " 12 — 6?" "How many 
 more are 17 than 8?" &c., are questions in Subtraction. 
 
 "We are required, in the first, to subtract 4 from 9 ; in the 
 second, to subtract 6 from 12; and in the third, to subtract 8 
 from 17. 
 
 Subtraction is a process by which we find the difference between two 
 numbers, or the excess of one number over another. 
 
 The larger of the two given numbers is called the minuend, the 
 smaller, or one to be subtracted, is called the subtrahend, and 
 the answer is called the difference, or remainder. 
 
 B. We write large numbers for Subtraction, so that figures of 
 the same denomination shall come under each other, and subtract 
 as illustrated in the following examples : — 
 
 1. How much is 8436 — 6122? 
 
 Solution. — Writing the numbers as opposite, we have / 0/j 
 2 units from 6 units leave 4 units; 2 tens from 3 tens 
 
 leave 1 ten ; 1 hundred from 4 hundreds leaves 3 hun- S / Q9 
 dreds ; 6 thousands from 8 thousands leaves 2 thousands. 
 
 Therefore, the answer is 2 thousands, 3 hundreds, 1 ten, 
 
 and 4 units, or 2314. 23 if A 
 
62 coLB urn's first part. 
 
 Ill the same manner perform tlie following : — 
 
 2. 4893 — 1231? 5. 4867—1614? 
 
 3. 5987 — 3125? 6. 9318 — 2106? 
 4 8958 — 6713? 7. 6985 — 1401? 
 
 C. If a figure of the subtrahend is larger than the corresponi- 
 ing figure of the minuend, we take one of the next higher 
 denomination of the minuend, and reduce (t. e.. change) it to the 
 required denomination, as in the following example : — 
 
 1. How much is 947 — 458 ? 
 
 Solution. — As we cannot subtract 8 units from 7 units, wo. take one 
 of the 4 tens (leaving 3 tens), and reduce 
 
 V o y/'V _ f minuend changed in form to show 
 OVOj// I the reduction. 
 
 Q j^ y^ *=" minuend. 
 ji- S O ^ subtrahend. 
 
 ^ O p =• quotient. 
 
 to its value in units : 1 ten = 10 units, which, added to the 7 units, 
 gives 17 units : 17 units - 8 units = 9 units. 
 
 As we cannot take 5 tens from 3 tens, we take one of the 9 hundreds, 
 leaving 8 hundreds, and reduce it to its value in tens : 1 hundred = 10 
 tens, which, added to the 3 tens = 13 tens ; 13 tens — 5 t-ens = 8 tens. 
 
 4 hundreds from 8 hundreds leave 4 hundreds. Therefore, 947 — 
 458 = 4 hundreds, 8 tens, and 9 units, or 489. 
 
 To prove the correctness of the answer, add the subtrahend 
 and remainder together; if their sum is equal to the minuend, 
 the work is correct ; if not, there is an error in the subtraction or 
 the addition, and the work should be re-examined to detect it. 
 
 To THE Teacher. — For more full illustrations, see Arithmetic and 
 ITS Applications. 
 
LESSON TWENTY-EIGHTH. 63 
 
 2. 48G4 — 2579? 8. 5426 — 3987? 
 
 3. 8149 — 34G3? 9. 9943 — 4399? 
 
 4. 2769 — 1487? 10. 9333 — 8888? 
 
 5. 2144 — 1397? 11. 4634 — 2359? 
 
 6. 8432 — 3586? 12. 9257 — 4328? 
 
 7. 4374 — 5856? 13. 8642 — 5853? 
 
 LESSON XXVIII. 
 
 The method of writing numbers by figures is called the Arabic 
 Method. 
 
 There is a method of expressing numbers by letters, called the 
 Roman Method. The letter I stands for one, V for five, X for ten, 
 L for fifty, C for one hundred, D for five hundred, and M for one 
 thousand. 
 
 If a letter is repeated, it indicates that the number for which it 
 stands is repeated. Thus : I stands for one, II for two, III for 
 three, X for ten, XX for twenty, XXX for thirty, CC for two 
 hundred, &c., &c. 
 
 If a letter representing one number stands before a letter, 
 representing a larger number, the value of the formei is sub- 
 tracted from the value of the latter. Thus : IV = 1 from 5 = 4, 
 IX = 1 from 10 = 9, XL = 10 from 50 = 40, XC = 10 from 
 100 = 90, &c. 
 
 If a letter representing one number stand before a letter repre- 
 senting a smaller number, the value of the former is to be added to 
 the value of the latter. Thus: VI = 5+ 1 = 6, XI = lO-f- 1 = 11, 
 XV = 10 -I- 5 = 15, &c. CX = 100 -f 10 = 110. Hence — 
 
64 COLBURN'S i'lRST PART. 
 
 1=1 XI == 11 XXI = 21 
 
 II = 2 XII = 12 XXIV = 24 
 
 III = 3 XIII = 13 XXV = 25 
 
 IV = 4 XIV == 14 XXX = 30 
 V = 5 XV = 15 XXXIX = 39 
 
 VI == 6 XVI = 16 XLIV = 44 
 
 VII = 7 XVII = 17 LXX = 70 
 
 VIII = 8 XVIII = 18 LXXXIX = 89 
 
 IX == 9 XIX = 19 XC = 90 
 
 X = 10 XX = 20 CXXXIX = 139 
 
 LESSON XXIX. 
 Tables of Moneys, Weights, and Measures. 
 
 A. The money we use is called United States or Federal 
 Money. 
 
 TABLE OF UNITED STATES MONEY. 
 
 10 mills = 1 cent. 
 
 10 cents = 1 dime. 
 
 10 dimes = 1 dollar. 
 
 10 dollars = 1 eagle. 
 The coins of the United States are : the cent, the three-cent 
 piece, the half-dime, worth 5 cents; the dime, worth 10 cents; 
 the quarter-dollar, worth 25 cents; the half-dollar, worth 50 
 cents ; the dollar, worth 100 cents ; the three-dollar piece ; the 
 eagle, worth 10 dollars ; the double-eagle, worth twenty dollars ; 
 the half-eagle, worth five dollars ; the quarter-eagle, worth two 
 and a half dollars, and the fifty-dollar piece. 
 
 * In reciting these tables, let the pupils say " equal" in place of 
 '* make one," the phrase often used. 
 
LESSON TWENTY-NINTH. 65 
 
 The character $ placed at the left of figures, shows that they 
 represent dollars, or values in United States Money. The dollars 
 are alwa3*s placed at the left of the decimal point (See Lesson 
 XXV), and the cents and mills at the right. 
 
 Thus, to express 14 dollars 38 cents, we should write 14 at the 
 left of the decimal point, and 38 at the right of it. Thus : $14.38. 
 
 Illustration. — $ 8.27 = 8 dollars, 27 cents. 
 $15.06 = 15 dollars, 06 cents. 
 $2,327 •=» 2 dollars, 32 cents, 7 mills. 
 
 Read the following : — 
 
 1. $4.28. 3. $82.36. 5. $40.03. 
 
 2. $5.37. 4. $75.07. 6. $28.79. 
 
 B. The money used in England is called English or Sterling 
 Money. 
 
 TABLE OF STERLING MONEY. 
 
 PULL TABLE. ABBREVIATED TABLE. 
 
 4 farthings = 1 penny. 4 gr. = Id. 
 
 12 pence = 1 shilling. 12 d. = Is. 
 
 20 shillings = 1 pound. 20 s. = 1£. 
 
 The English pound is worth about $4.84. 
 
 The English shilling is worth about a quarter of a dollar, and the 
 English penny is worth about 2 cents. The term shilling is sometimes 
 used in New York, New England, and some other States of the Union, 
 but it does not mean an English shilling. A New York shilling is 
 worth just 12J cents. A New England shilling is worth just 16§ 
 cents. The ninepence of New England is the same as the shilling of 
 New York. 
 
 C. Iron, flour, sugar, wool, coal, and almost all articles except 
 gold, silver, and jewels, are weighed by Avoirdupois Weight. 
 
 "^ 6* E 
 
66 colburn's first part. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 16 drams = 1 ounce. 16 dr. = 1 oz. 
 
 16 ounces = 1 pound. 16 oz. = 1 lb. 
 
 25 pounds = 1 quarter. 25 lbs. = 1 qr. 
 
 4 quarters = 1 hundred weight. 4 qrs. = 1 cwt. 
 
 20 hundred weight = 1 ton. 20 cwt.= 1 T. 
 
 N3TE. — Formerly the quarter was reckoned at 28 lbs., the hundred- 
 weight at 112 pounds, and the ton at 2240 lbs., and they are so reck- 
 oned at the present time in Great Britain, and at the United States 
 Custom Houses. Merchants usually reckon them as given in the table. 
 
 D. Gold, Silver, and precious stones are weighed by Troy 
 Weight. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 24 grains = 1 pennyweight. 24 gr. = 1 dwt. 
 
 20 pennyweights = 1 ounce. 20 dwt. = 1 oz. 
 
 12 ounces = 1 pound. 12 oz. = 1 lb. 
 
 E. Apothecaries' Weight is used in compounding or mixing 
 medicines, but they are sold by Avoirdupois weight. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 20 grains = 1 scruple. 20 gr. =19. 
 
 3 scruples = 1 dram. 3 9 = 1 5. 
 
 8 drams = 1 ounce. 8 .:^ = 1 5. 
 
 12 ounces = 1 pound. 12 g == 1 ib. 
 
 F. COMPARISON OF AVOIRDUPOIS, TROY, AND APOTHE- 
 CARIES' WEIGHT. 
 A pound Avoirdupois is heavier than a pound Troy, but an 
 ounce Avoirdupois is not so heavy as an ounce Troy. 
 
 Their relative weights may be seen in the following table of 
 comparison, which expresses the value of each in grains Troy ; — 
 1 lb. Avoirdupois = 7000 grains Troy, 
 lib. Troy = 1 lb. = 5760 " 
 1 oz. Avoirdupois = 437 J ** 
 1 oz. Troy = Ig = 480 " 
 1 dr. Avoirdupois = 27J^^ " 
 1 3 Troy r= 60 " 
 
 19" = 20 *« 
 
 Idwt. = 24 " 
 
 1 gr. Apothecaries = 1 ** 
 
LESSON TWENTY-NINTH. 67 
 
 It follows, then, that — 
 
 144 lbs. Avoirdupois = 175 Troy. 
 192 oz. " = 175 oz. Troy. 
 
 1 lb. " = J-Jf of 1 lb. Troy. 
 
 1 oz. " = {.Jl of 1 oz. Troy. 
 
 G. Long Measure is used for measuring lengths and dis- 
 tances. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 12 lines = 1 inch. 12 1. =1 in. 
 
 12 inches = 1 foot. 12 iR.= 1 ft. 
 
 3 feet = 1 yard. 3 ft. = 1 yd. 
 
 5 J yards, or •. 5 J yds., or ^ 
 
 y =1 rod, or pole. V = 1 rd. or p. 
 
 lejfeet / ^ 16Jft. / ^ 
 
 40 rods = 1 furlong. 40 rds. = 1 fur. 
 
 8 furlongs = 1 mile. 8 fur. = 1 m. 
 
 3 miles = 1 league. 3 m. =1 le. 
 
 H. Cloth Measure is used for measuring cloths, silks, &c. 
 
 PULL TABLE. ABBREVIATED TABLE. 
 
 2J inches = 1 nail. 2J in. = 1 na. 
 
 4 nails = 1 quarter. 4 na. = 1 qr. 
 
 4 quarters = 1 yard. 4 qr. = 1 yd. 
 
 I. Square Measure. — This measure is used in measuring 
 land, and all kinds of surfaces. 
 
 Preliminary Defitiitions. — An anc/le is the diflference in direction 
 of two lines. The point where the lines meet is called tha vertex 
 of the angle. 
 
 When the two angles formed by one straight line meeting 
 another are equal to each other, they are called rfght ai^gles. 
 
68 colburn's first part. 
 
 One line i^ perpendicular to another when it makes 'right angles 
 with it. 
 
 ^- The angle A C B is equal to the angle 
 
 BCD, and hence they are right angles. 
 D. Therefore, B C is perpendicular to A D. 
 
 An angle greater than a right angle, is 
 called an obtuse angle, and an angle less than a right angle is 
 called an acute angle. 
 
 A RIGHT ANGLE. AN ACUTE ANGLE. AN OBTUSE ANGLE. 
 
 A four-sided figure having all of its angles right angles, is called 
 a rectangle. 
 
 A rectangle having all of its sides equal, is called a square. A 
 square, then, has four equal sides, and four equal angles. 
 
 A square foot is a square measuring one foot on everj side. A 
 square yard is a square measuring a yard on every side, &c. 
 
 TABLE OF SQUARE MEASURE. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 144 square inches = 1 square foot. 144 sq. in. = 1 sq. ft. 
 
 9 square feet = 1 square yard. 9 sq. ft. = 1 sq. yd. 
 
 30J- square yards, or >» 30J sq. yds. or >| 
 
 }. = 1 sq. rod. V =1 sq. rd. 
 
 272J square feet, i 272J sq. ft. J 
 
 40 square rods = 1 rood. 40 sq. rds. = 1 R. 
 
 4 roods = 1 acre. 4 R. = 1 A. 
 
 040 acres = 1 square mile. 640 A. = 1 sq. m. 
 
 J. Cubic Measure. — Cubic Measure is used in measuring 
 solids. 
 
 A solid is a magnitude which has length, breadth, and thickness. 
 
LESSON TWENTY-NINTH. 69 
 
 A cube is a rectangular solid, whose length, breadth, and 
 
 height, are equal. It may also be defined as a solid bounded by 
 six equal squares. 
 
 A cube 1 foot long, 1 foot wide, 1 foot high, would be a cubic 
 foot. 
 
 A cube 1 yard long, 1 yard high, and 1 yard wide, would be a 
 cubic yard. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 1728 cubic inches = 1 cubic /foot. 1728 cu. in. = 1 cu. ft. 
 
 27 cubic feet = 1 cubic yard. ^^ ^- ^^- ^^ ^ ^^- y^' 
 
 16 cubic feet = 1 cord foot. IG cu. ft. = 1 cd. ft. 
 
 8 cord feet, or ^ 8 cd. ft. or 
 I =1 cord wd. 
 
 128 cubic feet J 128 cu, 
 
 W V.^^. il. Ul -\ 
 
 1 cord wd. y =1 cd. wd. 
 
 . ft. / 
 
 K. Circular or Angular Measure. — Circular or Angular 
 Measure is used to measure angles, and the circumferences of 
 circles. 
 
 A circle is a surface bounded by a curved line, which is every- 
 where equally distant from a point within, called the centre. The 
 boundary line is called the circuTfiference of the circle. 
 
 The figure represents a circle, of which C. 
 is the centre. 
 
 The distance from the centre of a circle to 
 the circumference is called the radius. 
 
 The distance from a point on one side of a 
 circle through the centre to a point on the 
 opposite side is called the diameter. Any portion of the circum- 
 ference is called an arc. 
 
 Every circumference of a circle, whether large or small, is sup- 
 posed to contain 360 equal parts, called degrees. Each degree is 
 divided into 60 equal parts, called minutes, and each minute into 
 60 equal parts, called seconds. 
 
70 colburn's first part. 
 
 A degree may be considered simply as the 360th part of the 
 circumference of thz circle considered. Hence its length, as well 
 as that of its subdivisions, must vary with the size of the circle. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 60 seconds = 1 minute. 60'''' = V. 
 
 60 minutes = 1 degree. 60-^ = 1°. 
 
 360 degrees = 1 circumference. 860° = 1 circ. 
 
 L. Dry Measure is used for measuring grain, nuts, salt, &c. 
 
 PULL TABLE. ABBREVIATED TABLE* 
 
 2 pints = 1 quai|p 2 pts. = 1 qt. 
 
 8 quarts = 1 peck. 8 qts. = 1 pk. 
 
 4 pecks = 1 bushel 4 pks. = 1 bu. 
 
 The chaldron of 36 bushels is sometimes used in measuring coal. 
 Ch. is the sign for chaldron. 
 
 The bushel contains 2150| cubic inches, and the quart contains 
 67^ cubic inches. 
 
 M. All kinds of liquids are measured by Liquid Measure. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 4 gills = 1 pint. 4 gls. = 1 pt. 
 
 2 pints = 1 quart. 2 pts. = 1 qt. 
 
 4 quarts = 1 gallon. 4 qts. = 1 gal. 
 
 The hogshead of 68 gallons is used in estimating the contents 
 of reservoirs, or other large bodies of water ; but in all other cases 
 the term hogshead is not a definite measure. Casks containing 
 from 50 or 60, to 100 or 200 gallons, are called hogsheads. 
 A barrel of cider is usually reckoned at 81 J gallons. 
 The gallon contains 231 cubic inches. 
 
 The beer gallon is sometimes used in measuring beer, milk, and 
 ale. It contains 282 cubic inches, and the beer quart contains 
 70J cubic inches. 
 
LESSON TWENTY-NINTH. 71 
 
 N. COMPARISON OF DRY, LIQUID, AND BEER MEASURE. 
 1 qt. dry measure = 67^ cubic inches. 
 1 qt. liquid measure =r 57J cubic inches. 
 1 qt. beer measure == 70J cubic inches. 
 
 0. TABLE OF TIME. 
 
 FULL TABLE. ABBREVIATED TABLE. 
 
 60 seconds = 1 minute. 60 sec. = 1 min. 
 
 60 minutes = 1 hour. 60 min. = 1 h. 
 
 24 hours = 1 day. 24 h. = 1 d. 
 
 7 days = 1 week, 7 d. = 1 wk. 
 
 865 days, or 52 >. 3G5 d. or 52 wk, 
 
 >l ooo a. or i)^ WK. % 
 
 1=1 year. ^^^ | = 1 y. 
 
 weeks, IJ days. ^ l^ d. 
 
 To avoid the inconvenience of reckoning J day with each year, 
 every fourth year (called leap year) is reckoned at 366 days, and 
 the others at 365. 
 
 The year is divided into 12 months, which differ somewhat in 
 length, as is seen in the following 
 
 TABLE OF MONTHS. 
 
 January has 31 days. July has 31 days. 
 
 February has 28 days.* August has 31 days. 
 
 March has 31 days. September has 30 days. 
 
 April has 30 days. October has 31 days. 
 
 May has 31 days. November has 30 days. 
 
 June has 30 days. December has 31 days. 
 
 * Except in leap year, when it has 29. 
 
72 colburn's first part. 
 
 p. MISCELLANEOUS. 
 
 12 things = 1 dozen. 
 
 12 dozen = 1 gross. 
 
 12 gross = 1 great gross, 
 
 20 tilings = 1 score. 
 A barrel of beef or pork "weighs 200 lbs. 
 A barrel of flour weighs 196 lbs. 
 
 , PAPER. 
 
 24 sheets = 1 quire. 
 
 20 quires = 1 ream. 
 
 BOOKS. 
 
 A sheet folded in 2 leaves is called a folio. 
 
 " ♦* «• " 4 «* ** " quarto, or 4to. 
 
 " " «« «* 8 " " " octavo, or 8vo. 
 
 " «* " " 12 " " " duodecimo or 12mo. 
 
 " " " *< 18 " " " 18mo. 
 This book is a duodecimo. 
 
 Q. FRENCH MEASURES AND WEIGHTS. 
 
 The folio-wing measures and weights are often referred to in 
 this country, especially in scientific works : 
 
 rRENCH LONG MEASURE. 
 
 10 millimetres = 1 centimetre. 
 10 centimetres = 1 decimetre. 
 10 decimetres = 1 metre. 
 10 metres = 1 decametre. 
 
LESSON TWENTY-NINTH. 73 
 
 10 decametres = 1 hectometre. 
 
 10 hectometres = 1 kilometre. 
 
 10 kilometres = 1 myriametre. 
 The metre is regarded as the unit of measure, and equals 39.371 
 of our inches. It is the twenty-millionth part of the distance 
 measured on the meridian, from one pole to the other. 
 
 FRENCH WEIGHTS. 
 
 10 milligrammes = 1 centigramme. 
 10 centigrammes == 1 decigramme. 
 10 decigrammes = 1 gramme. 
 10 grammes = 1 decagramme. 
 
 10 decagrammes = 1 hectogramme. 
 10 hectogrammes= 1 kilogramme. 
 10 kilogrammes = 1 myriagramme. 
 
 The gramme is regarded as the unit of this weight, and equals 
 about IS.yyy*^ grains Troy. 
 
 The kilogramme is the weight most frequently used in business 
 transactions, and equals very nearly 2^ pounds Avoirdupois. 
 
 FRENCH MONEY. 
 
 10 centimes = 1 decime. 
 
 10 decimes = 1 franc. 
 
 The franc equals 18| cents, and the five-franc piece often seen 
 in the United States, is equal m value to 93 cents. 
 
74 
 
 colburn's 
 
 FIRST PART. 
 
 
 
 LESSON XXX. 
 
 
 
 A. TABLE 
 
 • 
 
 
 Add 10 twos together. 
 
 
 
 
 2 times 1, or once 2=2 
 
 2 times 6, or 6 times 2 = 
 
 12 
 
 2 times 2 =4 
 
 2 times 7, or 7 times 2 = 
 
 14 
 
 2 times 3, or 3 times 2=6 
 
 2 times 8, or 8 times 2 = 
 
 16 
 
 2 times 4, or 4 times 2 = 8 
 
 2 times 9, or 9 times 2 = 
 
 18 
 
 2 times 5, or 5 times 2 = 10 
 
 2 times 10, or 10 times 2= 
 
 .20. 
 
 B. 1. 
 
 * times 7 = 14? 
 
 6. 
 
 4 = * times 2 ? 
 
 
 2. 
 
 * times 2 = 14? 
 
 7. 
 
 20 = * times 2 ? 
 
 
 3. 
 
 •je times 4 = 8 ? 
 
 8. 
 
 16 = -jv times 2 ? 
 
 
 4. 
 
 * times 2 = 12 ? 
 
 9. 
 
 10 = •}«• times 5 ? 
 
 
 5. 
 
 * times 2 = 18? 
 
 10. 
 
 6 = * times 2 ? 
 
 
 C. 1. 
 
 4 times ^ = 8? 
 
 6. 
 
 10 = 2 times * ? 
 
 
 2. 
 
 2 times « = 8 ? 
 
 7. 
 
 4 = 2 times * ? 
 
 
 3. 
 
 2 times* = 10? 
 
 8. 
 
 20 = 2 times * ? 
 
 
 4. 
 
 2 times * = 6 ? 
 
 9. 
 
 12 = 2 times * ? 
 
 
 6. 
 
 2 times * = 14 ? 
 
 10. 
 
 18 = 2 times * ? 
 
 
 D. 1. 
 
 8 times 2, plus 4 = ^f 
 
 times 10? 
 
 
 2. 
 
 2 times 5, plus 8 = * 
 
 times 2 
 
 ,? 
 
 
 3. 
 
 7 times 2, minus 6 = 
 
 * times 4? 
 
 1 
 
 4. 
 
 2 times 9, minus 6 = 
 
 * times 2 ? 
 
 
 5. 
 
 2 times 4, plus 8 = * 
 
 times 2 ? 
 
 
LESSON THIRTIETH. 75 
 
 To THE Teacher. — The reasoning processes of the following exam- 
 ples are very important, and should be thoroughly understood by the 
 scholars. Not till, by much drill and many repetitions, they have 
 become perfectly familiar, can they safely be omitted or neglected. 
 Indeed, if the pupil must, in his first exercises, omit either, it is far 
 better to give the reasoning process, and omit the answer, than to 
 omit the process, and give the answer only. 
 
 1. 2 pks. = -jf qts ? 
 
 Solution. — Since 1 peck = 8 quarts, 2 pecks must equal 2 times 8 
 quarts, or 16 quarts. Therefore, 2 pks. = 16 qts. 
 
 2. 6 qts. = ^ pts. ? 5. 2 yds. = -x- qrs. 
 
 3. 2 wks.= ^ da. ? G. 2 ,^. == ^- 9 ? 
 
 4. 2 dimes = -x- cents? 7. 2 sq. yds. = ^ sq. ft.? 
 
 8. How much will 6 apples cost at 2 cents a-piece ? 
 
 Solution. — Since 1 apple costs 2 cents, 6 apples will cost 6 times 2 
 cents, or 12 cents. Therefore, 6 apples at 2 cents each will cost 12 
 cents. 
 
 9. How much will 8 books cost at 2 dollars a-piece ? 
 
 10. How much will 2 hats cost at 5 dollars a-piece ? 
 
 11. How far will a man walk in 2 hours, if he walks at the rate 
 of 4 miles per hour ? 
 
 12. How many bushels will 8 boxes hold, if each box holds 2 
 bushels ? 
 
 13. How many quarts of berries will George pick in 9 days, if 
 he picks 2 quarts per day ? 
 
 14. How much will 10 pairs of shoes cost at 2 dollars per pair? 
 
 F. 1. 16 .^ = * § ? 
 
 Solution. — Since 8 drams Apothecaries' equal one ounce, 16 drams 
 must be equal to as many ounces as there are times 8 in 16, which are 
 2 times. Therefore, 16 3 = 2 3. 
 
76 colburn's first part. 
 
 2. 6 ft. = * yds. ? 5. 16 qts. = * pts. ? 
 
 3. 20 pts. == * qts. ? 6. 8 qrs. = -5^ yds. ? 
 
 4. 14 da. = * wks. ? 7. 16 pts. = * qts. ? 
 
 8. How many apples at 2 cents a-piece can be bouglit for 12 
 cents ? 
 
 Solution. — If one apple can be bought for 2 cents, as many apples 
 can be bought for 12 cents as there are times 2 in 12, which are 6 
 times. Therefore, 6 apples at 2 cents a-piece can be bought for 12 
 cents. 
 
 9. How many oranges at 3 cents a-piece can be bonght for 6 
 cents ? 
 
 10. How many shawls, at $7 each, can be bought for $14? 
 
 11. How many boxes, holding 2 bushels each, will be required 
 to hold 20 bushels of apples ? 
 
 Solution. — If 1 box is required to hold 2 bushels, as many boxes 
 will be required to hold 20 bushels as there are times 2 in 20. There- 
 fore, 10 boxes, each holding 2 bushels, will be required to contain 20 
 bushels of apples. 
 
 12. How many hours will it take a man to walk 14 miles, if he 
 walk at the rate of 2 miles per hour ? 
 
 13. How many days would it take a man to earn 10 dollars, if 
 he earned 2 dollars per day ? 
 
 14. How many pieces 8 feet in length can be cut from a piece 
 of string 16 feet in length ? 
 

 LESSON THIRTY-FIRST. 
 
