UC-NRLF SB E7fl flfll iHBBBHMBMMi XZ/^T >> i m 1 b i n $ n ' s < tj n r t e,r ! $ &u x ; & ^ , FIRST BOOK AEITHMETIC, INCLUDING OEAL AND WRITTEN EXEECISES. BY DANIEL W. FISH, A.M., EDITOR OF ROBINSON'S SERIES OF PROGRESSIVE ARITHMETICS. IVISON, BLAKEMAN, TAYLOR & CO., NEW YORK AND CHICAGO. 1878 GHORTER COURSE. FIRST BOC ' IN ARITHMETIC. PRIMARY. COMPLE TE . -RITHME TIC. In One volume* COMPLETE ALGEBRA. ARITHMETICAL PROBLEMS. ORAL and WRITTEN. ALGEBRAIC PROBLEMS. KE YS to COMPLETE ARITHMETIC and PROBLEMS, and to COMPLETE ALGEBRA and PROBLEMS, in separate volumes, for Teachers. Arithmetic, on AT, and WRITTEN, usually taught in THREE books, is now offered, complete and thorough, in ONE book, " THE COMPLETE ARITHMETIC." * This COMPLETE ARITHMETIC is also published in TWO VOLUMES. TAUT J. and PART II. are each bound separately , in CLOTH. Copyright, 1874, by DANIEL W. FISH. Eiectrotyped by SMITH & McDoucAL, 82 Beekman St., N. Y. A RITHMETIC has been defined as "the science of numbers, and J-^- the art of computing by means of them." It embraces, 1st. The mode of representing numbers by figures and signs, in accordance with an accepted system ; 2d. The principles and methods of their combination in addition, subtraction, multiplication, and division ; and, 3d. The application of these principles and methods to the solu- tion of practical problems. Primary Arithmetic can do little more than put the pupil in pos- session of the alphabet of numbers, and make him familiar with the various tables, securing readiness and accuracy in their use, and neatness and skill in written exercises upon them. In this book, the object has been to secure this end by easy, gradual, and national steps, and in such a familiar manner, as to avoid the drudgery of memorizing the abstract tables, and at the same time inspire : "" Diligent thought in regard to the more simple processes that involve the use of numbers. The plan of this book is unique, and it is believed that it will supply a want long felt by primary-school teachers. The* natural and orderly development of the subject, the systematic arrangement, the copious illustrative exercises, and the provision for exhaustive drill exercises, cannot fail to meet the views of the most thorough and exacting teacher, and, at the same time, interest and attract the pupil. The first seventeen lessons are intended to present the numbers from one to ten, inclusive, as numbers, in such manner that each of them shall be intelligently apprehended, not merely as a name> but as an IDEA what it is, how it is represented, and what is its value relative to each of the others. fi r^ 7 1 : IV PREFACE. Succeeding this are familiar illustrations of the use of signs ; ana a series of easy and progressive exercises present the various com- binations that can be made by means of any two of all the numbers under 12 giving practical drill exercises on the tables, and illus- trating their use by means of problems. A new form of table, "equal parts of numbers" has been intro- duced and practically applied, consistent with primary operations upon whole numbers, naturally deduced from the multiplication table as is the ordinary table of division. The practical value of this will be apparent from the problems and examples given in illustration, and the drill exercises in connection with it. The exercises in notation and numeration are simple and pro- gressive, and may be further extended, at the option of the teacher. The latter part of the book makes a more thorough, but still pro- gressive and systematic, presentation of principles and methods in the fundamental ruleaj oral and written. The simple exercises here presented in Fractions and Measures (Denominate Tables) will furnish opportunity for more extended exercises, such as the judicious teacher may desire, or the require- ments of particular classes may demand. The Pictot^al Illustrations, designed more for use than ornament, will commend themselves to the taste and judgment of discrimi- nating teachers. In the preparation of this book, the author has kept constantly in view such a systematic arrangement and development of prin- ciples and methods as to present the subject in the most natural as well as the most comprehensive manner. The author desires to make special acknowledgment of the valu- able services rendered in the plan, arrangement, and compilation of this book, by James Cruikshank, LL.D., a gentleman well known fco the educational world, whose large experience for a number of years as Superintendent of the Primary Schools of the city of Brooklyn has made him familiar with the needs of teachers of this grade of schools. With a desire to contribute to the facilities for elementary in- struction, this little work is confidently submitted to the public. D. w! F. BROOKLYN, April, 1875. HINTS TO r 'EACHERS rpHE division of this book into lessons is not at all intended to -*- mark out the limit of the daily exercises. Each lesson dis- cusses a separate topic, and many of them furnish or suggest mat- ter upon which several days may be profitably spent. Some of them present drill exercises that may be frequently repeated with profit returning to them from more advanced periods. Practically, the exercises embraced in this book cover three years or more of the primary school course as prescribed in most of our city schools. Advance slowly ; one step at a time, and always secure a perfect mastery of any principle or operation upon which another depends, before proceeding to the next. Go over only so much ground at any one exercise as may be thoroughly understood, and review daily. The greatest source of embarrassment to the teacher, and of dis- gust and waning interest on the part of the pupil, is found in the accumulation of imperfectly mastered lessons. Endeavor to secure the interest of the class, and never do for a pupil what he can be readily led to do for himself. Slate exercises are important from the first, and if judiciously conducted can never fail to please and instruct. The various combinations by addition, subtraction, multiplica- tion, and division, presented in the tables, furnish the instruments for all arithmetical operations. If the pupil is skillful in these, the only other thing needful is such a familiar knowledge of the relation of things as to know what process should be used in the solution of problems. In the early lessons in arithmetic, the judicious teacher will observe that the introduction of numbers and of the successive digits representing them should be gradual. Examples should at first contain only 1's and 2's ; then 1's, 2's, and 3's, until the pupil can add rapidly and correctly in whatever order they are combined ri HINTS TO TEACHERS. Then introduce 4's with the preceding, etc. This remark applies also to subtraction, multiplication, etc. In all cases where there is hesitation or forgetfulness, return to special drill, in series, to mas- ter the particular number upon which the fault occurs. The fundamental idea in all numerical combinations is found in counting in series. If each step as outlined in this book is mastered as indicated, progress will be easy and rapid, and the result most satisfactory. Thus, in counting by 6's, the following give all possible addi- tions of 6 with the units of any other number : + 6 + 6; 1 + 6 + 6; 2 + 6 + 6; 3 + 6 + 6; 4 + 6 + 6; 5 + 6 + 6; 6 + 6 + 6. Counting back gives all possible subtractions of 6. Multiplication and division, through the limit of each table are involved in count- ing by 6's to 72 ; as, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, and similarly for other tables. Observe carefully the models under the several lessons. At each step, as the pupil becomes familiar with the formal oper- ation of summing the series, he should be led to observe and state how many times the recurring number has been used ; thus, 6, 12, 18 (6 + 6 + 6) ; there are 3 sixes in 18, or 3 times 6 are 18, etc. Little real progress can be made even in memorizing until the name of each of the digits becomes to the pupil an intelligible sign of the number for which it stands. Thus, the figure 5, or the name five, should, upon being seen or heard, as clearly recall the idea of 5 units, singly, and together, as any common word brings up to the mind the idea which it represents. Care should be taken that the eye, as well as the ear, be addressed and cultivated. Skillful oral repetition of the tables does not necessarily produ'ce rapid and correct results, when the pupil has need to perform operations silently. This is an important consid- eration, inasmuch as the practical use of arithmetic is not oral, but mental, and the eye and the hand, rather than the ear and the tongue, become the instruments. BLACKBOARD DRILL. The blackboard should be a constant accessory in school-room instruction. A few among the many methods which should be used from time to time are suggested : 1. An exercise having been written upon the board, let a pupil, as called upon, go through the exercise as rapidly as is consistent HIKTS TO TEACHERS. HI with accuracy, pupils or teacher indicating errors in such a way as may be deemed expedient. Generally the pupil should be required to correct the error himself, when attention is called to it. Another may then take up the work, and so on. 2. Proceed as before, except that each pupil in turn should name one step or result, and any error being made, the next should correct it, or, failing to do so, any member of the class may raise his hand, and make the correction. No blunder should be allowed to pass unnoticed. It is generally advisable that each class exercise illustrated upon the board be also made a slate exercise for silent work. 3. When a little familiarity with any class of exercises is secured, the pupils should be encouraged to do the work upon the black- board themselves, without the intervention of the teacher. 4. From the very first no carelessness or slovenliness in making figures, or in the general form of the written exercises should ever, under any pretense, be allowed. Time spent in securing neatness will be regained tenfold in the pupils' subsequent progress, and in the culture in which it will result. This remark applies also to slate exercises. All slate exercises should be carefully examined, aod the errors pointed out, and then corrected by the pupil. Primary arithmetic does not involve any complicated processes of analysis or of reasoning. It deals chiefly with facts, and considers only the simplest and most evident relations of things. It is, therefore, recommended that for mal analyses be used but sparingly. Those given upon pages 41, 45, 46, and elsewhere, are only sugges- tive, and after the process (that is, the nature of the operation) in any given case is understood, they may be discontinued, or varied, or used only occasionally. They are not in themselves an end, but only a means of determining the operation to be performed. It is recommended that wherever problems are introduced, the relations of the things to which they refer be carefully explained, and then the relations of the numbers will be readily understood. The attention of the pupils may be called to the several steps by judicious questions, and they may also be encouraged to make problems suited to numbers given in any case ; as, given 5x4; we may say, " What is the cost of 4 yards of tape at 5 cents a yard ? " etc. Till HINTS TO TEACHERS. When problems involve more than one operation, the pupil's attention should be called to the reasons and necessity for each. See example, page 90, Ex. 7. Most of the lessons may and should be very much extended by additional examples and illustrations, always, however, observing to keep icithin the scope and spirit of the lesson. As, on page 10, no exercise must embrace any number beyond 6, nor any combination whose result is greater than 6 ; on page 50, no result greater than 36, and no number greater than 6 used in producing the result. In the review of each lesson, all smaller numbers should be used. The exercise on page 46, and others similar following, do not belong to Fractions, but exhibit a simple form of deduction from multiplication ; thus, since 3 times 4 are 12, it follows that 4 is contained 3 times in 12, and also that one-third of 12 is 4. It will be well generally to teach these in connection, as on page 101. Counting in series orally, and as illustrated on the board, should be often repeated, and many exercises may be given besides those contained in the lessons. FRACTIONS. No attempt has been made in this book to do more than present the simplest elementary ideas of fractions. Every exercise should be carefully illustrated by objects, and by lines or figures upon the board. MEASURES. So far as practicable, each of the tables should be illustrated by actual objects, presenting to the senses the values named, and the relations to each other of the different units. Exer- cises may be much extended, so that intelligent skill shall be acquired. Even though a class, upon taking up this book, have already acquired some knowledge of numbers, it will be found profitable to review carefully most of the lessons from the beginning, or at least to ascertain that each pupil is skillful in the exercises. In all oral exercises, so called, the aim should be not to make the pupil simply flippant and ready in repeating a number of similar exercises after a model has been given, for this often requires little or no thought, and is practically useless ; but what we may call mental msion should be cultivated to bring before the mind all the numbers and conditions involved, and arrange or group them so that the result may become apparent. The first lesson in numbers to be taught the child is to count. He cannot learn the yalue of figures from 1 to 10 until he can count ten objects. The Numeral Frame is one of the most convenient aids in teaching to count, and the primary operations in the use of numbers. When this cannot be had, a box of beans, marbles, or similar objects may be used. Holding up the frame before the pupils, move the balls on the first wire, one at a time, while the children count one, two, three, four, five, six, seven, eight, nine, ten. Then the exercise may be extended to twenty, by moving the balls on two wires ; then to thirty, by moving the balls on three wires, and so on to one hundred. When the pupils can count by ones to one hundred, they may be taught to count by twos. Move two balls on one wire, and two more on the next, and so on, the pupils saying, two, four, six, eight, ten, twelve, etc. When the pupils can count by tivos as far as fifty, and have also learned to write figures as far as twenty, write a column of 2's on the board, and train them to add the figures in the same manner as the balls were counted. Next, move one ball on the first wire, and two balls on each succeeding wire, while the pupils say, one, three, five, seven, nine, eleven, etc. After sufficient exercise on these combinations, require them to write a short column of 2's on their slates with a 1 at the bottom, and add them as before, writing the sum below the column. 10 IK A HITH,METIC. When sufficiently drilled &TL..iwos, the same method may be pursued in teaching them to count by threes. Thus, three, six, nine, twelve, fifteen, etc. Then com- mencing with one ; thus, one, four, seven, ten, thirteen, etc. Then with two ; thus, two, five, eight, eleven, fourteen, etc. Follow these drills with the same exercises on the slate or board as with the 2's. The same method should be pursued in teaching them to count and add by fours, fives, sixes, etc. At the same time that the pupil is thus taught to count, and to know figures as symbols, he should also be taught their value, and the value of numbers as associated with the number of objects counted. He should be required to perform the same operations on the slate or blackboard with figures that he has performed orally with objects. The apt teacher, by a judicious use of this frame, may easily teach a child to count from one to a hundred, and to add, subtract, multiply, and divide with facility. The counting and other operations should be done silently by the eye, and results only given by the voice. It is not intended to make a small work like this take the place of the living teacher. The sample lessons can contain but a few hints and methods to aid the teacher in giving the pupil something to do; therefore, the number and variety of the exercises on each page, both for oral and written drill, should be increased as the cir- cumstances and the capacity of the child seem to require. The how and the why can be much better explained by the teacher -than by the author, in so limited a space. One Nest Two Birds Three Leaves 3 III. Steeples ^ IY. Sheep ff V. LESSON I. 1 I. 2 II. Boys Seven Girls Eif/ht Acorns 6 VI. 7 vii. 8 VIII. Berries 9 IX. Posts 10 X. FIRST BOOK LESSON II. Here is a picture of a frame, with ten wires, and ten balls on each wire. It is called a Numeral Frame. 1. On the second wire at the right is one ball. One. Written 1 2. How many balls are one ball and one ball more ? Two. Written 2 3. Two balls and one ball? Three. Written 3 4. Three balls and one ball? Four. Written ^ 5. Four balls and one ball? Five. Written 5 6. Five balls and one ball? Six. Written 6 7. Six balls and one ball? Seven. Written / 8. Seven balls and one ball ? Eight. Written 8 9. Eight balls and one ball ? Nine. Written 9 10. Nine balls and one ball ? Ten. Written 10 11. How many balls on each wire ? Ten. 12. ' Is there any figure that stands for ten ? 13. What is the largest number expressed by one figure ? 14. What does 1 mean when it stands alone ? 15. What does it mean when it has a on the right of it ? Figures. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Naught, One, Two, Three, Fonr, Five, Six, Seven, Eight, Nine. COUNTING. AEITHMETIC. 13 LESSON III. 1. What do you see in the picture ? 2. How many horses ? How many dogs ? 3. What is in the dog's mouth ? How many baskets ? 4. Show me one book. Point to one boy. 5. Hold up one hand. How many hands have you ? 6. Hold up one finger. Hold up one more. 7. One finger and one more are how many fingers ? ''8. Two fingers are how many more than one finger ? 9. One finger is how many less than two fingers ? 10. How many ears has the horse ? How many, the dog ? 11. Make one short line on your slate. / 12. One line and one more are how many lines ? / / 13. How many ones make two ? 14. Kub out one line and how many are left ? 15. Two lines less one line are how many lines ? 16. What is a single thing called ? 17. Write the word one on your slate. &ne. 18. Make the figure for one on your slate. _/. W. How many are one and one more ? Two. FIEST BOOK LESSON IV. 1. How many girls in this picture ? How many birds ? 2. How many roses has one girl in her hand ? 8. How many roses has the other in her apron ? 4. One bird and one bird are how many birds ? 5. Two are how many more than one ? 6. How many ones make two ? 7. How many eyes have you ? How many ears ? 8. Hold up two fingers. Hold up one more. 9. Make two short lines on your slate. // 10. Make one more. How many are there now ? // / 11. Two lines and one line are how many lines ? 12. One and two are how many ? 13. Two lines are how many less than three lines ? H. Three lines are how many more than two lines ? 15. Eub out one line ; how many are left ? 16. Rub out two more lines ; how many are left ? 17. Write the word two on your slate. Q/tvo. 18. Make the figure for tivo on your slate. 2. 19. How many are two and one more ? Three* ARITHMETIC. 15 LESSON V. 1. In the picture, how many squirrels ? How many bees ? How many squirrels with a nut ? 2. Two squirrels and one squirrel are how many ? 3. Three bees are how many more than 2 bees ? 4. Three less one are how many ? Three less 2 ? 5. How many ones make three ? 6. How many are 2 and 1 ? 1 and 2 ? 7. Make three short lines on your slate. 8. Eub out one line ; how many are left ? 9. Rub out one more ; how many are left ? 10. Three less three are how many ? 11. Make three lines again. Now one more. 12. Three lines and one line are how many lines ? 13. Hold up one finger. Two fingers. Three fingers. 14. Count three. How many ones in three ? 15. How many 2's in three ? Ans. One 2 and 1 over. 16. Write the word three on your slate. dffllee. 17. Make the figure that stands for three. 3. 18. How many are three and one more ? Four. Ill III I I 16 FIEST BOOK LESSON VI. 1. How many birds in the picture ? How many eggs ? 2. How many are 2 birds and 2 birds ? & How many ones in four ? 4. Make 2 short lines ; then 2 more. // // How many 2's in four ? 2 and 2 are how many ? 6. Count four by ones. Count four by 2's. 7. Two birds taken from four birds leave how many? 8. Four less one are how many ? Four less 3 ? 9. How many are 3 and 1 ? 1 and 3 ? 10. How many ones in four ? How many 2's ? 11. How many 3's in four ? Ans. One 3 and 1 over. 12. Four are how many more than three ? How many more than two ? 18. How many eggs must be taken from the nest to leave 1 egg ? How many to leave 2 ? To leave 3 ? 14. Write the word four on your slate. ffikut,. 15. Make the figure that stands for four. Jf. 16. Count from 4 back to one. Four, three, two, one. 17. How many are four and one more ? Five. ARITHMETIC. 17 LESSON VII, 1. In this picture how many birds ? How many peaches ? How many birds on the limb ? 2. Five birds are how many more than 4 ? Than 3 ? 3. Three birds and 2 birds are how many birds ? ^. Two peaches and 3 peaches are how many ? 5. Five peaches less 3 peaches are how many ? 6. Five peaches less 2 peaches are how many ? 7. Two fishes and how many more make five fishes ? 8. Three fishes and how many more make five ? 9. Make 3 short lines. Now 2 more. /// // 10. Three and 2 are how many ? 2 and 3 ? .7L Make 4 short lines. Now one more. III! I 12. Four and 1 are how many ? 1 and 4 ? -7$. If 2 birds fly away, how many are left ? If 2 more ? 14. How many 2's in five ? .4^5. Two 2's and 1 over. 15. Write the word^zm (ffiwe. 16. Write the figure that stands for five. 5. 17. Count five. Count from five back to one. 18. How many are five and one more ? Six. 18 FIEST BOOK LESSON VIII. 1. There are three boats on the water and three on the land. How many in all ? 3 and 3 are how many ? 2. How many boys on the ice ? How many 2's in six? 3. Three boats from six boats leave how many boats ? 4. Six boys are how many more than 4 boys ? 5. How many 2's make six ? How many 3's make six ? 6. Count six by ones. By 2's. By 3's. 7. There are 2 sails on 1 boat ; how many on 3 boats ? 8. Three 2's are how many ? Two 3's are how many ? 9. How many are six boats less 5 boats ? 10. Four boats and how many more make six boats ? 11. Six boys less 2 boys are how many boys ? Six boys less 4 boys are how many ? 12. Three boats are how many less than six boats ? 13. Make six lines by 2's. ////// By 3's. ////// 14. Write the word six on your slate. <3&&. 15. Make the figure that stands for six. 6. 16. Count six. Count from six back to one. 17. How many are six and one more. Seven. IKAEITHMETIC. 19 LESSON IX. L How many boats are 2 boats and 2 boats ? 2. How many boys are 2 boys, 2 boys, and 2 boys ? #. Count by 2's to 4. Count by 2's to 6. 4. How many 2's in 6 ? How many 3's in 6 ? 5. Six cents are how many more than 3 cents ? How many more than 4 cents ? Than 2 cents ? 6. Kepeat this table. 1 and 5 are 6. 4 and 2 are 6. 2 and 4 are 6. 5 and 1 are 6. 3 and 3 are 6. 6 and are 6. 7. 6 from 6 leaves 0. 3 from 6 leaves 3. 5 from 6 leaves 1. 2 from 6 leaves 4. 4 from 6 leaves 2. 1 from 6 leaves 5. 8. A boy had 6 marbles and lost 3 ; how many mar- bles had he left ? 9. Six marbles less 4 marbles are how many marbles ? 10. Mary had 4 cents and Henry gave her 2 cents more. How many cents had she then ? 11. Two cents and 4 cents are how many cents ? 12. How many balls put with 2 balls will make 6 balls ? How many with 4 ? How many with 3 ? 13. How many balls taken from 6 balls will leave 3 balls ? Will leave 2 balls ? Will leave 5 balls ? How many are 14. 2 boys and 4 boys ? 15. 3 books and 3 books ? 16. 5 pins and 1 pin ? 17. 6 men less 2 men ? 18. >5 caps less 3 caps ? ./#. 6 pinks less 5 pinks ? 20 FIRST BOOK LESSON X. 1. Four trees and 3 trees are how many trees ? 2. If 3 trees be cut down, how many will be left ? 3. How many apples on the tree ? 3 apples and 2 apples and 2 apples are how many apples ? 4- If 3 apples fall from the tree, how many are left ? If 2 more fall, how rminy are left ? 5. Three girls and how many more make seven girls ? 6. Seven bunches of grain are how many more than 5 bunches ? Than 3 bunches ? Than 2 bunches ? 7. How many girls are shown in the picture ? If 1 girl leave, how many will remain ? If 3 leave ? If 4 leave ? If 2 leave ? If 6 leave ? 8. Make 3 lines on your slate. 3 more. / / / / / / 9. How many more will make seven ? 10. How many 3 ? s in seven and how many over ? 11. Write the word seven on your slate. &even. 12. Make the figure that stands for seven. 7. 13. Count seven. Count from seven back to one. 14. How many are seven and one more ? Eight* IK ARITHMETIC. 21 LESSON XL 1. How many ones in seven ? 1 1 1 1 1 1 1 2. How many 2's, and how many over ? 3. How many 3's, and how many over ? 4. Seven girls are how many more than 2 girls ? Than 5 girls ? Than 1 girl ? Than 3 girls ? Than 4 girls ? 5. Three and how many make 7 ? 4 and how many ? 2 and how many ? 5 and how many ? 6. Kepeat this table. 1 and 6 are 7. 4 and 3 are 7. 2 and 5 are 7. 5 and 2 are 7. 3 and 4 are 7. 6 and 1 are 7. 7. 7 from 7 leaves 0. 3 from 7 leaves 4. 6 from 7 -leaves 1. 2 from 7 leaves 5. 5 from 7 leaves 2. 1 from 7 leaves 6. 4 from 7 leaves 3. from 7 leaves 7. . James had 7 cents, and gave 5 cents for a pencil. How many cents had he left ? 9. George gave 4 peaches to his brother and 3 to his sister. How many did he give to both ? 10. How many books put with 2 books will make 7 books ? How many books are 3 books and 4 books ? 11. How many yards of ribbon cut from 7 yards will leave 5 yards ? Will leave 1 yard ? Will leave 6 yards ? How many are 12. 4 girls and 3 girls ? 18. 2 horses and 5 horses ? 14. 3 boxes and 4 boxes ? 15. 7 trees less 3 trees ? 16. 6 houses less 4 houses? 17. 7 figs less 5 figs ? 22 FIRST BOOK LESSON XII. 1. How many sheep are shown in the picture ? 2. There are 4 sheep in one place and 4 in another. How many in all ? 4 and 4 are how many ? 3. Eight cars are how many more than 7 ? How many more than 6 ? Than 5 ? Than 4 ? Than 3 ? Than 2 ?" ^. Make eight lines on your slate by 2's. II II II II By 4's. //// fill. 5. Count eight by 2's. Count eight by 4's. 6. How many 2's in eight ? How many 4's ? 7. Eight sheep less 4 sheep are how many sheep ? 8. If 2 cars are taken from eight cars, how many cars are left ? If 5 are taken ? If 6 are taken ? 9. Four sheep, 3 sheep, and 1 sheep are how many sheep ? 10. How many are eight less 6 ? Eight less 5 ? 11. Write the word eight on your slate. 12. Make the figure that stands for eight. 8. 13. Count eight. Count from eight back to one. 14. How many are eight and one more. Nine. IK ARITHMETIC. 23 LESSON XIII. 1. How many feet has a sheep ? How many feet have 2 sheep ? How many 4's in 8 ? 2. How many hands has one boy ? How many have 2 boys ? 3 boys ? 4 boys ? 3. How many boys must hold up both hands, to show 8 hands ? How many 2's in 8 ? 4. Eight are how many more than 7 ? Than 5 ? Than 2 ? Than 6 ? Than 4 ? 5. Eight less 6 are how many ? 7 less 3 are how many ? 6. Eepeat this table. and 8 are 8. 4 and 4 are 8. 1 and 7 are 8. 5 and 3 are 8. 2 and 6 are 8. 6 and 2 are 8. 3 and 5 are 8. 7 and 1 are 8. 7, 8 from 8 leaves 0. 7 from 8 leaves 1. 6 from 8 leaves 2. 5 from 8 leaves 3. 4 from 8 leaves 4. 3 from 8 leaves 5. 2 from 8 leaves 6. 1 from 8 leaves 7. 8. There are 2 red cars, 1 blue car, and 5 yellow cars in a train. How many in all ? 9. If there are 8 horses in a stable and 3 be taken out, how many will be left ? If 5 be taken ? If 1 be taken ? How many are 10. 4 cows and 4 cows ? 11. 6 hens and 2 hens ? 12. 1 pail and 7 pails ? IS. 3 words and 5 words ? 14- 8 men less 1 man ? 15. 8 figs less 5 figs ? 16. 8 rings less 7 rings ? 17. 8 dogs less 3 dogs ? 24 FIKSTBOOK LESSON XIV, 1. In the picture, how many roses are shown ? How many acorns ? 2. How many acorns on the upper branch ? How many on the lower ? How many on both branches ? 3. Four and 5 are how many ? 5 and 4 are how many ? 4. Nine less 5 are how many ? Less 4 are how many ? 5. How many 3's in 6 ? How many 3's in nine ? 6. Nine are how many more than 6 ? 6 than 3 ? 7. If 5 acorns drop from the branch, how many are left ? If 4 more drop, how many are left ? 8. If 3 roses are picked, how many are left ? If 3 more ? If 3 more ? 9. Make nine lines on your slate by 3's. /// /// /// 10. Count nine by ones. Count nine by 3's. 11. How many 4 ? s in nine, and how many over ? 12. Write the word nine. 13. Make the figure for nine. 9. IJ/.. Count nine. Count back from nine to one. 15. How many are nine and one more ? Ten. IK ARITHMETIC. 26 LESSON XV. 1. Make nine short lines. on your slate. /// /// /// 2. How many ones in 9 ? How many 3's in 9 ? 3. If 3 girls have 3 roses each, how many have they all? 4. How many must be taken from 9 to leave 6 ? To leave 4? To leave 5? To leave 1? To leave 7? To leave 2? 5. Repeat this table. and 9 are 9. 5 and 4 are 9. 1 and 8 are 9. 6 and 3 are 9. 2 and 7 are 9. 7 and 2 are 9. 3 and 6 are 9. 8 and 1 are 9. 4 and 5 are 9. 9 and are 9. 6. 9 from 9 leaves 0. 4 from 9 leaves 5. 8 from 9 leaves 1. 3 from 9 leaves 6. 7 from 9 leaves 2. 2 from 9 leaves 7. 6 from 9 leaves 3. 1 from 9 leaves 8. 5 from 9 leaves 4. from 9 leaves 9. 7. There are 9 leaves on two branches. If 4 leaves are on one of the branches, how many are on the other ? 8. A boy had 5 peaches in one pocket, and 4 in another. How many had he in both ? 9. If he give away 2 peaches out of each pocket, how many will he have left ? How many are 10. 7 tops and 2 tops ? 11. 3 dollars and 6 dollars ? 12. 8 pencils and 1 pencil ? 18. 4 nuts and 5 nuts ? 14. 9 eggs less 2 eggs ? 15. 7 birds less 5 birds ? 16. 8 cats less 7 cats ? 17. 9 mice less 4 mice ? 26 FIEST BOOK WESSON XVI. 1. Make 5 short lines ; then 5 more lines. I III! II III 2. How many ones make ten ? How many 5's make ten? 3. In the picture, how many books are on the upper shelf of the bookcase ? How many on the lower ? 4. How many cherries on the branch ? 5. If 2 cherries are picked, how many remain ? 6. If 2 more are picked, how many are left ? If 2 more ? 7. Ten cherries less 8 cherries are how many cherries ? 8. Make ten short lines by 2's. // // // // // 9. Count ten by 2's. Count ten by 5's. 10. How many flower-pots are on the ground ? How many are on the stand ? How many in all ? 11. Six pots and 4 pots are how many pots ? 12. Ten pots less 6 pots are how many ? IS. Name each of the numbers that can be expressed by a single figure. H. How is the number ten expressed ? 15. Write the word, ten. Q/en. 16. Make the figures to express ten. 10. ARITHMETIC. 27 LESSON XVII. 1. Make ten short lines on your slate. 1 1 1 1 1 1 1 1 1 1 2. Ten are how many ones ? How many 2's ? 5's ? 3. How many 4's in 10, and how many over ? 4. How many must be taken from 10 to leave 5 ? To leave 4 ? To leave 7 ? To leave 3 ? To leave 2 ? 5. John had 6 marbles and bought 4 more. How many had he then ? 6. Mary had 10 cents and gave 3 cents for a pencil. How many had she left ? 7. Eepeat this table. and 10 are 10. 1 and 9 are 10. 2 and 8 are 10. 3 and 7 are 10. 4 and 6 are 10. 8. 10 from 10 leaves 0. 9 from 10 leaves 1. 8 from 10 leaves 2. 7 from 10 leaves 3. 6 from 10 leaves 4. 5 and 5 are 10. 6 and 4 are 10. 7 and 3 are 10. 8 and 2 are 10. 9 and 1 are 10. 5 from 10 leaves 5. 4 from 10 leaves 6. 3 from 10 leaves 7. 2 from 10 leaves 8. 1 from 10 leaves 9. How many are 9. 5 pins and 5 pins ? 10. 3 trees and 7 trees ? 11. 6 birds and 4 birds ? 12. 2 sheep and 8 sheep ? 13. 9 plums less 5 plums ? 14. 10 pears less 7 pears ? 15. 8 pens less 5 pens ? 16. 10 figs less 9 figs ? In all these lessons of counting in series, the teacher should use objects, or the numeral frame, until the pupil thoroughly under- stands the process. 28 FIRST BOOK LESSON XVIII. REVIEW. 1. In the picture, how many blocks in each row ? 2. In the first row, how many blocks in the larger part? 3. Mne blocks and 1 more are how many ? 1 and 9 ? 4. Eight blocks and 2 more are how many ? 2 and 8 ? 5. Ten blocks less 8 blocks are how many ? 10 less 2 ? 6. Ten blocks less 3 blocks are how many ? 10 less 7 ? 7. Seven blocks and 3 more are how many ? 3 and 7 ? 8. Six blocks and how many more make 10 ? 4 and how many make 10 ? 9. Ten blocks less 6 blocks are how many ? 10 less how many are 6 ? 10. Seven and how many make 10 ? 3 and how manj make 10 ? 11. Ten blocks less 5 blocks are how many ? 5 from 10 leave how many ? 12. How many and 5 make 10 ? How many from 10 leave 5 ? 18. How many 5's in 10 ? How many 6's in 10 ? Ans. One 6, and 4 over. ARITHMETIC. 29 LESSON XIX. REVIEW. 1. Count by ones to ten. 2. Count by 2's to 4. 8. Count by 2's to 6. 4. Count by 3's to 6. 5. Count by 2's to 8. 6. Count by 4's to 8. 7. Count by 3's to 9. 8. Count by 2's to 10. 9. Count by 5's to 10. 10. How many 2's in 4 ? 11. How many 3's in 6 ? 12. How many 4's in 8 ? IS. How many 3's in 9 ? 14. How many 5's in 10? 15. Jane picked 4 pinks from one stem, 3 from an- other, and 2 from another. How many pinks had she ? 16. Oscar had 10 cents, and gave 2 cents for a pen and 5 cents for a pencil. How many cents had he left ? 17. If there are 6 birds in one cage and 4 in another, how many birds in both cages ? How many more in one than in the other ? 18. Willie caught five fishes ; how many more must he catch to have 8 ? 