 77 
 
 
 LESSON XXXI. 
 
 
 A. 
 
 Add 10 threes together. 
 
 
 Note.— The pupil should supply the missing part of this and the 
 subsequent tables, by his knowledge of the preceding ones. 
 
 
 TABLE. 
 
 
 3 times 3 = 9. 
 
 3 times 7, or 7 times 3 = 
 
 21. 
 
 3 times 4, or 4 times 3 = 12. 
 
 3 times 8, or 8 times 3 = 
 
 24. 
 
 3 times 5, or 6 times 3 = 15. 
 
 3 times 9, or 9 times 3 = 
 
 27. 
 
 3 times 6, or 6 times 3 = 18. 
 
 3 times 10 or 10 times 3 = 
 
 30. 
 
 B. 1. 
 
 * times 3 = 15 ? 
 
 5. 9 = ^^ times 3 ? 
 
 
 2. 
 
 * times 3 = 6 ? 
 
 6. 12 = 4ftimes4? 
 
 
 3. 
 
 * times 7 = 21 ? 
 
 7. U = ^ times 7 ? 
 
 
 4. 
 
 * times 3 = 30 ? 
 
 8. 24 = * times 8 ? 
 
 
 C. 1. 
 
 6 times * = 18? 
 
 5. 12 = 3 times * ? 
 
 
 2. 
 
 3 times * = 21 ? 
 
 6 24 = 3 times * ? 
 
 
 3. 
 
 3 times* = 27? 
 
 7. 15 = 5 times ^ ? 
 
 
 4. 
 
 8 times * = 16 ? 
 
 8. 30 == 3 times -^ ? 
 
 
 D. 1. 
 
 5 times 3, plus 5 = * 
 
 times 2 ? 
 
 
 2. 
 
 3 times 8, plus 6 = * 
 
 times 3 ? 
 
 
 3. 
 
 9 times 3, minus 9 = 
 
 * times 6 ? 
 
 
 4. 
 
 4 times 3, plus 4 = * 
 
 times 8 ? 
 
 ; 
 
 5. 
 
 2 times 9, plus 6 = * 
 
 times 3 ? 
 
 
 7* 
 
 
 
 
78 colburn's first part. 
 
 E. 1. 3 wks. = * da. ? 4. 3 yds 3 qrs. == * qr. ? 
 
 2. 2 bu. 5 pks. = * pks. ? 6. 3 pks. 7 qts. = ^ qts. ? 
 
 3. 9 yds. 2 ft. = * ft. ? 6. 7 qts. 1 pt. = * pts. 
 
 7. Francis says that he has money enough to buy 3 cocoa-nuts 
 at 9 cents a-piece, and still have 6 cents left. How much money 
 has he ? 
 
 8. William has 9 three-cent pieces, and 8 cents besides. How 
 many cents has he in all ? 
 
 9. Arthur has 3 half-dimes, and 2 three-cent pieces. How 
 much money has he ? 
 
 10. How many pen-holders, at 3 cents a-piece,*can be bought 
 for 15 cents ? 
 
 11 Willie had 27 cents, which he exchanged for their value in 
 three-cent pieces. How many three-cent pieces did he get ? 
 
 12. Amelia had 20 very nice apples. She ate 2, and divided 
 the rest among her playmates, giving 3 to each. Among how 
 many did she divide them ? 
 
 13. If Augustus has 37 apples, how many will he have left after 
 giving 3 of his companions 7 apples a-piece ? 
 
 14. Sarah bought 9 spools of thread at 3 cents a-piece, and 
 then had money enough left to buy 2 skeins of silk at 3 cents per 
 skein. How much money had she at first ? 
 
 15. Simon had 42 cents. He gave 10 cents for a writing book, 
 and 5 for an inkstand, and then exchanged the rest of his money 
 for three-cent pieces. How many three-cent pieces did he get ? 
 
LESSON THIRTY-SECOND. 79 
 
 LESSON XXXII. 
 
 TABLE. 
 
 A. Add 10 fours together. 
 
 4 times 4 = 16. 4 times 8, or 8 times 4 = 32. 
 
 4 times 5, or 6 times 4 = 20. 4 times 9, or 9 times 4 = 36. 
 
 4 times 6, or 6 times 4 = 24. 4 times 10, or 10 times 4= 40. 
 4 times 7, or 7 times 4 = 28. 
 
 B. Explanations and Definitions. — To multiply a number by 4, 
 is the same as to find 4 times that number; to multiply a number by 
 7 is the same as to find 7 times that number, <fcc., Ac. Thus, to mul- 
 tiply 6 by 4 is the same as to find 4 times 6, which is 24. 
 6 multiplied by 3 = 3 times 5 = 15. 
 8 multiplied by 4 = 4 times 8 = 32. 
 Multiplication, theriy is the process of finding any number of times 
 a given number. 
 
 The number to be taken some number of times is called the multi- 
 plicand ; the number showing how many times it is to be taken is 
 called the multiplier ,• the answer is called the product. 
 
 Thus, in " 8 times 3 = 24," 8 is the multiplier, 3 the multiplicand, 
 and 24 is the product, 
 
 Name the multiplier, multiplicand, and product in each of the 
 following examples : — 
 
 1. 9 times 4 = 36. 4. 4 times 6 = 24. 
 
 2. 3 times 8 = 24, 6. 3 times 3 = 9. 
 
 3. 7 times 4 s=r 28. 6. 4 times 4—16. 
 
 The multiplier and multiplicand are called factors of the 
 product. 
 
 Thus, in 9 times 4 = 36, 9 and 4 are factors of 36. 
 Name the factors in the above examples. 
 
80 
 
 
 colburn's first part. 
 
 Two oblique lines crossing thus, X > form the sign of multipli- 
 
 cation. 
 
 It may be read either as *' times" or as " multiplied by." 
 
 Thus, 
 
 "6X3 = 18" may be read either as " 6 times 3 = 18," or 
 
 " 6 multiplied by 3 = 18." 
 
 To 
 
 the Teacher. — It will probably be well to have the pupils at 1 1 
 
 first 
 
 reac 
 
 the sign of multiplication as though written "times;" but 
 
 they 
 
 should learn to read and use it in either way, as occasion may 1 1 
 
 require. 
 
 
 c. 
 
 1. 
 
 *X4 = 16? 9. 40 = *x4? 
 
 
 2. 
 
 * X 3 = 18 ? 10. 28 = * X 4 ? 
 
 
 3. 
 
 * X 4 = 36? 11. 24 = * X 4? 
 
 
 4. 
 
 *X5 = 20? 12. 32 = * X 8? 
 
 
 5. 
 
 4 X * = 40 ? 13. 32 = 4 X * ? 
 
 
 6. 
 
 6x* = 20? 14. 36 = 4 X*? 
 
 
 7. 
 
 4x* = 28? 15. 24 = 6 X*? 
 
 
 8. 
 
 2x* = 12? 16. 16 = 4 X*? 
 
 D. 
 
 1. 
 
 7 times 4, plus 8 = * times 9 ? 
 
 
 2. 
 
 9 times 8, plus 5 = * times 4 ? 
 
 
 3. 
 
 5 times 3, plus 3 times 7 = * times 4 ? 
 
 
 4. 
 
 2 plus 7, plus 6 times 3 = -x- times 8 ? 
 
 E. 
 
 1. 
 
 3 pk. 6 qt. = * times 5 qt. ? 
 
 Solution. — 3 pk. 6 qt. = 30 qt. ; and 30 qt. contains 5 qt. as many 
 
 times as 30 contains 5, which are 6 times. Hence 3 pk. 6 qt. = 6 times 
 
 5qts 
 
 
 
 Abbreviated Solution. — 3 pk. 6 qt. = 30 qt. ; and 30 qt. = 6 times 1 1 
 
 5 qt. 
 
 Hence 3 pk. 6 qt. = 6 times 5 qt. 
 
 
 2. 
 
 3 wk. 3 da. = * times 4 da. ? 
 
 
 3. 
 
 7 gal. 2 qt. = •}«• times 3 qt. ? 
 
 
 4. 
 
 2 wk. 4 da. == * times 6 da. ? 
 
 
 5. 
 
 6^23= times 2 9 ? 
 
 
 6. 
 
 8 yd. = ^t times 1 yd. 1 ft. ? 
 
 
 7. 
 
 4 gal. 2 qt. = ^ times 1 gal. 2 qt. 
 
LESSON THIRTY-SECOND. 81 
 
 8. Edward can -walk 4 miles per hour, and Herma-n can walk 
 3. How far can Edward walk in 6 hours ? Can Herman ? 
 
 9. Richard bought 8 newspapers at 2 cents a-piece, and scld 
 them for 4 cents a-piece. How much did he gain on them ? 
 
 10. If Daniel has 50 chestnuts, how many wiU he have left 
 after giving 4 of his companions 9 chestnuts a-piece ? 
 
 11. I bought 9 yards of cloth at $4 per yard, but it being 
 damaged, I was obliged to sell it for $12 less than it cost me. 
 For how much did I sell it ? 
 
 12. 1 bushel = * quarts? 
 
 13. 1 yard r= -x- nails ? 
 
 14. 4 pt. = ^ gills ? 
 
 15. Arthur had a basket which held just 4 qt., and he picked 
 nuts enough to fill it 6 times. How many quarts did he pick ? 
 How many pecks ? 
 
 16. How many oranges at 4 cents each can be bought for 6 
 three-cent pieces and 2 cents ? 
 
 17. How many apples at 3 cents a-piece can be bought for 6 
 oranges at 4 cents each ? 
 
 18. How many pairs of boots at $5 a pair, can be bought for 
 10 yards of cloth at $3 per yard ? 
 
 19. A man bought 10 quarts of berries, which he put into boxes 
 each holding 5 pints. How many pints of berries did he buy ? 
 How many boxes did he fill ? 
 
 20. Lucius is shelling corn into a three-peck measure, which he 
 empties into a bin large enough to hold 6 bushels 3 pecks. How 
 many pecks must he shell to fill the bin ? How many measure- ' I 
 fuls? 
 
 21. A newsboy sold 9 papers at 8 cents a-piece, and after 
 spending 9 cents, gave the rest of his money for papers at 2 cents 
 a-piece. How many papers did he get ? 
 
82 COLBURN*S FIRST PART. 
 
 22. A man gave 8 hats at $4 a-piece, and $8 in money for coats 
 at $10 a-piece. How many coats did he receive ? 
 
 23. A man put 7 gallons 2 quarts of molasses into jugs each 
 holding 3 quarts. How many jugs did he fill ? 
 
 24. Rufus had a string 5 yards 1 foot long, which he cut into 
 pieces just 2 feet long. How many pieces did it make ? 
 
 25. An apothecary put 6 .^ 1 9 of powders into papers, each 
 holding 2 9. How many papers did he fill ? 
 
 LESSON XXXIII. 
 
 A. Add 10 fives together. 
 
 5 times 5 = 25. 
 
 6 times 6, or 6 times 5 = 30. 
 6 times 7, or 7 times 6 = 35. 
 6 times 8, or 8 times 6 = 40. 
 6 times 9, or 9 times 5 = 45. 
 
 5 times 10, or 10 times 5 = 50. 
 
 B. Add 10 sixes together. 
 
 6 times 6 = 36. 
 
 6 times 7, or 7 times 6 == 42. 
 6 times 8, or 8 times 6 = 48. 
 6 times 9, or 9 times 6 = 54. 
 6 times 10, or 10 times 6 = 60. 
 
 CI. *X6==36? 6. 42=r*x7? 
 
 2. *X8 = 48? 6. 54 = *x9? 
 
 8. *x5 = 45? 7. 54 = *x6? 
 
 4. *X6 = 30? 8. 25=:*x5? 
 
LESSON THIRTY-THIRD. 83 
 
 D. 1. 9x* = 54? 6. 60 = 6 X*? 
 
 2. 7X*=35? 6. 40 = 8 X*? 
 
 3. 6 X ^ = 48? 7. 86 = 6 X *? 
 
 4. 8X* = 40? 8. 42 = 7 X*? 
 
 Explanation. — To diride a nnmber by 2 is to find how many times 
 
 2 equal it. 
 
 To divide a number by 6 is to find how many times 6 equal it 
 Hence, 35 divided by 5 = 7i for 35 = 7 times 5; 48 divided by 
 
 8 = 6, for 48 = 6 times 8. 
 
 Division, then, is the process of finding how many times one number 
 must be taken to equal another number 
 
 The number to bo divided is called the dividend. The number br 
 Trhich we divide is called the divisor, and the answer is called the 
 
 QUOTIENT. 
 
 Thus, in 35 divided by 7 == 5, 35 is the dividend, 7 is the divisor, 
 5 is the quotient. 
 
 In 54 =» * times 6, 54 is the dividend, & is the divisor, and 9, the an- 
 swer, is the quotient. 
 
 Name the divisor, dividend, and quotient of the following 
 examples : — 
 
 28 divided by 7 = 4. 86 = * x 6 ? 
 
 48 divided by 6 = 8. 20 = * x 5 ? 
 
 85 divided by 7 = 5. 28 = * X 7 ? 
 
 By these illustrations it appears that division is just the reverse 
 of multiplication. 
 
 A horizontal line with one dot above, and another below it, 
 forms the sign of division, thus : —. It may be read *' divided by." 
 28 -r 7 = 4 may be read, "28 divided by 7 equal 4." 
 
 F. What is the quotient of — 
 
 1. 16 ~- 4? 4. 25 — 5? 7. 18-^3? 
 
 2. 24—3? 6. 48-7-6? 8. 24—8? 
 
 3. 32 -f- 8 ? 6. 42 -T- 7 ? 9. 64 -7- 9 ? 
 
84 colburn's first part. 
 
 Ct. 1. 5 wk. Id. -f- 1 wk. 2 da. ? 
 
 Solution. — 5 wk. 1 da. — 36 daj'^s ; 1 wk. 2 da. = 9 days j and 36 
 da. = 4 times 9 da. Hence, 5 wk. 1 da. -r 1 wk. 2 d. = 4. 
 
 2. 3 pk. 6 qt. -r- 1 pk. 2 qt. 
 
 3. 6 yd. 2 ft. -^ 1 yd. 2 ft. 
 
 4. 9 gal. -i- 2 gal. 1 qt. 
 
 5. 6 wk. 3 da. -r 1 wk. 2 da. 
 
 6. 3g63-M§23. 
 
 7. 3 sq. yd. 6 sq. ft. ~ 8 sq. ft. 
 
 8. I sold 7 quarts of chemes at 6 cents per quart, and one 
 quart for 8 cents. How much did I receive ? I expended the 
 money thas received for rice at 5 cents per pound. How many 
 pounds of rice did I buy ? 
 
 9. If George walks at the rate of 15 rods per minute, and Wil- 
 liam walks at the rate of 21 rods per minute, how many more rods 
 per minute does William walk than George ? How many more 
 rods in 9 minutes. 
 
 10. If Susan gains 8 merit-marks per day, and loses 2 per day, 
 how many will she have at the end of 8 days ? 
 
 11. A man sold 5 pecks of chestnuts at the ratie of one dime 
 per quart. How many dimes did he receive ? 
 
 12. I sold 6 quarts of blackberries at the rate of 10 cents per 
 quart, and received in payment 5 three-cent pieces, and the rest 
 in half-dimes. How many half-dimes did I receive ? 
 
 13. Abner and Lemuel were in a store together, and their father 
 tt)ld them that they might each have 8 oranges worth 6 cents a- 
 piece, or 9 worth 5 cents a-piece. Abner chose the former, and 
 Lemuel the latter. How much more were Abner's oranges worth 
 than Lemuel's ? 
 
 14. How many bags, each containing 1 bu. 1 pk., can be filled 
 from 8 bu. 3 pk. of meal ? 
 
LESSON THIRTY- FOURTH. 
 
 15. How many house-lots, each containing 1 A. 2 R., can be 
 made from a piece of land containing 10 A. 2 R. ? 
 
 16. How many pictures at 2d. 1 qr. each can be purchased for 
 6d. 3 qr. ? 
 
 17. How many bushels in 8 bags, each containing 3 pecks ? 
 
 18. A furniture-dealer gave 6 bureaus, worth 7 dollars a-piece, 
 and 3 dollars in money, for chairs at 9 dollars per dozen. How 
 many dozen chairs did he buy ? 
 
 19. A fur-dealer gave 8 caps, at 5 dollars a-piece, and 2 dollars 
 in money, for muffs at G dollars a piece. How many muffs did he 
 receive ? 
 
 20. A boy earned 12 cents by doing some errands, and invested 
 the money in papers at 2 cents a-piece. He sold the papers at 
 4 cents each, and with the money received for them he bought 
 papers at 3 cents a-piece. He sold C of the papers for 5 cents 
 a-piece, and the rest for 2 cents a-piecc. He then spent 4 cents 
 for crackers, and gave the rest of his money for some very nice 
 Havana oranges at 6 cents a-piece. He gave 1 of the oranges to 
 his mother, and sold the rest at 8 cents a-piece. How much did 
 he receive for them ? 
 
 LESSON XXXIV. 
 
 7 times 7 = 49. 
 
 7 times 8, or 8 times 7 = 56. 
 
 7 times 9, or 9 times 7 = 63. 
 
 7 times 10, or 10 times 7 = 70. 
 
 8 times 8 = 64. 
 
 8 times 9, or 9 times 8 = 72. 
 8 times 10, or 10 times 8= 80. 
 
86 COLBURN*S FIRST PART. 
 
 9 times 9 = 81. 
 
 9 times 10, or 10 times 9 = 90. 
 
 10 times 10 = 100. 
 
 B. 66==*x8? 90 = *xl0? 
 81 = * X 9? 49_ ^e >< 7^ 
 
 63 = ^X7? 100=*xlO? 
 
 64 = ^X8? 72 = *x8? 
 
 C. 81-7-9? 56-^8? 100—10? 
 36-7-4? 21 -j- 7? 24 -r 8? 
 48 -i- 8? 45 -f- 6? 72 H- 9? 
 72 -f- 9? 81-7-9? 63-7-7? 
 
 D. 1. 7 times 8, plus 7, divided by 7, multiplied by 4 = -k- 
 times 6 ? 
 
 Solution. — 7 times 8 = 56, plus 7 =« 63, divided by 7 =* 9, multi- 
 plied by 4 == 36, which equals 6 times 6. 
 
 Note. — Let the pupil learn to call only the results, thus : 56, 63, 
 9, 36, 6. Let them also perform the work as the Teacher reads the 
 example. 
 
 2. 8 times 10, minus 8, divided by 9, plus 1, multiplied by 6, 
 minus 5 = * times 7 ? 
 
 3. 6 times 8, minus 12, divided by 4, multiplied by 3, plus 10 
 times 4, minus 3 = * times 8 ? 
 
 4. 7 times 6, plus 3, divided by 9, multiplied by 5, plus 3, 
 divided by 7, multiplied by 6, minus 2 = * times 2 ? 
 
 5. 3 times 8, plus 4 times 9, plus 21, divided by 9, multiplied 
 by 2, plus 7, divided by 5 ? 
 
 E. 1. How many are 9 times 2 yds. 2 ft. ? 
 
 Solution. — 9 times 2 yd. = 18 yd.; 9 times 2 ft. == 18 ft. =- 6 
 yds., which, added to 18 yd. = 24 yd. Hence, 9 times 2 yd. 2 ft. 
 24 yd. 
 
LESSON THIRTY-FOURTH. 87 
 
 2. 4 times 7 gal. 3 qt. ? 6. 9 times 3 sq. yd. 4 sq. ft. ? 
 
 3. 7 times 8 wk. 4 da. ? 6. 6 times 7 bu. 6 pk. ? 
 
 4. 6 times 9 yd. 2 ft. ? 7. 6 times 8 gal. 2 qt. ? 
 
 8. Bought 6 bags, each containing 8 bu. 2 pk. of peanuts, and 
 put them into casks each holding 3 bushels. How many casks 
 did they fill? 
 
 9. Austin had a basket which held just 3 pk. 4 qt., and an- 
 other which held just 6 pecks. He gathered nuts enough to fill 
 the smallest basket 10 times. How many times would they fill 
 the large basket? 
 
 10. A lady bought cotton sheeting enough to make 8 sheets, 
 each containing 6 yd. 1 qr., but afterwards concluded to put 6 
 yards in a sheet. How many sheets could she make ? 
 
 11. I bought a vessel of oil containing 32 quarts, and after 
 using 1 gal. 3 qts. of it, I sold the rest at the rate of 1 dollar for 
 1 gal. 1 qt. How much did I get for it ? 
 
 12. Sarah's mother oflfered to give her 8 large oranges worth 5 
 cents a-piece if she would tell her how many seven-quart baskets 
 could be filled from 5 pk. 2 qt. of berries. Sarah answered cor- 
 rectly. What was her answer ? She exchanged the oranges for 
 their value in drawing-pencils worth 10 cents a-piece. How many 
 pencils did she get ? 
 
 13. A fruit-peddler paid 48 cents for cherries at 6 cents a quart, 
 which he put into papers, each containing 1 gill. How many 
 papers did it take ? 
 
 14. By bujdng a lot of wood at $4 per cord, and selling it at $6 
 per cord, I gained $18. How many dollars did I gain on each 
 cord ? How many cords did I buy ? How many dollars did I 
 pay for the entire lot ? 
 
 15. By buying flour at $5 per barrel, and selling it for $8 per 
 
88 colburn's first part. 
 
 barrel, I gained $24. How many barrels did I buy, and what did 
 
 1 pay for the lot ? 
 
 16. By buying a lot of cloth for $5 per yard, and selling it at 
 $3 per yard, I lost $20. IIow many dollars did I pay for the 
 lot? 
 
 17. A man bought a lot of coal at $4 per ton, and sold it for 
 $8 per ton, by which he gained $36. How many dollars did he 
 pay for it ? 
 
 18. A laborer worked 6 weeks for 9 dollars per week, and put- 
 ting 6 dollars with the money thus earned, he bought coal at 6 
 dollars per ton. How many tons did he buy ? After laying aside 
 
 2 tons for his own use, he sold the remainder for 7 dollars per 
 ton, receiving in payment 2 dollars in money, and the rest in 
 flour at 9 dollars per barrel. How many ban*els of flour did he 
 receive ? 
 
 19. Robert had a basket holding 2 quarts 1 pint, and he 
 gathered chestnuts enough to fill it 8 times. How many four- 
 quart baskets could he fill with what he gathered ? 
 
 20. Augustus had 8 quarts of blackberries. He sold 1 quart 
 for 11 cents, and the rest for 10 cents per quart. With the money 
 received for them he bought cocoa-nuts at 9 cents a-piece. How 
 many cocoa-nuts did he buy ? He divided 2 of them among his 
 companions, and sold the rest for 10 cents a-piece. He gave 7 
 cents to a poor woman, and spent the rest of his money for fire- 
 crackers, at 7 cents a bunch. How many bunches did he buy ? 
 
LESSON THIRTY-FIFTH. 89 
 
 LESSON XXXV. 
 
 BiUs. 
 
 A. 1. Mr. Edward Crane keeps a store in Boston. On the 
 21st of February, 1856, lie sold to Mr. Alfred Hall 7 yards of 
 broadcloth at $5 per yard, 9 yards of cassimere at $2 per yard, 
 and 8 yards of doeskin at $3 per yard. Mr. Hall paid for them, 
 and asked Mr. Crane to make out a hillf i. e.j a written statement 
 of the transaction. He did it as follows : — 
 
 y yc/^j., SSzoac/ciomj a ^5 . . . ^35 
 P "^ad, %addime^ej a j&S . . ^o 
 
 
 J^77 
 
 If the goods had not been paid for, Mr. Crane would not have 
 receipted the bill, i. «., he would not have put his name after the 
 words "received payment." 
 
 Note to the Teacher. — A more full explanation may be found 
 in "Arithmetic and its Applications." 
 
90 colburn's first part. 
 
 B. Make out proper bills for each of the following transactions, 
 observing to give : — 
 
 1st. The date, t. «., the place and time. 
 2d. The names of the parties. 
 
 3d. The articles bought, with their prices and amount. 
 4th. The words ** received payment,** 
 
 6th. 1/ the good* art paid for , or bought for cash, the name of the 
 seller. 
 
 Note. — It may add to the interest and value of the following pro- 
 blems, to have each pupil write his own name, and that of some com- 
 panion, in the place of those here written. 
 
 1. John Brown, of New York, sold to Martin Draper, for cash, 
 July 16th, 1856, 9 bbls. of flour at $10 per barrel, and 8 bushels 
 of wheat at $2 per bushel. 
 
 2. William Fuller, of Chicago, sold to Ai-thur Simmons for cash, 
 Jan. 8th, 1856, 9 silk hats at $4 a-piece, 8 cloth caps at $2 
 a-piece, 6 beaver hats at $5 a-piece, and 11 di-ab hats at $1 
 a-piece. 
 
 3. Henry Mitch el & Co. sold to Francis Baker, on credit (t. «., 
 Mr. Baker at the time did not pay for them), May 1st, 1855, 6 
 bbls. of apples at $3 per barrel, 9 bbls. of potatoes at $2 per 
 barrel, 7 boxes of raisins at $3 per box, and 5 drums of figs at 
 $2 per drum. 
 
 Note. — Let the students now make out bills for several imaginary 
 transactions. 
 
 C. If in the transaction described under letter A., Mr. Hall had 
 paid $2 in money and delivered to Mr. Crane 3 coi*ds of wood at 
 $9 per cord, the bill would have been made out as follows : 
 
LESSON THIRTY-FIFTH, 
 
 91 
 
 ^o.^lon, c%/ i^/. 
 
 *^B56. 
 
 
 
 .^35 
 
 yc/d. S)o6d4cro^ a p3 
 
 . SA 
 
 ^77 
 
 i' 9;. 
 
 
 i ^ "^a^A 
 
 ps 
 
 
 . 2A 
 
 ^36 
 
 /4/ 
 
 Make out a bill for the following transaction : — 
 
 
 1. Moses White, a trader of Philadelphia, sold to Joseph Aus- 
 tin, May 17th, 1856, 7 ploughs at $9 each, 5 hay-cutters at $8 
 a-piece, 4 doz. scythes at $9 per dozen, and 8 doz. rakes at $3 
 per dozen ; and in part payment, Mr. Austin gave him 8 cords of 
 wood at $7 per cord, 4 cords of wood at $6 per cord, and $13 in 
 
 money. 
 
 
 Note. — Ld-t thf scholars make out bills for several imaginary trans- 
 *'^tions. 
 
 .' .: • . 
 
92 
 
 COLBURN S FIRST PART. 
 