19. There are seven cherries on one part of a twig, and 3 on another. How many cherries on both parts ? 20. If 5 cherries are picked, how many will be left? 30 FIRST BOOK LESSON XX. 1. How many balls on the upper wire of the Nu- meral Frame. 2. How many are ten balls and one ball more ? Eleven. Written 11 3. How many are eleven balls and one ball ? Twelve. Written 12 4. Twelve balls and one more are Thirteen. Written 13 5. Thirteen balls and one more are Fourteen. Written l' 6. Fourteen and one more are 7. Fifteen and one more are 8. Sixteen and one more are 9. Seventeen and one more are 10. Eighteen and one more are 11. Nineteen and one more are 12. Count from one to twenty. 13. How many units in the n number ten ? Write the figures for ten. H. What does the 0, or cipher, denote ? Ans. No units. 15. What does the 1 denote ? Ans. One ten. 16. Then, what do the figures 10 denote ? Ans. 1 ten and units, or ten. COUNTING. Written Fifteen. 15 Sixteen. 16 Seventeen. 17 Eighteen. 18 Nineteen. 19 Twenty. 20 mber one ? In the ten. 10 INAEITHMETIC. 31 LESSON XXI. 1. When two figures are written side by side, what does each figure denote ? 2. The one on the right denotes units, and the one on the left denotes tens. 3. What do the figures 11 denote? Ans. 1 ten and 1 unit, or eleven. 4. What do the figures 12 denote ? Ans. 1 ten and 2 units, or twelve. 5. What do the figures 15 denote ? Ans. 1 ten and 5 units, or fifteen. 6. Write the figures that stand for twenty. 20 7. How many tens are there in twenty? 8. What do the figures 20 denote ? Ans. 2 tens and units, or tiventy. 9. What then do the figures 21 denote ? Ans. 2 tens and 1 wmY, or twenty-one. Written ./#. Two tens and two units are Twenty "two. 22 11. Two tens and three units are Twenty-three. 23 12. Two tens and four units are Twenty -four. 1 cents. J> 5 cents. 5 cents. SECOND. By Division. 5)15 3 IK ARITHMETIC.' 53 LESSON XLIII. 1. How many times 2 cents in 10 cents ? In 12 cents ? 2. How many times 3 days in 12 days ? In 15 days ? 8. How many times 4 plums in 12 plums? In 16 plums? 4. How many times 5 hours in 15 hours ? In 20 hours ? 5. At 5 cents each, how many toys can be bought for 20 cents ? SOLUTION. As many toys as 5 cents are contained times in 20 cents, which are 4 times. Hence 4 toys can be bought for 20 cents. 6. At 2 cents each, how many pears can be bought for 10 cents ? 7. How many quarts of milk at 4 cents a quart, can be bought for 16 cents ? 8. If 1 lemon cost 3 cents, how many lemons can be bought for 15 cents ? 9. To how many boys can you give 12 apples, if you give them 4 apples apiece ? 10. How many hats can be bought for 25 dollars, at 5 dollars apiece ? 11. Ella paid 20 cents for some ribbon, at 5 cents a yard. How many yards did she buy ? 12. Clarence gave 16 cents for some tops, at 4 cents a piece. How many tops did he buy ? Division at Sight. 8-1-2=:? 8-5-4=? 15-f-5=? 15-^-3=? 6-^3=? 10-r-5=? 10-4-2=? 16-4-4=? 4)_20 5)^5 4)_16 5)J20 3)15 2)_12 54 FIRST BOOK LESSON XLIV. EQUAL PARTS OF NUMBERS. 1. If 6 oranges are divided equally between 2 girls, into how many equal parts are the oranges divided? 2. When a number is divided into 2 equal parts, whal is one of the parts called ? One-half of the number. 8. How many oranges are one-half of 6 oranges ? 6 ~- 2 = 3, or one-half of 6 is 3. 4. How many boys are one-half of 8 boys ? 5. How many peaches are one-half of 10 peaches ? 6. If you put 10 bushels of apples into 2 boxes, what part of the whole do you put into 1 box ? 7. Instead of finding how many times one number is contained in another of the same kind, it is sometimes required fco divide a number into equal parts. The opera- tion in both cases is the same, but the reasoning is some- what different. Thus, 8. At 3 cents each, how many pears can be bought for 6 cents ? SOLUTION. As many pears as 3 cents are contained times in 6 cents, which are 2 times. Hence 2 pears can be bought for 6 cents. 6-5-3 = 2. 9. Again, if 2 pears cost 6 cents, what will 1 pear cost ? SOLUTION. If 2 pears cost 6 cents, 1 pear will cost one-half of 6 cents, which is 3 cents. 6-^2 = 3. Itf ARITHMETIC. 55 10. If 12 cherries are divided equally among 3 boys, into how many equal parts are they divided ? 11. When a number is divided into 3 equal parts, what is one of the parts called ? One-third of the number. 12. How many cherries are one-third of 12 cherries ? 12 -r- 3 = 4, or one-third of 12 is 4. ./#. How many cents are one-third of 9 cents ? H. How many marbles are one- third of 15 marbles ? 15. If 12 pinks grow upon 4 stems, each containing an equal number, how many grow upon each stem ? 16. When a number is divided into 4 equal parts, what is one of the parts called ? One-fourth of the number. 17. How many pinks are one-fourth of 12 pinks ? 12 4 = 3, or one-fourth of 12 is 3. 18. How many dollars are one-fourth of 8 dollars ? 19. What is one-fourth of 12 ? Of 16 ? Of 20 ? 20. When a number is divided into 5 equal parts, what is one of the parts called ? One- fifth of the number. 21. How many balls are one-fifth of 20 balls ? @^ j@ QUO a 20 -T- 5 = 4, or one-fifth of 20. #. How many cents are one-fifth of 15 cents ? 56 FIRST BOOK LESSON XLV. 1. One-half is written thus, . of 6 is 623. 2. One-third is written %. % of 6 is 63=2. 8. One-fourth is written }. of 8 is 84=2. 4. One-fifth is written -. of 15 is 155=3. 5. How many halves in any thing ? How many thirds in any thing? How many fourths in any thing? How many fifths in any thing ? 6. What is \ of 4 books? \ of 10 miles? of 8 houses? 7. What is of 6 sheep ? J. of 9 weeks ? of 12 dollars? 8. What is of 4 men ? J of 1 6 pounds ? of 20 chairs ? 9. What is 4 of 10 barrels ? of 15 trees ? -J- of 25 dollars ? .70. How do you obtain one-half of a number ? One- third of a number ? One-fourth of a number ? One- fifth of a number ? 1.7. If 20 marbles be divided equally among 4 boys, how many marbles will each boy receive ? SOLUTION. Since 20 marbles are divided equally among 4 boys, one boy will receive one-fourth of 20 marbles, or 5 marbles. 12. If 3 books cost 15 cents, what is the cost of 1 book ? 13. Write on your slates, in a column, all the num- bers from 10 to 20. Eead them. 14. Write the numbers from 20 to 30, and read them. I/). Write the numbers from 30 to 40, and read them. 16. In a similar manner write and read the numbers from 40 to 50. From 50 to 60. From 60 to 70. From 70 to 80. From 80 to 90. From 90 to 100. 17. How many figures are required in writing each number from 9 to 99 ? IK ARITHMETIC. 5? LESSON XLV1. 1. How many are 5 and 1 ? 3 and 3 ? 4 and 2 ? 2. How many are 6 and 6 more ? 12 and 6 more ? 3. How many are 18 and 6 ? 24 and 6 ? 4. How many are three 6's ? Four 6's ? Five 6's ? 5. Count by 6's to 12. To 18. To 24. To 30. 6. How many are 30 less 6 ? 24 less 6 ? 18 less 6 ? 12 less 6 ? 6 less 6 ? 7. Begin with 1 and count .by 6's to 13. To 19. To 25. 8. Begin with 2 and count by 6's to 20. To 26. To 32. 9. How many are 6 + 6 + 6? 4 + 6 + 6? 2 + 6 + 6? 5 + 6 + 6? 3 + 6 + 6? 1 + 6 + 6? 10. How many are 18 6? 15-6? 9-6? 8-6? .12 6? 11 6? 13 6? 6 6? 11. How many are 3 times 6 ? 4 times 6 ? 6x5? 3x6? 5 times 6 ? 2 times 6 ? 1x6? 6x6? 12. How many 6's in 18 ? 6's in 30 ? 4's in 16 ? 6's in 24 ? 5's in 20 ? 5's in 25 ? EXERCISES FOR THE SLATE AKD BOARD. IS. Copy and add or count the following ; 6666666542 6666665664 6666664356 6543212435 58 FIRST BOOK LESSON XLVII. 1. If 18 figs are equally divided among 6 boys, into how many equal parts are the figs divided ? 2. When a number is divided into 6 equal parts, what is one of the parts called ? One-sixth of the number. 8. How many figs are one-sixth of 24 figs ? 24 -f- 6 = 4, or one-sixth of 24 is 4. 4. If 24 plants are set in 6 equal rows, what part of 24 plants is set in 1 row ? How many plants ? 5. One-sixth is written thus, . | of 24 is 24 -s- 6 =4. TABLES. ADDITION. SUBTRACTION. MULTIPLICATION. DIVISION. + 6= 6 66=0 6x0= 06=0 1 + 6= 7 7-6=1 6x1= 6 6-6=1 2 + 6= 8 86=2 6x2 = 12 126=2 3 + 6= 9 96=3 6x3 = 18 186=3 4 + 6 = 10 10-6=4 6x4=24 246=4 5 + 6 = 11 116=5 6x5=30 306=5 6 + 6=12 12-6=6 6x6=36 36-6=6 EQUAL PARTS OF NUMBERS. J- of 6 = 1 of 18 = 3 | of 30 = 5 | of 12 = 2 of 24 = 4 of 36 = 6 When the regular Jorm of each table has been thoroughly learned, require the pupil to reverse the order of the numbers ; thus, in Addition, for 2 + 6=8, say 6 + 2=8; in Subtraction, for 8-6=2, say 8-2=6; in Multiplication, for 6x2=12, say 2x6 =12 ; and in Division, for 12-*-6=2, say 12-^-2=6, etc. IK ARITHMETIC. 59 LESSON XLVIII. 1. If a man have 18 dollars, and he earn 6 more, how many dollars will he then have ? 18 + 6= ? 2. A man having 18 dollars gave 6 dollars for a barrel of flour. How many dollars had he left ? 18 6= ? 3. A laborer received 3 dollars a day for 6 days woi k. How many dollars did he receive in all ? 3 x 6 = ? 4. At 3 dollars apiece, how many chairs can be bought for 18 dollars ? 18-=-6= ? EXERCISES FOE THE SLATE AND BOARD. 5. Write the proper numbers in place of (?): 10 + 6=? 13 + 6=? 15 + 6=? 23 + 6=? 106= ? 13 6= ? 15 6= ? 22 6=? 4x6=? 5x6=? 3x6=? 6x6=? 12-f-6=? 18-f-6=? 30-^-6=? 36^-6=? Addition. 3 4 443 643 4 6 4 5 645 434 3 4 6 3 556 456 5 3 Subtraction. 9 10 12 10 7 12 11 10 13 9 ^J^_SJ>_1-1_1_1-1_?. Multiplication. 6356445666 4655646346 Division. 6)24 5)25 4)16 6)18 4)24 6)36 60 FIRST BOOK LESSON XLJX. 1. How many are 3 and 4 ? 4 and 3 ? 5 and 2 ? 2. How many are 7 and 1 ? 7 and 2 ? 7 and 3 ? 3. How many are 7 and 5 ? 7 and 6 ? 7 and 7 ? 4. How many are 14 and 7 more ? 14 + 7= ? 5. Count 21 by 3's. Count 21 by 7's. 6. How many are 21 and 7 more ? 21 + 7= ? 7. Count 28 by 2's. Count 28 by 4's. Count 28 by 7's. 8. Count 35 by 5's. Count 35 by 7's. P. How many are 35 less 7 ? 28 less 7 ? 21 less 7 ? 10. How many are 35 and 7 more ? 42 and 7 more ? 11. Count 42 by 2's. By 3's. By 6's. By 7's. 12. How many are 7 + 7 + 7? 7 + 7 + 7 + 7? 7 + 7 + 7 + 7 + 7? IS. How many are 35-7? 28-7? 21-7? 14-7? 77? 14- How many are 7x2? 7x3? 7x4? 7x5 7x6? 15. How many 7's in 21? 7's in 28? 7's in 35? 7's in 42? EXERCISES FOR THE SLATE AND BOARD. 16. Copy and add or count the following : 7777777756 7777777672 7654321137 When the pupil can rapidly and accurately add three figures in a column, the number of figures should be increased to four, then five, then six, and then to seven. INABITHMETIC. 61 LESSON L. 1. If 21 yards of cloth are made into 7 coats of the same size, into how many equal parts must the 21 yards be cut ? 2. When a number is divided into 7 equal parts, what is one of the parts called? One-seventh of the number. 3. How many yards are one-seventh of 21 yards ? O O O O O 00 00 00 00 00 00 00 21 -f- 7 = 3, or one-seventh of 21 is 3. 4. How many birds are one-seventh of 7 birds ? Of 14 birds ? Of 21 birds ? 5. If 21 bushels of apples are put into 7 barrels, what part of the 21 bushels is put into 1 barrel ? How many bushels ? 6. One-seventh is written thus, -3J-. | of 28 is 28-;- 7 =4. TABLES. ADDITION. + 7=7 3 + 7=10 6 + 7=13 1 + 7=8 4+7=11 7 + 7=14 2 + 7=9 5 + 7=12 8 + 7=15 SUBTRACTION. 7-7=0 10-7=3 13-7=6 87=1 11 7=4 147=7 9-7=2 12-7=5 15-7=8 MULTIPLICATION. 7x0= 7x3=21 7x6=42 7x1= 7 7x4=28 7x7=49 7x2=14 7x5=35 7x8=56 62 FIRST BOOK LESSON LI. TABLES Continued. DIVISION. 07=0 21-^7=3 42^7=6* 7-7=1 28^-7=4 49-f-7 = 7 14-n7= 2 35-^-7=5 56-i-7=8 EQUAL PARTS OF NUMBERS. | of 0=0 } of 21=3 | of 42 = 6 | of 7=1 | of 28=4 | of 49 = 7 | of 14=2 | of 35=5 i of 56=8 How many are 7 times 5 boys ? 5 times 7 coats ? 3 times 7 pears ? 7 times 7 cents ? 7 times 6 hats ? 4 times 7 horses ? How many are | of 14 roses ? | of 25 miles ? | of 28 dollars ? | of 42 oranges ? i- of 21 pounds ? | of 49 days ? Addition at Siyht. 4657773077 7773677754 Subtraction at Sight. 7 10 13 12 7 14 21 9 7 7 J^_Z_!J^J:_l_!_!JlJl Multiplication at Sight. 7267477757 3775724677 ARITHMETIC. 63 LESSON LIL 1. John gave 5 cents for a pencil, 4 cents for a top, and had 2 cents left. How many cents had he in all ? 2. There were 35 sheep in a lot and 7 jumped out. How many remained ? 8. What will be the cost of 6 papers of needles, at 7 cents a paper ? 4. How many pounds of rice can be bought for 42 cents, at 7 cents a pound ? EXERCISES FOR THE SLATE AND BOARD. 5. Write the proper numbers in place of (?). 21 + 7=? 21-7=? 3x7=? 21-7-7- ? 28 + 7=? 28-7=? 5x7=? 28-f-7=? 14 + 7=? 427= ? 6x7=? 35-7-7==? 29 + 7=? 297= ? 7x7=? 42-=-7= ? Addition. 7 5 4 3 6 7 4 7 6 7 7 7 6 7 1 5 3 7 7 6 4 7 7 7 6 6 6 7 7 9 10 11 14 13 12 17 11 6 12 5 Multiplication. 7740 3'577 7 6 7)14 6)24 7)28 5)30 7)35 6)36 64 FIKSTBOOK LESSON LIIL L How many are 7 and 1? 6and2? 5and3? 4and4? 2. How many are 8 and 5? 8 and 6? 8 and 7? 8and8? S. Count 16 by 2's. By 4's. By 8's. 4. How many are 16 and 8 more ? 16 + 8= ? 5. Count 24 by 2's. By 4's. By 6's. By 8's. 6. How many are 24 and 8 more ? 24 + 8= ? 7. Count 32 by 2's. By 4's. By 8's. 8. How many 4's in 32 ? How many 8's ? P. How many are 32 less 8 ? 24 less 8 ? 16 less 8 ? 10. How many are 32 and 8 more ? 32 + 8= ? 11. Count 40 by 2's. By 4's. By 5's. By 8's. 12. How many are 40 and 8 more ? 48 and 8 more ? 13. How many 8's in 40 ? How many 5's ? 14* How many are 8+8+8? 8+8+8+8? 8+8+8+8+8? IB. How many are 408? 328? 248? 168? 88? 16. How many are 8x2? 8x3? 8x4? 8x5? 8x6? 8x7? 17. How many are 8-5-8? 16-f-S? 24-5-8? 32-~8? 40-^8? EXEECISES FOR THE SLATE AND BOARD. 18. Copy and add or count the following : 8888888843 8888888858 8888888862 8765432187 IN AEITHMETIC. 65 LESSON LIV. 1. If 32 pounds of tea are put into 8 boxes, an equal number of pounds into each box, into how many equal parts are the 32 pounds divided ? 2. When a number is divided into 8 equal parts, what is one of the parts called ? One-eighth of the number. 3. How many pounds are one-eiglith of 32 pounds ? @J9 J9 ^ @ @ @J9 @ @JB "a <^B ^D 65 @ GOB . 6/B 32 -f 8 = 4, or one-eighth of 32 is 4 ^. How many are one-eighth of 16 men ? Of 24 men ? 5. If 40 boys sit upon 8 benches, on each an equal number, what part of the 40 boys sit upon 1 bench ? How many boys ? 6. One-eighth is written thus, . $ of 48 is 488=6. TABLES. ADDITION. 0+8= 8 3 + 8 = 11 6 + 8 = 14 1 + 8= 9 4 + 8=12 7 + 8 = 15 2 + 8=10 5 + 8=13 8 + 8 = 16 SUBTRACTION. 8-8=0 11_8= 3 148 = 6 9-8=1 12-8=4 15-8 = 7 108=2 13_8 = 5 16-8=8 MULTIPLICATION. 8x0= 8x3=24 8x6=48 8x1= 8 8x4=32 8x7=56 8x2=16 8x5=40 8x8=64 66 FIEST BOOK LESSON LV. TABLES Continued. DIVISION. 0-8=0 24^-8=3 48^-8=6 8-8=1 32^-8=4 56^-8=7 16-8=2 40-8=5 648=8 EQUAL PAKTS. to* 0=0 of 24=3 ^ of 48=6 iof 8=1 of 32=4 -i of 56 = 7 4 of 16=2 4 of 40=5 4 of 64=8 How many are 8 times 3 eggs ? 6 times 8 hours ? 5 times 8 nuts ? 8 times 8 cents ? 8 times 4 pins ? 7 times 8 quarts ? How many are -J of 24 miles ? -J- of 48 men ? of 16 dollars ? i of 56 bushels ? % of 32 ounces ? of 40 cents ? Addition at Sight. 8858473868 6788888584 Subtraction at Sight. 8 10 12 16 9 8 8 24 14 11 8888831888 Itf ARITHMETIC. 67 LESSON LVI. 1. Kobert found 16 ripe peaches under a tree ; he ate 3 and gave away 5. How many had he left ? 2. George had 24 cents, which was 8 more than Ella had. How many cents had Ella ? 3. At 8 cents each, what is the cost of 6 writing-books ? 4. At 8 cents a yard, how many yards of ribbon can be bought for 48 cents ? EXERCISES FOE THE SLATE AND BOAKD. 5. Write the proper numbers in place of (?): 16 + 8=? 15 + 8=? 32 + 8=? 22 + 8=? 168=? 158=? 328=? 228=? 7x8=? 5x8=? 8x8=? 6x8=? 24-^-8= ? 40-r-8= ? 48^-8= ? 56-r-8= ? Addition. 8 7 5 6 4 4 8 8 8 7 5 1 8 8 6 3 6 8 7 3 4 6 8 8 7 2 7 8 8 Subtraction. 9 11 12 8 16 14 24 15 12 10 _?_^_?_^_l_l_iJLA_i Multiplication. 8765487788 3548766778 Division. 6)18 8)56 8)40 7)49 6)48 68 FIRST BOOK LESSON LVII. 1. How many are 8 and 1? 7and2? 6and3? 5and4? 2. How many are 9 and 7 ? 9 and 8 ? 9 and 9 ? S. Count 18 by 2's. By 3's. By 6's. By 9's. 4. Count 27 by 3's. Count 27 by 9's. 5. How many 3's in 27 ? How many 9's in 27 ? 6. How many are 27 less 9 ? 18 less 9 ? 9 less 9 ? 7. Count 36 by 2's. By 3's. By 4's. By 6's. By 9's. 8. How many are 36 less 9 ? 27 less 9 ? 18 less 9 ? 9. Count 45 by 3's. By 5's. By 9's. 10. How many 5's in 45 ? How many 9's ? 11. How many are 45 and 9 more ? 45 + 9= ? 12. How many are 54 and 9 more ? 54+ 9= ? 18. Count 63 by 3's. By 7's. by 9's. 14. How many 7's in 63 ? How many 9's ? 15. How many are 9+9+9? 9+9+9+9? 9+9+9+9+9? 16. How many are 63-9? 54-9? 45-9? 36-9? 18-9? 17. How many are 9x3? 9x4? 9x5? 9x6? 9x7? 18. How many are 18-^9? 27-5-9? 36^9? 45^-9? 54-^-9? 63-^-9? EXERCISES FOR THE SLATE AND BOARD. 19. Copy and add or count the following : 999-9999999 9999999999 98 76543 2. IQ IK ARITHMETIC. LESSON LVIII. 1. If 45 trees are set in 9 rows, an equal number in each row, into how many equal parts are the 45 trees divided ? 2. When a number is divided into 9 equal parts, what is one of the parts called ? One-ninth of the number. 8. How many trees are one-ninth of 45 trees ? 45 -7- 9 = 5, or one-ninth of 45 is 5. 