 LESSON XXXVI. 
 
 A. 1. 29 = * times 6 ? 
 
 First Solution.— 29 = 24 + 5^ and 24 = 4 times 6. Hence 29 
 = 4 times 6 with 5 remainder. 
 
 Second Solution. — 29 = 4 times 6 with 5 remaining, for 4 times 
 6 = 24, and 5 = 29. 
 
 Note. — The remainder is really an undivided part of the dividend, 
 and might be subtracted from it without affecting the quotient. In 
 reality, but a part of the dividend is divided. 
 
 2. 46 = * times 5 ? 
 
 3. 31 = ^ times 9 ? 
 
 4. 67 = * times 7 ? 
 
 5. 88 = -Jf times 9 ? 
 
 6. 61 = * times 8 ? 
 
 7. 37 = * times 7 ? 
 
 B. 1. 42 — 4? 
 
 2. 83-^9? 
 
 3. 75 -^ 8 ? 
 
 4. 24 -f- 7? 
 
 13. 
 
 6. 19 ~ 8 ? 
 
 6. 47—7? 
 
 7. 39 ~- 6 ? 
 
 8. 51-^8? 
 
 8. 63 = * times 6 ? 
 
 9. 27 = * times 8? 
 
 10. 62 = # times 9 ? 
 
 11. 48 = -5^ times 7 ? 
 
 12. 69 = * times 9 ? 
 23 = * times 11 ? 
 
 9. 43 -f- 9? 
 
 10. 43 -r- 4 ? 
 
 11. 47 -r 8? 
 
 12. 47 -T- 6 ? 
 
 C. The comma, when used in connexion with Arithmetical signs, 
 shows that the result of all the preceding operations is to be con- 
 sidered in connexion with the sign following it. 
 
 Thus : "4 X 9, -r- 7," means that 4 times 9, or 36, is to be divided 
 by 7. I 
 
 **6 X 7 4- 18, -7- 9," means that the sum of 6 times 7 plus 18, »= 1 
 42 -|- 18 = 60, is to be divided by 9. 
 
LESSON THIRTY-SIXTH. 93 
 
 1. 6x9,-8? 5. 9x6+10, = *X 8? 
 
 2. 4x6, -T-T? 6. 4x9+17, = *X 6? 
 
 3. 5x9 + 8,-7-6? 7. 7 X 7 + 13, = * X 10? 
 
 4. 8 X 8 — 13, -r 9 ? 8. 9 X 9 -- 43, = * X 4 ? 
 
 D. 1. 41 da. = * wk.? 
 
 Solution. — Since 7 days = 1 wk., 41 da. must equal as many wk. 
 as there are times 7 in 41, which are 5 times with 6 remainder. Hence 
 41 da. = 5 wk. 6 da. 
 
 2. 
 
 33 qr. == * yd. ? 
 
 7. 
 
 46 fur. = * m. ? 
 
 3. 
 
 37,:^ =^§? 
 
 8. 
 
 69 da. = * wk. ? 
 
 4. 
 
 70 qt. = * pk. ? 
 
 9. 
 
 37 cd. ft. = * cd. ? 
 
 5. 
 
 19 ft. = * yd. ? 
 
 10. 
 
 60 sq. ft. = * sq. yd. ? 
 
 6. 
 
 23 ft. = ^ yd. ? 
 
 11. 
 
 20 m. = * le. ? 
 
 12. Moses has 35 cents, with which he wishes to buy oranges 
 at 6 cents a-piece. How many oranges can he buy ? 
 
 Solution. — If he can buy 1 orange for 6 cents, he can buy as many 
 oranges for 35 cents as there are times 6 in 35, which are 5 times and 
 5 remainder. Therefore, he can buy 5 oranges, and have 5 cents 
 remaining. 
 
 13. How many barrels of flour, at $6 per baiTel, can be bought 
 for $58 ? 
 
 14. How many shawls, at $7 a-piece, can be bought for $39 ? 
 16 How many books, at $3 a-piece, can be bought for $29 ? 
 
 16. How many terms tuition, at $7 per term, will $27 pay for? 
 
 17. How many hats, at $4 each, can be bought for $39? 
 
 18. How many cans, each holding 1 gal. 1 qt., can be filled 
 from 3 gal. 3 qt. of milk? 
 
94 colburn's first part. 
 
 19. A person who owes $39, wishes to pay as much as possible 
 m five-dollar bills, and the rest in one-dollar bills. How many 
 bills of each kind must he pay ? 
 
 20. William did 8 errands, for each of which he received 3 
 cents, and he had 11 cents before he did the errands. He bought 
 as many writing-books at 9 cents a-piece as he could pay for, and 
 spent the rest of his money for pens at 2 cents a-piece. How 
 many writing-books did he buy ? How many pens? 
 
 21. A tailor paid $35 for silk velvet at $5 per yard. He made 
 it into vests, putting 3 quarters into each vest. How many vests 
 did he make, and how many quarters had he remaining ? 
 
 22. One " Fourth of July" Thomas had 29 cents. He bought 
 as many bunches 'of crackers at 10 cents per bunch, as he could 
 pay for, and then spent the rest of his money for cherries, at the 
 rate of 7 for a cent How many bunches of crackers did he buy ? 
 How many cherries ? 
 
 23. A person who had 20 cents, said to a boy: **If you will 
 tell me how many loaves of bread, at 6 cents per loaf, I can buy 
 with my money, I will give you what there is left after paying for 
 the bread." The boy answered right. What was his answer? 
 How many cents ought the person to give him ? 
 
 21. If it requires 3 yards of broadcloth to make a coat, how 
 many coats can be made from a piece containing 29 yards of 
 broadcloth ? How many yards will be left after making the coats ? 
 If one yard of cloth will make 3 vests, how many vests can be 
 made from what remains, after making the coats ? 
 
 25. Lyman has 28 cents, Horace has 50. Chester has 63, and 
 Isaac has 47. Each bought as many pencils at 8 cents a-piece as 
 he could pay for, and gave the rest of his money to a poor wo- 
 man. How many pencils did each buy, and how many cents had 
 each to give the poor woman ? How many pencils were bought 
 in all ? How many cents did the woman receive ? 
 
 I - I 
 

 LESSON 
 
 THIRTY-SEVENTH 
 
 95 
 
 
 LESSON XXXVII 
 
 
 
 
 A. 1. 2 times 3 tens 
 
 ? Then 8 times 30 ? 
 
 
 
 2. 6 times 7 tens 
 
 ? Then 5 times 70 ? 
 
 
 
 3. 8 times 4 tens ? Then 8 times 40 ? 
 
 
 
 4. 7 times 3 tens ? Then 7 times 30 ? 
 
 
 B. 1, 
 
 4X8? 
 
 4. 
 
 7x9? 
 
 7. 
 
 8X7? 
 
 2 
 
 4x 30? 
 
 6. 
 
 7X90? 
 
 8. 
 
 8x70? 
 
 3. 
 
 40x3? 
 
 6. 
 
 70 X 9? 
 
 9. 
 
 80 X 7? 
 
 C 1. 
 
 6 X 40? 
 
 5. 
 
 6x 30? 
 
 9. 
 
 9x 60? 
 
 o 
 
 9 X 30? 
 
 6. 
 
 60 X 3? 
 
 10. 
 
 8x 90? 
 
 3. 
 
 6x 40? 
 
 7. 
 
 9 X 70? 
 
 11. 
 
 7 X 70? 
 
 4. 
 
 40 X 4? 
 
 8. 
 
 90 X 7? 
 
 12. 
 
 8 X 80? 
 
 D. 1. 
 
 12- 
 
 -4? 
 
 4. 
 
 64 ~ 8? 
 
 7. 
 
 54 -r- 9? 
 
 2. 
 
 120 H 
 
 -4? 
 
 5. 
 
 640 -f- 8? 
 
 8. 
 
 640 -T- 90 ? 
 
 3. 
 
 120- 
 
 - 40? 
 
 6. 
 
 640 -r 80? 
 
 9. 
 
 540 -r 9? 
 
 E. 1. 
 
 630- 
 
 r 9? 
 
 
 8. 420- 
 
 ^60? 
 
 
 2. 
 
 720- 
 
 -8? 
 
 
 9. 90- 
 
 - 30? 
 
 
 3. 
 
 560- 
 
 -7? 
 
 
 10. 270- 
 
 - 30? 
 
 
 4. 
 
 250 H 
 
 -6? 
 
 
 11. 420- 
 
 - 6? 
 
 
 6. 
 
 240- 
 
 - 3? 
 
 
 12. 810- 
 
 - 9? 
 
 
 6. 
 
 720- 
 
 - 80? 
 
 
 13. 810 - 
 
 - 90? 
 
 
 7. 
 
 180- 
 
 r 3? 
 
 
 14. 360 - 
 
 -40? 
 
 
 F. 1. 6 X 47? 
 
 
 
 
 
 S0LUTION.—6 times 40 = 240 ; 
 = 282. Therefore, 6 times 47 = 
 
 6 times 7 = 42 
 
 = 282, 
 
 , which, 
 
 added to 240 
 
r^^ ^ 
 
 ,96 colburn's iirst part. 
 
 Note. — As soon as the principle is understood, the pupil should 
 solve such problems by naming only the results. Thus: 6 times 
 47 = 240 4- 42 = 282. 
 
 2. 7 X 96 ? 7. 4 X 27 ? 12. 3 x 37 ? 
 
 3. 8 X 34? 8. 9 X 82? 13. 6 x 28 ? 
 
 4. 9 X 37 ? 9. 6 X 97 ? 14. 6 X 43 ? 
 
 5. 6x94? 10. 4x23? 15. 9x81? 
 G. 8x23? 11. 7x94? 16. 7x63? 
 
 G. "When we wish to write the work, we may, if we choose, 
 solve exaipples in multiplication, as explained in the following 
 solution of the first question under F. 
 
 A7 
 
 Explanation. — 6 times 7 units =-= 42 units = 4 tens, 
 and 2 units. "Writing the 2 units, and reserving the 4 tens 
 A to add to the product of the tens, we have 6 times 4 tens= 
 
 24 tens, to which, adding the 4 tens from the former pro- 
 duet, gives 28 tens, which wo write. The answer, then, 
 is 282. 
 
 Note. — It will be seen that when we do not write the work, we 
 begin at the left hand to multiply; and when we do write it, we begin 
 at the right hand. 
 
 Perform in this way the examples under letter F. 
 
 H. 1. 6 times 498 ? 
 
 Solution. — 6 times 8 units = 48 units == 4 tens and 8 units. 
 Writing the 8 units, and reserving the 4 tens to add to the product of 
 the tens, we have 6 times 9 tens = 54 tens, and 4 tens added, equal 
 58 tens = 5 hundreds and 8 tens. "Writing 8 
 tens, and reserving the 5 hundreds to add to 
 the product of the hundreds* column, we have 
 6 times 4 hundreds = 24 hundreds, and 5 hun- 
 dreds added = 29 hundreds == 2 thousands 
 ^^ and 9 hundreds, which being the last product 
 
 we write, the answer then is 2988. 
 
 ^j?^ 
 
 ^ 
 
 ^0 
 
LESSON THIRTY-EIGHTH. 97 
 
 Note. — When the reductions are fully mastered, abbreviated forms 
 
 like the following may be introduced with advantage. 
 
 6 times 8 = 48. Write 8, and add 4 to the next product. 6 times 
 
 9 are 54 and 4 are 58. 
 4 are 24 and 5 are 29, 
 
 W^rite 8 and add 5 to the next product. 6 times 
 which we write. Hence, 6 times 498 »=- 2988. 
 
 The following form of naming only results should finally be 
 
 adopted. 48 units ; 64, 58 tens ; 24, 29 hundreds, ^n*.-— 2988. 
 
 2. 9 X 847 ? 
 
 8. 
 
 6 times $2.75? 
 
 8. 8 X 298 ? 
 
 9. 
 
 4 times $8.76? 
 
 4. 4x746? 
 
 10. 
 
 9 times $32.75? 
 
 6. 8 X 327 ? 
 
 11. 
 
 8 times $27.84 ? 
 
 6. 4 X 238 ? 
 
 12. 
 
 5 times $97.83 ? 
 
 7. 6 X 379 ? 
 
 13. 
 
 4 times $28.59 ? 
 
 LESSON XXXVIII. 
 
 A. 1. 476 ^ 
 
 7? 
 
 
 Solution. — 7 is contained in 47 tens, 6 tens* 
 
 times, with 5 teni re- 
 
 maining. 5 tens «= 50 
 
 units, and 6 units added, are 56 units, 7 is 
 
 contained in 56 units 8 
 
 units' times. Hence, 476 -r 7 «« 60 -f- 8 — « 68. 
 
 Note. — Not till the reductions are fully understood, should the 
 pupil be allowed to abbreviate this explanation to the common one : *' 7 
 is contained in 47, 6 times, with 5 remainder. 7 is contained in 56, 8 
 times. Hence the quotient is 68." 
 
 2. 315 — 8? 
 
 6. 672 -^ 8? 
 
 10. 429 -7- 8 ? 
 
 8. 216 — 4? 
 
 7. 144 -H 6? 
 
 11. 413 — 5? 
 
 4. 392 -. 8 ? 
 
 8. 279 -~ 9 ? 
 
 12. 673—7? 
 
 5. 217—7? 
 
 9. 137-^2? 
 
 13. 528 ~- 9 ? 
 
98 
 
 COLBtlRN S FIRST PART. 
 
 Examples in division are performed and explained in the same 
 manner when we write the work as when we do not. The work 
 of the first example, letter A., would usually be written as in the 
 annexed model : 
 
 7y76 
 
 6B 
 
 B. 1. 2738 -4- 8 ? 
 
 Solution. — 8 is contained in 27 hundreds, 3 hundred times with 3 
 hundreds remaining. "VVe therefore write 3 as the hundreds* figure 
 ' Q Q ^ of the quotient. The 3 hundreds re 
 
 })27t 
 
 -2 
 
 maining = 30 tens, and 3 tens added 
 
 r= 33 tens. 8 is contained in 33 tens, 
 
 *^ //Q) 4r tens times, with 1 ten remaining. 
 
 We therefore write 4 as the tens figure 
 of the quotient. The 1 ten remaining = 10 units, and 8 units added = 
 IS units. 8 is contained in 18 units 2 units' times and 2 units re- 
 maining. Therefore, 2738 -*- 8 == 3 hundreds, 4 tens, and 2 units, or 
 342 with a remainder of 2. 
 
 Note. — The remainder is written after the quotient with the sign 
 of subtraction, to show that it is an undivided part of the dividend. 
 
 2. 4756 -T- 4 ? 
 
 8. 3297 -7- 6 ? 
 4. 4347-7-9? 
 6. 2981 ^ 11 ? 
 
 6. 3297 -r 6 ? 
 
 7. '43G1 -^ 5 ? 
 a 2459 -f 8 ? 
 
 9. 4272 ~ 12 ? 
 10. 8943 ^ 9 ? 
 
 11. 
 
 2137- 
 
 -5? 
 
 12. 
 
 4264 H 
 
 - 13? 
 
 13. 
 
 8375- 
 
 - 12? 
 
 14. 
 
 2986- 
 
 r 4? 
 
 15. 
 
 3176 H 
 
 - 8? 
 
 16. 
 
 4327 H 
 
 - 9? 
 
 17. 
 
 2052- 
 
 - 3? 
 
 18. 
 
 1379- 
 
 -2? 
 
 19. 
 
 7436- 
 
 - 8? 
 
LESSON THIRTY-NINTH. 99 
 
 
 
 LESSON 
 
 XXXIX. 
 
 The following tables, if thoroughly learned, will save a yast 
 
 deal of labor in the Arithmetical operations of life. A distin- 
 
 guished educator, 
 
 now Superintendent of Schools in one of the 
 
 principal cities of the Union, says 
 
 that in his opinion, a knowledge 
 
 of these tables would save hours of valuable time, not only to the 1 1 
 
 student. 
 
 but to the business man. 
 
 With most classes, the Teacher 
 
 will find it desirable to give additional exercises similar to those 1 1 
 
 of the preceding Lessons. 
 
 
 11 X 
 
 2, or 2 X 
 
 11 = 22. 
 
 16 X 2, or 2 X 16 = 32. 
 
 12 X 
 
 2, or 2 X 
 
 12 == 24. 
 
 17 X 2, or 2 X 17 = 34. 
 
 13 X 
 
 2, or 2 X 
 
 13 = 26. 
 
 18 X 2, or 2 X 18 = 36. 
 
 14 X 
 
 2, or 2 X 
 
 14 = 28. 
 
 19 X 2, or 2 X 19 = 38. 
 
 15 X 
 
 2, or 2 X 
 
 15 == 30. 
 
 20 X 2, or 2 X 20 = 40. 
 
 11 X 
 
 3, or 3 X 
 
 11 = 33. 
 
 16 X 3, or 3 X 16 = 48. 
 
 12 X 
 
 3, or 3 X 
 
 12 = 36. 
 
 .17 X 3, or 3 X 17 = 51. 
 
 13 X 
 
 3, or 3 X 
 
 13 = 39. 
 
 18 X 3, or 3 X 18 = 54. 
 
 14 X 
 
 3, or 3 X 
 
 14 = 42. 
 
 19 X 3, or 3 X 19 = 57. 
 
 15 X 
 
 3, or 3 X 
 
 15 = 45. 
 
 20 X 3, or 3 X 20 == 60. 
 
 11 X 
 
 4, or 4 X 
 
 11 = 44. 
 
 16 X 4, or 4 X 16 = 54 
 
 12 X 
 
 4, or 4 X 
 
 12 = 48. 
 
 17 X 4, or 4 X 17 = 68. 
 
 13 X 
 
 4, or 4 X 
 
 13 = 52. 
 
 18 X 4, or 4 X 18=: 72. 
 
 14 X 
 
 4, or 4 X 
 
 14 == 5G. 
 
 19 X 4, or 4 X 19 = 76. 
 
 15 X 
 
 4, or 4 X 
 
 15 = 60. 
 
 20 X 4, or 4 X 20 = 80. 
 

 100 
 
 colburn's 
 
 FIRST PART. 
 
 
 11 X 
 
 5, or 5 
 
 X 
 
 11 = 55. 
 
 16 X 5, or 5 
 
 X 16 = 80. 
 
 
 12 X 
 
 5, or 5 
 
 X 
 
 12 = 60. 
 
 17 X 5, or 5 
 
 X 17 == 85. 
 
 
 13 X 
 
 5, or 5 
 
 X 
 
 13 = 65. 
 
 18 X 5, or 5 
 
 X 18 = 90. 
 
 
 14 X 
 
 5, or 5 
 
 X 
 
 14 = 70. 
 
 19 X 5, or 5 
 
 X 19 = 95. 
 
 
 15 X 
 
 5, or 5 
 
 X 
 
 15 = 75. 
 
 20 X 5, or 5 
 
 X 20 = 100. 
 
 
 11 X 
 
 6, or 6 
 
 X 
 
 11 = 66. 
 
 16 X 6, or 6 
 
 X 16 = 96. 
 
 
 12 X 
 
 6, or 6 
 
 X 
 
 12 = 72. 
 
 17 X 6, or 6 
 
 X 17 = 102. 
 
 
 13 X 
 
 6, or 6 
 
 X 
 
 13 = 78. 
 
 18 X 6, or 6 
 
 X 18 = 108. 
 
 
 14 X 
 
 6, or 6 
 
 X 
 
 14 = 84. 
 
 19 X 6, or 6 
 
 X 19 = 114. 
 
 
 15 X 
 
 6, or 6 
 
 X 
 
 15 = 90. 
 
 20 X 6, or 6 
 
 X 20 =r 120. 
 
 
 11 X 
 
 7, or 7 X 
 
 11 = 77. 
 
 16 X 7, or 7 
 
 X 16 = 112. 
 
 
 12 X 
 
 7, or 7 
 
 X 
 
 12 = 84. 
 
 17 X 7, or 7 
 
 X 17 = 119. 
 
 
 13 X 
 
 7, or 7 
 
 X 
 
 13 = 91. 
 
 18 X 7, or 7 
 
 X 18 = 126. 
 
 
 14 X 
 
 7, or 7 
 
 X 
 
 14 = 98. 
 
 19 X 7, or 7 
 
 X 19 = 133. 
 
 
 15 X 
 
 7, or 7 
 
 X 
 
 15 = 105. 
 
 20 X 7, or 7 X 20 = 140. 
 
 
 11 X 
 
 8, or 8 
 
 X 
 
 11 = 88. 
 
 16 X 8, or 8 
 
 X 16 = 128. 
 
 
 12 X 
 
 8, or 8 
 
 X 
 
 12 = 96. 
 
 17 X 8, or 8 
 
 X 17 = 136. 
 
 
 13 X 
 
 8, or 8 
 
 X 
 
 13 = 104. 
 
 18 X 8, or 8 
 
 X 18 = 144. 
 
 
 14 X 
 
 8, or 8 
 
 X 
 
 14 == 112. 
 
 19 X 8, or 8 
 
 X 19 = 152. 
 
 
 15 X 
 
 8, or 8 
 
 X 
 
 15 = 120. 
 
 20 X 8, or 8 
 
 X 20 = 160. 
 
 
 11 X 
 
 9, or 9 
 
 X 
 
 11 == 99. 
 
 16 X 9, or 9 
 
 X 16 = 144. 
 
 
 12 X 
 
 9, or 9 
 
 X 
 
 12 = 108. 
 
 17 X 9, or 9 
 
 X 17 = 153. 
 
 
 13 X 
 
 9, or 9 
 
 X 
 
 13 = 117. 
 
 18 X 9, or 9 
 
 X 18 = 162. 
 
 
 14 X 
 
 9, or 9 
 
 X 
 
 14 = 126. 
 
 19 X 9, or 9 
 
 X 19 = 171. 
 
 
 15 X 
 
 9, or 9 
 
 X 
 
 15 = 135. 
 
 20 X 9, or 9 
 
 X 20 = 180. 
 
LESSON FORTIETH. 
 
 101 
 
 11 X 10, or 10 X 11 = 110. 
 
 12 X 10, or 10 X 12 = 120. 
 
 13 X 10, or 10 X 13 = 130. 
 
 14 X 10, or 10 X 1^ =■ 140. 
 
 15 X 10, or 10 X 15 = 150. 
 
 16 X 10, or 10 X 10 ^ 160. 
 
 17 X 10, or lOx 17===: 170. 
 
 18 X 10, or 10 X 18 = 180. 
 
 19 X 10, or 10 X 19 = 190. 
 
 20 X 10, or 10 X 20 == 200. 
 
 LESSON XL. 
 
 Note. — Should the Teacher deem it best, the class may omit this 
 nnd the next three Lessons, till after some of the first Lessons on 
 Fractions have been learned. 
 
 A. From the exercises of Lesson XXXVII , B. and C, we may 
 infer that 4 times 30 = 40 times 8 ; that 70 times 9 = 7 times 
 90, &c., &c. In like manner, 40 times 27 = 4 times 270, or, 4 
 tens' times 27 = 4 times 27 tens ; 80 times 436 = 8 times 4360, 
 or, 8 tens' times 436 = 8 times 436 tens, &c., &c. 
 
 1. What is the product of 70 times 389 ? 
 
 Solution. — 70 times 389 = 7 times 
 3890, which may be found by the method 
 explained in Lesson XXXVIL, G. and 
 H. Thus : 7 times units = units. 7 
 times 9 tens = 63 tens = 6 hundreds and 
 3 tens, Ac, &o. 
 
 70 
 
 2. 
 
 20 times 64 ? 
 
 3. 
 
 80 times 29 ? 
 
 4. 
 
 40 times 36 ? 
 
 5. 
 
 60 times 94 ? 
 
 6. 
 
 90 times 37 ? 
 
 7. 
 
 20 times 93 ? 
 
 8. 
 
 30 times 84 ? 
 
 9. 
 
 90 times 72 ? 
 
 26p30 
 
 10. 30 times 979 ? 
 
 11. 40 times 832 ? 
 
 12. 70 times 697 ? 
 
 13. 20 times 443 ? 
 
 14. 60 times 927 ? 
 
 15. 80 times 423 ? 
 
 16. 50 times 975 . 
 
 17. 30 times 476? 
 
 9* 
 
lOi: 
 
 COLBURN S FIRST PART. 
 
 B. Since 24 == 20 + 4, 24 times any number must equal 20 
 times that number plus 4 times that number. Since 86 = 80 -}- 6, 
 86 times any number must equal 80 times that number plus 6 
 times that number, &c., &c. 
 
 1. What is the product of 29 times 863 ? 
 S63 
 
 2p 
 
 Solution. — Since 29 = 20 + 9, 29 times 
 863 must equa^20 times 863, plus 9 times 863. 
 We first multiply by 9, and then by 20, by the 
 methods before explained, and add the products 
 together as in the written work at the left. 
 
 7767 
 
 ^726 
 
 250S7 
 
 2. 38 times 481 ? 
 
 3. 27 times 936 ? 
 
 4. 68 times 427 ? 
 
 5. 43 times 268 ? 
 
 6. 31 times 492 ? 
 
 7. 68 times 946 ? 
 
 8. 79 times 368 ? 
 
 9. 42 times 427 ? 
 
 10. 54 times 329? 
 
 11. 61 times 428? 
 
 LESSON XLI. 
 
 A. When the divisor is a large number, it is often convenient 
 or necessary to use the nearest number of tens, hundreds, or thou- 
 sands, as a trial divisor , to determine the probable quotient figure. 
 
 12. 
 
 89 X 2796 ! 
 
 13. 
 
 38 X 9582 ? 
 
 14. 
 
 22 X 4858 ? 
 
 15. 
 
 56 X 9375 ? 
 
 16. 
 
 4^ X 2401 ? 
 
 17. 
 
 63 X 2485? 
 
 18. 
 
 81 X 3258? 
 
 19. 
 
 69 X 2846 ? 
 
 20. 
 
 44 X 8132 ? 
 
 21. 
 
 74 X 9123 ? 
 
LESSON FORTY-FIRST. 
 
 103 
 
 Illustrations. — In dividing by 31, 32, 33, or 34, we may make 30 
 or 3 the trial divisor. In dividing by 36, 37, 38, or 39, we may make 
 40 or 4 the trial divisor. In dividing by 35, we may make either 30 
 or 40 the trial divisor. 
 
 B. 1. What is the quotient of 178 -^ 53? 
 
 Solution. — "We may make 50 or 5 the trial divisor, for 53 is con- 
 tained in 178 about the same number of times that 50 is ; or, that 5 is 
 contained in 17, which is 3 times. To test the correctness of this con- 
 clusion, we must find 3 times ^. It is 159, which, subtracted from 
 178, leaves 19, thus showing that 178 -^ 53 = 3 with 19 remainder. 
 
 The work would be written 
 by placing the divisor at the 
 left of the dividend, the quo- 
 tient at the right, and the pro- 
 duct with the remainder be- 
 neath the product. 
 
 63 
 
 W78 f3 
 /5p = 3x53 
 
 /p ^e, 
 
 Note. — The Teacher should illustrate and explain the method of 
 proceeding when the above process gives a trial quotient figure either 
 too large or too small. [See "Arithmetic and its Applications," 91st 
 article, and solution to 2d example, 113th page, and to 4th example,. 
 114th page.] . -^ 
 
 2. 96 
 
 3. 127 
 
 4. 228 • 
 6. 683 • 
 
 6. 281 
 
 7. 469 
 
 8. 356 
 
 9. 429 
 
 24? 
 31? 
 64? 
 82? 
 29? 
 48? 
 61? 
 67? 
 
 10. 
 
 256- 
 
 - 38? 
 
 11. 
 
 124- 
 
 - 19? 
 
 12. 
 
 387- 
 
 -45? 
 
 13. 
 
 621 - 
 
 -84? 
 
 14. 
 
 438- 
 
 - 62? 
 
 45. 
 
 279- 
 
 - 94? 
 
 16. 
 
 349- 
 
 - 82? 
 
 17. 
 
 624- 
 
 -79? 
 
104 
 
 COLBURN S FIRST PART. 
 
 C. What is the quotient of 2856 -h 59 ? 
 
 6pj 
 
 2856 ^AS 
 236 
 
 Explanation. — 69 is eo 
 near 60, that we make 6 the 
 trial divisor. Since 6 is con- 
 tained 4 times in 28, we make 
 4 the first figure of the quo- 
 tient, and infer that 59 is con- 
 tained 4 tens' times in 285 
 tens. Multiplying 59 by 4 
 tens, gives 236 tens. Hence 
 ^ we write 236 under the 285, 
 and subtract the former from the latter. It leaves a remainder of 49 
 or 49 tens, 490 units, to which, adding the 6 units, gives 496 units. 
 Since 6 is contained 8 times in 49, we make 8 the next figure of the 
 quotient, and infer that 59 is contained 8 times in 496. Multiplying 
 59 by 8, gives 472; hence we write 472 under the 496, and subtract 
 the former from the latter. It leaves a remainder of 24. Hence, 
 2856 -^59-48 with 24 remainder. 
 
 Proof. — 48 times 59, plug 24, equalf 2856. 
 
 A96 
 A72 
 
 ~YA=e^t 
 
 em. 
 
 2. 
 
 843 H 
 
 - 31? 
 
 8. 
 
 579 H 
 
 -43! 
 
 4. 
 
 827- 
 
 -15» 
 
 5. 
 
 1748 - 
 
 - 42? 
 
 6. 
 
 3947- 
 
 -49? 
 
 7. 
 
 8246- 
 
 - 91? 
 
 8. 
 
 4217- 
 
 - 88? 
 
 9. 
 
 8321 - 
 
 r 94? 
 
 10. 
 
 6735- 
 
 - 83? 
 
 11. 
 
 2317 -i 
 
 - 88? 
 
 12. 
 
 7635- 
 
 - 82? 
 
 13. 
 
 1749- 
 
 -22? 
 
 14. 
 
 2175- 
 
 r 25? 
 
 15. 
 
 4802- 
 
 - 49? 
 
 16. 
 
 6237- 
 
 - 74? 
 
 17. 
 
 4238- 
 
 - 52? 
 
 18. 
 
 6947- 
 
 r75? 
 
 19. 
 
 8286- 
 
 -47? 
 
LESSON FORTY-SECOND. 105 
 
 LESSON XLII. 
 
 A. 1. The multiplier and multiplicand are called factors of 
 their product. (See Lesson XXXII., B.) 
 
 2. The FACTORS of any number are the numbers which, multi- 
 plied together, will give that number for a product. 
 
 Illustrations. — 6 and 3 are factors of 15, because 15 = 5 X 3. 
 
 Again, 6 and 2, 3 and 4, or 3, 2, and 2, are factors of 12, because 12 
 = 6X2, = 3X4 = 3X2X2. 
 
 Again, 2 is a factor of 4, 6, 8, 10, &c. 3 is a fiictor of 6, 12, 15, 
 18, &c, 
 
 3. From the above illustrations, we see that the factors of a 
 number are divisors of it, i. <?., they are such numbers as will 
 divide it without a remainder. 
 
 4. A prime number is a number which has no factors except 
 itself and 1. 
 
 Illustrations. — 1, 2, 3, 5, 7, and 11, are each prime numbers. 
 
 5. A composite number is a number which has other factors 
 besides itself and 1. 
 
 Illustrations. — 4, 6, 8, and 9, are each composite numbers, for 
 4«=2X 2, 6 = 3X2, 8 = 2X4 = 2X2X2, and 9 = 3X3. 
 
 6. A number is divided into factors, when any factors which 
 will produce it are found. 
 
 Illustrations. — In '< 12 =* 6 X 2," 12 is divided into the factors 6 
 and 2, but in " 12 -= 2 X 2 X 3," it is divided into the factors 2, 2, 
 and 3. 
 
 7. A number is divided into its prime factors when it is divided 
 into factors which are all prime numbers. 
 
 Illustrations.— 18 =-2X3X3. 80 = 2 X 3 X 5. 
 
8. The product of a number taken any number of times as a 
 factor, is called a power of that number. 
 
 Illustrations. — 8, which is the product of 2 X 2 X 2, i. e., of 2 
 taken 3 times as a factor is the third power of two; 25, which is the 
 product of 5 X 5, t. e., of 5 taken 2 times as a factor is the second power 
 of five. 
 
 9. We may indicate the power of a number by writing a small 
 figure, called an exponent, above it and a little to the right. 
 
 Illustrations.— 3 ' = 3 X 3 X 3, or 3 to the third power. 2 » = 2 
 X 2 X- 2 X 2 X 2 , or 2 to the fifth power. 
 
 10. The second power of a number is sometimes called its 
 square, and the third power its cube. Thus, 8, or 2 ', is the cube 
 of 2 ; 25, or 5*, is the square of 5. 
 
 B. Write the prime factors of the numbers from 1 to 100, as 
 in the following model : — 
 
 1 = Prime. 6 = 2x3. 
 
 2 = Prime. 7 = Prime. 
 
 3 = Prime. 8 = 2x2x2 = 2' 
 
 4 = 2x2 = 2'. 9 = 3 X 3 = 3^ 
 
 5 = Prime. 10 = 2 X 5. 
 
 C. A factor is common to two or more numbers when it is a 
 factor of each of them. 
 
 Illustrations. — 2 is a common factor of 4, ^ and 14, for it is a 
 factor of each. 
 
 1. What prime factors are common to 24, 36, 
 and 48 ? 
 
 Dividing each number into its prime factors, gives — 
 Solution.— 24 =^=2x2x2x3=2^X3. 
 36 = 2 X 2 X 3 X 3 = 2' X 3*. 
 48 = 2x2x2x2x3 = 2* X 3. 
 
LESSON FORTYrSECOND. 107 
 
 By inspecting these, we see that 2' and 3 are factors of each 
 number, and that there is no other common factor. Hence 2, 2, 
 and 3, or 2 ** and 3 are the prime factors required. 
 
 What prime factors are common — 
 
 3. 
 
 To 12 and 18 ? 
 
 10. 
 
 To 6, 8, and 10? 
 
 4. 
 
 To 15 and 25? 
 
 11. 
 
 To 12, 18, and 30 ? 
 
 5. 
 
 Toll and 20? 
 
 12. 
 
 To 20, 30, and 50? 
 
 6. 
 
 To 30 and 40 ? 
 
 13. 
 
 To 42, 56, and 84 ? 
 
 7. 
 
 To 36 and 54 ? 
 
 14. 
 
 To 63, 81, and 99? 
 
 8. 
 
 To 7 and 9 ? 
 
 15. 
 
 To 8, 9, and 25 ? 
 
 9. 
 
 To 39 and 54 ? 
 
 Ifc 
 
 To 7, 49, and 84 ? 
 
 D. A Common Divisoii of two or more numbers is any number 
 which will exactly divide each of them. 
 
 Illustration. — 4 is a common divisor of 4, 8, 12, and 32. 
 
 The Greatest Common Divisor of two or more numbers is the 
 largest number which is a divisor of each of them. It is also the 
 product of all their common prime factors. 
 
 1. What is the greatest common divisor of 24, 
 36, and 60 ? 
 
 Solution. — The greatest common divisor of 24, 36, and 60, is the 
 product of all the pfime factors common to these numbers. 
 
 24 = 2x2x2x3 = 2^x3. 
 
 36 = 2 X 2 X 3 X 3 = 22 X 32. 
 
 60 = 2x2x3x5 = 2^x3x5. 
 
 We see that the only common prime factors are 2, 2, and 3. 
 Hence 2 X 2 X 3, or 12, must be the greatest common divisor 
 required. 
 
108 colburn's first part. 
 
 What is the greatest common divisor — 
 
 2. Of 30 and 42? 9. Of 4, 6, and 12? 
 
 3. Of 4 and 12? 10. Of 3, 9, and 15? 
 
 4. Of 8 and 20 ? 11. Of 18, 27, and 45 ? 
 6. Of 35 and 49 ? 12. Of 14, 28, and 56 ? 
 
 6. Of 63 and 72 ? 13. Of 30, 48, and 54 ? 
 
 7. Of 42 and 63 ? 14. Of 24, 60, and 84 ? 
 
 8. Of 72 and 96 ? 15. Of 45, 75, and 90 ? 
 
 LESSON XLIII. 
 
 A. The product of any numbers is sometimes called their mul- 
 tiple. Thus, 12 is a multiple of 1, of 2, of 3, of 4, of 6, and of 
 12, for it equals 1 X 12, or 2 x 6, or 3 x 4. 
 
 Hence, any number is a multiple of all its factors and divisors, 
 and a factor of all its multiples. 
 
 Every multiple of a number must contain all the prime factors 
 of that number. 
 
 1. What prime factors must every multiple of 
 18 contain ? 
 
 Solution. — Since 18 = 2 X 3', every multiple of 18 must contain 
 the factors 2 and '6 \ or 2, 8, and 3. 
 
 What prime factors must be contained in every 
 multiple — 
 
 2. Of 12? 5. Of 33? 8. Of 48? 
 
 3. Of 9? 6. Of 28? 9. Of 60? 
 
 4. Of 21? 7. Of 75? 10. Of 36? 
 
 B. A Common Multiple of several numbers is a number which 
 is a multiple of all of them. 
 
LESSON FORTY-THIRD. 109 
 
 The Least Common Multiple of several numbers is the least 
 number which is a multiple of all of them, and is therefore the 
 smallest number which contains all the prime factors of each of 
 them. 
 
 1. What is the least common multiple of 36 and 
 
 48? 
 
 Solution. — Tho least common multiple of 36 and 48 is the smallest 
 number which contains all their prime factors. 
 
 36 == 2 X 2 X 3 X 3 == 2^ X 3'. 
 48 = 2x2x2x2x3 = 2* 3. 
 
 Solution. — "We must have 48 or its factors, which are 2 * X 3. We 
 must also have the factors of 36, which are 2* X 3 2, but as we have 
 already taken 2 ' X 3, we have only to introduce the remaining factor 
 3, which gives 48 X 3, or 2 * X 3 * = 144, as the L. C. M required. 
 
 2. What is the least common multiple of 9, 24, 
 and 30? 
 
 Solution. — The least common multiple of these numbers is the least 
 number which contains all the prime factors of each of them. 
 
 9 = 3X3. 
 
 24 = 2X2X2X3. 
 
 30 = 2 X 3 X 5. 
 
 We must have 30 or its factors, which are 2 X 3 X 5. We must also 
 have the factors of 24, which are 2 X 2 X 2 X 3 j but as we have 
 already taken 2 X 3, we have only to introduce the remaining factors 
 2x2, which will give 2 X 3 X 5 X 2 X 2, or 30 X 2 X 2. We must 
 have the factors of 9, which are 3 and 3, but as we have already taken 
 one S, we have only to introduce another 3, which gives as the L. C. M. 
 required, 2X 3 X5x2x2x 3, or 30 X2X2X3 = 360. 
 
 What is the least common multiple — 
 
 8. Of 8 and 12 ? 6. Of 2, 4, and 6 ? 
 
 4. Of 6 and 9? 6. Of 9, 12, and 18? 
 
110 
 
 colburn's first part. 
 
 10. 
 
 7. Of 4 and 12 ? 
 
 8. Of 7 and 8 ? 
 
 9. Ofl2andl5? 
 Of 16 and 20? 
 
 11. Of 10, 25, and 30? 
 
 12. Of 4, 6, and 12 ? 
 
 13. Of 3, 4,5, and 6? 
 
 14. Of 8, 10, 12, and 20? 
 
 XoTE.— If the class have time for it, the Teacher will do well to 
 give them the method explained in Arithmetic and its ArPLiCATiONs, 
 page 154. 
 
 LESSON XLIV. 
 
 Oral Exercise. — Exhibit any convenient thing, as an apple, to 
 the class, and, cutting it into two equal parts, ask, "What have 
 I done to the apple?" Ans. — "You have cut it." "Into how 
 many parts have I cut it?" A7is. — " Into two parts." " How do 
 the parts compare in size ?" Ans. — " They are equal." " Then 
 how have I divided the apple ?" Ajis. — " You have divided it 
 into two equal parts." "When anything is divided into two equal 
 parts, the parts are called halves of the thing. What, then, 
 
LESSON FORTY-FOURTH. Ill 
 
 will you call these parts of an apple?" Ans. — " Halves of an 
 apple." " What will you call one part ?" Ans. — " One-half of 
 an apple." "What will you call both parts?" Am. — "Two 
 halves of an apple." *' What do both together equal ?" Ans. — 
 '• \ whole apple." *' Then how many halves of an apple equal a 
 whole one ?" Ans. — " Two halves of an apple." 
 
 Continue and extend these illustrations by exercises similar in 
 character to these suggested below : — 
 
 " How shall I divide this apple (showing another) into halves?" 
 Dividing it, ask, " How many halves have I from it ? How many 
 halves did I have from the first apple ? How many halves are 
 there in all ? Then two halves and two halves are how many 
 halves ? If I should give away one-half, how many halves should 
 I have left ? Then one-half from four halves leaves how many 
 halves ?" 
 
 Vary these exercises, dividing apples, strings, pieces of paper, 
 lines, &c., till the class understand fully the value of halves, 
 thirds, &c., and see' clearly that they can be added, subtracted, 
 multiplied, and divided as whole numbers are. Such a course 
 will save much hard labor afterwards, both to Teacher and pupil. 
 
 A. Explanations. — 1. Such parts as are obtained by dividing any- 
 thing or any number into two equal parts, are called halves of that 
 thing or number. One such part is called one half; two such parts 
 are called two halves ; three such parts are called three halves, 
 &c., <fcc. 
 
 2. Such parts as are obtained by dividing anything or number into 
 3 equal parts, are called thirds of the thing or number. One such 
 part is called one third, two such parts are called two thirds, three 
 such parts are called three thirds, four such parts, four thirds, 
 <tc., &c. 
 
 3d, In like manner such parts as are obtained by dividing anything 
 or number into four equal parts are called fourths of the thing or 
 number; such as are obtained by dividing it into five equal parts, are 
 
112 colburn's first part. 
 
 called fifths; into six, are called sixths; into seven, are called 
 
 SEVENTHS, Ac, <tc. 
 
 4. Parts like these are called fractional parts. 
 
 B. 1. What are sevenths? 
 
 Ans. — Sevenths of any thing or number are such fractional parts 
 as would be obtained by dividing it into seven equal parts. 
 In the same way explain each of the following : — 
 
 2. 
 
 Fifths ? 
 
 6. Twelfths ? 
 
 8. 
 
 Fourths ? 
 
 3. 
 
 Thirds? 
 
 6. Halves ? 
 
 9. 
 
 Twentieths ? 
 
 4. 
 
 Ninths ? 
 
 7. Sixths? 
 
 10. 
 
 Tenths? 
 
 C. 1. Instead of the word sixihsy we may write the figure 6 
 with a line above it, thus : r. Instead of the word halves^ we may 
 write 2, &c., &c. 
 
 Hence f tj means tenths, which are fractional parts of such kinds 
 as are obtained by dividing a unit into 10 equal parts. 
 
 ■5 means eighths, which are, &c. 
 z means thirds, which are, &c. 
 i\ means twenty-firsts, which are, &o. 
 s means fifths, which are, &c. 
 
 The number which thus shows into how many parts a unit is 
 divided, is called a denominator, because it gives a name or 
 denomination to the parts. To indicate that a number is a denomi- 
 nator, draw a little line over it. 
 
 D. 1. How many sixths does it take to equal the 
 whole of anything ? 
 
 Ans. — Six, because sixths are such parts as are obtained by 
 dividing a unit into six equal parts. 
 
LESSON FOKTY-FOURTH. 113 
 
 2. How many ninths does it take to equal the 
 whole of anything ? 
 
 3. How many twelfths ? 6. How many -ju ? 
 
 4. How many halves ? 7. How many t? ? 
 
 5. How many thirds ? 8. How many ? ? 
 
 E. In order to show how many fractional parts are taken or 
 considered, a figure is written above the denominator. 
 
 Illustrations. — To express three fourths, the figure 3 may be 
 written above the denominator 4, thus ; |. 
 
 Five sixths may be written ^ 
 
 Eight fifteenths may be written ^j. 
 
 In like manner, J = 4 ninths ; { | = 13 seventeenths ; and 
 f f = 29 thirty-firsts. 
 
 The figure which thus enumerates or numbers the parts, is called 
 the NUMERATOR, and shows how many parts are taken or consi- 
 dered. The numerator is written above the denominator, and 
 separated from it by a line. 
 
 F. Such expressions as "two-thirds," "three-elevenths," 
 
 **T7»" "|»" ^^-J ^^•» ^^® called FRACTIONS. 
 
 1. What does the fraction | express ? 
 Ans. — The fraction three-fourths expresses the value of 3 such 
 parts as are obtained by dividing a unit into 4 equal parts. 
 
 In the same manner explain each of the following fractions : — 
 
 2. f. 4. |. 6. ^. 8. if. 
 
 G. Fractions may be explained after the following model : — 
 The fraction three fourths expresses the value of three equal 
 parts of such kind that four of them would equal a unit. 
 
114 colburn's first part. 
 
 1. Explain each of the fractions under F. according to the last 
 model. 
 
 2. What is the numerator and what the denominator of each ? 
 
 H. A mixed number is a whole number and a fraction. 
 Illustrations. — 4§, which is read "four and two-thirds." 
 
 Read each of the following : — 
 
 1. 5 J. 2. 5?. 3. 4f. 4. 28 J. 
 
 I. These illustrations suggest the following definitions : — 
 
 1. Fractional parts of any thing^ quantity or number^ are such 
 parts as are obtained by dividing it into equal partB. Or — 
 
 2. Fractional parts of any thing, qttantify or nmnber, are equal 
 partit of such kind that a given number of them- will equal that thing, 
 quantity^ or number, 
 
 3. A FRACTION expresses the value of one or more such parts 
 as are obtained by dividing a unit into equal parts. Or — 
 
 4. A FRACTION expresses the value of one or more such equal 
 parts that a given number of them will equal a unit. 
 
 5. The number which shows into how many parts the thing is 
 divided, or how many of the parts are equal to a unit, is called 
 the DENOMINATOR of thc fractiou. 
 
 6. The number which shows how many parts are taken or consi- 
 dered, is called the numerator of the fraction. 
 
 7. The denominator is so called Jbecause it gives the name or 
 denomination to the parts. 
 
 8. The numerator is so called because it enumerates the parts 
 taken or considered. 
 
 .] Write ench of the following in fijruros: — 
 
LESSON FORTY-FIFTH. 115 
 
 1. Two- thirds. 5. Four and seven- tenths. 
 
 2. Eight-ninths. 6. Ten and four-fifths. 
 
 3. Thirteen-nineteenths. 7. Twelve and eleven-twelfths. 
 
 4. Six twenty-firsts. 8. Six and two-thirds. 
 
 LESSON XLV. 
 
 A. Fractions may arise from division as in the following 
 examples : — 
 
 29 = * times 6 V 
 
 Solution. — 29 == 4 times 6, with 5 remaining, or it equals 4| 
 tirnes 6. 
 
 Note. — The^ first part of the above solution should be omitted as 
 soon as the pupil is prepared to give the final answer without it. The 
 entire dividend is here divided, and ihQ fraction Jive sixths is apart of 
 the quotient, and not, like the remainder 5, a part of the dividend. 
 Hence it is wrong to say " 29 = 4 times 6, with |. remaining." These 
 distinctions are important, and should be observed in the solutions. 
 See Lesson XXXVL, Note under A. 
 
 Perform by the above solution the examples under Letter A. and 
 B., Lesson XXXVL 
 
 B. 1. IIo-w many quarts of vinegar at 6 cents 
 per quart, can be bought for 53 cents ? 
 
 Solution. — Since 1 quart of vinegar can be bought for 6 cents, as 
 many quarts can be bought for 63 cents as there are times 6 in 53, 
 which are 8| times. Hence, 8| quarts of vinegar at 6 cents per quart 
 can be bought for 63 cents. 
 
 2. How many pounds of sugar at 8 cents per pound, can be 
 bought for 68 cents? 
 
 8. How many yards of cloth at $4 per yard, can be bought for 
 $31? 
 
116 COL burn's first part. 
 
 4. How many bags, each containing 3 bushels, can be filled 
 from 29 bushels of grain ? 
 
 5. How many hours will it take a horse to trot 33 miles, if he 
 trots 7 miles per hour ? 
 
 6. How many weeks will it take a man to earn $78, if he earn 
 $9 per week? 
 
 7. How many hours will it take a ship to sail 63 miles, if she 
 sail 8 miles per hour ? 
 
 8. How many months wUl it take a man, -who earns $12 per 
 month, to earn $105? 
 
 9. A man put 9 bu. 3 pk. of grain into bags, each holding 1 
 bu. 3 pk. How many bags could he fill? 
 
 10. If a man can earn enough in one day to buy 1 gal. 2 qts. 
 of oil, how many days will it take him to earn enough to buy 13 
 gal. 1 qt. ? 
 
 LESSON XLVI. 
 
 A. Fractions may be added, subtracted, multiplied, and divided 
 as whole numbers are. Thus : — 
 
 2 4 = 6 Just as 2 days + 4 days = 6 days. 
 
 1 J = I, just as 5 qts. — 3 quarts = 2 quarts. 
 
 9 times | == |^, just as 9 times 3 pecks = 27 pecks. 
 
 S> are contained 4 times in ||, just as 6 lb. are contained 4 
 times in 24 lb. 
 
 B. 1. 7 = * fourths ? 
 
 Solution. — Since 1 = 4 fourths, 7 must equal 7 times 4 fourths, 
 which are 28 fourths. Therefore, 7 =< ^s. 
 

 
 LESSON FORTY 
 
 -SIXTH. 117 
 
 2. 
 
 81 
 
 = * fifths ? 
 
 
 
 
 Solution. - 
 are 40 fifths, 
 
 - Since 1 =- 5 fifths, 
 and 4 fifths added an 
 
 8 must equal 
 ) 44 fifths. 
 
 8 times 5 fifths, which 
 Hence 8| =Y- 
 
 NOTE.- 
 
 -Compare these solutions with those of E., Lesson XXX 
 
 3. 9 
 
 
 ^ tenths ? 
 
 7. 
 
 4-^ 
 
 = * seventeenths? 
 
 4. 5 
 
 
 : * thirds ? 
 
 8. 
 
 2A 
 
 = * nineteenths ? 
 
 5. 8 
 
 
 ^ nineteenths ? 
 
 9. 
 
 7}?- 
 
 = * seventeenths ? 
 
 6. 4J=: 
 
 ^ fourths ? 
 
 10. 
 
 6/r 
 
 = ^ elevenths? 
 
 c. 
 
 1. 
 
 5^8 =: Hi ones ? 
 
 
 
 
 SOLUTION.- 
 
 as there are t 
 
 —Since 9 ninths =< 1, 58 ninths must equal as many ones 
 imes 9 in 58, which are 6J times. Hence, -''-^ = 6|. 
 
 NOTE.- 
 
 -Compare this solution with th 
 
 )se of F., Lesson XXX. 
 
 2. 
 
 V 
 
 = ^ ones? 
 
 
 7. 
 
 %« = * ones ? 
 
 3. 
 
 27 
 5 
 
 = ^ ones ? 
 
 
 8. 
 
 f r = -3^ ones? 
 
 4. 
 
 V 
 
 = * ones? 
 
 
 9. 
 
 Y/ = * ones? 
 
 5. 
 
 33 
 
 = ^- ones? 
 
 
 10. 
 
 V = -5^ ones ? 
 
 6. 
 
 V 
 
 = * ones ? 
 
 
 11. 
 
 II = -jf ones? 
 
 D. 
 
 1. 
 
 What is the sum of 
 
 A + t\? 
 
 1st Solution.— -j9- -f JL = js 
 
 = 1tV 
 
 
 2d Solution. — Observing that 
 ,-"t + t\ = tV + T> + T'f = 
 
 
 /t = 
 
 ll, or 1, we may have 
 
 Note. — The second forms of solution to the problems under D. and J 
 G., and the third form under E., will often be found much easier than 
 the first. 
 
 , - .... 1 
 
118 
 
 COLBURN 
 
 'S FIRST PART. 
 
 
 2. 5 + 1? 
 
 5. ? + 
 
 f+4+i? 
 
 
 3. t'+}J? 
 
 6. 1 + 
 
 f+l + l? 
 
 
 4. I'l+iJt 
 
 7. A + 
 
 A+iV+ii? 
 
 E. 
 
 1. 4+3 + i? 
 
 Then 4 + 3 i ? 
 
 
 
 2. 8 + 7 + 1 ? 
 
 Then 8 + 7i? 
 
 
 
 3. 12+21 + f? 
 
 Then 12 + 21, 
 
 J» 
 
 4. 
 
 What is the sum 
 
 0f6| + 8|? 
 
 
 1st Solution.— 6| and 8 an 
 
 1 14 J, and ' are 14J = 15| 
 
 2d Solution. — 6 4-8 = 14; j + | = J = 
 14 = 15 f . 
 
 1|, which, added to 
 
 3d Solution. — Observing 
 
 that 6J + J = 7, we have 6j + 8^ = 
 
 6| + 
 
 • + 8| = r + 8| = 
 
 16|. 
 