4. flow many rods are one-ninth of 9 rods ? Of 18 rods ? Of 36 rods ? 5. If 27 bushels of grain be put into 9 bags of equal size, what part of the 27 bushels .will 1 bag contain ? How many bushels ? 6. One-ninth is written thus, ^. ^ of 54 is 549=6. TABLES. ADDITION. 1 + 9=10 4 + 9=13 7 + 9 = 16 2 + 9 = 11 5 + 9 = 14 8 + 9=17 3 + 9 = 12 6 + 9 = 15 9 + 9 = 18 ' SUBTRACTION. 109 = 1 13 9=4 16 9 = 7 119=2 149=5 17 9=8 129=3 15 9 = 6 189 = 9 MULTIPLICATION. 9x1= 9 9x4=36 9x7=63 9x2 = 18 9x5=45 9x8 = 72 9x3=27 9x6=54 9x9=81 70 FIRST BOOK LESSON LIX. TABLES Continued. DIVISION. 9-r-9 = l 36-f-9=4 63-f-9 = 7 18-r-9=2 45-^-9=5 72-5-9=8 27-f-9=3 54-r-9 = 6 81 9=9 EQUAL PARTS. |of 9=1 i of 36=4 i of 63 = 7 i of 18=2 i of 45 = 5 | of 72=8 of 27=3 i of 54=6 i of 81 = 9 How many are 9 times 4 balls ? 8 times 9 miles ? 6 times 9 boxes ? 9 times 7 feet ? 9 times 5 hens ? 9 times 9 cents ? How many are i of 27 sheep ? | of 63 gallons ? J- of 36 acres ? $ of 72 girls ? $ of 54 boys ? | of 81 marbles ? Addition at Siylit. 9978499959 5699937899 Subtraction at Sight. 10 9 11 12 9 18 27 26 14 15 In all the foregoing tables, the pupil should be required to invert the order of the numbers, and to repeat them backward and pro- miscuously until they are thoroughly memorized. IN ARITHMETIC. 71 LESSON LX. 1. In a garden are 18 pear trees and 9 peach trees. How many of both ? How many more pear trees than peach trees ? 2. There are 6 peaches on each tree. How many peaches on the 9 trees ? 3. At 9 cents a pound, how many pounds of sugar can be bought for 45 cents ? For 54 cents ? For 27 cents ? 4. How many are one-ninth of 18 books? Of 36 melons? EXERCISES FOE THE SLATE AND BOARD. 5. Write the proper numbers in place of (?). 27 + 9=? 36 + 9= ? 63 + 9=? 279=? 36-9=? 63-9=? 3x9=? 5x9=? 8x9=? 27^9=? 45-7-9=? 72-5-9=? c Addition. 7 8 7 9 9 8 9 6 8 9 3 2 9 5 9 7 3 6 8 9 9 9 4 2 4 9 9 8 8 9 Subtraction. 12 35 9 16 18 27 29 17 16 9 _!_^_!_^J^_^Ii_^_!_i Multiplied tion. 7896798899 8659879869 Division. 9)27 6)54 9)63 9)72 9)81 7)63 FIKSTBOOK LESSON LXI. 1. How many are 9 and 1 ? 7 and 3? 6 and 4? 5 and 5? 2. How many are 10 and 5 ? 10 and 7 ? 10 and 8 ? 8. Count by 2's to 20. By 4's. By 5's. By 10's. If. How many 2's in 20 ? How many 4's ? 5's ? 10's ? 5. How many are 20 and 10 more ? 20 + 10= ? 6. How many are 30 and 10 ? 40 and 10 ? 50 and 10 ? 7. How many 10's in 30 ? In 40 ? In 50 ? . In 60 ? 8.. Count 40 by 2's. By 4's. By 5's. By 8's. By 10's. 9. Count by 10's From 1 to 91. From 6 to 96. From 2 -to 92. From 7 to 07. From 3 to 93. From 8 to 98. From 4 to 94. From 9 to 99. From 5 to 95. From to 100. 10. How many are 10 + 10 + 10 -f 10? 10 + 10 + 10 + 10 + 10? 11. How many are 10010? 9010? 7010? 4010? 12. How many are 10 x 5 ? 10 x 7 ? 10 x 6 ? 10 x 8 ? 10 x 10 ? 18. How many are 40-4-10? 504-10 i 704-10? 804-10? 904-10? IJf. When a number is divided into 10 equal parts, what is one of the parts called ? One-tenth of the number. 15. How many barrels are one-tenth of 70 barrels ? 70 4- 10 = 7, or one-tenth of 70 is 7. 16. How many sheep are one-tenth of 80 sheep ? 17. One-tenth is written thus, -fa. -^ of 90 is 904-10=9. IN- ARITHMETIC. LESSON LXII. TABLES. ADDITION. + 10=10 4 + 10 = 14 8 + 10=18 1 + 10 = 11 5 + 10 = 15 9 + 10=19 2 + 10=12 6 + 10 = 16 *10 + 10=20 3 + 10=13 7 + 10=17 11 + 10=21 SUBTRACTION. 1010=0 1410 = 4 18-10= 8 11-10=1 15_10 = 5 19-10= 9 12-10=2 16 10 = 6 20 10=10 13 10= 3 17_10 = 7 2110=11 MULTIPLICATION. 10x0= 10x4=40 10 x 8= 80 10x1 = 10 10x5 = 50 10 x 9= 90 10x2=20 10x6 = 60 10x10=100 10x3=30 10x7 = 70 10x11 = 110 DIVISION. 010=0 40-10=4 80-f-10= 8 1010=1 5010=5 90-^-10= 9 20-10=2 60-10=6 100-r-10=10 30-10=3 7010=7 EQUAL PARTS. 3*5- of 0=0 -^ of 40=4 iVof 80= 8 ^o- of 10=1 ^ of 50 = 5 ^of 90= 9 ^ of 20=2 ^ of 60 = 6 -iV of 100=10 ^ of 30=3 ^ of 70=7 74 FIRST BOOK LESSON LXIII. 1. How many are 10 and 1 ? 8 and 3 ? 9 and 2 ? 7 and 4 ? 6 and 5 ? 2. How many are 11 and 4 ? 11 and 6 ? 11 and 9 ? 3. How many are 11 and *7 ? 11 and 10 ? 11 and 11 ? 4. Count by 2's to 22. How many 2's in 22 ? How many ll's in 22? 5. How many are 22 and 11 more ? 22 + 11= ? 6. How many are 33 and 11 more ? 33 + 11= ? 7. How many are 44 and 11 more ? 55 and 11 ? 8. How many ll's in 33 ? In 44 ? In 55 ? In 66 ? 9. Count 44 by 2's. By 4's. By ll's. 10. How many are 66 + 11 ? 77 + 11 ? 88 + 11 ? 99 + 11 ? 11. How many are 99-11? 88-11? 77-11? 66-11? 77-11? 12. How many are 11x2? 11x3? 11x4? 11x5? 11x6? 11x7? 13. How many are 88-f-ll? 77-r-ll? * 66-7-11? 55-hll? 4411? 14. When a number is divided into 11 equal parts, what is one of the parts called ? One-eleventh of the number. 15. How many boys are one-eleventh of 44 boys ? 44 -^ 11 = 4, or one-eleventh of 44 is 4. 16. How many days are one-eleventh of 55 days ? 17. One-eleventh is written thus, ^. -^ of 66 is 66 ~- 11 = 6. IS. If 88 cents are divided among 11 boys, what part of the whole does each boy receive ? How many cents ? ARITHMETIC. 75 LESSON LXIV. TABLES. ADDITION. + 11 = 11 4 + 11 = 15 8 + 11 = 19 1 + 11=12 5 + 11 = 16 9 + 11 = 20 2 + 11 = 13 6 + 11 = 17 10 + 11=21 3 + 11=14 7 + 11 = 18 11 + 11=22 SUBTRACTION. 11-11=0 15 11=4 1911= 8 1211=1 16-11 = 5 20-11= 9 13-11=2 17-11 = 6 21 11 = 10 1411=3 18 11 = 7 22-11=11 MULTIPLICATION. 11x0= 11x4=44 11 x 8= 88 11x1 = 11 11x5 = 55 11 x 9= 99 11x2=22 11x6 = 66 11x10=110 11x3=33 11x7 = 77 11x11=121 DIVISION. 011=0 4411 = 4 8811= 8 11-11 = 1 55 11 = 5 9911= 9 2211=2 66 11 = 6 11011=10 33-11=3 7711 = 7 12111=11 EQUAL PARTS. ^of 0=0 ^ of 44=4 fr of 88= 8 ^ of 11 = 1 j^- of 55=5 -^ of 99= 9 -^ of 22 = 2 -& of 66 = 6 T V of 110=10 A of 33=3 TV of 77=7 A- of 121=11 FIKST BOOK LESSON LXV. 1. How many are 11 and 1 ? 8 and 4 ? 9 and 3 ? 2. What numbers multiplied together will produce 12 ? 3. How many are 12 and 4 ? 12 and 6 ? 12 and 8 ? 4. How many are 12 and 9 ? 12 and 10 ? 12 and 12 ? 5. Count by 2's to 24. By 4's. By 6 ? s. By 8's. By 12's. 6. How many are 24 and 12 more ? 24 + 12= ? 7. How many are 36 and 12 more ? 36 + 12= ? 8. How many squares in one row ? How many 12's ? 9. How many squares in two rows ? How many 12's ? 10. How many squares in three rows? How many 12's ? 11. How many squares in 4 rows ? 12. In 5 ? In 6 ? In 7 ? In 8? In 10? In 11 ? In 12? 13. What is | of 12 ? of 12 ? of 12 ? $ of 12 ? 14. How many 12's in 24 ? In 36 ? In 48 ? In 60 ? 15. Count 60 by 3's. By 5's. By 6 ? s. by 10's. By 12's. How many are 16. 36 + 12? 48 + 12? 60 + 12? 72 + 12? 84 + 12? 17. 96-12? 84-12? 72-12? 60-12? 48-12? 18. 12x2? 12x3?- 12x4? 12x5? 12x6? 19. How many are 24-^12? 36-^12? 48-f-12? 60-7-12? 20. When a number is divided into 12 equal parts, what is one of the parts called ? One-twelfth of the number. 21. How many eggs are one-tivelfth of 60 eggs ? 60 -r- 12 = 5, or one-twelfth of 60 is 5. 22. One-twelfth is written ^. ^ of 72 is 72-4-12 = 6. ARITHMETIC. LESSON LXVI. TABLES. ADDITION. 1 + 12=13 2 + 12=14 3 + 12 = 15 5 + 12 = 17 6 + 12 = 18 7 + 12 = 19 8 + 12=20 9 + 12=21 10 + 12=22 11 + 12=23 12 + 12=24 SUBTRACTION. 13-12=1 1412=2 16-12=4 17_12=5 18-12 = 6 19 12 = 7 20-12=8 21-12= 9 2212=10 23-12 = 11 24-12=12 MULTIPLICATION. 12x1 = 12- 12x2=24 12x3=36 12x4=48 12x5 = 60 12x6 = 72 12x7 = 84 12x8 = 96 12 x 9 = 108 12x10=120 12x11 = 132 12x12 = 144 DIVISION. 12-4-12 = 1 24-r-12=2 36-^12=3 48-4-12=4 60-^-12=5 72-4-12 = 6 84-5-12== 7 96-=-12=8 108-4-12= 9 120-4-12=10 132-4-12=11 144-^12 = 12 fg of 12 = 1 -^ of 24=2 ^ of 36=3 & of 48=4 EQUAL PARTS. -^ of 60 = 5 & of 72 = 6 A- of 84=7 A of 96=8 fy Of 108= 9 & of 120 = 10 & of 132=11 A- of 144=12 78 FIBST BOOK ONE HUNDRED. LESSON LXVII. 1. How many rows of blocks are shown in this picture ? 2. How many blocks in each row ? How many blocks in all? 3. Ten times 10 blocks are how many ? 10 x 10 ? 4. How many 10's in one hundred ? How many units ? 6. Count from 1 to 100 by 10's. From 5. From 7. 6. Write the figures that stand for one hundred. 100 7. How then do we express one hundred in figures ? 8. What does 1 denote when it stands alone ? 9. What does it denote with one on the right of it ? 10. What does it denote with two 00 on the right of it ? 11. Hence, we represent 2 tens or twenty, by 20 ; and 12. 20 tens or Two hundred, by 200. 18. 30 tens or Three hundred, by 300. H. 40 tens or Four hundred, by 400. 15. 50 tens or Five hundred, by 500. 16. 60 tens or Six hundred, by 600. 17. 70 tens or Seven hundred, by 700. 18. 80 tens or Eight hundred, by 800. 19. 90 tens or Nine hundred, by 900 ARITHMETIC. ONE HUNDRED and FIFTY- SIX. 156. LESSON LXVIII. 1. Any figure standing alone is units; thus, 6 is 6 units. 2. When two figures are written together, the one on the right is units, the one on the left is tens; thus, 56 is 5 tens or fifty, and 6 units, and is read, fifty-six. 3. When three figures stand together, the one on the right is units, the next figure is tens, and the third on the left is hundreds ; thus, 156 is 1 hundred, 5 tens and 6 units, read one hundred fifty -six. 4. What do the figures 243 denote ? Ans. 2 hundreds, 4 tens, and 3 units, and is read two hundred forty-three. 5. What do the figures 427 denote ? 6. Copy and ^ead the following, naming the hundreds, tens, and units in each. 341 184 537 782 872 462 265 673 394 935 80 FIRST BO OK LESSON LXIX. 1. How many units in 100 ? How many tens 9 How many hundreds 9 2. How many units in 200 ? How many tens ? How many hundreds ? 3. How many units in 300 ? In 400 ? In 500 ? In 600 ? 4. How many tens in 300 ? In 400 ? In 500 ? In 600 ? 5. How many hundreds in 300 ? In 400 ? In 500 ? 5. How many tens in 210 ? In 220 ? In 340 ? In 450 ? 7. How many units in 225 ? How many te/zs and units 9 How many hundreds, tens, and %mYs ? 6*. 10 units = 1 ten : 10 tens = 1 hundred. 9. When no number is named for any place, fill the place with a cipher; thus, seven hundred six is 7 hun- dred, no tens, and 6 units, and is written 706. 10. Seven hundred eighty is written 780. 11. Copy and read the following, naming the number of hundreds, tens, and units in each. 563 287 301 804 203 409 640 711 650 105 12. Write the following numbers in figures, and name the hundreds, tens, and units in each. 18. Seven hundred eight. 14. Five hundred sixty. 15. Three hundred eighty-seven. 16. One hundred ninety-five. 11. Eight hundred seven. 18. Six hundred fourteen. 19. Four hundred sixteen. Itf ARITHMETIC. 81 LESSON LXX. 1. In what place are units written ? Tens ? Hun- dred? 2. How many units make one ten ? 3. How many tens make one hundred ? 4- How many units are 7 tens and 9 units ? 5. How many tens are 4 tens and 5 tens ? 6. How many hundred are 3 hundred and 4 hundred ? How many tens ? How many units ? Express in one number, by figures, each of ths following : 7. 4 tens, 6 units, and 5 hundred. 8. 5 units, 8 hundred, and 4 tens. 9. Six hundred, eight units, and five tens. G 10. Seven tens, no hundreds, and six units. 11. Four units, no tens, and five hundred. 5 12. Nine tens, seven hundred, and no units. ( 7 Write the following numbers in columns, placing units under units, tens under tens, and hundreds under hundreds : 18. 366, 48, 104, 261, 407, 39, and 7. 14. 59, 116, 204, 16, 320, 40, and 10. 15. What is the greatest number that can be expressed by one figure ? 16. What is the greatest number that can be expressed by two figures ? 17. The greatest number that can be expressed by three figures is 999. 18. 999 and 1 more equal One thousand. FIEST BOOK ONE THOUSAND. LESSON LXXL 1. How many are 10 times 10 ? 2. In the picture there are 100 small blocks in the top row or layer. How many hundred blocks are there in 2 rows or layers ? 8. How many hundred blocks in 3 rows ? In 4 rows ? 4. How many in 5 rows ? In 6 rows ? In 7 rows ? In 8 rows ? In 9 rows ? In 10 rows ? 5. Ten hundred equals one thousand. 6. One thousand is written thus : 1,000 7. What does 1 with three OOO's on the right denote ? 8. In like manner are represented, Two thousand, by 2,000. Three thousand, by 3,000. Four thousand, by 4,000. Five thousand, by 5,000. Six thousand, by 6,000. Seven thousand, by 7,000. Eight thousand, by 8,000. Nine thousand, by 9,000. 9. In any number expressed by four figures, the figure at the right is units, the next is tens, the next hundreds, and the fourth figure from the right is thousands. Thus, 2345, is 2 thousands, 3 hundreds, 4 tens, 5 units, and is read two thousand three hundred forty-five. ARITHMETIC,, 83 LESSON LXXII. L What do the figures 1040 denote ? Ans. 1 thousand, no hundreds, 4 tens, and no units, and is read one thousand forty. 2. In the same manner copy and read the following : 2406 4051 3007 1904 1572 5200 3333 6070 Write in figures the following numhers : 3. Three thousand five hundred seven. 4. One thousand two hundred ten. 5. Two thousand one hundred three. 6. Four thousand thirty. 7. Five thousand forty-six. 8. Seven hundred eight. 9. Five hundred ninety. 10. The greatest number that can be expressed by four figures is 9999. 11. 9999 and 1 more equal ten thousand. 10 ,000 12. What does 1 with four OOOO's at the right denote ? In like manner are represented, 13. 2 ten-thousands, or twenty thousand, by 20,000. 14. 3 ten-thousands, or thirty thousand, by 30,000. 15. 9 ten-thousands* or ninety thousand, by 90,000, etc. 16. 10 units make 1 ten. cc . j 2 02 i^ O '3 H a HP 3 5 7 1 2 1 2 1 3 4 3 5 4 6 7 8 5 9 77. 10 tens 18. 10 hundred 19. 10 thousand 0. 10 ten-thousands 1 hundred. 1 thousand. 1 ten-thousand. 1 hundred thousand. 100,000 84 FIRST BOOK LESSON LXXIII. 1. It has been shown that the same figure has a dif- ferent value, according to its place from the right ; thus, 6 is 6 units, 60 is 6 tens, 600 is 6 hundreds, etc. 2. The different places are sometimes called orders "of units; thus, 324 represents 4 units of ike first order, 2 units of the second order, or 2 tens, and 3 units of the third order, or 3 hundreds. 3. The different orders of units are grouped into periods of 3 figures each. 4. The first group on the right is called the period of units, the second, the period of thousands, the third, the period of millions, as shown in the following TABLE. PERIODS. 3d. 2d. 1st. f ri NAME. _? & 15 ^ tD ORDERS or 3 a | 2 S 53 3 S S UNITS. [ a H t> H H P MnP NUMBER. 127, 3^4, 549 5. This number is read one hundred and twenty-seven million, three hundred and sixty-four thousand, five hundred forty-nine. Each period is read like a number of three figures, giving it the name of the period ; thus, 120, 120, 120 is 120 million, 120 thousand, 120. IN ARITHMETIC. 85 LESSON LXXIV. cc +2 '3 2 2 1 4 205 040 307 460 084 602 700 To assist the pupil in learning to write and to read numbers readily, he may be required to prepare on slate or paper, or the blackboard, exercises similar to the following. L The first number is read, 3 hundred 7. 2. The second is read, 2 thou- sand 4 hundred 60. 3. The third is read, 14 thousand 84. 4. The fourth is read, 205 thousand 6 hundred 2. 6. The 5th is read, 2 million 40 thousand 7 hundred. When proficiency in smaller numbers is attained, this exercise may be extended to higher periods. Copy, point off into periods, and read the following : 6. 3472; 5060; 17043; 20304; 600317; 108300. 7. 500037 ; 2405037 ; 910307 ; 76301 ; 30406. 8. Write the preceding numbers in columns, placing units under units, tens under tens, etc. Write in figures, arrange in columns, point off and read, 9. Nine thousand five hundred twelve; twenty-two thousand nine hundred forty; sixty thousand four hun- dred eight; ten thousand one hundred fifteen. 10. One hundred twenty-five thousand three hun- dred eleven ; three hundred seven thousand five hundred four; five hundred and eleven thousand fifteen. 86 FIRST BOOK a AUDI lox LE LXXV. .7. How many are 7 hats and 5 hats ? 6 boys and 7 boys ? 8 men + 9 men ? 9 units -f 7 units ? 2. Can you add 6 books and 4 dollars ? 3. Why not ? .^s. Only numbers representing things of the same Icind can be added. 4- How many units in 9 tens ? In 7 tens ? In 8 tens ? 5. How many tens are 7 tens and 5 tens ? How many units ? 6. How many hundreds are 2 hundred and 6 hundred ? How many tens ? How many units ? 7. How many tens are 10 units ? 30 units ? 50 units ? 8. How many hundreds are 10 tens ? 20 tens ? 40 tens ? EXERCISES FOR THE SLATE AND BOARD. 1. What is the sum of 34 and 53 ? 4 = 3 tens 4 units. Numbers to be added, \ 3 = 5 tens 3 units. > w i > 7 = < Each of these numbers is made up of tens and units. Adding each, the sum is 8 tens and 7 units, or 87 units. Sum, 87 = 8 tens 7 units. In similar manner add the following : (2.) (3.) (4.) (5.) (6.) (7.) (8.) 23 44 52 17 26 75 18 55 32 27 71 52 23 60 ARITHMETIC. 87 LESSON LXXVI. 1. I paid 7 dollars for a hat and 9 dollars for a vest ; how many dollars did I pay for both ? ANALYSIS. I paid the sum <^jj^f^^ and 9 dollars, which is 16 dollars. 2. A farmer sold a sheep fo^BJ^j^and a calf for 8 dollars ; what did he receive for 1 3. A hoy had 16 marbles and his mother gave him 5 more ; how many marbles had he then ? 