 
 5. 
 
 9? +6?? 
 
 9- Si%+'!i%+&tV 
 
 6. 
 
 3I + 4J? 
 
 10. 5| + 7| 
 
 + 4J+5I? 
 
 7. 
 
 4A+3/tT 
 
 11. 8J + 2| 
 
 + 7J+3f? 
 
 8. 
 
 Hi + ^V 
 
 12. 5J + 6| 
 
 + 55 + 45? 
 
 F. 
 
 1. 6?-3i? 
 
 
 
 Solution.— 6J — 3 == 3 J ; 
 
 3J — |=:3|. Hence, 67— 3| = S|. 
 
 2. 
 
 8f —2\1 
 
 5. 
 
 16J -9|? 
 
 3. 
 
 14| -7|? 
 
 6. 
 
 1513 _4j\? 
 
 4. 
 
 Wt-4t'5? 
 
 7. 
 
 38ii - 292J ? 
 
 G. 
 
 1. What is the value of 1 — 
 
 t\? 
 
 SoLUTios.— 1 =• if, and j| 
 
 -TV = tV 
 
 
LESSON FORTY-SIXTH 
 
 119 
 
 2. What is the value of 8 
 
 -^u 
 
 ? 
 
 
 Solution. ~ 8 - -^^ « 7|J - /^ = 
 
 = Ui 
 
 
 
 3. i — i? 
 
 
 6. 
 
 3-1? 
 
 4. 1-if? 
 
 
 7. 
 
 S-fVf 
 
 5. 1-ii? 
 
 
 8. 
 
 9-1? 
 
 9. 8^^,-13? 
 
 
 
 
 1st Solution.— 8-''^ = 7 + l^"^^ — 
 
 i? = 
 
 m- 
 
 i5 = HI. 
 
 2d Solution. -8-^^ — 13 — sJ^^ ~ 
 
 •r\- 
 
 ^% = 
 
 ^-i%=^m- 
 
 10. 23y5^^16A? 
 
 
 
 
 1st Solution.~23-j:^^ — 16 = 7^^ 
 
 = 6|B. 6|| 
 
 - r% = «tV 
 
 2d Solution. — 23^5^ — 10 = 7/^ 
 
 ; Vj 
 
 -t\ 
 
 = h'i - A - 
 
 11. 9i -7|? 
 
 15. 
 
 23A 
 
 -13/,? 
 
 12. 4A-/,? 
 
 IG. 
 
 8/f 
 
 -3ii? 
 
 13. 16xV-5f2? 
 
 17. 
 
 64J 
 
 + 4J - gp 
 
 14. 43f — 17|? 
 
 18. 
 
 23| 
 
 + 17| - 8J ? 
 
 H. 1. George had a very large apple. 
 
 He gave William J of 
 
 it, Joseph 1 of it, and ate the rest. 
 
 What 
 
 part of it did he eat ? 
 
 2. Edward earned | of a dollar by picking blackberries, | of a || 
 
 dollar by picking strawberries, and 
 
 J of a 
 
 dollar 
 
 by picking blue- 
 
 berries. How much did he earn in 
 
 all? 
 
 
 
 3. Who can tell whether the sum of S 
 .greater or less than 24|, and how much ? 
 
 1 + 5 
 
 1 + 8| + 7| is 
 
 4. From a lot containing 8^ acres, there were 5| acres sold. 
 
 How many acres were left ? 
 
 
 
 , 
 
120 COLBURN*S FIRST PART. 
 
 5. Mr. Stone gave -f-^ of his money for a lot of land, and ^ for 
 a horse. What part of it had he left ? 
 
 6. Isaac caught three nice trout. The first weighed 3-j-'^^ lb., 
 the second -weighed 2j| lb., and the third weighed lj| lb. How 
 much did they all weigh ? 
 
 7. Mr. Davis owns -^^^ of a vessel, Mr. Mason owns ^®^, and 
 IsIt. Allen owns the rest. What part of the vessel does Mr. Allen 
 own? 
 
 8. Julia's father gave her || of a dollar, her mother gave her 
 ] J, her brother gave her ^|, and her uncle gave her enough to 
 make up 2 dollars. How much did her uncle give her ? 
 
 9. Farmer Brown had VI-^^ tons of hay in his barn, and 15^'^^ 
 tons in his stacks. How many tons had he in both ? He moved 
 o\^ tons from his stacks into his barn. How many tons were 
 then in his bam ? How many in his stacks ? 
 
 10. A man paid 11 J dollars for a coat, 3J dollars for a pair of 
 pants, and 2| dollars for a vest, giving in payment a twenty-doUar 
 bill. How much ought he to receive back ? 
 
 LESSON XLVII. 
 A. 1. 9 times JJ ? 
 
 Solution. — 9 times ^ = Y> which, since ^ = 1, must equal as 
 many ones as there are times 7 in 64, which are 7| times. Hence, 9 
 times ^ «x 75, 
 
 Abbreviated Solution. — 9 times ^ =. *j4 js- 75. 
 
 2. 4 times /^ ? 6. 9 times | ? 
 
 7. 4 times /^ ? 
 ? 
 
 5. 7 times jl ? 9. 6 times 11 ? 
 
 4. 4 times «? 8. 5 times f? 
 
LESSON FORTY-SEVENTH. 121 
 
 10. 
 
 6 times 7| ? 
 
 SoLUTipN.— 6 times 7 = 42, and 6 times ^ = l^ =. 33,Tvhich, added 
 to 42, gives 453. Hence, 6 times 7| == 45|. 
 
 11. 
 
 9 times 8f? ' 15. 6 times 8^9^? 
 
 12. 
 
 8 times 93? 16. 5 times 9f? 
 
 13. 
 
 7 times 4|? 17. 4 times 8jJ ? 
 
 14. 
 
 4 times 6}f ? 18. 6 times 14/g ? 
 
 B. 
 
 1. 12| = * times 2i ? 
 
 Solution. — 12| = ^^ ', 2|- = | ; and ^j contains ^' as many times 
 as 51 contains 9, which are 5| times. Hence, 123. = 5| times 2^, 
 
 2. 81-^3? 
 
 Solution.— 8| « %« ; 3 =. f ; and 2^« -^ | = 26 -^ 9 ==2|. Hence 
 8| -r- 3 = 23. 
 
 3. 
 
 7i = * times 1}? 8. TJ — IJ? 
 
 4. 
 
 4| = ^ times If? 9. 8f~2J? 
 
 6. 
 
 84. == * times If ? 10. 9 J H- 3 J ? 
 
 6. 
 
 9§== ^ times 2|? 11. Gf-f-la? 
 
 7. 
 
 58 = * times J? 12. 8|-MJ? 
 
 C. ] 
 yard 
 
 I How much will 7 yards of cloth cost at 9^ dollars per 
 
 2. A 
 
 $7 per 
 
 man gave 6 cords of wood at $42 per cord for raisins at 
 cask. How many casks did he buy ? 
 
 3. How many yards of cloth, at $3 per yard, can be bought for 
 8 bbls. of cider at $3| per barrel ? 
 
 10 
 
122 colburn's first part. 
 
 4. I gave 9 firkins of butter at $4| per firkin for flour at $7 
 per barrel. How many baiTels did 1 buy ? 
 
 6. How many shade trees, worth | of a dollar a-piece, can be 
 bought for 51 dollars? 
 
 6. How many skeins of silk, worth -J of a dime per skein, can 
 be bought for 6| dimes ? 
 
 7. How many baskets, each containing ^ of a bushel, can be 
 filled from 8^ bushels of peaches ? 
 
 8. How many boxes, each holding J of a quart, can be filled 
 from 7 1 quarts of blackberries ? 
 
 9. A man paid 6J dollars for grain at J of a dollar per bushel. 
 How many bushels did he buy? He put the grain into bags 
 each holding IJ bushels. How many bags did he fill? 
 
 10. Ralph paid 5| dimes for parched corn at | of a dime per 
 quart. How many quarts did he buy ? After giving away 1 of 
 a quart, he put the rest into paper bags each holding | of a quart. 
 How many bags did it take ? 
 
 11. A farmer exchanged 5 barrels of apples at 1§ dollars per 
 barrel, for oil at lij dollars per gallon. How many gallons of oil 
 did he get? 
 
 12. How many pounds of tea, worth ^ of a dollar per pound, 
 should be given for three books worth 2^ dollars a-piece ? 
 
 13.^ A man who had lOJ bushels of potatoes, used 2 J bushels, 
 and then sold the rest at J of a dollar per bushel, receiving in 
 payment cloth at J of a dollar per yard. IIow many yards of 
 cloth did he receive ? 
 
 14. I bought 8 barrels of flour at $7^ per barrel, and gave in 
 payment 12 cords of wood at $4§ per cord, and the rest in apples 
 at 4 of a dollar per bushel. How many bushels of apples did I 
 sive? 
 
LESSON FORTY-EIGHTH. 123 
 
 LESSON XLVIII. 
 
 A. 1. What part of 5 is 1 ? 
 ^7w. — 1 is ^ of 6, because 5 times 1 = 5. 
 
 What part — 
 
 2. Of 7 is 1? 6. Of 10 is 1? 
 
 8. Of 2 is 1? 6. Of Sis 1? 
 
 4. Of 9 is 1? 7. Of Sis 1? 
 
 B. 1. What part of 8 is 3 ? 
 
 Solution. — Since 1 = J of 8, 3 must equal f of 8. 
 
 What part — 
 
 2. Of 12 is 7 ? 6. Of 9 is 10 ? 
 
 3. Of 9 is 4? 7. Of 13 is 11? 
 
 4. Of7is2? 8. Of8is5? 
 
 5. Of 10 is 9? 9. Of 5 is 8? 
 
 C. 1. What part of 9 quarts is 4 quarts ? 
 
 Solution, — i quarts is the same part of 9 quarts that 4 is of 9, which 
 
 What part — 
 
 2. Of 8 yd. is 3 yd. ? 6. Of $5 is $3? 
 
 3. Of 11 lb. is 7 lb. ? 6. Of £12 is £5? 
 
 4. Of 7 lb. is 11 lb.? 7. Of £5 is £12? 
 
 8. "V^Tiat part of the cost of 7 yd. is the cost of 4 yd. ? 
 
 9. What part of the cost of 3 acres is the cost of 4 acres ? 
 
124 colburn's first part. 
 
 10. What part of the cost of 10 bushels is the cost of 9 bushels? 
 
 11. When flour is 9 dollars per barrel, what part of a barrel can 
 be bought for 1 dollar ? For 5 dollars ? 
 
 12. Mr. Edwards and Mr. Boyden bought a cask of oil, contain- 
 ing 8 gallons, which they so divided that Mr. Edwards had 3 
 gallons, and Mr. Boyden had 6. What part of the cost should 
 each pay ? 
 
 13. Mr. Avery, Mr. Leavens, and Mr. Congdon together bought 
 17 bushels of corn, which they so divided that Mr. Avery took 4 
 bushels, Mr. Leavens took 6 bushels, and Mr. Congdon took the 
 remainder. What part of the cost should each pay ? 
 
 LESSON XLIX. 
 A. 1. What is i of 63? 
 
 Ans. — 1 of 63 is 7, because 9 times 7 = t)3. 
 
 2. 1- of 24 ? 4. J- of 49 ? 6. | of 64 ? 
 
 3. ^of25? 6. 4 of 42? 7. ^jt of 81 ? 
 
 8. What is I of 72 ? 
 
 1st Solution.— i of 72 = 3 times t of 72 ; J of 72 is 9, and 3 times 
 9 = 27. Hence | of 72 = 27. 
 
 2d Solution. — J of 72 = 9, and f of 72 must equal 3 times 9, which 
 are 27. Hence, § of 72 = 27. 
 
 9. fof40? 12. 7 of 56? 15, J of 36? 
 
 10. fofl8? 13. 3?^ of 80? 16. I of 63? 
 
 11. I of 48? 14. 5 of 72? 17. |of54? 
 
 B. 1. How many are 4| times 9 ? 
 
LESSON FORTY-NINTH. 125 
 
 l«t Solution. — 4§ times 9=4 times 9 + § of 9 = 4 times 9 + 2 
 times i of 9. 4 times 9 = 36 ; J of 9 = 3, and 2 times 3 = 6, which, 
 added to 36 == 42. Hence, 4§ times 9 = 42. 
 
 2d Solution.— 4 times 9 = 36. J of 9 = 3, and § of 9 must equal 
 2 times 3, or 6, which, added to 36 = 42. Hence, 4^ times 9 = 42. 
 
 2. 71 times 6 ? 8. | of 54 == * times 3 ? 
 
 3. 9 J times 8 ? 9. f of 42 = * times 9 ? 
 
 4. 8J times 10 ? 10. | of 40 = * times 7 ? 
 
 5. 6§ times 9 ? 11. 8§ times 9 = ^ times 10? 
 
 6. 8| times 12 ? 12. 5| times 12 = * times 8 ? 
 
 7. 5^ times 14 ? 13, 7i times 8 = * times 5 ? 
 
 14. I of 30 + f of 56 = ^ times 5 ? 
 
 15. « of 45 + I of 25 = * times 9 ? 
 
 16. 4 of 49 + I of 36 = * times 6 ? 
 
 17. 3 of 72+ 3 of 63 ==* times 9? 
 
 18. f of 64 + 4 of 28 = * times 7? 
 
 C. 1. If 8 pictures cost 72 cents, how muchwill 
 5 cost ? 
 
 Solution. — If 8 pictures cost 72 cents, 1 picture will cost J of 72 
 cents, or 9 cents, and 5 pictures will cost 5 times 9 cents, or 45 cents. 
 Hence, if 8 pictures cost 72 cents, 5 will cost 45 cents. 
 
 2. If 7 sheep cost $49, how many dollars will 3 sheep cost ? 
 
 3. If 8 papers of candy cost 66 cents, how much will 5 papers 
 cost? 
 
 4. If a girl receives 45 merit-marks for 9 perfect lessons, how 
 many will she receive for 5 perfect lessons ? 
 
 5. If 3 milk cans will hold 24 quarts of milk, how many quarts 
 will 7 milk cans hold ? 
 
 6. How much will 3 quarts of molasses cost at 32 cents per 
 gallon ? 
 
 11* 
 
126 colburn's first part. 
 
 Solution. — If 1 gal or 4 qt. cost 32 cents, 1 quart, or i of a gal., 
 wil. cost i of 32 cents, or 8 cents, and 3 quarts will cost 3 times 8 cents, 
 or 24 cents. Therefore, 3 quarts of molasses, at 32 cents per gallon, 
 would cost 24 cents. 
 
 7. How mucli will 7 qt. of meal cost at 24 cents per pk. ? 
 
 8. How much will 3 gills of oil cost at 48 cents per qt. ? 
 
 9. If a yard of cloth is worth 24 cents, how much is a piece 2 
 feet in length worth ? 
 
 10. If 1 acre 3 roods (or 7 roods) of land cost 63 dollars, how 
 many dollars will 1 acre cost ? 
 
 11. If 1 gal. 1 qt. of burning fluid cost 81 cents, how much 
 will 1 qt. cost? How much will 1 gallon cost? 
 
 12. If a pound of coffee cost 42 cents, what will ^ of a pound 
 cost? 
 
 Solution. — If a pound of coffee cost 42 cents, I of a pound will cost 
 I of 42 cents, which is 6 cents, and |- of a pound will cost 6 times 6 
 cents, or 36 cents. Therefore, ^ of a pound of coffee, at 42 cents per 
 pound, will cost 36 cents. 
 
 13. If a man can perform a piece of work in 72 minutes, in how 
 many minutes could he perform ^ of it ? 
 
 14. What will y of a yard of linen cost at 64 cents per yard ? 
 
 15. Brass is composed of 6opper and zinc. If J of it is zinc, 
 and the rest copper, how many pounds of copper will there bo in 
 a bar of brass weighing 25 lbs. ? 
 
 16. How much will 5| lb. of sugar cost at 8 cents per lb. 
 
 17. How much will 8 J barrels of flour cost at 6 dollars per 
 barrel ? 
 
 18. How many square rods of land are there in a piece 9 rods 
 long and 6§ rods wide ? 
 
 Solution. — Since a piece of land 1 rod long and 1 rod wide contains 
 1 sq. rd., a piece 9 rods long and 1 rod wide must contain 9 sq. rd., 
 and a piece 9 rods long and 6§ rods wide must contain 65 times 9 sq. 
 
LESSON rORTY- NINTH. 127 
 
 rd., which equal 60 sq. rd. Hence, a piece of land 9 rods long and 6§ 
 rods wide conUiins 50 sq. rd. 
 
 19. How many sq. ft. are there in a blackboard 12 ft. long and 
 2 1 ft. wide? 
 
 20. How many sq. yd. are there in a floor 5 yd. long and i^ 
 yd. wide ? 
 
 21. How much will it cost to paint a surface 8 ft. long and 4 J 
 wide, at 3 cents per sq. ft. ? 
 
 22. How much will it cost to plaster a wall 12 ft. long and 8J 
 ft. wide, at 4 cents per sq. ft. ? 
 
 23. If a man can walk 32 miles in 8 hours, how far can he walk 
 in one hour ? How far in 9J hours ? 
 
 24. If 8 tons of meadow-hay cost 72 dollars, how much will 5J 
 tons cost ? 
 
 25. If 7 yd. of cloth are worth 49 lb. of butter, how many 
 pounds of butter ought 4| yd. of cloth to be worth ? 
 
 26. Olive has 40 picture-books, Ella has | as many as Olive, 
 and Ada has J as many as Ella. How many has Ada? How 
 many has Ella ? 
 
 27. John is J as old as his father, who is 36 years old, and 
 William is f as old as John. How old is John? How old is 
 William ? 
 
 28. Edward had 3 cents, and Robert had 5. They put their 
 money together and bought 72 filberts. How many filberts ought 
 each to have ? 
 
 29. Harriet, Maria, and Caroline sent some berries to market, 
 for which they received 63 cents. Now, if Harriet sent 3 quarts, 
 Maria 2, and Caroline 4, how many cents ought each to receive ? 
 
 30. If for two three-cent pieces and 2 cents 64 marbles can be 
 bought, how many marbles can be bought for 1 three-cent piece 
 and 2 cents? How many for 3 three-cent pieces ? 
 
128 colburn's first part. 
 
 31. Julia's basket holds 3 pt. 1 gill. If she can fill it with ber- 
 ries in 45 minutes, how many minutes would it take her to fill 
 Susan's basket, which holds but 1 pint ? How long to fill Jose- 
 phine's basket, which holds but 1 gill ? 
 
 LESSON L. 
 
 A. 1. Whatis Jof 3? 
 
 Solution. — i of 3 is | of 1, for if 3 equal things should be divided 
 into 4 equal parts, one of those parts would equal | of one thing. 
 
 Note. — This may be illustrated to the eye by taking 3 equal lines 
 
 and dividing them into 4 equal parts, arranged as in 
 
 the figure at the left. One part will then contain i 
 
 — — ZI HI of 2 lines, which, as will be seen, is equivalent to | 
 of a line. 
 
 2. iof7? 4. iof3? 6. l^ofS? 
 
 3. Jof3? 5. +ofl? 7. J of 4? 
 
 B. From the preceding exercises, it follows that |, or § of 1 
 = ^ of 3 ; that J, or J of 1 = J of 7, &c. Hence, § of any num- 
 ber equal 3 times ^ of that number, and also J of 3 times that 
 number; | of any number equal 4 times ^ of that number, and 
 also i of 4 times that number. 
 
 1. What is ? of 5 ? 
 
 1st Solution.— I of 5 «= 7 times J of 5 ; J of 6 — t, and 7 times f 
 = \s ^ 4|. Hence | of 5 r= 4^. 
 
 2d Solution. — I of 5 = J of 7 times 5 ; 7 times 5 = 35, and J of 
 35 = 4|. Hence, ^ of 5 = 4f. 
 
 Note. — The pupil should master the first solution, and then the 
 second, and afterwards be required to use in each example the one best 
 adapted to that example. 
 
LESSON FIFTIETH. 129 
 
 2. J of 2? 4. fofS? 6. y\of7? 
 
 3. 5 of 6? 6. j^ofS? 7. 3 of 4? 
 
 C. 1. What is ^ of 6T ? 
 
 1st Solution.— i of 67 =- J of 64 -|- J of 3 ; | of 64 — 8 ; J of 3 = 
 I, which, added to 8 = 8f . Hence, i of 67 — 8f. 
 
 2d. Solution.— i of 67 = J of 64 -f- i of 3 «= SJ. 
 
 3d Solution.— i of 67 = 8f . 
 
 Note. — The first and second solutions are chiefly valuabla as a pre- 
 paration for the third. 
 
 2. iof43? 5. fofl7? 8. f^ofSQ? 
 
 3. ^of28? 6. 4 of 20? 9. J of 43? 
 
 4. J of 17? 7. I of 80? 10. J of 27? 
 
 D. 1. i of 52| ? 
 
 Solution. — i of 52f = ^ of 48 -f J of 4| ; J of 48 — 8 ; J of 4« 
 or of 3^0 -= 5^ which, added to 8 = 8^. Hence, J of 52 2 „. 8 5. 
 
 2. iofl7i? 8. J of 17 qt. 1 pt. ? 
 
 3. iof41|? 9. lof41pk. 5qt. ? 
 
 4. ; of 66| ? 10. 1 of 66 sq. yd. 8 sq. ft. ? 
 6. |of26f? 11. |of49bu. 2pk.? 
 
 6. f of 86i ? 12. i of 58 yd. 2 ft. ? 
 
 7. I of 75 ? 13. I of 26 wk. 4 days ? 
 
 E. 1. g of 36 = * times 5 ? 4. » of 49 == * times 8? 
 
 2. f of25=r *times2? 5. ^ of 45 = * times 8? 
 
 3. j\ of 70 == ^ times 5? 6. f of 63 = * times 6 ? 
 
 F. 1. I of 45 = * times J of 42 ? 
 
130 colburn's first part. 
 
 Solution. — | of 45 = 8 times l. of 45 ; ^ of 45 = 5, and 8 times 
 5 = 40 ; I of 42 = 7, and 7 is contained in 40, 5 1 times. Therefore, 
 I of 45 = 5| times i of 42. 
 
 2. j\ of 80 = * times J of 64 ? 
 
 3. I of 36 == * times J of 24? 
 
 4. I of 72 = * times } of 32 ? 
 
 5. I of 40 = * times | of 12 ? 
 
 6. I of 16 = * times J of 9? 
 
 7. ^ of 25 = * times ^ of 9 ? 
 
 8. I of 19 = ^ times J of 7 ? 
 
 9. J of 14 = * times i of 8? 
 
 G. 1. If 7 inkstands cost 45 cents, what will 3 cost ? 
 
 2. K 9 melons cost 77 cents, what will 5 cost ? 
 
 3. If 8 weeks* board cost $27, what will 7 cost? 
 
 4. If 4 men eat 23 pounds of meat in a month, how many 
 pounds will 7 men eat in the same time ? 
 
 6. If 8 horses eat 37 cwt. of hay in a month, how much will 5 
 horses eat in the same time ? 
 
 6. If 1 pk. of cranberries cost 63 cents, how many cwt. will 3 
 qt. cost? 
 
 7. If a gallon of burning fluid is worth 79 cents, what are 1 qt. 
 1 pt. worth ? 
 
 8. f of a furlong = how many rods ? 
 
 Suggestion.— Since 40 rods = 1 furlong, |- of a furlong equal | of 
 40 rods. 
 
 9. f of a qr. = how many pounds t 
 
 10. 5 of a cu. yd. = how many cubic feet ? 
 
 11. |. of an hour = how man^ minutes ? 
 
LESSON FIFTIETH. 131 
 
 12. ^ of a day = how many hours ? 
 
 13. f of a bu. = how many pk., qt., &c. ? 
 
 \ 
 
 Solution. — Since 1 bu. = 4 pk., § of a bu. must equal f of 4 pk., 
 or 2§ pk. But since 1 pk. = 8 qt, § of a pk. must equal ^ of 8 
 qt., or 5 J qt. Since 1 qt. = 2 pt., J of a qt. must equal J of 2 pt., 
 or § of a pt. Since 1 pt. = 4 gills, § of a pt. must equal § of 4 gills, 
 or 23 gills. Therefore, § of a bu. = 2 pk. 5 qt. pt. 2§ gills. 
 
 14. ^ of a £ = how many s., d., and qr. ? 
 
 15. ^ of a lb. = how many oz. and dwt. ? 
 
 16. ^ of a ton = how many cwt., qr., lb. ? 
 
 17. I of a sq. yd. = how many sq. ft, sq. in. ? 
 
 18. ^ of a lb. = how many oz., dwt., gr. ? 
 
 19. I of a wk. = how many da., h., &c. ? 
 
 20. 3 of a lb. = how many g., g., &c. ? 
 
 21. J of a £. = how many s., d. qr. ? 
 
 22. Frederic and Benjamin gathered some nuts, of which Fre- 
 derick gathered 4 qts., and Benjamin 2 qts. They sold them for 
 39 cents. How many cents ought each to receive ? 
 
 23. Mr. Ames and Mr. Clapp bought the apples on 2 large trees 
 for 9 dollars. Mr. Ames paid 5 dollars, and Mr. Clapp paid 4 
 dollars. There proved to be 87 pecks of apples on the trees. 
 How many pecks ought each to have ? How many bushels ? 
 
 24. If 4 pounds of rice are worth 24 cents, how many poundo 
 )f rice ought to be given for 7 J pounds of sugar worth 8 cents per 
 pound ? 
 
 25. How many half-pounds of coffee, worth 16 cents per 
 pound, would be given for 2J bushels of oats, worth 44 cents per 
 bushel ? 
 
 26. I gave 4 pk. 3 qt. of nuts, worth 24 cents per peck, for eggs 
 at 12 cents per dozen. How many dozen eggs did I receive ? 
 
132 colburn's first part. 
 
 27. I worked 3J weeks, at $18 per week, and received in pay- 
 ment $30 in money and the balance in shoes at $3 per pair. 
 How many pairs of shoes did I receive ? 
 
 28. James asked his father for some drawing-pencils, to which 
 his father replied, ** 4 good drawing-pencils will cost as much as 
 3 writing books at $1 per dozen. Now, if you will tell me how 
 much 6 drawing-pencils will cost, I will buy them for you.'' 
 What should have been James's answer ? 
 
 LESSON LI. 
 
 A. 1. What part of ^ is f ? 
 
 Solution. — 2 \s the same part of | that 2 is of 5, which is |. 
 
 What part — 
 
 2. OfAisJ? 4. Of 2V is aV- ^' 
 
 3. Of^is/^? 5. Of J is 4? 7. Of f is J ? 
 
 8. What part of 2| is 10| ? 
 
 Solution. — 2^ = 1 ; 10 J, and ^ is the same part of | that 31 is of 8, 
 which is 3^1 or 3J. Hence, lOJ = 3^1 of 2§, or it equals 3| times 2^. 
 
 What part — 
 
 9. Of 2 J is 4 J ? 14. Of 3 da. is 1 wk. ? 
 
 10. Of 71 is 2f ? 15. Of 4 sq. ft is 1 sq. yd. 
 
 11. Of9|is2|? 16. Of 1 pk. 1 qt. is 3 pk. 7 qt. ? 
 
 12. Of J is 1? 17. Of3yd. 1ft. is8yd. 2ft.? 
 
 13. Of I is 1 ? 18. Of 9 d. 1 qr. is 3 d. 2 qr. ? 
 
 B. 1. 3 = ^ of what number ? 
 
2. 
 
 9 = 1 of * ? 
 
 3. 
 
 7 = iof*? 
 
 4. 
 
 3 = Jof*? 
 
 6. 
 
 | = iof^? 
 
 6. 
 
 | = iof^? 
 
 LESSON FIFTY-FIRST. 133 
 
 1st Solution. — 3 = ^ of 8 times 3, or 24. 
 
 2d Solution. — If 3 is ^ of some number, 1 of the number must be 8 
 times 3, or 24. Hence, 8 -= i of 24. 
 
 7. f = itof^e? 
 
 8. 9 J = i of * ? 
 
 9. 7| = i of * ? 
 
 10. 2bu. 3pk. = Jof *? 
 
 11. 5wk. 3da. == Jof *? 
 
 C. 1. 17 = I of what number ? 
 
 Solution. — If 17 = | of some number, I of that number must be i 
 of 17, which is 5|, and J of the number must be 4 times 5§, which is 
 22t. Therefore, 17 = I of 22f . 
 
 Note.— To prove this, see if | of 22§ = 17. 
 
 2. 36 = 4 of * ? 12. 3| times 10 == -J of *? 
 
 3. 32 = |of^? 13. 91 times7 = |of *? 
 
 4. 40 = I of * ? 14. 5| times 9 = | of * ? 
 
 5. 42=«of*? 16. 8Jtimes8 = -jPg of *? 
 
 6. 81 = j\ of * ? 16. 8J times 9 = 13 times * ? 
 
 7. 27 = I of ^ ? 17. 9J times 6 = 2} times * ? 
 
 18. 8| times 10 = If times * ? 
 
 19. 3f times 9 = 3J times * ? 
 
 10. 40 = I of * ? 20. 6} times 8 = 1| times * ? 
 
 11. I of 45 = 5 of * ? 21. 8| times 6 = 2J times * ? 
 
 D. 1. If I of a gallon of molasses cost 25 cents, what will 1 
 gallon cost ? 
 
 12 ~~ ~ ' -' 
 
134 colburn's first part. 
 
 Solution. — If £ of a gallon cost 25 cents, ^ of a gallon will cost J 
 of 25 cents, which is 8i cents ; and J of a gallon will cost 4 times 8J 
 cents, which are 33^ cents. Therefore, 1 gallon of molasses will cost 
 33i cents, if | of a gallon cost 25 cents. 
 
 2. If I of a yard of muslin cost 37 cents, "wliat will 1 yard 
 cost? 
 
 3. If |- of a yard of linen cost 53 cents, what will 1 yard cost ? 
 
 4. If J of a month's wages amount to 23 dollars, what will 1 
 month's wages amount to ? 
 
 5. John is 17 years old, and his age is | of his teacher's age. 
 How old is his teacher ? 
 
 6. Deborah says that she has 41 cents. " Then," said Lavinia, 
 " you have only -^^ as many as I have." How many cents had 
 Lavinia ? 
 
 7. David told George that ^ of his money would buy 5J pounds 
 of raisins at 9 cents per pound. ** Then," replied George-, " you 
 have 6 cents more than I have." How many cents did each of 
 the boys have ? 
 
 8. Seth's father gave him a half-dollar to buy a pound of tea 
 with, saying to him, "4 of a pound of tea will cost 80 centn, and 
 if you will tell me how much a pound will cost, you may have the 
 money which will be left after paying for the tea." Seth an- 
 swered correctly. What was his answer ? How many cents would 
 be left after paying for the tea ? 
 
 9. A farmer gave 7f dozen of eggs at 9 cents per dozen for J 
 of a gallon of oil. What was a gallon of the oil worth ? 
 
 10. A butcher received 35 shillings for f of a hundred weight 
 of beef. What would ho receive for a hundred weight at the same 
 price. What would he receive for -^^ of a hundred weight ? 
 
 11. A schoolmaster being asked his age, replied, *' f of my life 
 have been spent in teaching. I have taught in Boston 25 years, 
 which is f of all the time I have taught." What was his age ? 
 
LESSON FIFTY-FIRST. 135 
 
 12. If a yard of muslin costs 70 cents, and ^ of a yard of mus- 
 lin costs I as much as a yard of cambric, what will a yard of 
 cambric cost? 
 
 13. If a yard of linen costs 56 cents, and | of a yard of linen 
 costs I as much as a yard of lawn, how much will a yard of lawn 
 cost ? How much will J of a yard of lawn cost ? 
 
 14. If Joseph can earn 54 cents in one day, and it takes William 
 y^j of a day to earn as much as Joseph can earn in | of a day, 
 how much can William earn in one day ? 
 
 15. Mr. Battles gave 25 dollars for a cow, and 1| times what 
 he gave for the cow is equal to 2 J times what he gave for a heifer. 
 What did he give for the heifer ? 
 
 16. After Arthur had given -| of his money for a blank book, 
 and I of it for a grammar, he had 18 cents left. How many cents 
 had he at first ? 
 
 17. I of Mr. Ball's farm is tillage, * of it is pasturage and the 
 rest, 3 acres, is orchard. How many acres does he own ? 
 
 18. William and Henry were trying to puzzle each other about 
 their ages. William told Henry that he had spent J of his life in 
 Philadelphia, f of it in New York, and the rest in Boston, and 
 that he had lived 5 years more in New York than in Boston. 
 Henry found out his age. What was it ? 
 
 19. Henry then said to William, *'I have lived in Hartford, 
 Providence, and Boston. I spent -^^ of my life in Hartford ; I 
 lived in Providence twice as long as in Hartford, and have livei 
 in Boston 2 years more than 3 times as long as in Providence." 
 William found out Henry's age. What was it ? 
 
136 colburn's first part. 
 
 LESSON LII. 
 
 A. 1. What is the effect of multiplying the nu- 
 merator of the fraction j% by 3 ? 
 
 Ans. — Multiplying the numerator of the fraction y*^ by 3, gives 
 1^ for a result, which expresses 3 times as many parts, each of 
 the same .size as before, and is, therefore, 3 times as large. 
 Hence, multiplying the numerator of j\ by 3, multiplies the frac- 
 tion by 3. 
 
 What is the effect of multiplying the numerator 
 of— 
 
 2. T\by2? 4. 4 by 3? 6. 4 by 5? 
 
 3. 2\by6? 6. 8 by 7? 7. | by 9 ? 
 
 8. What is the effect of dividing the numerator 
 of the fraction j| by 6 ? 
 
 Ans. — Dividing the numerator of || by 6, gives j^j for a result, 
 which expresses J as many parts, each of the same size as before, 
 and is therefore J as large. Hence, dividing the numerator of 
 If ^y 6, divides the fraction by 6. 
 
 Note. — Tho first of the above solutions is equivalent to "3 times 
 y^y = XZj just as 3 times 4 apples = 12 apples;" and the second is 
 equivalent to " J of j | == J^^, just as J of 12 apples = 2 apples." 
 They are necessary as a preparation for the exercises which follow. 
 
 What is the effect of dividing the numerator of — 
 
 9. 14 by 5? 11. I by 4? 13. |fby7? 
 
 10. I? by 2? 12. §by8? 14. ||by9? 
 
LESSON FIFTY-SECOND. 137 
 
 Hence, multiplying the numerator of a fraction multiplies the frac- 
 tion, and dividing the numerator divides the fraction, 
 
 B. To THE Teacher. — Should the pupils find any difficulty in under- 
 standing the following exercises, illustrations should be given by divi- 
 ding visible objects, such as apples, lines, <fcc., into various kinds of 
 fractional parts. 
 
 From the nature of fractional partr, it follows that — 
 
 1st. The larger the number of fractional parts into which any 
 
 unit is divided, or which it takes to equal that unit, the smaller 
 
 each part will be. 
 
 2d. The smaller the number of fractional parts into which any 
 
 unit is divided, or which it takes to equal that unit, the larger 
 
 each part will be. 
 
 1. Whicli parts are larger in size, halves or 
 
 fourths ? 
 
 Ans. — Halves, because it takes a less number of them to equal 
 a unit. 
 
 Which parts are larger in size — 
 
 2. Halves or thirds ? 6. Halves or tenths ? 
 
 8. Fourths or eighths ? 6. Fourths or twelfths ? 
 
 4. Thirds or ninths ? 7. Fifths or twentieths ? 
 
 C. 1. Each half equals how many sixths ? 
 
 1st Form of Answer. — Each half equals J of 6 sixths, which 
 is 3 sixths. Hence, each half equals 3 sixths. 
 
 2d Form of Answer. — Each half equals 3 sixths, for if a unit 
 should be divided into 6 equal parts, J of the unit would contain 
 3 of them. 
 
138 colburn's first part. 
 
 2. Each half equals how many fourths ? 
 
 3. Each third equals how many ninths ? 
 
 4. Each fourth equals how many twelfths ? 
 
 5. Each fifth equals how many tenths ? 
 
 6. Each sixth equals how many eighteenths ? 
 
 D. From the foregoing exercises we may infer that — 
 
 1st. Multiplying the number of fractional parts into which a unit 
 
 is divided, or which it takes to equal a unity divides each part. 
 
 2d. Dividing the number of parts into which a unit is divided, or 
 
 which it takes to equal a unit, multiplies each part. 
 
 1. What is the eifect of multiplying the denomi- 
 nator of the fraction | by 4 ? 
 
 Ans. — Multiplying the denominator of the fraction J by 4, 
 gives -^^ for a result, which expresses the same number of parts 
 each J as large as before. Hence, -^^ = J of J, or multiplying 
 the denominator of ^ by 4, has divided the fraction by 4. 
 
 What is the effect of multiplying the denominator 
 of— 
 
 2. §by5? 4. fby2? 6. |by3? 
 
 3. §by6? 5. 4 by 3? 7. fby4? 
 
 What is the effect of dividing the denominator of 
 
 Ans. — Dividing the denominator of the fraction -f^ by 2, gives | 
 for a result, which expresses the same number of parts each twice 
 as large as before. Hence, | = two times ^^, or the fraction ^- 
 has been divided by 2. 
 
LESSON FIFTY-SECOND. 139 
 
 What is the effect of dividing the denominator — 
 9. Of 5 by 3? 11. Of 2^ by 8? 13. Of ^^ by 7? 
 
 10. Of I by 2? 12. Of 2^^ by 5? 14. Of |J by 9 ? 
 
 E. The numerator and denominator are called terms of the 
 fraction. 
 
 I. What is the effect of multiplying both terms 
 of I by 6 ? 
 
 Ans. — Multiplying both terms of the fraction f by 6, gives -}-| 
 for a result, which expresses 6 times as many parts, each J as 
 large as before. Hence, the value of the fraction is not altered, 
 
 What is the effect of multiplying both terms of — 
 
 2. J by 3? 6. /ffby3? 8. |by8? 
 
 3. J by 2? 6. I by 4? 9. /^byS? 
 
 4. I by 4? 7. fby7? 10. fby6? 
 
 II. What is the effect of dividing both terms of 
 if by 5. 
 
 Ans. — Dividing both terms of the fraction -15 by 5, gives | for 
 a result, which expresses J as many parts each 5 times as large 
 as before. Hence, the value of the fraction is not altered, or 
 
 15 3 
 
 25 — 5- 
 
 What is the effect of dividing both terms of — 
 
 12. -i%by3? 15. T\by3? 18. i4by7? 
 
 13. 4 by 4? 16. /^by3? 19. J| by 5 ? 
 
 14. -{-«by5? 17. i|by7? 20. f J by 9 ? 
 
140 colburn's first part. 
 
 F. From the foregoing, it appears that — 
 
 1. Multiplying the numerator multiplies the fractiony hy multiply- 
 ing the number of parts considered^ without affecting their size. 
 
 2. Dividing the numerator divides the fractiony hy dividing the num- 
 ber of parts considered^ without affecting their size. 
 
 3. Multiplying the denominator divides the fractiony by dividing 
 each party without affecting the number of parts considered. 
 
 4. Dividing the denominator multiplies the fraction, by multiplying 
 each party without affecting the number of parts considered. 
 
 6. A fraction may be multiplied either by multiplying the numerator 
 or by dividing the denominator. 
 
 6. A fraction may be divided either by dividing the numerator or 
 by multiplying the denominator. 
 
 7. Multiplying both numerator and denominator of a fraction by 
 the same number both multiplies and divides the fraction by that num- 
 ber, andy therefore, does not alter its value. 
 
 8. Dividing both numerator and denominator of a fraction by the 
 same number, both divides and multiplies the fraction by that number, 
 and, thereforey does not alter its value. 
 
 LESSON LIII. 
 
 A. A fraction is said to be in its Lowest Terms when its 
 numerator and denominator are the smallest entire numbers which 
 will express its value. 
 
 When a fraction is in its Lowest Terms, there is no entire num- 
 ber greater than one which will divide both numerator and deno- 
 minator without a remainder. 
 
LESSON FIFTY-THIRD. 141 
 
 Hence, a fraction may he reduced to its lowest terms hy dividing 
 both numerator and denominator by the same numbers, 
 
 1. Reduce sf to its lowest terms. 
 
 Solution. — Both terms of jL| can be divided by 6. Dividing 
 them, gives | for a result, which expresses ' i as many parts, each 
 6 times as large as before. Hence, ±| — |, and as | admits of no fur- 
 ther reduction, it is the fraction required. 
 
 The number by which we divide in reducing fractions to their 
 lowest terms, are said to be canceled. Thus, in the first solu- 
 tion, we canceled the factor 6 ; in the second, we canceled the 
 factor 2, and then the factor 3. 
 
 Note. — The pupil should not only master the explanation, but 
 should also learn to give the results without the explanation. Let him 
 also observe that a fraction can always be reduced to its lowest terms 
 by dividing both terms by their greatest common divisor. 
 
 Reduce each of the following fractions to its lowest 
 terms : — 
 
 2. tV 5. if. 8. Jf. 11. A8. 
 
 3. j%. 6. if. 9. If. 12. 4|. 
 
 4. ^\. 7. J|. 10. if. 13. ^, 
 
 B. A fraction is sometimes expressed by the factors of its 
 numerator and denominator. 
 
 4X9 
 Example.— The fraction — 1_ which may be read, " The fraction 
 15 X 8, 
 
 having 4 times 9 for its numerator, and 15 times 8 for its denomina- 
 tor,'^ or, " The fraction 4 times 9, divided by 15 times 8." 
 
 Such fractions should be reduced to their lowest terms before 
 muhiplying their factors together. 
 
 Reduce ^^ ^ to its lowest terms. 
 
 Solution.— Canceling 4 from the factor 4 of the numerator and S 
 
142 colburn's first part. 
 
 of the denominator, gives 1 in place of the former, and 2 in place of 
 
 the latter.* Cancelling 3 from the factor 9 in the numerator, and 15 
 
 of the denominator, gives 3 in place of the former, and 5 in place of 
 
 the latter. ■}• As no further division can be made, we multiply the re- 
 
 4X9 1X3 3 
 maining factors together, which gives == = — 
 
 In writing out the work, it is customary to draw a line through 
 the numbers from which factors have been canceled, and to write 
 the quotients above the dividends of the numerator, and below 
 those of the denominator. 
 
 Reduce each of the following to its lowest terms : 
 
 ^6X4 Q 28 X 36 ^^ 3x4x6 
 
 ' 24 X 14 * 6 X 8 X 3 
 
 ^ 49 X 25 ^j 4 X X 21 
 
 * 35 X 35 * 14 X G X 3 
 
 « 2x3x4 .., 15 X 7X 13 
 
 9X8 
 
 10 
 
 X 
 
 9 
 
 21 
 
 X 
 
 6 
 
 12 
 
 X 
 
 7 
 
 35 
 
 X 
 
 18 
 
 16 
 
 X 
 
 18 
 
 12. 
 
 6x5x8 13x35x9 
 
 9 4x7x 9 ^3 24 X 18 X 25 
 
 45 X 24 '7x6x8 ' 36 x 45 x 56. 
 
 LESSON LIV. 
 A. 1. How can J of a fraction be found ? 
 
 Ans. — J of a fraction can be found by dividing its numerator 
 by 2 ; or by multiplying its denominator by 2. For, dividing the 
 numerator by 2, will give for a result a fraction expressing J as 
 many parts, each of the same size as those of the given fraction ; 
 
 *This makes the fraction express i as many parts, each 4 times as 
 large as before, and hence does not alter its value. 
 
 f This makes the fraction express J as many parts, each 3 times as 
 large as before, and therefore does not alter its value. 
 
LESSON PIFTY-FOURTH. 143 
 
 or, dividing the denominator, will give for a result a fraction 
 expressing the same number of parts, each J as large as those of 
 the given fraction. 
 
 Note. — The statement of the reasons should not be omitted till it i« 
 certain that the pupil fully understands them. 
 
 State the method of finding — 
 
 2. J of a fraction. 5. -jJ^ of a fraction. 
 
 3. )j of a fraction. 6. J of a fraction. 
 
 4. I- of a fraction. 7. J^ of a fraction. 
 
 B. 1. What is the effect of multiplying the nu- 
 merator of a fraction by 3, and the denominator by 
 4? 
 
 Ans. — Multiplying the numerator of a fraction by 3, and the 
 denominator by 4, gives for a result f of the original fraction, for 
 it gives 3 times as many parts, each J as large as before. 
 
 What is the effect of multiplying the numerator 
 of a fraction — 
 
 2. By 4, and the denominator by 7 ? 
 
 3. By 8, and the denominator by 3 ? 
 
 4. By 11, and the denominator by 6 ? 
 
 5. By 12, and the denominator by 10 ? 
 
 6. By 24, and the denominator by 17 ? 
 
 C. 1. How can | of a fraction be found ? 
 
 1st Method. — ^ of a fraction can be found by getting | of the nu- 
 merator for a new numerator, without altering the denominator. 
 
 2d Method. — f <>^ * fraction can be found by multiplying the nu- 
 merator by 5 and the denominator by 6. 
 
144 
 
 COLBURN S FIRST PART. 
 
 Note. — The pupil should observe that the first method gives | as 
 many parts of the same size as before, and that the second gives 5 
 times as many parts, each ^ as large as before. He should observe, 
 further, that the first method is identical with that of Lesson L. 
 
 Explain the methods of finding — 
 
 2. 
 
 1^ of a fraction. 
 
 
 6. 
 
 1 of a fraction. 
 
 3. 
 
 ^ of a fraction. 
 
 
 7. 
 
 IJ of a fraction. 
 
 4. 
 
 y^ of a fraction. 
 
 
 8. 
 
 j^ of a fraction. 
 
 6. 
 
 y'*j of a fraction. 
 
 
 9. 
 
 J of a fraction 
 
 D. 
 
 1. What 
 
 is \ 
 
 Ofy^? 
 
 
 
 2. 
 
 J off? 
 
 1 
 
 3 
 
 2 
 6. }of}? 
 
 
 10. |of||» 
 
 3. 
 
 tofA^ 
 
 
 7. fjof§|? 
 
 
 11. /,off? 
 
 4. 
 
 T^.0f|? 
 
 
 8. Ifofil? 
 
 
 12. |of|f? 
 
 5. 
 
 |ofJ? 
 
 
 9. ioff? 
 
 
 18. ^\otJ^1 
 
 E. 1. What is the product of f times ^f ? 
 
 3 
 
 Solution.- f times if - 1 of if ^ ^ ^^ ^ tV 
 
 2 5 
 
 9 X 28 6 
 
 2. What is the product of y^ x || ? 
 
 1st Solution.— y\ X f f , or fV times f|=A of §|= jj^^^ 
 
 2d. Solution. — /^ X ff, or j\ multiplied by f|, -= ||of ^9^=» 
 
 28 X 9 _ 6 
 
 33 X 14 ^^' 
 
LESSON FIFTY-FOURTH. 145 
 
 Note. — The slight difference between the first and second solution 
 results from the different reading of the sign of multiplication. We 
 recommend the first as being the most simple. 
 
 5. |X ^L? 8. ^fXi?? 11. 4 X JX ax VV? 
 
 12. 4J X 21? 
 
 Solution. — 4ix 2| = | X y== V = ^2^* 
 
 13. 2|x4J? 15. 2JX3J? 17. If X IJ? 
 
 14. 5JxH? 16. 4fxl-|? 18. 8JX7J? 
 
 G. 1. f = I of what number ? 
 
 Solution. — If f = « of some number, i of that number must be i 
 
 3 
 
 of i, which is = J ; and ^ of the number must be 7 times J, 
 
 which are ^. Hence, | = 5 of f . 
 
 2. |=:40f^-? 6. ^==:^of^t 
 
 3. |4 = j% of ^- ? ^. 3J = 2} times ^ ? 
 
 4. /g = 1^ times -)f ? S. 2J = 2f times * ? 
 6. i=2i times * ? 9. 4 J = f of * ? 
 
 H. 1. How much will f of a yard of cloth cost at | of a dollar 
 per yard ? 
 
 2. How much will f of a quart of filberts cost at | of a dime 
 per quart ? 
 
 3. George gathered | of a bushel of cranberries, and sold f of 
 what he gathered. What part of a bushel did he sell ? 
 
 4. Rufus earned J of a dollar, and then spent | of what he had 
 earned. What part of a dollar did he spend ? 
 
 j3 - 
 
146 colburn's first part. 
 
 5. If 1 pound of tea is worth | of a dollar, what is ^ of a pound 
 w or til ? 
 
 6. If a man can hoe | of an acre of corn in 1 day, what pnrt 
 of an acre can he hoe in J of a day ? 
 
 7. If 5 pounds of soap cost J of a dollar, what will one pound 
 cost? 
 
 8. If 4 oranges cost J of a dollar, what will 1 orange tost ' 
 What will 3 cost ? 
 
 9. If 6 pine apples cost J of a dollar, what will 6 cost ? 
 
 10. If 5 lb. of coffee cost | of a dollar, what will 10 lb. cost? 
 
 11. If § of a yard of silk velvet cost 5 J dollars, what will 1 
 yard cost ? 
 
 Solution. — If § of a yard of silk velvet cost 5J dollars, ^ of a yard 
 will cost i of 5i dollars. ^ of 5i = i of 4 -[- i of U, or of |, which is 
 2|. If J of a yard cost 2§ dollars, | of a yard will cost 3 times 2^ 
 dollars, which are 7J dollars. 
 
 12. If I of an acre cost 30J dollars, what will 1 acre cost ? 
 
 13 If J of a cask of oil is worth 64 J dollars, what is the cask 
 worth ? 
 
 14. If 2 J cords of wood are worth $18J, what is 1 cord worth? 
 
 15. If a wood-cutter can cut 6 J acres of wood in 2J days, how 
 much can he cut in 1 day? 
 
 16. If Rufus can shell 2f bushels of corn in 1 hour, how many 
 bushels can he shell in 2 J hours ? 
 
 17. If Rufus can shell 7J bushels of corn in 2J hours, how many 
 bushels can he shell in 1 hour ? 
 
 18. If Albert can walk 13 J miles in 4 hours, how far can he 
 walk in 1 hour ? How far in 2 J hours ? 
 
 19 If Albert can walk 7 J miles in 2 J hours, how far can he 
 walk in 1 hour ? How far in 4 hours ? 
 
LESSON FIFTY-FOURTH. 147 
 
 20. When 4^ bushels of corn can be bought for $2|, how many 
 bushels can be bought for 1 dollar? How many for y^^ of a 
 dollar ? 
 
 21. When 1|J- bushels of corn can be bought for y^^ of a dollar, 
 how many bushels can be bought for 1 dollar ? How many for 
 $2|? 
 
 22. If J of a yard of linen is given in exchange for f of a yard 
 of silk worth | of a dollar per yard, what ought the linen to be 
 worth per yard ? 
 
 23. If I of a yard of silk is given in exchange for | of a yard 
 of Unen worth f of a dollar per yard, what ought the silk to be 
 worth per yard ? 
 
 24. What part of 1 rod is 4 yd. 2 ft. 1| in. ? 
 
 Solution. — Since 1 in. =j^^ of a foot, 1| in. or 1^2 of an inch must 
 equal ^ of ^^ of a ft. = 1 ft., to which, adding the 2 ft., gives 21 
 ft,, or y ft. Since 1 ft. = J of a yd., y of a ft. must equal y of i of 
 a yd.= I yd., to which, adding the 4 yd., gives 4-| yd. = 3_3 yd. Since 
 1 yd. = 2^ of a rd. ^^ of a yd. must equal "^^^ of Jt rd. = | rd. 
 Hence, 4 yd. 2 ft. 1| in. = ^ of a rod. 
 
 Prove by Solution to 13th problem, page 131. 
 
 25. What part of 1 bu. is 3 pk. 1 qt. 1} pt. ? 
 
 26. Of 1 gal. is 2 qt. 1 pt. 26 gi. ? 
 
 27. Of 1 lb. is 8 oz. 14f dr. ? 
 
 28. Of 1 wk. is 5 da. 10 h. 40 m. ? 
 
 29. Of 1 £. is 2 s. 10 d. 1^ qr. 
 
 30. Of lib. is 9 §.45.29. 8gr.? 
 
 31. Of 1 lb. is 4 oz. 5 dwt. 17J- gr.? 
 
 32. Of 1 T. is 2 cwt. qr. 22 lb. 3 oz. 8f 02. 
 
148 colburn's first part. 
 
 LESSON LV. 
 
 A. 1.1 = how many times I ? 
 
 Solution. — 1 = |, and | = 5 times ^. Hence, 1 «= 5 times i. 
 
 2. What is the quotient of 1 - i ? 
 
 Solution.— 1 = .7, and ;f - *- i = 7 -f- 1 = 7. Hence, 1—1 = 7. 
 
 3. l = ^timesj? 7. 1 -r J? 
 