4* James paid 6 cents for a pencil, 5 cents for an orange, and 8 cents for a ball ; what did he pay for all ? Find the sum of 5. 9, 7, and 2. 6. 5, 8, and 4. 7. 16, 4, and 7. 8. 8, 5, and 7. 9. 15, 5, and 8. 10. 21, 7, and 5. 11. 9, 10, and 6. 12. 8, 9, and 10. 18. 18, 6, and 7. EXERCISES FOR THE SLATE AND BOARD. 1. What is the sum of 324, 213, and 431 ? Numbers ,324 = 3 hunds. 2 tens 4 units. Each of thege to be J 2 1 3 = 2 hunds. 1 ten 3 units. numbers is made added, (431 = 4 hands. 3 tens 1 unit. up of hundreds, a Q * "Q tens and unit s. 968 = 9 hunds. o tens o units. Adding each, the sum is 9 hundreds 6 tens 8 units, or 968 units. In a similar manner add the following : (*) (^) (-*) (6.) (6.) 213 304 503 104 504 425 123 172 302 670 160 562 223 493 125 88 FIRST BOOK LESSON LXXVII. The following are all the combinations that can be made with the nine digits, except with the unit 1, up to 9 and 9 are 18. These thoroughly com^d to memory, will very much facili. fcate the adding of long columns with ease and accuracy. 2 and 2 are 4 * 4 and 7 are 11 2 and 3 are 5 4 and 8 are 12 2 and 4 are 6 4 and 9 are 13 2 and 5 are 7 5 and 5 are 10 2 and 6 are 8 5 and 6 are 11 2 and 7 are 9 5 and 7 are 12 2 and 8 are 10 5 and 8 are 13 2 and 9 are 11 5 and 9 are 14 3 and 3 are G 6 and G are 12 3 and 4 are 7 G and 7 are 13 3 and 5 are 8 6 and 8 are 14 3 and 6 are 9 6 and 9 are 15 3 and 7 are 10 7 and 7 are 14 3 and 8 are 11 7 and 8 are 15 3 and 9 are 12 7 and 9 are 16 4 and 4 are 8 8 and 8 are 1G 4 and 5 are 9 8 and 9 are 17 4 and 6 are 10 9 and 9 are 18 Let the above combinations be repeated also in the reversed order ; thus, 2 and 5 are 7, 5 and 2 are 7. etc. The above may also be used for subtraction, the third column being the minuend, and either of the others the subtrahend. Copy the above on slate, or board, in each form, using the signs. ARITHMETIC. 89 LESSON LXXVIII. How many are 2 + 22, 2 + 32, etc., to 2 + 92? 3 + 2, 3 + 12, 3 + 22, 3 + 32, etc., to 3 + 92? 8. 4 + 2, etc., to 4 + 92? 4. 5 + 2, etc., to 5 + 92? 5. 6 + 2, etc., to 6 + 92? 6. 7 + 2, etc., to 7 + 92? 7. 8 + 2, etc., to 8 + 92? 8. 9 + 2, etc., to 9 + 92? How many are P. 2 + 3 y 2 + 13, etc., to 2 + 93? 10. 3 + 3, 3 + 13, etc., to 3 + 93? 11. The same also, with 4, 5, 6, 7, 8, and 9. 12. Then 2 + 4, etc., 3 + 4, etc., 4 + 4, etc., as above. 18. In the same way, 2 + 5, etc., 3 +'5, etc., to 9 + 5, etc., till the sum is 9 + 85. The foregoing is given only as a sample of a series of exercises, which will help to secure rapidity and accuracy in all possible com- binations of the 9 digits with any number. 1. Name all the numbers in combinations of two each, that make 4. Ans. 3 + 1=4, 2 + 2=4, 1 + 3=4. 2. Name all that make 5. Ans. 4 + 1, 3 + 2, 2 + 3, and 1 + 4. 8. That make 6. Ans. 5 + 1, 4 + 2, 3 + 3, 2 + 4,andl + 5. In the same manner, the numbers that make, 7 9 11 13 15 17 19 21 23 8 10 12 14 16 18 20 22 24 4. Write on the slate or board a table of each ; thus of 7, 6 + 1 = 7, 5 + 2 = 7, 4 + 3 = 7, 3+4=7, 2 + 5 = 7, 1 + 6 = 7. 7-1 = 6, 7-2=5, 7-3=4, 7-4=3, 7-5=2, 7-6=1. 90 FIRST BOOK LESSON LXXIX. 1. How many tens in 34 units ? Ans. 3 tens and 4 units. 2. How many tens in 37 units ? In 56 units ? 3. How many hundreds in 30 tens ? In 50 tens ? 4. How many hundreds in 3G tens ? EXERCISES FOB THE SLATE A^D BOARD. 1. What is the sum of 524, 315, and 472 ? 524 Write units of the same order in the same 345 column. AH** Begin at the bottom of the units column, and add each column separately, and instead of say- Sum 1342 ing 3 units and 5 units are 8 units, and 4 units are 12 units, name the successive results only ; thus, 3, 8, 12, the sum of the units, equal to \ ten and 2 units. Write the 2 units in units' place, and add the 1 ten to the lower number in the tens' column ; then, 1, 8, 12, 14, the sum of the tens, equal to 1 hundred and 4 tens. Write the 4 tens in tens' place and add the 1 hundred to the hundreds' column ; then, 1, 5, 8, 13, the sum of the hundreds, equal to 1 thousand and 3 hundreds, which Write in the hundreds' and thousands' places. In the (2.) 423 542 365 (*.) 134 250 675 same manner copy, (8.) (4.) 304 210 718 634 532 184 and add : (5.) (6.) 514 75 301 610 198 393 716 84 205 men. (9.) 384 92 807 boys. (10.) 300 480 78 pounds. (6 (11.) 2036 462 84 days. ARITHMETIC. LESSON LXXX, 91 Count 1. By 2's from to 36. 9. By 5's from to 40. 2. By 3's from to 27. 10. By 5's from 1 to 36. 3. By 3's from 2 to 29. 11. By 5's from 2 to 37. 4. By 3's from 1 to 31. 12. By 5's from 3 to 38. 5. By 4's from to 40. 13. By 6's from to 36. 6. By 4's from 1 to 37. 14. By 6 ? s from 2 to 38. 7. By 4's from 2 to 38. 15. By 6's from 4 to 40. 8. By 4's from 3 to 39. 16. By 6's from 5 to 41. Count back 17. By 2's from 30 to 0. 26. By 5's from 40 to 0. 18. By 2's from 29 to 1. 27. By 5's from 41 to 1. 19. By 3's from 30 to 0. 28. By 5's from 42 to 2. 20. By 3's from 28 to 1. 29. By 5's from 43 to 3. 21. By 3's from 29 to 2, 30. By 6's from 37 to 1. 22. By 4's from 40 to 0. 31. By 6's from 39 to 3. 23. By 4's from 41 to 1. 32. By 6's from 38 to 2. 24. By 4's from 42 to 2. 33. By 6's from 40 to 4. 25. By 4's from 43 to 3. 34. By 6's from 41 to 5. EXERCISES FOR THE SLATE AND BOARD. Copy and add 3 (*) 3 3 3 3 (ft) 3 3 3 3 (S.) 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 1 4 3 2 1 5 4 3 2 92 FIRST BOOK LESSON LXXXI. 1. Count alternately by 2's and 3's to 35. * Written, 2 + 3 + 2 + 3 + 2 + 3 + 2 + 34-2 + 3 + 2 + 3 + 2 + 3. Counted, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27, 30, 32, 35. Count 2. By 2's and 3's to 40. 3. By 3's and 4's to 28. 4. By 2's and 4's to 36. 5. By 2's and 5's to 28 6. By 3's and 5's to 32, 7. By 4's and 5's to 36. EXERCISES FOR THE SLATE AISTD BOARD. Copy and add (1.) (2.) (ft) 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 1 4 4 3 3 9 8 7 6 UO (*) (4) 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 3 2 1 5 4 3 2 8 7 6 4 (7.) (8.) w 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 1 2 3 6 3 2 1 4 7 6 8 6 * This exercise may at first be written partially on the board and counted at sight, and then omitted entirely. IN ARITHMETIC. 93 LESSON LXXXII. 1. Count by 7's, from to 49 ; from 1 to 43 ; from 2 to 44 ; from 4 to 46 ; from 5 to 47 ; from 6 to 48. 2. Count by 8's, fromOto48; froml to49; from2 to 50; from 4 to 52 ; from 5 to 53 ; from 6 to 54 ; from 7 to 55. 8. Count by 9's, from to 54; from 1 to 46; from 2 to 47; from 4 to 49 ; from 5 to 50 ; from 6 to 51 ; from 7 to 52 ; 4. Count back by 7's from 49 to ; 46 to 4 ; 43 to 1. 5. By 8's from 48 to ; from 42 to 2 ; from 47 to 7. 6. By 9's from 45 to ; from 40 to 4 ; from 41 to 5. EXERCISES FOR THE SLATE A:ND BOARD. Copy and add, (L) (2.) (3.) 7777 7777 7777 7777 7777 7777 7777 7777 7777 2100 3210 7654 8 8 ) 8 8 8 (5.) 8 8 8 8 (6.) 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 3 2 1 6 5 4 3 9 8 7 9 (7 9 9 9 9 (*) 9 9 9 9 (9.) 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 4321 8765 9876 94 FIRST BOOK LESSON LXXXIII. Addition at Sight. 1. Any two numbers less than 100, one of which con- tains only tens, may be added at sight (or orally). Thus, 34 + 20=54. Observe, that 3 tens and 2 tens are 5 tens or 50, and the 4 units make 54. 2. Name the results to the following : 20+ 8 40 + 14 50 + 18 80 + 17 30 + 12 40 + 26 60 + 33 70 + 27 30 + 16 40 + 42 60 + 25 20 + 36 30 + 20 50 + 27 70 + 14 50 + 44 34 + 40 56 + 30 17 + 60 28 + 50 ADDING UNITS AND TENS OKALLY. 1. What is the sum of 23 and 45 ? ANALYSIS. 23 is equal to 2 tens and 3 units ; 45 is equal to 4 tens and 5 units ; 2 tens and 4 tens are 6 tens or 60, and 3 units and 5 units are 8 units ; 60 and 8 are 68. Hence 23 and 45 are 68. In like manner find the sum of 2. 12 + 49. 3. 23 + 64. 4. 38 + 17. 5. 51+26. 6. 29 + 30. 7. 48 + 34. 8. 72 + 16. 9. 37 + 26. EXERCISES FOB THE SLATE AND BOARD. 1. What is the sum of 56 feet, 450 feet, and 680 feet? 2. A farmer sold 48 bushels of wheat, 36 bushels of corn, 27 bushels of rye, and 28 bushels of barley ; how many bushels of grain did he sell in all ? 8. What is the sum of 1942 dollars, and 685 dollars ? IN ARITHMETIC. 95 LESSON LXXXIV. Direct the attention of the pupil to the fact, that the same figures when added, always give the same unit figure in the result. That is, 3 and 5 are 8 ; 3 and 25 are 28 ; 3 and 45 are 48, etc. If, in adding any of the higher numbers, he hesitates, refer him to the primary sum of those numbers. Thus, if the pupil hesitates on 68 and 5, ask: "What unit figure do 8 and 5 give?" (Am. 3.) Then 68 and 5 must give the next 3 above 68, that is 73. The pupil should be required to perform all operations on the. slate or board, without moving the lips, pronouncing or writing the results only. BLACKBOARD DRILL. Place an example upon the blackboard, as the following: 267189 ^ a ^ u P on some P u pil> wn will begin with the right hand column and say, " 9, 17, 24, 27, 33, 42 wilts, equal to 4 tens and 2 units ; write the 2 units 918753 tinder the column added, and add the 4 tens to the 592847 next column.'* The next pupil will without any 703928 delay take up the process, beginning with the 4 tens 564789 reserve( l, and say, "4, 12, 14, 18, 23, 26, 34 tens, equal to 3 hundreds and 4 tens ; write the 4 tens 3533242 under the column added and add the 3 hundreds to the next column." So in quick succession let each column be added upwards, then downwards, then from right to left, and from left to right, until the whole class have been exer- cised upon this example. Very young children, if properly drilled on preceding lessons, may easily be taught, in this way, to add long columns of figures with astonishing rapidity and correctness. To vary the above, let each pupil in order give one result only, the next pupil, immediately pronouncing the following, etc. 96 FIRST BOOK LESSON LXXXV. EXERCISES FOR THE SLATE AND BOARD. (f) men. 542 176 628 (2.) (3.} feet. days. 820 153 507 208 418 759 (4.) (5.) feet. miles. 1450 2157 1234 1528 2357 1372 (6.) pounds, 1740 2031 1507 473 256 380 1567 2143 1423 (7.) (8.) (9.) (10.) (11.) (12.) 1247 5020 1305 3172 1526 4214 2072 1513 6040 1094 5017 2030 4289 3156 3708 7165 2157 5327 3070 1208 1159 2082 1215 1456 Exercise the class on the following, as suggested ir I esson LXXXIV. (13.) (^) (15.) (16.) (17.) 3678 8786 78982 37987 216677 2766 5968 69771 66789 569911 8345 8789 68339 44321 543344 3875 9896 56234 91389 576677 (1ft (19.) (20.} (21.) (22.) 32507 23412 35092 275143 1214187 10325 57638 52803 413100 2742361 47018 15421 47524 650406 1275142 53106 62732 60832 210350 2020560 61007 54298 11462 132415 1603915 27589 17323 74260 113765 1846084 ARITHMETIC. 97 LESSON LXXXVI. 1. If 8 cents be taken from 14 cents, how many cents remain ? Ans. 6 cents. Since 14 is diminished, or made less, by subtracting 8 trom it, 14, the greater number, is named the Minuend, which means to be diminished, and 8, the less number, is named the Subtrahend, v/iiich means to be subtracted. Since 6 shows how many cents remain after subtracting 8 cents from 14 cents, it is named the Remainder, or the Difference between 14 and 8. 2. What is the difference between 13 horses and 9 horses ? 3. Sixteen men are how many more than 8 men ? 4. Five cents are how many less than 12 cents ? 5. How many must be added to 9 to make 12 ? 6. Can you take 6 boys from 11 sheep ? Why not ? Ans. Only numbers representing things of the same kind can be subtracted. 7. Harry bought 12 peaches, and gave 5 of them to his sister ; how many had he left ? 12 6= ? ANALYSIS. He had left the difference between 12 and 5, which is 7. 12-5=7. 8. On a tree were 1G pigeons, and 9 flew away ; how many were left on the tree ? 16 9~ ? 9. How many are 19_ 6? 24 less 6? 4 from 11? 2010? 15 less 9? 7 from 19? 1711 ? 21 less 7 ? 8 from 24 ? 98 FIBSTBOOK LESSON LXXXVII. 1. How many are 10 less 2 ? 11 less 2 ? 12 less 2 ? 13 less 2 ? etc., to 20 less 2 ? 2. How many are 10 less 3 ? 11 less 3 ? 12 less 3 ? 13 less 3 ? etc., to 30 less 3 ? 3. How many are 10 4? 114? 12 4? 134? etc. 4. Subtract 7 from 10 : from 11 ; from 12 ; from 13 ; etc. 5. Take 8 from 10 ; from 11 ; from 15 ; from 16 ; etc. 6. How many are 109 ? 11 9 ? 12 9 ? 149 ? 15_9? 16-9? etc., to 35-9? 7. George had 17 marbles, and gave five to one boy and 4 to another ; how many marbles had he left ? 17 9= ? ANALYSIS. He liad left the difference between the sum of 5 mar- bles and 4 marbles or 9 marbles, and 17 marbles which is 8 marbles. 8. Jane had 18 cents, and she bought a yard of ribbon for 6 cents, and a spool of thread for 5 cents ; how many cents had she left Y 18 11= ? 9. James bought a ball for 10 cents and a pencil for 6 cents. How much change should he receive for 25 cents ? * 10. Copy and write the result in place of ( ? ) : 147= ? 218=? 2410=? 18-5= ? 19-7=? 21-9=? 123=? 154=? 17 8= ? 11. Also the following : 6 + 104=? 12 + 58=? 1410 + 12=? 7__ 4 + 8=? 209 + 12=? 25+ 7 2= ? 9 + 5_6 ? 16 7 + 13=? 189 + 20=? IK ARITHMETIC. 99 LESSON LXXXVIII. When numbers are so large that the difference cannot be found at once, units may be taken from units, tens from tens, and hundreds from hundreds, etc. EXERCISES FOR THE SLATE AND BOARD. 1. Subtract 644 from 968. Minuend, 968 = 9 liunds. 6 tens 8 units. Subtrahend, 644 6 " 4 " 4 " Difference, 324 3 " 2 " 4 " PROOF. Add the difference to the subtrahend. If the work is right, the sum will equal the minuend. Thus, 968644=324, the difference; and 324 + 644968, the minuend. In like manner subtract and prove the following : (2.) (3.) (4.) (5.) (ft) Minuend, 835 769 578 274 857 Subtrahend, 423 634 453 121 524 Difference, 412 (7.) (8.) (9.) (10.) (11.) 936 792 478 891 527 214 351 436 170 204 Subtract and prove: 12. 623 from 944. 19. 899 -673. 18. 431 from 862. 20. 777 -504. 14. 354 from 798. 21. 999 -657. 15. 256 from 579. 22. 886 273. 16. 170 from 382. 23. 709 207. 17. 322 from 694. 24. 686 475. 18. 4206 from 9876. 25. 8989 - 4736, 100 FIRST BOOK LESSON LXXXIX, 1. One ten is how many more than 1 unit ? 2. One ten is how many more than 3 units ? 3. Two tens are how many more than 7 units ? 4. Twenty-five are how many more than 6 ? than 7 ? 5. Two tens and 6 units less 8 units are how many ? ANALYSIS. 2 tens and 6 units are 26 units, or 1 ten 16 units ; and 8 units from 1 ten 16 units leaves 1 ten 8 units, or 18 units. 6. Three tens 5 units less 1 ten 6 units are how many ? 3516? ANALYSIS. 3 tens 5 units equal 2 tens 15 units ; and 1 ten 6 units from 2 tens 15 units leaves 1 ten 9 units, or 19 units. EXERCISES FOE THE SLATE AND BOARD. 1. Subtract 27 from 84. i H 84 is 8 tens and 4 units, 27 is 2 tens and 7 Minuend, $ & units. Since 7 units cannot be subtracted Subtrahend 2 7 from 4 units, increase the 4 units by 1 ten or 10 units from the next higher order, making Remainder, 57 14 units. 7 units from 14 units leave 7 units, which write in units' place. Since we have taken 1 of the 8 tens, there are only 7 tens left. 2 tens from 7 tens leave 5 tens, which write in tens' place. Hence, 27 from 84 leaves 57. PROOF: 57 + 27=84. Copy, subtract and prove the following : (*) (*-) (40 (*) (*0 Minuend, 439 674 523 834 423 Subtrahend,J-72 327 247 263 276 (7.) (8.) (9.) (10.) (11.) (12.) 4571 5274 7345 9876 6721 2925 2786 1548 5456 4894 3834 1673 IK ARITHMETIC. 101 LESSON XC. 1. A boy had 9 cents, he earned 10 more, then gave 7 to his sister. How many cents had he left ? 2. Five pounds, 7 pounds, and 4 pounds are how many less than 20 pounds? 3. Belle had 16 pinks, and gaye 3 to Mary, and 5 to Anna. How many had she left ? 4- Jennie had 25 cents, and bought some buttons for 8 cents, a pencil for 4 cents, and some thread for 6 cents. How many cents had she left ? 5. JSTed having 19 cents, lost 4, spent 5, earned 3, and gave away 6. How many cents had he then ? 6. How many are 5, 7, and 4, less 2 and 3 ? 7. How many are 10, 3, and 6, less 5 and 7 ? 8. How many are 15 less 7 added to 4*and 3 ? 9. How many are 12 less 5, added to 10 less 3 ? 10. How many are 13 and 7, less 6 and 9 ? 11. How many are 10, 7, and 5, less 9 and 3 ? EXERCISE FOR THE SLATE AKD BOARD, OR ORALLY. Copy and write the proper number in place of ( ? ) : 12. 18 7 + 7 = ? 13. 20 + 18 - ? == 25. U. 9 + 11 = 10 + ? 15. 24 ? = 7 + 9. 16. 21 - 9 + ? = 19. 17. 24 + ? = 12 + 18. 18. ? + 16 = 26 6. 19. 10 + 13 = ? + 11. 20. 27 7 = 17 + ? 21. 36 + 4 = ? + 20. 22. 11 + 7 = 26 ? 23. 23 5 = ? + 2. The teacher should increase the number of examples in every lesson, as the capacity of the class may require, or the time allow. 102 FIKST ,BOOK LESSON XCI. How many are 1. 2 from 2, 2 from 12, etc., to 2 from 92 ? 2. 2 from 3, 2 from 13, etc., to 2 from 93 ? 5. 2 from 4, 2 from 14, etc., to 2 from 94 ? 2 from 5, 2 from 15, etc., to 2 from 95 ? 6. 2 from 6, 2 from 16, etc., to 2 from 96 ? 6. 2 from 7, 2 from 17, etc., to 2 from 97 ? 7. 2 from 8, 2 from 18, etc., to 2 from 98 ? & 2 from 9, 2 from 19, etc., to 2 from 99 ? 9. 2 from 10, 2 from 20, etc., to 2 from 90 ? 10. In the same manner 3 from 10, 3 from 11, 3 from 12, and 3 from 19, etc., to 3 from 99. 11. Also, 4 from 10, etc., 5 from 10, etc., until all the 9 digits have been used in the same way. 12. One ten is equal to how many units ? 13. One hundred is equal to how many tens ? H. Two hundred is how many tens more than 2 tens ? 15. Three hundred is how many tens more than 4 tens ? than 5 tens ? 16. Three hundred is equal to 2 hundred and how many tens ? Ans. 300=200 and 10 tens. 17. If 2 tens or 20 units he taken from 3 hundred, what remains ? Ans. 300=200 and 10 tens : 2 tens from 200 and 10 tens leave 200 and 8 tens, or 280. 18. Four hundred and 3 tens, less 1 hundred and 4 tens are how many ? ANALYSIS. 4 hundred and 3 tens,(equal 3 hundred and 13 tens) less 1 hundred and 4 tens, equal 2 hundred and 9 tens, or 290. AKITHMETIC. 103 LESSON XCII. EXERCISES FOR THE SLATE AXD BOARD. L Subtract 279 from 800. Since 9 cannot be taken from units, and since there are no tens, we cannot take 1 from that order. Going on to the order of hundreds, take 1 hundred, equal to 10 tens leaving 7 hundreds ; of these 1 tens take 1 ten or 10 units, leaving 9 tens, and the minuend 800 is equal to 7 hundred 9 tens and 10 units (700 + 90 + 10^800). Subtract as before, 7 9 10 Minuend, $00 Subtrahend, 279 Remainder, 521 In like manner copy, subtract, and prove: 2. 267 from 500. 7. 2241 from 7000. 3. 672 from 740. 8. 127 from 4300. 4. 504 from 820. 9. 32 from 1000. 5. 1260 from 3005. 10. 3030 from 10200. *6. 4521 from 6206. 11. 237 from 8040. Find the difference between 12. 524 and 376. 17. 907 and 2340. 13. 427 and 806. 18. 89 and 1472. U. 900 and 679. 19. 1200 and 164. 15. 321 and 450. ,20. 2040 and 320. 16. 784 and 508. 21. 3672 and 1075. What is the result $%. Of 3416 2040. 27. Of 12364 - - 1400 $3. Of 4006 - 844. 28. Of 8070 -4105 24. Of 6400 - 640. 29. Of 7346 - 6462 25. Of 2706 1371. 30. Of 20371 - 8106 . 26. Of 7120 762. 31. Of 16070 - 1284. 104 FIRST BOOK LESSON XCIII. Subtraction at Sight. Any number may be subtracted from another at sight, if the difference is not greater than 10. Thus, 1. 95=4; 19 15=4; 29 25=4; 3935=4, etc. 2. 146 = 8; 2416 = 8; 3426 = 8; 44 36 = 8, etc, 3. How many are 49 43 ? 36 29 ? 63 58 ? 92 84 ? 52 47? 78-70? 83-76? 81-73? Any number not greater than 10 can be subtracted at sight from any other number. Thus, 8 3=5; 18 3 = 15; 283=25; 383=35, etc. 5. 15 7=8; 25 7=18; 35 7=28; 45 7=38,etc. 6. How many are 27-8? 49-3? 51-4? 81-5? 61-2? 36 7? 649? 73 6? 94-8? 75^-9? ORAL AND WRITTEN EXERCISE. (^.) Write the two numbers 344 and 579, and add them 344 without drawing a line, and write their sum 923 as the g w Q third number ; then add the three numbers and write their sum, 1846, as the fourth number ; then add the four numbers, and write their sum, 3692, as the fifth, and so on. The same example may be used for exercise in sub- 3692 traction, by subtracting from the last result each of the 7384 preceding numbers in succession, until nothing remains. In the same manner, copy, add, and prove the follow- ing, extending each to the sixth number : (*) (*) (-*) (5.) (&) 327 674 384 540 1257 918 241 609 703 720 IK ARITHMETIC. 105 Ml 'LICATION LESSON XCIV. 1. If 5 boys can sit upon one bench, how many boys can sit upon 3 benches ? 5 and 5 and 5 are how many? 2. If a man earn 3 dollars a day, how many dollars will he earn in 5 days? 3 + 3 + 3 + 3 + 3, are how many? Five 3's, or 5 times 3, are how many ? 8. Count to 15 by 3's. By 5's. How many 3's in 15 ? How many 5's ? 4. There are 7 days in 1 week ; how many days are there in 4 weeks ? 7 + 7 + 7 + 7=? Four 7's, or 7 x 4= ? 5. Count by 4's to 28. By 7's. How many 4's in 28 ? How many 7's ? 6. Is the result of 4 times 7 and of 7 times 4 the same ? 7. At 4 dollars a barrel, what will 6 barrels of apples cost ? Six 4's, or 6 times 4, are how many ? 8. What is the difference between six 3's and three 6's ? 9. How many are eight 3's ? 3 times 8 ? 8x3? 10. Eepeat the table from times 2 to 12 times 2. Thus, times 2 is 0, once 2 is 2, twice 2 are 4, 3 times 2 are 6, etc. 11. Repeat back from 12 times 2 to times 2. Thus, 12 times 2 are 24, 11 times 2 are 22, 10 times 2 are 20, etc 12. From times 3 to 12 times 3, and back to times 3. 13. From times 3 to 12 times 4, and back. 14. From times 5 to 12 times 5, and back. 106 FIESTBOOK LESSON XCV. 1. If one hat cost 7 dollars, what will 4 hats cost? ' SOLUTION. Since 1 hat costs 7 dollars, 4 hats will cost 4 times 7 dollars, or 28 dollars. 2. At 6 cents each, what- will 5 pencils cost? 3. At 8 cents a pound, what will 6 pounds of soap cost ? 4. If a man earn 4 dollars a day, how many dollars can he earn in 8 days ? 5. If 5 boys have 7 marbles each, how many marbles have they all ? 6. What is the cost of 6 books, at 4 dollars each ? 7. If a boy receives 5 merit marks a day for 5 days, how many marks does he receive in all ? 8. If a horse troo 8 miles an hour, how far will he trot in 6 hours ? 9. Bessie had 3 rose-bushes, and there were 9 roses on each bush. How many roses upon all ? Eepeat the table 10. From times 6 to 12 times 6, and back to times 6. 11. From times 7 to 12 times 7, and back. 12. From times 8 to 12 times 8, and back. 13. How many units are 4 times 9 units ? How many tens ? Ans. 3 tens and 6 units. 14. How many are 5 times 12 units ? How many tens ? 15. How many tens are 6 times 5 tens ? How many hundreds ? How many units ? 16. How many are 4 times 6 tens ? How many hun- dreds and tens ? How many units ? 17. How many hundreds are 3 times 7 tens ? IN ARITHMETIC. 107 LESSON XCVI. 1. Taking one of two numbers as many times as there are units in the other is called Multiplication. 2. The number taken or multiplied is named the Multiplicand. 3. The number to multiply by, or that shows how many times the multiplicand is to be taken, is named the Multiplier. 4. The result obtained by the multiplication is named the Product. 5. The multiplicand and multiplier are Factors of the product 6. Thus, 8x6 = 48. 8 is the multiplicand, 6 is the multiplier, 48 is the product, and 8 and 6 are the factors. EXEECISES FOB THE SLATE AKD BOAED. 1. How many are 3 times 42 ? BY ADDITION. BY MULTIPLICATION. The result may be _ . . ,,,. Multiplicand, 4 2 otoined b y ^Uion. Numbers Write 42 three times to be \ 42 Multipher, 3 andadd; thesis 126. added, Product, 12 6 But it may be done Bum, 126 much shorter and easier by writing the multiplicand 42 but once ; and as it is to be taken 3 times, write the multiplier 3 under it in units' place and multiply. Thus, 3 times 2 units are 6 units, which write in units' place ; and 3 times 4 tens are 12 tens, or 1 hundred and 2 tens, which write in tens' and hundreds' place. Hence the product 126 is the same as the sum, In like manner solve by both methods : 2. 3 times 53. 3. 5 times 61. 4. 4 times 220. 5. 2 times 643. 6. 3 times 4032. 7. 6 times 610. 8. 4 times 3102. 9. 2 times 7234. 108 FIRST BOOK LESSON XCVII. 1. At 5 dollars a cord, what will 6 cords of wood cost ? 7 cords ? 8 cords ? 2. If there are 8 trees in ooe row, how many trees in 5 rows ? In 8 rows ? In 9 rows ? In 10 rows ? 3. If you earn 12 cents an hour, how many cents can you earn in 3 hours ? In 5 hours ? In 6 hours ? In 7 hours ? In 10 hours ? 4. How many are 5 times 10 cents, and 8 cents more ? 5. How many are G times 7 dollars, less 5 dollars ? 6. James gave 4 cents apiece for 5 oranges, and had 10 cents left. How much money had he at first ? 7. Belle paid G cents a yard for 3 yards of ribbon. How much change should she receive for 25 cents ? EXERCISES FOR SLATE AKD BOARD. 1. How many are 4 times 48 ? Multiplicand, 4 8 The result may be obtained by addition, but Multiplier, 4 tne shorter method is to write the multiplier under the multiplicand in units' place and Product, 192 u . , , T multiply ; thus, 4 times 8 units are 32 units, or 3 tens and 2 units. Write the 2 units in units' place, and reserve the 3 tens to add to the product of the tens. Next, 4 times 4 tens are 16 tens, and the 3 tens reserved added make 19 tens, or 1 hundred and 9 tens, which write in hun- dreds' and tens' places. Hence, 4 times 48 are 192. (2.) (A) (4.) (5.) Multiply 72 136 247 1265 By _4 _j> _3 _J> 6. Multiply 2436 hy 3 ; by 5 ; by 4 ; by 6. 7. Multiply 3057 by 2 ; by 3 ; by 4 ; by 5 ; by 6. INAEITHMETIC. 109 LESSON XCVIII. The following table contains all the products from 12 to 144 in- clusive that can be produced by any two factors, not less than 2 nor greater than 12. These combinations should be thoroughly committed, and re- peated in the reverse, as well as in the direct order. 2x 6 = 12) 4x 8=32 7x 9= 63 V 3x 4=12 \ 3x11=33 8x 8= 64 2x 7=14 5x 7=35 6x11= 66 3x 5 = 15 3 x 12=36 \ 7x10= 70 2x 8=16) 4x 9=36 V 8x 9= 72 4x 4=16) 6x 6=36 ) 6x12= 72 2x 9 = 18) 4x10=40) 7x11= 77 > ( 3x 6 = 18 \ 5x 8=40 ) 8x10= 80 2x10=20) 6x 7=42 9x 9= 81 4x 5=20) 4x11=44 7x12= 84 3x 7=21 5x 9=45 8x11= 88 2x11=22 4x12=48) 9x10= 90 2x12=24 \ 6x 8=48 ) 8x12= 96 3x 8=24 V 7x 7=49 9x11= 99 4x 6=24) 5x10=50 10x10=100 5x 5=25 6x 9=54 10x11=110 3x 9=27 5x11=55 10x12=120 4x 7=28 7x 8=56 11x11 = 121 3x10=30) 6x10 = 60) 11x12=132 5x 6=30 ) 5x12 = 60 ) 12x12=144 The above may also be used as a division table, the numbers in the third column being used as dividends, and those either in the first or second as divisors. Thus, 2 times 6 are 12 ; 6 times 2 are 12. 6 in 12, 2 times ; 2 in 12, 6 times, of 12 is 6 ; -J- of 12 is 2. 110 FIKST BOOK LESSON XCIX. The following are all the products in the multiplica- tion table to 132. Let the pupil at sight, or upon hearing a number, name at least one set of factors. Thus, 21. Am. 7 times 3, or 3 times 7. 60. Ans. 6 times 10, or 5 times 12, etc. 4 20 35 55 81 6 21 36 56 88 8 22 40 60 90 9 24 42 63 96 1G 25 44 64 99 12 27 45 66 100 14 28 48 70 108 15 30 49 ' 72 110 16 32 50 77 121 18 33 54 80 132 EXERCISES FOR THE SLATE AXD BOARD. (1.) (2.) (3.) (4.) 1026 4150 6703 14037 _8 _6 _5_ 4 5. If there are 52 weeks in 1 year, how many weeks are there in 8 years ? 6. If a steamer run 265 miles a day, how many miles will she run in 7 days ? 7. A man bought 4 houses, and paid 4385 dollars for each. What did he pay for all ? 8. In 1 mile are 5280 feet. How many feet in 3 miles ? ARITHMETIC. Ill LESSON C. EXERCISES FOR SLATE AND BOARD, OR ORALLY. Give the result of 3x7 5x3 6x3 9x7 6x 7 7x5 8x8 9x9 4x6 8x 3 4x9 7x6 8x7 5x8 3x 9 9x6 4x7 6x9 7x5 3x10 The pupil may make oral problems for the above. Thus, for "3x7." " What will 7 lemons cost at 3 cents each?" Another pupil may solve the same by a simple analysis ; thus, 7 lemons will cost 7 times 3 cents, or 21 cents. Copy, and write the result in place of (?): 8x7+ 4=? 7x9 + 10= ? 6x6+ 8=? 4x4+ 9= ? 9x8-12=? 8x6- 9=? 9x910= ? 10x8- 7=? 7x 9 + 10=? 8x11 8= ? 7x10 + 12=? 11 x 6- 9=? FOR THE SLATE AND BOARD. In the following, the numbers given as multiplicands may each be multiplied, separately, by each of the multipliers following it ; or first by one of them and the product by the next, and so on. 1. 9704x2, 4, 6, 8. 2. 658x3, 5, 7, 6. 3. 1463x4, 2, 3, 5. 4. 2789x3, 4, 2, 6. 5. 3596x2, 4, 3, 5. 6. 2043x4, 5, 2, 7. 7. 8. 9. 10. 11. 12. 3804 x 5,- 2964 x 3, 4070 x 5, 5164 x 7, 3005 x 8, 10804 x 6, 6, 8, 9, 8, 9, 7, 7. 6. 4. 9. 5. 8, 9. 8x12=? 6x?=72 ? x7 9x 6==? ?> ;5=r45 12 x? 12 p 12 p 7 12 8 8 p 9 p p T 96 96 72 63 84 112 FIRST BOOK LESSON CI. Multiplication at Sight. 14 9x ? =81 iO ?xll = 77 ? ? 11 _9 12 J> 81 60 99 This exercise may be continued at the option of the teacher. EXERCISES FOR THE SLATE A:N T D BOARD. 1. Multiply 362 by 24. Multiplicand, 362 Write the multiplicand, and Multiplier, 24 under U the multiplier, the units in units' place, and the 4 times, tens in tens' place, and multi- 724 20 times. . ply by each figure separately. Multiply 362 by the 4 units, Product, o o o o /e 4 times. and tlien by the ~ tens ; add the products, and the sum is the entire product, 8688. In multiplying by the units, write the first figure of the product in units' place. In multiplying by the tens, write the first figure of the product in tens' place. In the same manner, multiply 2. 487 by 18. 3. 618 by 23. 4. 3241 by 16. 6. 894 by 22. 5. 2046 by 25. 7. 709 by 36. 8. Multiply 30426 by 17 ; by 28 ; by 35 ; by 44 ; by 42. 9. Multiply 24305 by 52 ; by 48 ; by 65 ; by 72 ; by 66. 10. Multiply 426173 by 61; by 43; by 53; by 19; by 34. The teacher should increase these examples according to the wants and capacity of the class ; also instruct them how to multi- ply by 10, or 100. ARITHMETIC. 113 DTVISIOIV LESSON GIL 1. How many times 3 peaches are 18 peaches ? How many 3's in 18 ? 18-7-3= ? 2. How many times can 4 oranges be taken from 16 oranges? How many 4's in 16 ? 16-7-4= ? 3. How many times 4 pounds. are 20 pounds? How many 4's in 20 ? 4. How many times can 4 be taken from 20 ? 4 in 20 how many times ? 20-7-4= ? 5. If a box will hold 5 pounds of honey, how many such boxes will hold 30 pounds ? 6. How many times 5 pounds are 30 pounds ? How many 5's in 30 ? How many times can 5 be taken from 30 ? 7. From a pile of 24 marbles, how many groups of 4 marbles each can be made ? 4 in 24, how many times ? 8. How many groups of 6 each ? How many 6's in 24 ? 9. How many groups of 3 each ? How many 3's in 24 ? 10. How many groups of 8 each ? How many 8's in 24 ? 11. How many groups of 2 each ? How many 2 ? s in 24 ? 12. How many times can 8 dollars be taken from 24 dollars ? From 40 dollars ? From 56 dollars ? 13. How many times 8 in 24 ? In 40 ? In 56 ? In 64 ? 14. How many times 9 boys cfre 27 boys ? Are 36 boys ? 15. How many 9's in 18 ? In 27 ? In 54 ? In 72 ? In 81 ? FIEST BOOK LESSON CIII. 1. At 4 cents each, how many lemons can be bought for 20 cents ? SOLUTION. As many lemons can be bought for 20 cents, as 4 cents are contained times in 20 cents, which are 5 times. Hence, at 4 cents each 5 lemons can be bought for 20 cents. 2. If a man earn 3 dollars a day, in how many days can he earn 12 dollars? 18 dollars? 27 dollars? 30 dollars? 3. If a horse travel 7 miles an hour, in how many hours can he travel 28 miles ? 42 miles ? 63 miles ? 70 miles ? 4. If 5 lemons cost 20 cents, what is the price of each ? SOLUTION. Since 5 lemons cost 20 cents, 1 lemon will cost 1 -fifth of 20 cents, or 4 cents. 5. Do you find how many times 5 lemons are contained in 20 cents, or do you find one of 5 equal parls of 20 cents ? 6.' How do you find one of 5 equal parts of a number ? One of 4 equal parts ? One of 7 equal parts ? 7. How many times 5 miles are 40 miles ? 8. What is 1 of 5 equal parts, or 1 fifth of 40 miles ? 9. If 18 boys sit on 3 benches, how many boys sit on each bench ? 1 third of 18 boys are how many boys ? 10. If 6 tons of coal cost 3G dollars, what is the cost of 1 ton ? 11. If 8 cords of wood cost 40 dollars, what is the cost of 1 cord ? The teacher will observe and illustrate that both forms of divi- sion are deductions from multiplication. Thus, since 8 times 4 are 12, it follows that 4 is in 13 three times, and that 1 third of 12 is 4. IN ARITHMETIC. 115 LESSON CIV. 1. Finding how many times one number is contained in another of the same kind, or finding one of the equal 'parts of a number, is called Division. 2. The number divided is named the Dividend. 3. The number used to divide by is named the Divisor. 4. The result obtained by the division is named the Quotient. 5. The divisor and quotient are the factors of the dividend. 6. When the division is not exact, the part of the dividend re- maining is called the Remainder, and it must always be less than the divisor. 7. Thus, in the example, 72 -s- 8 = 9, 72 is the dividend, 8 is the divisor, and 9 is the quotient. PROOF : 8 x 9 72. EXEECISES FOR SLATE ASTD BOARD. 1. Divide 4680 by 2. Divisor, Dividend. Write the divisor at the left of the dividend, 2 ) 4680 with a curved line between them, and draw a ~~^~ line under the dividend. Begin with the highest order and divide, thus: 2 in 4, 2 times ; Quotient, 23 2 in 6, 3 times ; 2 in 8, 4 times ; 2 in 0, times, writing each quo- tient figure under the figure divided, since it is the same order of units as that figure. PROOF: 2340 x 2 = 4680. In the same manner, divide and prove 2. 396 by 3. 8. 8088 by 8. 4. 5050 by 5. 5. 4884 by 4. 6. 7007 by 7. 7. 8462 by 2. 8. 6248 by 3. 9. 9603 by 3. 10. 8008 by 8. 11. 7777 by 7. 12. 8408 by 4. 18. 5550 by 5. 14. 7007 by 7. 15. 8642 by 2. 16. 9000 by 9. 116 FIRST BOOK LESSON CV. EXERCISES ox THE TABLES. 1. Divide and prove ; thus, 2 in 16, eight times ; 8 times 2 are 16. 18-5-6 27-5-9 28-5-4. 28-7-7 36-7-6 42-5-7 36 9 45 5 48-6 40-^8 66-7-7 54-5-9 In proving division, multiply the divisor by the quotient, not the quotient by the divisor. 2. In ,the same manner, give the quotient and proof of the following : 64 8= '77 11= 819= 84 7 = 72 9= 568= 728= 80-8 = 63 7 = 70 10 = 60 12 = 63 9 = 3. In the following, find the product, and then each factor, thus: Given 8x4. 4 times 8 are 32 ; 8 in 32, four times ; 1 fourth of 32 is 8. 