 4. l = *timesi? 8. l-f-JL? 
 
 5. 1 = * times -^L ? 9. 1 -f- 4 ? 
 
 6. 1 = 4t times J^? 10. 1 -^ i? 
 
 Inference.— Since 1 = 2 times i, = 3 times J, &c., it follows that 
 there will be 2 times as many halves, 3 times as many thirds, <fcc., as 
 there are times 1 in any number. 
 
 B. 1. 5 = * times J? 
 
 Solution. — Since 5 contains 1, 5 times, it must contain J, 3 times 5 
 times, or 15 times. Hence, 5 = 15 times J. 
 
 2. What is the quotient of 8 -^ J ? 
 
 Solution. — Since the quotient of 8 -r- 1 = 8, the quotient of 8 -r J 
 must equal 3 times 8 or 24. Hence, 8 -7- J = 24. 
 
 3. 7 = ^timesi? 9. 8-f--L? 
 
 4. 3 = ^ times ^ ? 10. 2 -f- J- ? 
 
 5. 5 = *tiiPes^V- 11- 10 -r J? 
 
 6. 9 = * times J? 12. 4~J? 
 
 7. 4 = * times J ? 13. 12 -^ ^ ? 
 
 8. 6==^ times J? 14. 9-rJ? 
 
 
 
LESSON FIFTY-FIFTH. 149 
 
 C. From all the preceding exercises, it must be obvious that 
 the quotient of a number divided by 1 equals that number. Thus 
 3 A. 1 = 3 ; 7 -r 1 = 7, &c. So f —1 = J; | -^- 1 = |, &c. 
 
 1. I = how many times J ? 
 
 Solution. — Since | contains 1, | times, it must contain i, 4 times 
 ? times, which are ^^ times == 2| times. Hence, 3 ___ 23 times I, 
 
 2. What IS the quotient of f -^ 7 ? 
 
 Solution. — Since the quotient of § divided by 1 = f , the quotient 
 of I divided by ^ must equal 7 times f, which are y = 4^. Hence, 
 A -^ 1 = u_ 
 
 3. 4 = ^ times J ? 8. | -f- J 
 
 4. j\ = * times -J ? 9. 
 
 5. § = ^ times i ? 10. 
 
 6. f == -sf times J? 11. 
 
 7. I = -x- times J? , 12. 
 
 ^? 
 
 
 D. From the nature of division, it is obvious that, while the 
 dividend remains the same, the larger the divisor is, the smaller will 
 be the quotient, and the smaller the divisor is, the larger will be 
 the quotient. 
 
 Thus the quotient of 8 divided by 2 is 4, which is J of the quo- 
 tient of 8 divided by 1 ; the quotient of 15 -f- 3 is 5, which is J 
 of the quotient of 15 divided by 1. So the quotient of a number 
 divided by | must be J of its quotient divided by j ; the quotient 
 of a number divided by -? must be J of its quotient divided by J, 
 &c., &c. 
 
 1. 8 = * j? 
 
 Solution.— Since 8 contains -J, 5 times 8 times, it must contain 3 
 i of 5 times 8 times, or | of 8 times, which are 13J times. Hence, 
 8 = 13J times 5. 
 
 13* ■ ■ 
 
150 colburn's first part. 
 
 2. What is the quotient of 4 -r- | ? 
 
 Solution.— Since the quotient of 4 divided by l = 7 times 4, the 
 quotient of 4 divided by 3 must equal i of 7 times 4, or 1 of 4, which 
 is 9J- Hence, 4 H- ^ = 9^. 
 
 3. 7 = ^^ times f ? 9. 8 -f- /^ ? 
 
 4. 5 =: If times | ? 10. 2 -^ JL ? 
 
 5. 8 = * times J ? 11. 6 -f- § ? 
 
 6. 1 = * times | ? 12. 1 -i- | ? 
 
 7. 1 = * times J ? 13. 1 _i. | ? 
 
 8. 4 = * times ^t 14. 9 ~ 4 ? 
 
 E. 1. § = how many times | ? 
 
 Solution.— Since | contains ^, 7 times | times, it must contain 3 
 J of 7 times | times or | of | times = i^ times. Hence, 2^14 
 times 3. 
 
 2. What is the quotient of f -^ 4 ? 
 
 Solution. — Since the quotient of | divided by l = 7 times |, the 
 quotient of | divided by « must equal J of 7 times |, or 1 of |, which, 
 by canceling the factor 3, equals ^. Hence, | -^ ^ __ j^ 
 
 3. | = *times/g? 8. |-f-f? 
 
 4. } = ^times|? 9. 91 ~- |? 
 6. f = * times /^ ? 10. /^ -r |? 
 
 6. 5 = * times I? 
 
 7. f = * times -^p ? 
 
 F. The preceding exercises and solutions make it evident that 
 the quotient of a number divided by f = J of 3 times the num- 
 ber sr= 1 of the number ; the quotient of a number divided by 
 i = J of 9 times the number = | of the number, and generally 
 that — 
 
LESSON FIFTY-FIFTH. 161 
 
 The quotient of a number divided by a fraction equals the product 
 of that number multiplied by the fraction inverted. 
 
 1. What is the quotient of ^ ~- 2| ? 
 
 Ans.-i- -^ 2f = 4 ^ I = 4 >^ t = i. 
 
 2. 4-7-lJ? 5. 2i~4J? 8. 8J~4J? 
 
 3. |~7i? 6. f|~3i? 9. /,~|? 
 
 4. 5i~-4f? 7. 93-7-14? 10. 8i-r5|? 
 
 G. Examples in division of fractions sometimes appear in the 
 form of fractions. They are then called complex fractions. 
 
 Illustration. — _ which equals 4§ -^ 3 \. 
 
 A complex fraction, then, has a fraction in one or both its 
 terms. They may be explained after the following model : — 
 
 4| 
 
 — expresses the value of 4| equal parts of such kind that 3 J 
 
 of them will equal a unit. 
 
 Complex fractions may be reduced to simple ones by merely 
 performing the indicated division. 
 
 Thus : -I = 4f -■ 31 = 1 4 -^ 1 6 _ ^^ X 5 35 ^ 
 
 Reduce each of the following to simple fractions : 
 1. II. 3. !i 5 ^^ 
 
 2 J 
 
 3. 
 
 Si 
 41 
 
 4. 
 
 ^ 
 
 2. !l. 4. £i. 6. 
 
 2| 8J li 
 
 H. 1. How many melons at f of a dime each can be bought 
 for 5 dimes ? 
 
152 colburn's first part. 
 
 Solution.— Since 1 melon can be bought for | of a dime, as many 
 melons can be bought for 5 dimes as there are times | in 5, which are 
 I of 5 times or 2_o times == 6| times. Hence, G§ melons at | of a dime 
 a-piece can be bought for 5 dimes. 
 
 2. How many bushels of corn at f of a dollar per bushel can 
 be bought for 7 dollars ? 
 
 3. How many hours will it take a scholar who learns ^ of a 
 page per hour to learn 3 pages ? 
 
 4. When tea is worth J of a dollar per lb., how many pounds 
 can be bought for $5. 
 
 5. If a man can walk -^^ of a furlong in 1 minute, how many 
 minutes will it take him to walk -J of a furlong? 
 
 G. If a man can gather | of the apples on a certain tree in 1 
 hour, how many hours will it take him to gather -f-^ of them ? 
 
 7. Edward divided | of a rood of land into flower-beds, each 
 containing -^^ of a rood. How many beds did he make ? 
 
 8. A man who had $5, gave J of his money for grass seed at 
 $2i per bushel. How many bushels did he buy ? 
 
 9. How many pounds of pearlash at -^-^ of a dime per pound 
 can be bought for § of a pound of chocolate at 3f dimes per lb.? 
 
 10. If Josephine can learn J of a lesson in an hour, how many 
 hours will it take her to learn 1 lesson ? 
 
 11. Albert has a cord 28 feet long, which he wishes to cut into 
 pieces each 2| feet long. How many pieces will it make ? 
 
 12. A man who liad but $9, invested f of his money in cloth 
 at IJ dollars per yard, and the rest of it in cloth at 1 J dollars per 
 yard. How many yards of each kind did he buy ? 
 
 13. When a bushel of potatoes can be bought for | of a dollar, 
 how many bushels of potatoes can be bought for 9 bushels of 
 corn at i of a dollar per bushel ? 
 
LESSON FIFTY-SIXTH. 153 
 
 14. How many bottles, eacli holding ^ of a. quart, can be filled 
 from f of a gallon of wine ? 
 
 15. A farmer has 2^ tons of hay in one stack, and 3| tons in 
 another. He carries it to market in loads each weighing 1| tons. 
 How many loads will both stacks make ? 
 
 16. How many tiles f of a foot long and i of a foot wide will it 
 take to cover 18 sq. ft. of surface ? 
 
 17. How many square yards in a floor 12 feet long and 9 feet 
 wide, and how many yards of carpeting f of a yard wide, will it 
 take to cover it ? 
 
 18. When $1 is received for f of a sq. ft. of land, how many 
 dollars will be received for a strip 16 feet long, and j^^ o^ ^ ^^ot 
 wide ? 
 
 LESSON LVI. 
 
 A. 1. f = how many twelfths ? 
 
 Solution. — Since 1 =: l|, S of 1 must equal | of {-|, which are ^^. 
 Hence, S = ^\. 
 
 In a similar manner reduce — 
 
 9. f and J to twelfths. 
 
 10. I and |- to thirty-sixths. 
 
 11. f and -p^ to fortieths. 
 
 12. -^^ and f to twenty-fourths. 
 
 13. -^^ and :^^ to sixtieths. 
 
 14. I and l to fifty -sixths. 
 
 15. |, ^ and rp^ to seventieths. 
 
 Let the pupil now solve the above questions by the following 
 form : — 
 
 2. 
 
 f to sixths. 
 
 3. 
 
 J to eighths. 
 
 4. 
 
 1 to tenths. 
 
 5. 
 
 J to twentieths. 
 
 6. 
 
 5 to forty-fifths. 
 
 7. 
 
 1 to twenty-firsts. 
 
 8. 
 
 IJ to thirty-sixths. 
 
154 colburn's first part. 
 
 Solution to problem 1st. — Since the required denominator, 12, is 3 
 times the given denominator, 4, we multiply both terms of f by 3, 
 which gives | == -j9-. 
 
 B. 1. Fractions have a COMMON DENOMINATOR when they have 
 the same denominator. 
 
 Illustration. — | and J do not have a common denominator, but 
 i, If and 1 have the common denominator 9. 
 
 2. Fractions having different denominators can be reduced to 
 a common denominator, t. e., to equivalent fractions having a 
 common denominator. This is illustrated in the last 7 examples 
 under A. • 
 
 3. In reducing fractions to a common denominator — 
 
 1st. Select a convenient number for the commoii denominator, 
 2d. Reduce the given fractions by the method explained in A. 
 
 4. It will usually be most convenient to select the least com- 
 mon multiple of the denominators of the given fractions for a 
 common denominator. 
 
 C. 1. Reduce |, |, and \^ to a common deno- 
 minator. 
 
 Partial Solution. — We first find the least common multiple of the 
 given denominators 6, 8, and 12. It is 24, which we therefore select 
 for the common denominator. The problem is now equivalent to the 
 following: *' Reduce 5 |, and 11 to twenty-fourths," and may be 
 solved by one of the methods explained under A. 
 
 Reduce the fractions in each of the following 
 examples to a common denominator : — 
 
 2. i and J. 6. 1, J, and |i. 
 
 3. fandi, - 7. /,, -j-V and f. 
 
 4. iand3. 8. ^^ f, and /,. 
 
 5. I and |. 9. -4, J, and ^\. 
 
I 
 
 LESSON FIFTY-SEVENTH. 155 
 
 10. I J, and i. 
 
 11. |, i, and |. 
 
 12. |, I, and f 
 
 13. -rV/p andi. 
 U. I, 5, and i 
 
 15. 
 
 3, -5j, and i. 
 
 16. 
 
 l^iVli^^^di 
 
 17. 
 
 i, i^iand/,. 
 
 18 
 
 |, J, i, and /^. 
 
 19. 
 
 iVA^i.^^^il- 
 
 LESSON LVII. 
 
 A. In order that fractions mav be added or subtracted, they 
 must be simple fractions, and have a common denominator. 
 Hence — 
 
 Complex and compound fractions must be reduced to simple 
 fractions, and simple fractions to a common denominator, before 
 they can be added or subtracted. 
 
 1. What is the sum of | of | + |f 4- i + j^. 
 
 Solution. — Reducing the compound and complex fractions to sim- 
 
 pie ones, we have, |. of | = f, and— = |. Hence, the problem be-- 
 
 comes § -f- -| -f" t 4" tV Selecting 24 as the common denominator, 
 and reducing the fractions to twenty-fourths, as explained in Lesson 
 LVL, gives t + 1 4- § 4- ^7_ _ 16 + |o ^. .1 5 +_ 14 _2 ij. 
 
 2. What is the value of t^ — M of U ? 
 
 5^ 
 
 Abbreviated Solution. — Keducing the given fractions to simple ones 
 and then to a common denominator, we have,— i — JLg. of 11 = |1 — 
 
 cj 2^ lo 24 
 
 2¥ — 2i 
 
 3. Add the fractions in each of the examples under C, Lesson 
 LVI. 
 
 4. Find the difference of the fractions in the second, third, 
 fourth, and fifth examples under C, Lesson LVI. 
 
156 colburm's first part. 
 
 5. In each example following tlie fifth under C, Lesson LVI.. 
 subtract the last fraction from the sum of the others. 
 
 6. l+t? 21. l+i + l? 
 
 7. 2i + 3i? 22. f+4 + i?' 
 
 8. 3J + 4|? 23. ^ 4-15 + 1? 
 9- 7f + 2§? 24. ^s,+-;,+|4-i? 
 
 10. 4|-4-7i? 25. iofj + l? 
 
 11. 5i+4§? 26. |of,?, + -t|? 
 
 12. G J — 2t ? 27. 3i + 8 J — 6/^ ? 
 
 13. 7i — 2J? ^ 28. 4|+ 64—7^3? 
 
 14. 5J+3i? * 29. |of|4 + |ofJ? 
 
 15. 85-8}? 80. £i+|ofj?,? 
 
 H 
 
 16. 0J-8|? 81. §+ll+i? 
 
 "7 
 
 17. 
 
 4i — 2f? 82. 1* , ?f , ?i? 
 
 ^ 3 + 6i + 5 
 
 18. 5« + 2§? 33. 2of«of|i+4? 
 
 19. 7i-4i? 84. 8i + 34 + 7J? 
 
 20. 9i + 6J? 35. ?i + |of|? 
 
 ^5 
 
 B. 1. Rufus bought a slate for J of a dollar, a writing book 
 for J of a dollar, a geography for J of a dollar, and an atlas for 
 J, of a dollar. What was the cost of the whole ? 
 
 2. Edward spends IJ hours each day in studying history, 1| 
 hours in studying geograj^hy, and 1|- hours in studying grammar. 
 How many hours does he spend in studying all these branches ? 
 
 3. A man bought a large pine-apple and gave J of it to Sarah, 
 \ of it to Jane, ^^ of it to Susan, JL of it to Maria, and the rest 
 to Emma. What part of it did he give to Emma? 
 
 4. I spent I of my money for land, j\ of it for buildings, and 
 put the rest at interest. What part of it did I put at interest? 
 
LESSON FIFTY-SEVENTH. 157 
 
 5. I bought an umbrella for $lf, and a pair of shoes for $2|. 
 How much did both cost ? How much more did the shoes cost 
 than the umbrella ? 
 
 6. A farmer sold 6J tons of hay, and then had 8| tons left. 
 How many tons had he at first ? 
 
 7. A man walked from Dedham to Boston, a distance of 10 
 miles, in 3 hours. He walked 2| miles the first hour, and 3y'^j 
 miles the second. How far did he walk in the third hour ? 
 
 8. Mr. Wheelock bought a book for $1|, and a ream of paper 
 for $2|, giving in payment a five-dollar bill. How much money 
 ought he to receive back ? 
 
 9. Mr. Nichols's orchard contains 3f acres, and his house-lot 
 contains | of an acre. How many acres do both contain, and how 
 many more acres are there in his orchard than in his house-lot ? 
 
 10. Mr. Turner bought 4 loads of hay. The first weighed J 
 of a ton, the second weighed i of a ton, the third weighed ^ of a 
 ton, and the fourth weighed f of a ton. What did they all 
 weigh ? 
 
 11. I bought I of a yard of silk velvet at $7 per yard, and j 
 of a yard of satin at $6 per yard. What did both cost ? 
 
 12. I sold 7 barrels of apples at $2J per barrel, receiving in 
 payment 4 yards of cloth at $3g per yard, and the rest in money. 
 How much money did I receive ? 
 
 13. Thil^morning I had $10i, but I have since paid away 
 $6/^. How much have I left ? 
 
 14. A farmer gathered 7i barrels of russets, 8| barrels of pip- 
 pins, 6-| barrels of greenings, and 9 J barrels of sweetings. How 
 many barrels did he gather in all ? 
 
 15. Arthur, Richard, and Edwin were talking about their mo- 
 ney. Arthur said that he had $4f <' Then," said Richard, " I 
 have $2-5- more than you have." Edwin thought a moment, and 
 
 14~ '^ 
 
168 colburn's first part. 
 
 then said, " If I had $3 J more than I now have, I should have as 
 macn as both of you together. Hov?- many did Richard have ? 
 How many did Edwin have ? 
 
 16. Rufus spent i of his money for writing paper, J of it for 
 pens, and the rest, which was 4 cents, for a pencil. What part 
 of his money did he spend for a pencil ? How many cents did he 
 spend in all, and how many for each article ? 
 
 17. A farmer has i of his sheep in one pasture, | of them in 
 another, and the rest, 6 sheep, in another. How many has he in 
 all, and how many in each pasture ? 
 
 18. Benjamin being asked his age, replied, <*I have spent j^^ 
 of my life in Brooklyn, J of it in New York, ^ of it in Baltimore, 
 and the rest, 6 years, in Cincinnati. What was his age ? 
 
 19. A drover says that if he sells J of his sheep to one man, 
 and J of them to another, he shall sell 6 more to the first man 
 than to the second. How many sheep has he ? 
 
 20. James and George were talking about their ages. James 
 said that i of his age exceeded J of it by li years ; to which 
 George replied, ** Then you are only | as old as I am.'* What 
 was the age of each boy ? 
 
 21. 3i times a certain number exceeds 2 times the number by 
 12. What is the number ? 
 
 22. If I could sell my cow for 13 dollars more than 3 times 
 what she cost me, I should receive $100 for her. How^'much did 
 she cost me ? 
 
 23. There is an orchard in which | of the trees bear peaches, 
 J bear cherries, J bear apples, and the rest bear pears. Now, if 
 there are 7 more apple trees than peach trees, how many trees 
 are there in the orchard, and how many of each kind ? 
 
 24. Mr. Jones and Mr. French traded in company. Mr. Jones 
 put in $3 as often as Mr. French put in $4. When they. came 
 
LESSON FIFTY-EIGHTH. 159 
 
 to divide the gain, it was found that Mr. French's share was $8 
 more than Mr. Jones's. How much did they gain, and what was 
 the share of each? 
 
 LESSON LVIII. 
 
 A. 1. Of what denominations is the number 427 
 composed ? 
 
 Ans. — The number 427 is composed of 4 hundreds, 2 tens, and 
 7 units. 
 
 In the same way tell of what denominations each 
 of the following numbers is composed : — 
 
 2. 678. 5. 5276. 8. 2008. 
 
 3. 
 
 982. 
 
 6. 
 
 3028. 
 
 9. 
 
 3254. 
 
 4. 
 
 201. 
 
 7. 
 
 1406. 
 
 10. 
 
 6897. 
 
 11. Explain the use of the figures of the above 
 numbers, as in the following — 
 
 Model. — In 427, the 7 marks the units* place, and shows that 
 there are 7 units ; tjfie 2 marks the tens' place, and shows that 
 there are 2 tens ; the 4 marks the hundreds' place, and shows that 
 there are 4 hundreds. 
 
 12. Kjive the value of each figure of the above 
 numbers, as in the following — 
 
 Model. — In 427, the 7 = 7 units ; the 2 = 2 tens, or 20 
 units ; the 4 = 4 hundreds, or 40 tens, or 400 units. 
 
 B The foregoing illustrations show that — 
 
 The value of each figure is ten times the value it would have if it 
 stood one place farther to the right, and one-tenth of the value it would 
 have if it stood one place farther to the left. 
 
160 colburn's first part. 
 
 ==1 
 
 1. Compare the values expressed by the 2's of 222. 
 
 Ans. — The first, or right hand 2, expresses ^-^ the value of the 
 second 2, and y|^ the value of the third 2 ; the second, or middle 
 2, expresses 10 times the value of the first 2, and Jg- the value 
 of the third ; the third, or left-hand 2, expresses 10 times the 
 value of the second 2, and 100 times the value of the first. 
 
 Compare in the same way the figures of each of 
 the following numbers: — 
 
 2. 333. 5. 5555. 8. 808. 
 
 3. 
 
 111. 
 
 6. 
 
 9909. 
 
 9. 
 
 7777 
 
 4. 
 
 444. 
 
 7. 
 
 6006. 
 
 10. 
 
 2202 
 
 C. Marking the places by a period, or decimal point (see 
 Lesson XXII., C), we may make new places at the right of the 
 point, by calling the first tenths, the second hundredths, the third 
 thousandths, &c. 
 
 Thus: 42.37 = 4 tens, 2 units, 3 tenths, and 7 hundredths; 
 .348 = 3 tenths, 4 hundredths, and 8 thousandths. 
 
 Name the denominations of the figures of the fol- 
 lowing numbers : — 
 
 1. 
 
 23.47 
 
 4. 
 
 1.46 
 
 7. 
 
 4.596 
 
 10. 
 
 2.7 
 
 2. 
 
 6.825 
 
 5. 
 
 .008 
 
 8. 
 
 1.037 
 
 11. 
 
 4.06 
 
 3. 
 
 .3698 
 
 6. 
 
 .06 
 
 9. 
 
 .027 
 
 12. 
 
 30.03 
 
 D. Such numbers are read by first reading the figures at the 
 left of the point, as though they stood alone, — and then reading the 
 figures at the right of the point, as though they stood alone^ naming 
 afterwards the denomination of the right-hand figure. 
 
 Illustrations. — 42.37 = 42 and 37 one-hundredths = 42-^-^^. 
 .348 = 348 thousandths = /_48^, &c. 
 
LESSON FIFTY-EIGHTH. 161 
 
 In the^same way, read each of the numbers under C. 
 
 Numbers expressed by figures written both at the right and 
 the left of the point, were, in the above form, read as mixed 
 numbers. They may, with equal propriety, be read as improper 
 fractions. 
 
 Illustrations.— 42.37 = 42-^^^= \2_3^7 . 5.5 ,_, 5.^^ _ |.j^ &c. 
 
 Read each number under C, which is greater than 1, as an 
 improper fraction. 
 
 E. 1. K the decimal point of any number be remored one place 
 farther towards the right, or, which is the same thing, the figures 
 be removed one place towards the left, each figure will represent 
 10 times as large a value as before ; while if the point be removed 
 one place farther towards the left, or, which is the same thing, the 
 figures be removed one place toward the right, each figure will 
 represent one-tenth as large a value as before. 
 
 2. A similar change of two places, would multiply or divide a 
 number by 100, — of three places, by 1000, &c., &;c. 
 
 3. Hence — To multiply a number by 10, it is only necessary to 
 remove the point one place farther towards the right; to multiply by 
 100, remove it two places, ^c, Sfc. 
 
 4. So, to divide a number by 10, remove the point one place towards 
 the left; to divide by 100, remove ii two places, ^c, ^'c. 
 
 5. If there are not figures enough at the right or left of the point 
 to make these changes, annex or prefix zeroes to make up the deficiency. 
 
 Illustrations. 
 
 46 X 10 = 460 46 -r 10 = 4.6 
 
 3.7 X 10 = 37 4 -. 10 = .4 
 
 5.86 X 10 = 58.6 067 -r 10 = .0067 
 
 234 X 100 = 23400 634 -^ 100 = 5.34 
 67.8 X 100 = 5780 .8& -r 100 == .0085 
 6.294 X 100 = 629.4 6.9 ~ 100 = .069 
 
 14* L 
 
162 colburn's first part. 
 
 Multiply each of the following numbers by 10 : — 
 
 1. 84 4. 6.24 7. 2847 10. 8246 
 
 2. 5.6 6. 63.7 8. 54.09 11. .9374 
 
 3. .63 6. 286 9. 3.275 12. 23.16 
 
 13. Multiply each of the above numbers by 100, and then by 
 1000. 
 
 14. Divide each of the above numbers by 10, then by 100, 
 then by 1000. 
 
 15. Find ^>g. of each of the above numbers, then ji^, then 
 
 F. 1. What is .03 of 145.6 ? 
 
 Solution.— Since .03 of a number equals 3 times ^J.^ of that num- 
 ber, it may be found by removing the point two places further towards 
 the left and multiplying by 3, which would give .03 of 145.6 = 3 times 
 1.456 = 4.368. 
 
 Note.— Probably it will be better to have the pupil perform most 
 
 of these questions on 
 
 his slate. 
 
 
 
 2. .4 of 6.8? 
 
 6. .07 of 5.6? 
 
 10. 
 
 .003 of 279? 
 
 3. .05 of 27? 
 
 7. .02 of 176? 
 
 11. 
 
 .004 of 8.27? 
 
 4. .3 of 56? 
 
 8. .2 of .06? 
 
 12. 
 
 1.2 of 43? 
 
 5. 1.3 of 6.7? 
 
 9. .25 of 183? 
 
 13. 
 
 1.42 of .687? 
 
 G. 1. What IS the quotient of 4.8 -^ .006? 
 
 Since the quotient of a number divided by .006 equals (Lesson LV.) 
 J of 1000 times that number, it may be found by removing the point 
 three places toward *he right and dividing by 6, which would give 
 4.8 -f- .006 = h of 4800 = 800. 
 
 2. 4.25 — 5? 6. 32 -r .008? 10. .325 — 25? 
 
 3* 25 ~ .05 ? 7. 2.76 ~ 1.2 ? H. 36 -^ .006 ? 
 
 4. .06 -^ .006 ? 8.^ 2.76 -^ .12? 12. 49 -^ 4.9 ? 
 
 5, .0144 -f- .12? 9. 2.76-^. .012? 13. 37 -r .037 ? 
 