12 x 3 9x6 8 x 11 9x9 4. Copy and write the correct number in place of ( ? ) : Quotients, 7 ? ? 8 9 6 ? Divisors, _? __7 J _? _9 _? _5 Dividends 70 70 63 64 ? 54 60 6x7 9x5 9x8 7x5' 8 x 7 7 x 9 8x6 6x9 10 x 7 7x6 9x4 8 x 10 ARITHMETIC. 117 LESSON CVI. EXERCISES FOR THE SLATE AND BOARD. 1. Divide 447 by 3. Divisor. Dividend. Quotient. 3 ) 447 ( 149 3_ 14 12 27 2_7 In this example, write the divisor at tiie left, and the quotient at the right of the dividend, and begin at the left to divide, thus : 8 is contained in 4, 1 time and a remainder ; write 1 for the first figure of the quotient and multiply the divisor 3 by it, and sub- tract the product 3 from 4 hundreds, the number divided, and the Remainder is 1 hundred, equal to 10 tens, to which add the 4 tens of the dividend, making 14 tens, expressed by bringing down the 4 to the right of the 1 hun- dred. Then 3 is contained in 14, 4 times and a remainder. Write the 4 in the quotient, multiply the divisor by it, and subtract the product 12 from 14, and the remainder is 2 tens, or 20 units, to which add the 7 units of the dividend, making 27. 3 is contained in 27, 9 times. Multiplying and subtracting as before, nothing remains. PROOF : 149 x 3 = 447. The work may be shortened very much by what is termed Short Division, as follows : 3 is contained in 4, 1 time and 1 remainder ; 1 pre- 3 ) 447 fixed or placed before 4, makes 14 ; 3 in 14, 4 times -^ 4 g and 2 remainder ; 2 prefixed to 7 makes 27 ; 3 in 27, 9 times. Hence the quotient is 149. In like manner, divide and prove 2. 8752 by 4. S. 7625 by 5. 4. 7122 by 6. 5. 5343 by 3. 6. 8561 by 7. 7. 9024 by 8. 8. 76344 by 6. 9. 90324 by 4. 10. 83210 by 5. 118 FIRST BOOK LESSON CVII. 1. If 6 men earn 24 dollars, what part of 24 dollars does 1 man earn ? How many dollars ? 2. At 9 cents a quart, how many quarts of milk can be bought for 63 cents ? 3. How do the solutions of these two examples differ ? 4. When the divisor and dividend are of the same name or kind, what do we do ? Ans. Find how many times the dividend contains the divisor. 5. What is the quotient ? Ans. Times. 6. When the divisor and dividend are not of the same name or kind, v/hat do we do ? Ans. Find a certain part of the dividend. 7. What is the quotient? Ans. A part of the dividend 8. How many yards of cloth, at 7 dollars a yard, can be bought for 70 dollars ? For 63 dollars ? For 84 dollars ? EXERCISES FOR THE SLATE AND BOARD. 1. How many times is 6 contained in 1834? Divisor. Dividend. Since 6 is not contained in 1, say, 6 in 6)1834 18, 3 times and no remainder ; 6 in 3, times " ' 6 in 34 ' Quotiento54 ' mes an re ' mamder, which write over the divisor 6, as a part of the quotient. PROOF : 305 x 6 + 4 = 1834. In like manner, 2. Divide 3324 by 2 ; by 4 ; by 6 ; by 7 ; by 8. 8. Divide 9106 by 3 ; by 4 ; by 5 ; by 8. 4. Divide 60530 by 5 ; by 6 ; by 7 ; by 8 ; by 9. 5. Divide 65625 by 3 ; by 4 ; by 7 ; by 8 ; by 9. IK ARITHMETIC. 119 LESSON CVIII. AKOTHER FOKM OF EXPKESSISTG DIVISION". 1. Division is sometimes indicated by placing the divi- sor "under the dividend, and separating them by a line. Thus, if means 12 divided by 4, or 1 fourth of 12, and is the same as 12-7-4, the quotient being 3. 2. Find the quotient, or value, of each of the following: V; V; ; ; ; ; --; ; ; V; ; ; ; This form of indicating division often simplifies two or more operations that are to be performed ; thus, 3. It is required to multiply 12 by 5, and divide the product by 6. Written, ^=10 ; for, 12 x 5 = 60, and 1 sixth of 60, or 60 -f- 6 ^ 10. 4. To 7 times 8 add 4 and divide the sum by 6. 8 v 74-4 Written, - ^r 1 = 10 ; for, 8 x 7 is 56, plus 4 is 60, and 1 sixth of 60, or 60 ~ 6 = 10. Express by signs each of the following : 5. Divide the difference between 17 and 5 by 4. 6. Divide the product of 12 and 4 by 8. 7. What is 1 seventh of the sum of 34 and 8 ? 8. From the sum of 25 and 10 subtract 5, and divide the remainder by 6. 9. From the product of 10 and 7 subtract 6, and divide the remainder by 8. 120 FIRST BOOK LESSON CIX. 1. What is 1 half of 12 ? Of 16 ? Of 18 + G ? 2. What is 1 third of 15 ? Of 18 ? Of 21 + 9 ? 8. What is 1 fourth of 1 6 ? Of 24 ? Of 30 + 6 ? 4. What is 1 fifth of 20 ? Of 30 ? Of 40 - 5 ? 5. What is 1 sixth of 24 ? Of 36 ? Of 38 + 10 ? 6. What is 1 seventh of 35 ? Of 42 ? Of 60 11 r 7. What is 1 eighth of 48 ? Of 32 ? Of 70 6 ? 8. What is 1 ninth of 54 ? Of 63 ? Of 60 + 12 ? 9. What is 1 tenth of 90 ? Of 87 7 ? Of 62 + 8 ?* 10. What is of 4 times 9 ? J of 5 times 8 ? 11. What is \ of 60 4? $ of 12 x 4 ? | of 66 + 6 f m What is t of 6x6? of 12x6? of 75 - 9 ? EXERCISES FOR THE- SLATE AND BOARD. Find the value of each of the follomng expressions in a single number : ('.) 21 9 (*) 43-7 (*) 12 x 5 (*) 9x8 6 6 (5.) 7x9 + 7 9 8x9-8 10 (*) 6 x 9 + 10 12 12 x 7 + 7 (ft) 13 + 34 8 ( 42 + 8 36 14 45 9 (1L) x 4 + 4 12 128 x 3 28 x 16 242 x 21 4 8 7 IK AEITHMETIC. 121 LESSON CX. 1. If 4 caps cost 12 dollars, what will 6 caps cost ? SOLUTION. Since 4 caps cost 12 dollars, 1 cap costs 1 fourth of 12 dollars, or 3 dollars ; and 6 caps will cost 6 times 3 dollars, or 18 dollars. 2. How many dollars are 6 times 1 fourth of 12 dollars ? 3. If 6 oranges cost 18 cents, what will 8 oranges cost? 4. If 7 yards of ribbon cost 56 cents, what will 3 yards cost ? 5 yards ? 8 yards ? 6 yards ? 9 yards ? 5. If a man earn 30 dollars in 6 days, how many dol- lars does he earn in 4 days ? In 3 days ? In 5 days ? 6. How many yards of cloth can I buy for 36 dollars, if 3 yards cost 12 dollars ? 7. If a man walk 27 miles in 9 hours, how far does he walk in 7 hours ? 8. How many are 8 times % of 42 ? 9 times of 35 ? 10 times | of 18 ? 6 times % of 72 ? EXERCISES FOE THE SLATE AKD BOARD. 1. If 6 acres of land are worth 366 dollars, what are 8 acres worth ? 2. If there are 174 lines on 6 pages of this book, how many lines on 1 6 pages ? 3. If 8 cords of wood are worth 32 dollars, what are 27 cords worth ? 4. What will 45 pounds of beef cost, if 8 pounds cost 96 cents ? 5. If 10 tons of coal are worth 70 dollars, what are 125 tons worth ? 122 FIESTBOOK LESSON CXI. EXERCISES FOE THE SLATE AND BOARD. 1. Divide 3180 by 15. 15)3180(212 When the divisor consists of two or more 3 Q figures, the operations of multiplying and subtracting cannot well be carried on in the mind, and so we write the result of each 1 5 separate operation. o r\ Since 15 is not contained in 3, we say 15 in 31, 2 times, and write the 2 at the right U of the dividend as the first figure of the quo- tient. Multiply the divisor by this quotient figure, and write the product 30 under the figures divided. Subtract, and to the remainder 1, annex 8, the next figure of the dividend, making 18 for a new dividend. Dividing, multiplying and subtracting as before, we have a remainder of 3, to which annex the of the dividend, and we have a new dividend of 30 ; which divide as before, and nothing re- mains. PROOF : 212 x 15=3180. 2. Divide 12708 by 28. Divisor. Dividend. Quotient. PROOF. 28)12708(453f| 453 Quotient. 112 28 Divisor. 150 3624 140 906 108 12684 84 24 Remainder. 2 4 Remainder. 12708 Dividend. If the product of the divisor by the quotient figure is greater than the part of the dividend divided, the quotient figure is too great and must be diminished. If the remainder after any subtrac- tion is greater than the divisor, the quotient figure is too small, and must be increased. IN ARITHMETIC. 123 LESSON CXII, 1. How many times is 5 contained in 10 x 6 ? 2. How many times is 3 x 3 contained in G3 ? 8. Divide 48 by 1 sixth of 24. By 1 fourth of 32. 4. A boy gave 5 peaches to each of 6 boys, and had 1 third as many left. How many had he left ? 5. Mary has 5 cents and Susie 7. If Willie has 4 times as many as both, how many cents has he ? 6. What is 1 fourth of 6 times 8 ? 1 third of 5 times 6 ? 7. A man sold 8 barrels of apples at 5 dollars a barrel. How many yards of cloth, at 4 dollars a yard, can he buy with the money ? 8. Three boys have some marbles. James has 10, Henry 9, and John 4. If they divide them equally, how many marbles will each have ? 9. How many days work, at 4 dollars a day, will pay for 3 tons of coal worth 8 dollars a ton ? EXEECISES FOB THE SLATE AND BOARD. 1. Divide 32760 by 12 ; by 13 ; by 14 ; by 15. 2. Divide 970640 by 23 ; by 34 ; by 25 ; by 36. 8. Divide 40320 by 12 ; by 24 ; by 15 ; by 16. 4. Divide 816480 by 24 ; by 27 ; by 35 ; by 36. 5. Divide 445280 by 32 ; by 41 ; by 28 ; by 33. 6. How many pounds of sugar, at 14 cents a pound, will pay for 13 pounds of butter, at 28 cents a pound ? 7. How many pounds of coffee worth 25 cents a pound, can be bought for 15 bushels of oats worth 65 cents a bushel? 124 FIRST BOOK LESSON CXIII. 1. A boy having 12 oranges bought C more, and then sold 7 ; how many had he left ? 2. James sold his sled for 45 cents, which was 9 cents more than it cost : what did it cost ? 3. Belle paid 8 cents a spool for 7 spools of thread, and 9 cents for some buttons : what did she pay for both ?- 4- Which is less, 7 times 8 or 6 times 9 ? Which is greater, 5 times 12 or 8 times 8 ? 5. What will be the cost of 7 coats at 12 dollars each ? 6. If Jane has 11 cents and Grace has G times* as many, how many cents has Grace ? How many have both ? 7. If John has 40 marbles and Dick has 1 fourth as many, how many marbles has Dick ? 8. If 6 quarts of milk cost 54 cents, what will be the cost of 4 quarts ? 7 quarts ? 9 quarts ? 8 quarts ? 9. Willie sold his knife for 36 cents, and received in payment 3 quarts of chestnuts at 8 cents a quart, and the remainder in money ; how much money did he receive? 10. If a man can chop 16 cords of wood in 8 days, how much can he chop in 3 days ? In 5 days ? In 7 days ? In 12 days ? 11. How many barrels of flour worth 8 dollars a barrel, will pay for 4 loads of hay worth 12 dollars a load ? 12. If 7 yards of cloth cost 28 dollars, what will 5 yards cost? 13. If 1 yard of ribbon cost 10 cents, how many yards can he bought for 60 cents? For 70 cents? For 90 cents ? For 120 cents ? IH ARITHMETIC, 125 LESSON CXIV. EXERCISES FOR THE SLATE AND BOARD. 1. The greater of two numbers is 590, and the less 364 ; what is their difference ? 2. The less of two numbers is 128, and their differ- ence is -75 ; what is the greater ? 3. The sum of two numbers is 405, and one of the numbers is 214 ; what is ike other ? 4. How many must be ;uiled to 24 to make 56 ? 5.*H6w many more than 24 men are 210 men ? 6. How many less than i:2 sheep are 27 sheep? 7. How many more than (::> + 36 is 124 ? 8. How many less than i^- -1 + 153 is 246 ? 9. How many must be aaciea to 74 + 127 to make 304 ? 10. A man paid 3146 dollars for a house, which was 380 dollars more than it was worth ; what was it worth ? 11. From 1705, take the sum of 540 and 603. 12. From the difference of 3242 and 646 take 845. 13. Bought a house for 5607 dollars, which was 825 less than it was worth ; what was it worth ? H. A man owed 756 dollars, and paid at one time 206 dollars, and at another time 324 dollars. How much did he still owe ? 15. Mr. Smith has 640 acres of land, and Mr. Jones has 124 acres less ; how many acres has Mr. Jones ? 10. From a bin containing 394 bushels of oats, 248 bushels were taken, and afterwards 86 bushels returned ; how many bushels were then in the bin ? 126 FIRST BOOK LESSON CXV, EXERCISES FO'R THE SLATE A^D BOARD. 1. If a steamer sail 265 miles a day, how far will she sail in 16 days ? In 18 days ? In 24 days ? 2. A man bought 136 barrels of flour at 8 dollars a : barrel, and sold the whole for 1248 dollars. What was his gain ? - 3. What is the value of each of the following articles : ' 25 pounds of sugar at 12 cents a pound ; 16 pounds of tea at 75 cents a pound ; and 38 pounds of ham at 14 cents a pound ? 4 What is the value of the whole ? How much more is the tea worth than the sugar ? 5. How many tons of hay worth 16 a ton, will pay for 24: cows worth 32 dollars each ? 6. If a clerk earn a salary of 950 dollars a year, and his expenses are 525 dollars a year, how much can he save in 6 years ? 7. A farmer sold 96 bushels of potatoes, at 52 cents a a bushel. How many pounds of coffee, at 26 cents a pound, will pay for the potatoes ? 8. If 24 yards of cloth cost 144 dollars, what will 56 yards cost at the same rate ? 9. A farmer sold 15 tons of hay,at 18 dollars a ton and 36 cords of wood, at 4 dollars a cord : what was the value of both ? 10. He divided the whole amount of money equally among 6 creditors : how much did each receive ? AEITHMETIC. LESSON CXVI. L Multiply 3417 by 9; 2. Multiply 10782 by 21; 8. Multiply 56043 by 32; 4. Multiply 28340 by 41 ; 5. Multiply 730081 by 44 ; 6. Multiply 186304 by 65 ; 7. Multiply 304071 by 46 ; 8. Multiply 415036 by 75 ; 9. Multiply 630400 by 46 ; 10. Multiply 297182 by 77 ; 11. Multiply 710345 by 68 ; by 12; by 14; by 16. by 18; by 26; by 19. by 25; by 36; by 27. by 29; by 46; by 37. by 53; by 38 ; by 54. by 67 ; by 55; by 49. by 72; by 84; by 66. by 63; by 54; by 82. by 57; by 68; by 74. by 94; by 95; by 89. by 77; by 81; by 93. LESSON CXVI I. Find the value of 1. 598467- 2. 541604- S. 639514- 4. 954632- -6. -8. -4. -7. 5. 985478- 6. 517401- 7.^378487- ."371561- -7. -8. -9. -8. 9. 6897583- 10. 4996504- 11. 1918575- 12. 5982039- -5. -8. -6. -9. How many times 18. Is 13 contained in 273 ? In 4550 ? In 36721 ? 14. Is 17 contained in 465? In 50571 ? In 12991 23? 15. Is 18 contained in 10404 ? In 7831 ? In 11052 ? . Divide 35280 by 25 ; by 32 ; by 27 ; by 24. 17. Divide 122764 by 36 ; by 24; by 28 ; by 26. 18. Divide 7462450 by 42 ; by 36 ; by 52 ; by 48. 123 FIEST BOOK LESSON CXV.III. 1. If a peach is divided into tivo equal parts, one of these parts is called one-half of a peach. 2. How many halves make one, or a whole thing ? 3. If a pear is divided into three equal parts, one of these parts is called one-third of a pear. 4. How many thirds make one, or a whole thing ? 5. How many thirds in one pear ? 6. If an apple is divided into four equal parts, one of these parts is called one-fourth of an apple. 7. How many fourths make one, or a whole thing ? 8. If an orange is divided into five equal parts, one of these parts is called one-fifth of an orange. 9. How many fifths make one, or a whole thing? 10. One-half is written . 11. One-third is written ; two-thirds, f . One-fourth is written ; two-fourths, f ; three-fourths, |. 12. One-fifth is written | ; two-fifths, f ; three-fifths, f ; four-fifths, | ; five-fifths, |. IN ARITHMETIC. 129 LESSON CXIX. 1. One whole thing is equal to how many "halves ? Thirds f Fourths ? Fifths f Sixths t Sevenths ? etc. unit. Two-halves. Five-fifths. Six-sixths. Seven-sevenths. Eight-eighths. .Nine-ninths. 2. When anything is divided into six equal parts, what is one part called? Two parts? Three parts? Four parts ? Five parts ? 3. What is one of the seven equal parts of anything called ? Three of seven equal parts ? Five of seven equal parts ? Six of seven equal parts ? 4. What is meant by one-eighth of a number or whole thing? Ans. One of the eight equal parts of it. 5. What is meant by one-ninth of a number or whole thing ? Two-ninths ? Five-ninths 9 6. What is meant by one-tenth of anything ? Six-tenths 9 7. How many sixths make a whole thing ? How many sevenths? Eighths? Ninths? Tenths? 13C FIRST BOOK LESSON CXX. 1. A Unit is one, or a single thing. 2. Numbers that express or represent equal parts of a unit are called Fractions; as, one-half () ; two- thirds (|), etc. 3. A fraction is expressed by two terms or numbers, and when written, one is placed below and the other above a short line ; as, , -f, -f, etc. 4- The number below the line is the Denominator. Thus, in the fraction f , 4 is the denominator, and shows that the unit is divided iufour equal parts, named fourths. 5. The number above the line is the Numerator, Thus, in f , 3 is the numerator, and shows that 3 of- the 4 equal parts are taken, or expressed, by the fraction. 6. The numerator and denominator of a fraction are called the Terms. Thus, 3 and 4 are the terms of the fraction f . 7. Eead the following fractions, and name the denom- inator and the numerator in each : i i i 4> i t, t, i, f, A, 1- . Write the following in figures : Five-ninths. Four-ninths. One-twelfth. Three-sevenths. Seven-ninths. Five-elevenths. Two-fifths. Seven-tenths. Nine-fourteenths. Four-sevenths. Five-sevenths. Eleven -twelfths. Five-sixths. Eight-ninths. Eight-fifteenths. Three-eighths. Nine-tenths. Seven-twentieths. IK ARITHMETIC. Ul LESSON CXXI. 1. How do you find one-half of any number? Am. Divide the number by 2. 2.. How do you find 1 third of any number ? ? | ? i? I? i? i.? j_? etc. 5. What is I- of 6? Of 8? Of 12? Of 18? Of 20? 4. What is | of 9? Of 15 ? Of 21 ? Of 24? Of 27 ? 5. What is i of 12 ? Of 16 ? Of 28? Of 32? Of 40? 6. What is ^ of 20 ? Of 35 ? Of 45 ? Of 50 ? Of 60 ? 7. What is | of 18? \ of 21 ? -|of30? of 36 ? 8. What is | of 28 ? of 54 ? of 48 ? } of 63 ? 0. What is of 40 ? ^ of 45 ? \ of 49 ? of 72 ? i of 56 ? 