LESSON FIFTY-EIGHTH. 163 
 
 H. The term per cent is often used in place of one hun- 
 dredths. 
 
 Thus, 6 per cent = .06, or y J^ ; 9 per cent = .09, or yf,, 
 &c., &c. 
 
 1. I gathered 43 bushels of apples, receiving 12 per cent of 
 them for my labor. How many bushels did I receive ? 
 
 2. I bought a sleigh for $16.20, and paid a sum equal to 8 per 
 cent of the cost for having it repaired. How much did 1 pay for 
 having it repaired ? 
 
 3. A man who had 87 bushels of apples, sold .7 of them and 
 kept the rest. How many bushels did he sell ? How many did 
 he keep ? 
 
 4. A father left, at his death, 97 acres of land, to be so divided 
 that his widow should have .4 of it, his oldest son .3, his youngest 
 son .2, and his daughter the rest. "What was the share of each ? 
 
 5. George received 9 per cent of $144, and William received 6 
 per cent of $216. Which received the most? 
 
 6. A trader bought a lot of goods for $36, and sold them so as 
 to gain 10 per cent of the cost. What was his gain, and for how 
 much did he sell them ? 
 
 7. Bought goods for $300, and sold them so as to gain 15 per 
 cent. What was my gain ? 
 
 8. I gave $28.60 for a lot of goods, but I was obliged to sell 
 them so as to lose 8 per cent. How many dollars did I lose, and 
 for how many dollars did I sell them ? 
 
 9. IMr. Brown bought a horse for $150, and sold him at an 
 advance, or gain, of 12 per cent. What was his gain? 
 
 10. I bought a carriage for $175, and, after paying 12 per 
 cent of the cost for repairing it, I sold it for $225. Did I gain 
 or lose, and how much ? 
 
164 COLBURN*S FIRST PART. 
 
 J. The money which men charge for their services in buying 
 or selling goods for others, is called commission, and is usually a 
 certain per cent of the cost of the goods bought, and of the 
 money received for those sold. 
 
 1. Mr. Clarke sold a lot of goods for Mr. Davis for $500, at a 
 commission of 3 per cent. What did his commission amount to, 
 and how much money would be left for Mr. Davis ? 
 
 2. I sold a lot of goods for $250, at a commission of 4 per cent. 
 What did my commission amount to, and what would be left for 
 the owner of the goods ? 
 
 3. A commission merchant sold 85 barrels of flour, at $8 per 
 barrel, receiving a commission of 2 per cent. What was his com- 
 mission ? 
 
 4. I bought $860 worth of cloth for Mr. Arnold, charging him 
 a commission of 2 per cent. What was my commission, and what 
 ought Mr. Arnold to pay me for the cloth and my commission ? 
 
 6. George bought a jack-knife for James for 75 cents, charging 
 a commission of 8 per cent. How much ought James to pay for 
 the knife and George's commission ? 
 
 6. Mr. Greene bought a lot of shoes for Mr. Gardner, for which 
 he paid $120, and charged 3J per cent commission. What 
 ought Mr. Gardner to pay for the shoes and Mr. Greene's com- 
 mission? 
 
 7. By selling a horse for 20 per cent more than he cost, I 
 gained $80. What did he cost, and for how much did I sell him ? 
 
 Suggestion. — The given per cent can often be reduced to lower 
 terms. Thus, 20 per cent. = -.?« = -l ; 16§ per cent = 151 -= |, «5;c. 
 
 8. By selling a lot of merchandise at an advance of 12 J per 
 cent, I gained $9.50. What did it cost me, and for how much 
 did I sell it ? 
 
 9. My commission of 3 per cent for selling a lot of goods was 
 $15. For how much did I sell them ? 
 
LESSON i^IFTY-NINTH. 165 
 
 10. I lost 25 per cent of the cost of a horse by selling him for 
 $120. What per cent of his cost did I receive ? How many dol- 
 lars did he cost me ? How many dollars did I lose ? 
 
 11. I gained 16f per cent of the cost of a horse by selling him 
 for $140. What was his cost, and how many dollars did I gain? 
 
 12. By selling some cloth at 24 cents per yard, I should gain 5 
 per cent more than 1 should by selling it at 28 cents per yard. 
 What was its cost ? 
 
 13. By selling cloth at 12 J cents per yard, I gain 25 per cent. 
 For how much should I sell it to gain 50 per cent ? 
 
 LESSON LIX. 
 
 A. 1. If I should have the use of another man's horse for a 
 day, or a week, I ought to pay for it ; or if I should occupy a 
 house or a store belonging to another, I ought to pay rent for the 
 use of it. In like manner, if I should borrow a sum of money, I 
 ought to pay for the use of it. 
 
 2. Money thus paid for the use of money, is called Interest. 
 
 8. The money lent or used is called the Principal, and the 
 principal and interest together, form the amount. 
 
 Illustrations. — If I should pay $3 for the privilego of using $100 
 for six months, the $3 would be the interest of the $100 for 6 months ; 
 tho $100 would be the principal, and $100 + $3, or $103, would be 
 the amount. 
 
 4. The interest is usually a certain number of one hundredths 
 of the principal for each year it is used. This number of one 
 hundredths is called the Rate per cent, or simply the Rate. 
 
 Illustration. — If a man is to pay a sum equal to -^ - of the prin- 
 cipal for each year he uses it, the rate is 6 per cent. 
 
 6. In computing interest, a month is reckoned at 30 days. 
 
166 colburn's first part. 
 
 B. 1. What is the interest of $8 for 2 years 9 
 mo., at 4 per cent. ? 
 
 Solution. — At 4 per cent per year, the interest for 2 yr. 9 mo., or 
 2| years, must be 2| times 4 per cent, or 11 per cent, of the principal. 
 11 per cent of $8 = 11 times 8 cents = 88 cents, or $.88 = the 
 answer. 
 
 What is tne interest — 
 
 2. Of $7 for 2 jr., at 6 per cent ? 
 
 3. Of $9 for 3 yr., at 5 per cent ? 
 
 4. Of $18 for 6 mo., at 6 per cent? 
 
 5. Of $248 for 4 mo., at 6 per cent? 
 
 6. Of $43.21 for 1 yr. 10 mo. at 6 per cent ? 
 
 7. Of $52.30 for 2 yr. 6 mo., at 4 per cent? 
 
 8. Of $132 for 1 yr., at 7 per cent? 
 
 9. Of $937 for 8 mo., at 6 per cent ? 
 
 10. Of $42.73 for 2 yr., at 4 J per cent? 
 
 11. Of $23.17 for 1 mo., at 6 per cent? 
 
 12. Of $24.36 for 9 mo., at 8 per cent? 
 
 13. Of $53.27 for 1 yr. 4 mo., at 6 per cent? 
 
 C. Interest is more frequently reckoned at 6 per cent per 
 year, than at any other rate. Hence, in all the following exam- 
 ples and explanations, interest should be reckoned at 6 per cent, 
 unless otherwise stated. 
 
 2 months being J of a year, interest for 2 months at 6 per cent, 
 must equal J of 6 per cent, or 1 per cent of the principal, which 
 may be found by removing the decimal point 2 places to the left, 
 and is as many cents as there are dollars in the principal. 
 
 At 6 per cent per year, what is the interest for 
 2 months of — 
 
 1. $37? 4. $657? 7. $85.75? 
 
 2. $58? 5. $938?- 8. $123.79? 
 
 3. $49? 6. $8238? 9. $437.28? 
 
LESSON FIFTY-NINTH. 167 
 
 10. What is the amount of each of the above ? 
 
 J). Interest for 2 months being 1 per cent of the principal, 
 interest for 100 times 2 months, or 200 months, or 1(5 years 8 
 months, must be 100 per cent of the principal, which is the prin- 
 cipal itself. 
 
 From this we compute the following table : — 
 At 6 per cent per year, interest for — 
 
 200 mo., or 16 yr. 8 mo. = principal. 
 
 J of 200 mo., or 8 yr. 4 mo. = J of prin. 
 
 J of 200 mo., or 66f mo., or 5 yr. (5 mo. 20 da. = J of prin. 
 
 J of 200 mo., or 50 mo., or 4 yr. 2 mo. = J of prin. 
 "" ^ of 200 mo., or 40 mo., or 3 yr. 4 mo. = J of prin. 
 
 J of 200 mo., or 33J mo., or 2 yr. 9 mo. 10 da. = J of prin. 
 
 J of 200 mo., or 25 mo., or 2 yr. 1 mo. = J of prin. 
 
 ^i_. of 200 mo., or 20 mo., or 1 yr. 8 mo. = -yL. of prin. 
 
 jL of 200 mo., or 16f mo., or 1 yr. 4 mo. 20 da. == -^L of prin. 
 
 j^-g of 200 mo., or 13 J mo., or 1 yr., 1 mo. 10 da. = y^ of prin, 
 
 j^ of 200 mo., or 12i mo., or 1 yr. 15 da. = ^'^ of prin. 
 
 1, What is the interest of $60 for each time men- 
 tioned in the table ? 
 
 ^n*. — The interest of $60 for 200 mo., or 16 yr. 8 mo. = 
 $60; for 100 mo., or 8 yr. 4 mo. — ^ of $60 = $30; for 66| 
 mo., or 5 yr. 6 mo. 20 da. = J of $60 = $20, &c., &c. 
 
 2. What is the interest of $36 for each time men- 
 tioned in the table ? 3. Of $48.72 ? 
 
 What is the interest of — 
 
 4. $40 for 100 mo. ? 6. $64 for 4 yr. 2 mo. ? 
 
 5. $48 for 12i mo. ? 7. $24.60 for 5 yr. 6 mo. 20 da. ? 
 
168 colburn's first part. 
 
 $66 for 16§ mo. ? 16, $16.64 for 1 yr. 15 da. ? 
 
 $24.36 for 66§ mo. ? 17. $25.75 for 3 yr. 4 mo. ? 
 
 $16.98 for 25 mo. ? 18. $44.36 for 1 yr. 8 mo. ? 
 
 $84.60 for 20 mo. ? 19. $16.24 for 8 yr. 4 mo. ? 
 
 $42 for 50 mo. ? 20. $44 for 2 yr. 1 mo. ? 
 
 $37 for 40 mo. ? 21. $60.45 for 1 yr. 1 rao. 10 da. : 
 
 $54.72 for 33J mo. ? 22. $43.78 for 16 yr. 8 mo. ? 
 
 $75.15 for 13 J mo. ? 23. $75 for 2 yr. 9 mo. 10 da. ? 
 
 24. What is the amount of each of the ahove ? 
 
 E. The interest for 20 mo., or 1 yr. 8rao., being -jJ^ of the 
 principal, may be found by removing the decimal point 1 place 
 to the left, and is as many dimes as there are dollars in the 
 principal. 
 
 Hence the interest for — 
 
 J of 20 mo., or 10 mo., = ^ of ^^ of principal. 
 
 J of 20 mo., or 6f mo., or 6 mo. 20 da. = J of ^ of prin. 
 
 J of 20 mo., or 6 mo. = J of -J^ of prin. 
 
 J of 20 mo., or 4 mo. = J of y'^ of prin. 
 
 J of 20 mo., or 3 J mo., or 3 mo. 10 da. = J of ^^ of prin. 
 
 i of 20 mo., or 2J mo., or 2 mo. 15 da. = J of ^^ of prin. 
 
 jJL of 20 mo., or If mo., or 1 mo. 20 da. = ^'^ of -j'^ of prin. 
 
 j^ of 20 mo., or IJ mo., or 1 mo. 10 da. = -jJ^ of ^^ of prin. 
 
 1. What is the interest of $24 for each time men- 
 tioned in the table ? 
 
 Ans.— The interest of $24 for 10 mo. = J^ of $2.40 = $1.20 ; 
 for 6| mo., or 6 mo. 20 da. = J of $2.40 = $.80, &c. 
 
 2. What is the interest of $120 for each time 
 
LESSON FIFTY-NINTH. 169 
 
 mentioned in the table? 3. Of $7.50? 4. Of 
 $4.86? 
 
 What is the interest of — 
 
 5. $72 for 2 J mo, ? 12. $483.60 for 1 mo. 20 da. ? 
 
 6. $60 for I J mo. ? 13. $27 for 5 mo. ? 
 
 7. $486 for If mo. ? 14. $7.50 for 2 mo. 15 da. ? 
 
 8. $15 for 10 mo. ? 15. $74.10 for 3 mo. 10 da. ? 
 
 9. $2.40 for 6f mo. ? 16. $55 for 1 mo 10 da. ? 
 
 10. $64.50 for 4 mo. ? 17. $1.86 for 6 mo. 20 da. ? 
 
 11. $36.60 for 3J mo. ? 18. $54.20 for 4 mo. ? 
 
 19. What is the amount of each of the above? 
 
 F. The interest for 2 months, or 60 days, "being ^-J^ of the 
 principal, it follows that the interest for — 
 
 J of 2 mo., or 1 mo., or 30 da. = J of jj^ of the principal. 
 
 J of 2 mo., or 20 da. = J of ^^^ of the prin. 
 
 J of 2 mo., or 15 da. = J of -j-i.^ of the prin. 
 
 J of 2 mo., or 12 da. = J of y|^ of the prin. 
 
 J of 2 mo., or 10 da. = J of y^^ of the prin. 
 
 J^ of 2 mo., or 6 da. = ^^ of j^^ or y^i^^ of the prin. 
 
 -jL of 2 mo., or 5 da. = ^^ of j.}^ of the prin. 
 
 J of 6 da., or 3 da. = J of y^*j^ of the prin. 
 
 J of 6 da., or 2 da. = J of y^J^^ of the prin. 
 
 I of 6 da., or 1 da. = J of y^^^^^ of the prin. 
 
 1. What is the interest of $432 for each time 
 mentioned in table ? 
 
 Solution.— The interest of $432 for 2 mo. is $4.32; for 1 mo. is I 
 of $4.32, which is $2.16, &c., &c., * * * for 6 days, is $.432; for 3 
 (lays is h of $.432, which is $.216, &c, &c. 
 _ 
 
170 colburn's first part. 
 
 2. What is the interest of $360 for each time 
 mentioned in the table ? 3. Of $60.30 ? 
 
 What is the interest of — 
 
 4. $42 for 20 da. ? 9. $192 for 5 da. ? 
 
 6. $36.24 for 15 da. ? 10. $43.50 for 12 da. t 
 
 6. $48 for 10 da. ? 11. $86.37 for 30 da. 
 
 7. $89 for 6 da. ? 12. $228 for 1 da. ? 
 
 8. $174 for 3 da. ? 13. $234 for 2 da. ? 
 
 14. What is the amount of each of the above ? 
 
 G. The foregoing principles furnish short and expeditious 
 methods of computing interest for any time whatever. 
 
 1. What is the interest of $72.60 for 8 mo. 20 da.? 
 
 1st Solution. — 8 mo. 20 da. = 6 mo. 20 da. -f- 2 mo. The interest 
 of $72.60 for 6 mo. 20 da. = i of $7.26 = $2.42, and the interest for 
 2 mo. = $.726, which, added to $2.42 = $3,146 = Ans. 
 
 2d Solution. — 8 mo. 20 da. == 10 mo. — 1 mo. 10 da. The interest 
 $72.60 for 10 mo. = i of $7.26 = $3.63, and the interest for 1 mo. 10 
 da. = ^ij of $7.26 = $.484, which, subtracted from $3.63 = $3,146 
 = Ans. 
 
 3d Solution. — 8 mo. 20 da. = 8 mo. -f- 20 da. The interest of 
 $72.60 for 8 mo. or 4 times 2 mo. = 4 per cent of $72.60 = $2,904, 
 and the interest for 20 da. = J of $.726 = $.242, which, added to 
 $2,904 = $3,146 = Ans. 
 
 The work can be wi'itten as follows: — 
 
 1st Solution. 2d Solution. 
 
 $72.60 =prin. $72.60 = prin. 
 
 2A2 = int. 6 mo. 20 da. 3.63 = int. 10 mo. 
 
 .726 = int. 2 mo. .484 = int. 1 mo. 10 da. 
 
 $3,146 = int. 8 mo. 20 da. $3,146 = int. 8 mo. 20 da. 
 
 The form for the third solution would be similar to these. 
 
LESSON FIFTY-NINTH. 171 
 
 What is the interest of — 
 
 2. $90 for 3 mo. 16 da. ? 11. $54.24 for 7 mo. 15 da. ? 
 
 3 $128 for 22 mo. 15 da. ? 12. $150 for 35 mo. 10 da. ? 
 
 4 $64 for 2 mo. 10 da.? 13. $184 for 2 yr. 3 mo. ? 
 
 5. $32 for 5 mo. 15 da. ? 14. $96 for 52 mo. 15 da. ? 
 
 6. $120.90 for 3 mo. 20 da.? 15. $186.60 for 7 mo. ? 
 
 7. $88.24 for 5 mo. 6 da.? 16. $28.16 for 2 yr. 6 mo. ? 
 
 8. $72.96 for 1 mo. 26 da. ? 17. $384 for 19 mo. 27 da.? 
 
 9. $500 for 9 mo. 24 da. ? 18. $30.24 for 6 mo. 17 da.? 
 10. $1000 for 3 mo. 29 da. ? 19. $450.36 for 3 mo. 15 da. ? 
 
 H. Business men often use such methods as the following in 
 connexion with those already explained : — 
 
 At 6 per cent, per year the interest of $2 for 1 month is 1 cent. 
 Hence — 
 
 The interest of $2 is 1 cent per month. 
 
 The interest of $20 is 1 dime per month. 
 
 The interest of $200 is 1 dollar per month. 
 
 1. What is the interest of $2 for each of the fol- 
 lowing times ? 
 
 3mo. ? 
 
 2 yr. 3 mo. ? 
 
 15J mo. ? 
 
 9 mo. ? 
 
 1 yr. 5 mo. ? 
 
 4 mo. 10 da. ? 
 
 15 mo. ? 
 
 2 yr. IJ mo. ? 
 
 2 yr. 7 mo. ? 
 
 What is the interest for each of the above times 
 of — 
 
 2. $1? 6. $5? 10. $500? 
 
 3. $6? 7. $10? 11. $14? 
 
 4. $8? 8. $200? 12. $80? 
 
 5. $20? 9. $50? 13. $800? 
 
172 colburn's first part. 
 
 At 6 per cent, per year — 
 The interest o/ $6 is 1 mill per day. 
 The interest of $60 is 1 cent per day. 
 The interest o/$600 is 1 dime per day. 
 The ifiterest of $6000 is 1 dollar per day. 
 
 What is the interest of $6 for each of the fol- 
 lowing times ? 
 
 3 da. ? 1 mo. 3 da., or S3 da. ? 3 mo. 6 da. ? 
 
 7 da. ? 1 mo. 17 da. ? 6 mo. 12 da. ? 
 
 19 da. ? 2 mo. 25 da. ? 4 mo. 9 da.? 
 
 What is the interest for each of the above times 
 
 of — 
 
 
 
 
 
 1. $60? 
 
 5. 
 
 $600? 
 
 9. 
 
 $6000? 
 
 2. $30 ? 
 
 6. 
 
 $300? 
 
 10. 
 
 $1000? 
 
 3. $20? 
 
 7. 
 
 $150? 
 
 11. 
 
 $1500? 
 
 4. $120? 
 
 8. 
 
 $1800? 
 
 12. 
 
 $500? 
 
 LESSON LX. 
 
 MISCELLANEOUS PROBLEMS. 
 
 1. 7 times 8, plus 4, divided by 5, multiplied by 3, minus 4, 
 minus 8, divided by 2, divided by 3, multiplied by 9, multiplied 
 by 2 = how many times ? 
 
 2. Multiply J of 18 by J of 8, add J of 27, divide by J of 28, 
 add ^j of 33, and square the number. 
 
 3. If I of a yard of cloth worth 14 cents per yard, are given 
 for § of a pound of chocolate, how many pounds of coffee at 12 
 cents per pound should be given for 3J pounds of chocolate ? 
 
LESSON SIXTIETH. 173 
 
 4. If I should expend the sum of $9 + $8 + $5 + $9 + $4 
 -|- $7 for flour at $7 per barrel, and sell the flour at $8 per 
 barrel, then expend the proceeds for cloth at $3 per yard, and 
 sell the cloth for $4 per yard, and then, after spending $10, and 
 losing $6, should expend the remainder for tea at the rate of 3 
 pounds for $2, how many pounds of tea should I buy ? 
 
 5. Find the cost of 1| lb. coffee at 16 cents per lb., 2i lb. 
 raisins at 8 cents per lb., 2| lb. figs at 15 cents per lb., 4 J lb. 
 sugar at 9 cents per lb., 1| lb. tea at 40 cents per lb., and J lb. 
 cotton at 32 cents per lb. Make out a bill on the supposition 
 that you sold the above articles to one of your school-mates. 
 
 6. What must be the length of the side of a square field con- 
 taining J^ as many square rods as a field 9 rods long and 8 rods 
 wide? 
 
 7. Arthur sold a certain number of apples at the rate of 2 for 
 a cent, and Robert sold as many at the rate of 3 for a cent. 
 Arthur received 12 cents more than Robert. How many apples 
 did each sell ? 
 
 8. A thief drew J of the wine out of a certain cask, and, to 
 escape detection, filled it with water. The next night he drew 
 out J of the contents of the cask, and again filled it with water. 
 How many gills of wine will there now be in each gallon of the 
 mixture ? 
 
 9. 2i times a certain number added to ^ of that number is 5^ 
 less than 3 times the number. What is the number ? 
 
 10. A lady being asked her age, replied, <' My father is 30 
 years older than my sister Sarah, and 8J times the difference 
 between their ages is 5 times my father's age. Now, if you will 
 tell how old my father and sister are, I will tell you how to find 
 my age ? " A correct answer having been given, the lady said, 
 " To 3 times my father's age, add 6 times my sister's age, and 
 
174 colburn's first part. 
 
 you will obtain a sum J of which is 9 yeai s more than 4^ times 
 my age ?" What was the age of each ? 
 
 11. A teacher wishing to obtain a bU.<;V board 15 ft. long and 
 6 ft. wide, bought boards for the purpose at 2i cents per square 
 foot. He hired a carpenter to make it, paying him 75 cents for 
 his work. He paid 11 cents per square yard to have it painted 
 and varnished, and it cost 25 cents to have it brought to the 
 school-room and put up. What was the whole cost ? 
 
 12. My parlor and sitting-room are each 5 yards wide, but 
 my parlor is 2 yards longer than my sitting-room. Yhe floor of 
 my sitting-room contains 30 square yards. What is the length 
 of my parlor floor, and how many square yards does it contain ? 
 
 13. David said to Harry, ** If i the sum of our ages be added 
 to i of your age, the same will equal | of my age, and I am 12 
 years older than you are. What was the age of each of the 
 boys? 
 
 14. Mr. Warren bought a cask of oil at $1.20 per gallon, but I 
 of it leaked out. For how much per gallon must he sell the rest 
 so as neither to gain nor lose ? 
 
 15. Mr. Allen owes Mr. Mason 62 cents, and the only coins 
 he has are 1 half-dollar, 1 quarter-dollar, 1 half-dime, and 2 
 three-cent pieces, while the only coins Mr. Mason has are 4 half- 
 dollars, 5 dimes, and 2 cents. How can change be made so that 
 the debt may be paid with these coins ? 
 
 16. What number added to J of itself equals 36 more than i 
 of the number ? 
 
 17. A man sold 6 barrels of apples and 2 barrels of pears for 
 $23, receiving twice as much per barrel for th,e pears as for the 
 apples. IIow many dollars did he receive for each ? 
 
 18. By selling cloth at $3.50 per yard, I lose 12i per cent of 
 its cost. How many dollars should I lose on each yard by selling 
 it at S3 per yard? 
 
LESSON SIXTIETH. 175 
 
 19. I sold i of a lot of grain for what | of it cost, thereby 
 gaining $16.. How much did the entire lot cost me ? 
 
 20. A. and B. traded in company. A. put in $360, and B. 
 put in § of J of I of 42 times i as much as A. They gained a 
 a sum equal to f of their joint stock. How much did they gain, 
 and what was the share of each ? 
 
 21. If Mr. Walton's blackboard were 2 ft. wider than it now 
 is, it would contain 26 more square feet, but if it were 2 feet 
 longer, it would contain 11 more square feet. How many square 
 feet does it contain ? 
 
 22. George has money enough to buy 2^ quarts of chestnuts, 
 Rufus has twice as much as George, and Edward has J as much 
 as Rufus. They all have 57 cents. How much are the chestnuts 
 worth per quart, and how many cents has each of the boys ? 
 
 23. The interest of Mr. Butler's money for 5 yr. 6 mo. 20 da., 
 at 6 per cent, will equal $8000. How much money has he ? 
 
 24. If a pound of rice is worth f as much as a pound of sugar, 
 and 6 lb. of rice and 10 lb. of sugar are worth $1.26, how much 
 are 5 lb. of rice and 7 lb. of sugar worth ? 
 
 26. Why is it that if we multiply any number whatever by 3, 
 add 7 to the product, add the first number taken to this, add 9 to 
 this, divide this by 4, add 3 to this, and then subtract from this 
 the first number taken, the result will always be 7 ? 
 
 26. By selling cloth at $1.25 per yard, I lose 16| per cent. 
 For how much per yard must I sell it to gain 20 per cent ? 
 
 27. There are f as many acres in my orchard as there are in 
 my pasture, and J as many in my garden as in my orchard. If 
 there are 17 acres in all, how many are there in each lot? 
 
 28. I bought a lot of goods for $600, and after keeping them 
 1 month 17 days, I sold them for $650. Now, allowing that I 
 had to pay interest on the money invested, at the rate of 6 per 
 cent, what was my net gain ? 
 
176 colburn's first part. 
 
 29. A man bought a cask of wine, but § of it leaked out. He 
 put in as much water as there was wine remaining, and sold the 
 mixture at the same price per gallon that he gave for it. What 
 part of the cost did he lose ? 
 
 30. After paying $3 more than ^ of my money to one man, and 
 §G more than i of what I had left to another, I had $7 left. How 
 much did I have at first ? 
 
 31. I sold 10 bushels of corn for Mr. Austin, and 8 bushels 
 for Mr. Brown, receiving $11 for the lot. Now, allowing that 
 Mr. Austin's corn is worth 20 per cent more per bushel than Mr. 
 Brown's, and that I am to receive $1 for my services, how much 
 money ought I to pay to each ? 
 
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 VALUABLE SCHOOL BOOKS. 
 
 WARREN'S SERIES OF GEOGRAPHIES. 
 
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 THE COMMON-SCIIODL GEOGRAPHY. 
 
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 JBER>ilFS HISTORY OF THE UNITED STATES. 
 
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