4 7(9. What do you understand by f of any number or tiling ? Ans. Two of the five equal parts into which the number or thing is divided. 11. What do you understand by f of any number or thing? By |? By i? f? By*? f? f? By ? I? I? 1? 12. What is meant by % of any number or thing ? By I? t? I? 1? By^? A? A? A? ^. What is | of 12? SOLUTION. Ofte-third of 12 is 4 ; 2 thirds of 12 are 2 times 4, or 8. Hence f of 12 are 8. ^. What is f of 16 ? f of 15 ? f of 20 ? f of 25 ? 15. What is f of 18? | of 30? f of 21 ? f of 28 ? | of 32 ? f of 36 ? 16. What is f of 27 ? f of 18 ? f of 36 ? ^ f 4 <> ? ft- of 30 ? T \ of 44 ? 132 FIRST BOOK LESSON CXXII. 1. How do you find how many halves there are in any whole number? Am. Multiply the whole number by 2. 2. How do you find how many thirds there are in any number? How many fourths? Fifths? Sixths? Sevenths? Eighths? Ninths? Tenths? etc. 3. How many halves are there in 3 peaches ? SOLUTION. Since in 1 peach there are 2 halves, in 3 peache* there are 3 times 2 halves, or 6 halves. Hence, in 3 peaches are 6 halves. * 4. How many halves are there in 4 ? In 8 ? In 9 ? 5. How many thirds are there in 5 ? In 7 ? In 8 ? 6. How many fourths are there in 7? fifths in 8 ? sixths in 6 ? sevenths in 9 ? eighths in 7 ? ninths in 5 ? tenths in 8 ? 7. How many halves in one and a half ? SOLUTION. In 1 are 2 halves, and 1 half added makes 3 halves. Hence, in 1 and 1 half there are 3 halves. 8. How many halves in 3 and 1 ffalf ? In 5 and 1 half ? 9. How many thirds in 2 and 2 thirds ? SOLUTION. Since in 1 there are 3 thirds, in 2 there are 2 times 3 thirds, or 6 thirds, and 2 thirds added make 8 thirds. Hence, in 2 and 2 thirds there are 8 thirds. 10. How many thirds in 4 and 1 third ? fourths in 3 and 1 fourth ? 11. How m%nj fift-hs in 2 and 3 fifths? sixths in 5 and 4 sixths ? sevenths in 3 and 2 sevenths ? J?#. How m&nj fourths in 6 and 3 fourths ? In 7 and 1 fourth ? In 8 and 3 fourths ? IK ARITHMETIC. 133 LESSON CXXIII. 1. When the numerator and denominator of a frac- tion are equal, the value of the fraction is equal to 1 ; as J=l; t=l; f=l, etc. 2. When the numerator is greater than the denomina- tor, the value is greater than 1; as -:=1-J; f 2, etc. S. A whole number and a fraction written together are called a Mixed Number ; as of, read 5 and 3 fourths*. 4* Read the following mixed members : 5. How m&aj fifths are 4f ? SOLUTION, Since in 1 there are 5 fifths, in 4 there are 4 times 5 fifths, or 20 fifths, and 3 fifths added make 23 fifths. Hence 4| are ^. 6?. How many fourths are 9 ? Are 7| ? Are 8J ? 7. How many Mr^s are 12f ? .M/Ms are 6-J- ? Sixths are 7| ? Fourths are lOf ? & Change 4f to eighths; 9-f to sevenths; 8 T V to tewMs ; 5| to ninths ; 4^- to tivelfths. 9. How many owes are ^ ? SOLUTION. Since 4 fourths equal 1, 12 fourths are as many Ts as 4 fourths are contained times in 12 fourths, or 3 times. Hence if are 3 ones or 3. Or \ 2 -=12-i-4=3. 10. Find the value of * ; of - 1 / ; of ^ 5 of 5 of ^. ^^. Change to a whole or a mixed number -?/- ; - 2 / ; y V; ; --; ^. What is the value of ^ ? 4^? ff? -%- ? f o ? ^. Find the value of ^ ; ff ; ^ ; -ff 134 FIEST BOOK LESSON CXXIV. 1. How many are % fifths and 3 fifths 9 Ans. 5 fifths, or f1. 2. Can you add 3 apples and 5 figs 9 Why not ? 3. Can you add 3 fourths and 5 sixths 9 Why not ? ^. Only fractions having the same denominator can be added. 5. How many are -J + f ? -4/w. -^=11. 0. What is the sum of f and ? of T V and -& ? Of | and*? Of^and^? 7. Add -^ and ^. T 7 T and T 8 T . f and -J. f and . 8. How many are 7 ninths less 2 ninths 9 Ans. 6 ninths, or f. 9. Can you take 6 books from 9 s?#to ? Why not ? 10. Can you subtract 6 sevenths from 9 twelfths 9 Why not ? 71. Only fractions having the same denominator can be subtracted. 12. How many are f-f ? ^-yV ? T V~ T V ? - A ? 1& Subtract f from |. T S T from &. T 4 T from |f . 1^. How many are U. Howmany 1's are 10. From | + | + 1 take |. 17. From 1 V + tV + ii subtract ^. 18. John spent f of his money and lost f of it. What part of his money had he left ? 19. James paid f of a dollar for a book, and f of a dol- lar for a slate. How much did he pay for both ? How much more for the book than for the slate ? IX ARITHMETIC. 135 LESSON CXXV. 1. If a cake is cut into 12 equal pieces, what part of the whole cake is each piece ? Ans. One-twelfth. 2. How many twelfths make a whole thing ? 3. How do we find 1 twelfth of any number ? 4. If a cake is divided into 3 equal parts, each part is called 1 third. 5. If each third of the cake is di- vided into 4 equal parts, into how many parts is the whole cake divided ? 6. How many twelfths in 1 third? In 2 thirds? 3 thirds ? 7. If a cake is cut into 4 equal parts, each part is called 1 fourth. 8. If each fourth of the cake is di- vided into 3 equal parts, into how many parts is the whole cake divided ? 9. How many tivelfths in 1 fourth ? In 2 fourths ? In 3 fourths ? In 4 fourths ? 10. If Edie has of a cake and Asa J of it, what part have both? SOLUTION. Since \ is equal to ^, and J is equal to T \, they have the sum of T \ and -f , or T 7 . .7^. How much more has Edie than Asa? -J- J= ? 12. How many twelfths in $ ? In? Inf? InJ? Inf? 13. Which is greater -J- or ? or | ? f or } ? ^. If Mary has , and Jennie of a melon, what part of the whole melon have both ? 136 FIBST BOOK LESSON CXXVI. 1, Draw lines upon the slate or board of equal length, and divide them into 12 equal parts. 2. By a heavier mark these lines may be divided into halves, thirds, fourths, and sixths, respectively. 8. Hence, it is plain that halves, thirds, fourths, sixths, and twelfths, may be added or subtracted when changed to parts of equal size or magnitude. What is the sum of 4. $ and | ? 6. i- and ? 6. 1 and 4- ? 7. | and f ? 8. and -^ ? 9. f and f ? What is the value of 10. 11. 12. and ? -& and | ? f and? ^. 4 ./. 4 .76. f | ? A- When fractions have different denominators, they must be changed to fractions of like value, having the same denominators, before they can be added or subtracted. IN ARITHMETIC. 13? LESSON CXXVII. 1. 4 is of what number ? SOLUTION. Since 4 is 1 half of a number, 2 halves or the nun^ ber are 2 times 4, or 8. Hence 4 is J of 2 times 4, or 8. 5 is of what number ? of what number ? & 7 is of what number ? of what number ? 4- 6 is of what number ? -^ of what number ? 5. 4 is of what number ? of what number ? #. 8 is of what number ? of what number ? 7. 3 is ^ of what number ? ^ of what number ? 8. If of a ton of coal cost 2 dollars, what will 1 ton cost ? 9. George gave a beggar 6 cents, which was - of all the money he had ; how much money had he ? 10. If of a pound of coffee cost 8 cents, what will of a pound cost ? SOLUTION. Since 1 fourth of a pound costs 8 cents, 3 fourths of a pound will cost 3 times 8 cents, or 24 cents. 11. If -J- of a bushel of pears cost 9 cents, what will f of a bushel cost ? 12. In -3*5- of a dollar are 10 cents ; how many cents in T V of a dollar ? 13. If a man walk 6 miles in of a day, how many miles can he walk in 1 day ? 14. A pencil cost 8 cents, which was % the cost of a book ; what was the cost of the book ? 15. If -J- of a melon cost 7 cents, what will f of it cost ? ./#. If -J- of a barrel of flour is worth 2 dollars, what is | of a barrel worth ? What is a whole barrel worth ? 138 FIRST BOOK LESSON CXXVIII. 1. If a barrel of sugar is worth 24 dollars, what is \ of it worth ? 2 thirds ? 1 fourth ? 3 fourths ? . If a hoy earns 10 dollars in a week, and spends 2 fifths of it, what part is left ? How many dollars ? 3. When coal is 7 dollars a ton, what part of a ton will 1 dollar buy ? 2 dollars ? 3 dollars ? 5 dollars ? 4. At | of a dollar a yard, what will 4 yards of cloth cost ? SOLUTION. Since 1 yard costs f of a dollar, 4 yards will cost 4 times | or f of a dollar, equal to 2| dollars. 5. At f of a dollar a pound, what will 6 pounds of butter cost ? 8 pounds ? 9 pounds ? 6. How many are 3 times f ? 2 times $ ? 4 times -J- ? 7. How many are 5 times f ? 6 times f ? 7 times -J- ? . "What is the difference between 3 fourths of 24 and 4 fifths of 30 ? 2 thirds of 27 and 1 sixth of 60 ? 9. If a ton of hay cost 21 dollars, what will 3 sevenths of a ton cost ? 2 thirds of a ton ? 10. A boy having 20 marbles, gave } of them to one companion, and \ to another ; how many had he left ? 11. Jane paid 25 cents for a book, and f as much for a slate ; what did she pay for the slate ? 12. Fanny is 14 years old, and her sister is f as old : how old is her sister ? 13. What will be the cost of 6 boxes of figs, at | of a dollar a box ? 14. If 1 pineapple cost f of a dollar, what will 4 pine- apples cost? What will G cost ? 8? 9? 10? IK ARITHMETIC. 139 LESSON CXXIX. 1. How many oranges in 6 thirds of an orange ? In 7 thirds ? In 9 thirds ? In 10 thirds ? 2. How many yards in -J of a yard ? In $ of a yard ? In -LA? In-^? In-^? 3. If 3 books cost 12 fif fchs of a dollar, what will 1 book cost? SOLUTION. Since 3 books cost 12 fifths of a dollar, 1 book will cost 1 third of 12 fifths, or 4 fifths of a dollar. 4. If 3 pounds of coffee cost -f$ of a dollar, what will 1 pound cost ? What will 5 pounds cost ? 6. If 4 yards of ribbon cost f of a dollar, what will 3 yards cost ? 5 yards ? 7 yards ? 10 yards ? 6. If a man walks f of 6 miles in 1 hour, how far can he walk in 3 hours ? How far in -J of an hour ? 7. If 5 bushels of oats are worth -^ of a dollar, what is 1 bushel worth ? 3 bushels ? G bushels ? 10 bushels ? 8. If 3 fourths of a bushel of cranberries are worth 3 dollars, what is -J- of 6 bushels worth ? 9. At 4 dollars a yard, what will 2% yards of cloth cost ? SOLUTION. Since 1 yard costs 4 dollars, 2 J yards will cost 2| times 4 dollars ; 2 times 4 dollars are 8 dollars, and | of 4 dollars is 2 dollars, which added to 8 dollars make 10 dollars. Hence 2J- yards will cost 10 dollars. 10. At 10 dollars a barrel, what will barrel of flour cost ? What will $ of a barrel cost ? 2| barrels ? 11. At f- of 16 cents a pound, what will \ of a pound of sugar cost ? | of a pound ? 1 pounds ? 2 pounds ? 140 J?iKST BOOK LESSON CXXX. 1. In measuring a quantity, some definite part of it is taken as a Unit of Measure ; as, a yard, a gallon, etc., by which all quantities of that kind are measured. Hence, the length of a piece of cloth is ascertained by applying a yard measure ; the capacity of a cask by the use of a gallon measure ; the weight of a body by the pound weight, etc. 2. Measures may be classified into six kinds : Money or Value, Extension, Capacity, Weight, Time, Angles or Arcs. MONEY. 3. Money is the measure of the value of things. 4. The legal money of this country is called United States Money. 5. The Gold Dollar is the unit of United States Money. 10 Mills (m.) = 10 Cents 10 Dimes 100 cents = 1 dollar ; 25 cents = J dollar ; TABLE. 1 Cent . . ct. 1 Dime . . d. 1 Dollar . $. 75 cents = f dollar ; 20 cents = J- dollar ; / 1000 m. ;1 = ] 100 ct. ( 10 d. 50 cents = | dollar ; 12| cents = J dollar. The money of the Dominion of Canada is the same as that of the United States. IK ARITHMETIC. 141 LESSON CXXXI. 1. How many milffe in 1 cent ? 3 cents ? 5 cents ? 2. How many cents are 30 mills ? 50 mills ? 60 mills ? 3. How many dimes are 20 cents ? 40 cents ? 4. How many cents are 3 dimes ? 4 dimes ? 6 dimes ? 5. How many dimes in 1 dollar ? How many cents ? & In half a dollar, how many dimes ? Cents ? 7. How many cents are 1 dollar ? 2 dollars ? 6*. How many cents in 1 dollars? 2 dollars? P. How many dollars in 200 cents ? In 150? In 450? 10. The sign $ signifies dollar or dollars, and is placed before the number. Thus $3 means three dollars ; $14 means 14 dollars, etc. 17. When dollars and 0%., 30 6. June, June, 30 12. December, Dec., 31 A sheet of paper folded in A sheet of paper folded in A sheet of paper folded in A sheet of paper folded in A sheet of paper folded in A sheet of paper folded in A sheet of paper folded in A sheet of paper folded in BOOKS. 2 leaves is called a folio. 4 leaves is called a quarto, or 4to. 8 leaves is called an octavo, or 8w. 12 leaves is called a duodecimo, or 12mo. 16 leaves is called a 16 mo. 18 leaves is called an 18mo. 24 leaves is called a Ql^mo. 32 leaves is called a 32mo. MONEY MENTIONED IN THE BIBLE. A Talent (gold) A Talent (silver) A Mancli or Mina A Pound (Mina) A Shekel (gold) A Shekel (silver) A Golden Daric or Dram A Piece of Silver (Stater) Tribute Money (Didrachm) A Bekah A Piece of Silver (Drachm) A Penny (Denarius) . A Farthing ( Assarium) A Mite value in U. S. Money, $26593.809 1602.024 11 27.699 15.715 8.861 0.547 5.312 0.628 " " " 0.314 0.263 0.157 0.152 " " " 0.008 " " " 0.002 IN ARITHMETIC. 165 LESSON CLI. 1. How many days in 1 week ? In 3 wk. ? In 4 wk. ? 2. How many hours in of a day ? In J ? In f ? 8. How many minutes in | of an hour ? In J? In ? 4. How many months in % of a year ? In ? In f ? 5. How many days has a leap year ? 6. How often does leap year occur ? 7. Name the months of the year. 8. Name the months that have 30 da. each. 31 da. 9. What month is it now ? How many days has it ? 10. How many days from April 20th to May 10th. 11. How many days from Aug. 1st to Sept. 5th. 12. How many buttons in 1 gross ? In -| of a gross ? 13. How many pens in J of a gross ? How many dozen ? H. Find the cost of -J of a gross of eggs, at 20 cents a dozen. 15. How many make 1 pair ? 2} pairs ? 5 pairs ? 16. How many in 1 set ? In J a set ? In 2| set ? 17. How many quires of paper in ^ of a ream ? 18. How many sheets of paper in 2 quires ? In -J- of a quire ? 19. Find the cost of of a ream of paper, at 20 cents a quire. #0. How many score in 40 ? In 60 ? In 100 ? #./. How many in of a score ? In 1 score ? 2 score ? 22. Which is greater, of a ream or 100 sheets of paper ? 2. What is the difference between J- of a dozen, and 6 dozen eggs ? &. Which is greater, 6 sets of chairs, or 2 dozen chairs? 166 FIRST BOOK LESSON CLII. 1. What is the difference in a foot long and a foot square ? In a square foot and a cubic foot ? 2. How many inches long is a block in the form of a cubic foot ? How many wide ? 8. What is measured by the fathom ? By the hand ? 4. How many feet deep is a river that measures 5 fathoms ? 5. How many feet high is a horse that measures 15 hands ? 6. How many square feet in a blackboard 27 ft. long and 3 ft. wide ? How many square yards ? 7. Which is greater, 10 square inches or a 10 inch square ? 8. Which is greater, 6 cubic inches or a 6 inch cube ? 9. How many squares, each equal to a square foot, are equal to a surface 12 ft. long and 8 ft. wide ? 10. How many cubes, each equal to a cubic foot, are equal to a block 5 ft. long, 4 ft. wide, and 3 ft. thick ? 1 L How is stone measured ? How is wood measured ? 12. Give the dimensions of a cord-foot. IS. How many pints of water will fill a vessel that holds 1 gallons ? 14. How many times can a peck measure be filled from 2i bushels ? 15. Which is heavier, a bushel of wheat or a bushel of corn ? A bushel of barley or a bushel of oats ? 16. Which is heavier, a barrel of flour or a barrel of pork ? A keg of nails or a cental of grain ? ARITHMETIC. 167 LESSON CLI. ROMAN NOTATION. 1. This method employs seven capital letters to repre- sent numbers. LETTEKS. I, V, X, L, C, D, M. VALUES. 1, 5, 10, 50, 100, 500, 1000. #. Eepeating a letter repeats its value. Thus, XX represents 20 ; CCC, 300 ; DD, 1000. 3. When a letter is placed after one of greater value, its value is to be added to that of the greater. Thus, VI represents 6 ; XV, 15 ; LXX, 70 ; DC, 600. 4. When a letter is placed before one of greater value, its value is taken from that of the greater. Thus, IV represents 4 ; IX, 9 ; XL, 40 TABLE OF ROMAN NOTATION. I = 1 XIV = 14 LX 60 II = 2 XV = 15 LXX = 70 III = 3 XVI = 16 LXXX = 80 IV = 4 XVII = 17 XC = 90 V = 5 XVIII = 18 C = 100 VI = 6 XIX = 19 CXIX = 119 VII = 7 XX = 20 CC = 200 VIII = 8 XXI = 21 ccx = 210 IX = 9 XXV = 25 D = 500 X = 10 XXX = 30 DCV = 605 XI = 11 XXXIV = 34 M = 1000 XII = 12 XL = 40 MDL = 1550 XIII = 13 L = 50 MDCLXVI = 1666 MDCCCLXXV=1875, one thousand eight hundred and seventy-five. 168 FIRST BOOK. MULTIPLICATION TABLE. 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12. 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 i 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 1O 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 i 1 13 14 15 ie!i7 18|19 2O 2l'22 23J24 2 20 23 30 32; 34 36 38 40| 42 44 46 69 92 115 48 _i? ! 96 120 3 4 39 42 45 60 48 51 54 57 eo ~80 100 63 66 52 56 64 80 68 72 76 84 88 5 65 70 75 85 90 95 105 110 6 78 84 90 96 112 102 119 108 114 i-?o 126 132 138 144 7 91 98 105 126 133 140 147 154 161 168 8 104 112 120 128*136 144 162 152 160 168 176 184 192 216! 9 117 126 135 144 153 171180189 198 207 10 130 140 150 160 170 187 180 198 216 190 209 228 200 220 240 210 220 230 240 264 ill 143 154 165 176 231 242 253 12 156 16g 180 192 204 252 264 276 288 UNIVERSITY OF CALIFORNIA LIBRARY