New rudiments of arithmetic: combining mental and slate exercises for intermediate departments
Thomson, James B. (James Bates), 1808-1883
Fc<A<..4 T"\ I V.74 . S 2n TIjOpON'S NEW GRpED SERIES. IN THREE BOOKS. I, NEW MENTAL ARITHMETIC. (For Primary Departments.) II. NEW RUDIMENTS OF ARITH- METIC. (For Intermediate Departments.) Ill, NEW PRACTICAL ARITHMETIC. (For Grammar Departments) THOJJSON'S SUPPLEl^ENT^RY COURSE. Fob Higheb iNsrmrflbKS. {In preparation,) Entered according to Act of Congress, in the year 1872, by JAMES B. THOMSON, In the Ofl3ce of the Librarian of Congrees, at Washington, D. C. Electrotyped by Smith & McDougal, 83 Beekman Street, N. Y. PREFACE. The "New Graded Series/' of which this is the second book, is divided into three parts. The object of this arrangement is convenience and economy. While there may be objections to an " indeterminate series of school-books/* it must be admitted that exer- cises in reading, arithmetic, etc., which are adapted to the capacity of beginners, are totally unfit for advanced classes. In view of this fact, it requires no arguments to show that a "limited series," adapted to the different capacities of learners, is a dictate of common sense. Each book in this Series is complete in itself. The defini- tions and principles, so far as each extends, are expressed in the same languagcy but the cxamjples are all different The present work consists of a course of Mental and Written Exercises combiiied. It is designed : ist. To develop the elems»tary prificiples of the science by oral examples. 2d. To familiarize the pupil with the application of these principles to the solution of problems requiring the use of the slate. 3d. To lead him to generalize the principles thus developed, and to put the steps of particular solutions into a concise statement, or General Rule, 4th. To secure accuracy and rapidity in the combina- tion of numbers. Finally, the work is specially adapted to intermediate classes, who are beginning to "cipher." The New Elements, it is hoped, may facilitate the progress of pupils, and merit the approval of teachers. James B. Thomson. New York, July, 1872. SUGGESTIONS 1. Pablicable attention should be paid to the assignment of Lessons. They should be neither too long, nor too short; but adapted to the capacity of the class, and the time they have for preparation. 2. Thoroughness should be insisted on, at every step. The acceptance of an imperfect lesson, whether from sympathy, or inattention, is a positive injury to the pupil. 3. The most effective auxiliaries of thoroughness are frequent reviews and Tabular drills. "Practice makes perfect." 4. A perfect recitation implies both promptitude and correctness. In reciting problems, the analysis should be logical, and the language correct. 5. Pupils should be encouraged to study out different solutions of the same problem, and to exercise their judgment in selecting the most simple, logical, and concise. 6. Care should also be taken to prevent the habit of adding by counting the fingers. Counting is not addition. Pupils should be taught to add numbers as a whole, and be able to name the sum of any two given digits, instantly. 7. The definitions should be carefully explained, and thoroughly committed to memory. Each principle and rule should be dwelt upon until the pupil comprehends it, and is able to give a correct account of it, in his own language, or that of the author. 8. Cultivate the Habit of self- Reliance in the solution of problems. It is better for the pupil to solve one example, independent of the answer and all extraneous aid, than a dozen by the help of a teacher, or a key. 9. Special pains should also be taken to cultivate the perceptive faculties, and correct the erroneous ideas of learners as to distance, surface, weight, etc. 10. In developing the idea of Fractions, and the Units of Weights and Measures, let the pupil divide some object into halves, thirds, etc., and, if possible, let him see and handle the actual standards of length, surface, capacity, and weight. These simple acts will give him a more exact idea of Fractions, and of Weights and Measures, than a score of pictures,  less than 10, - - 24 What numbers can be added together, - - 25 When the sum of a column is  larger than the one above it, - - 41 General Rule for Subtraction, - - 42 Drill  or more figures, - - 53 General Rule for Multiplication, - -  ----- - 79 Coasting,  . - - . loi Least Common Denominator^  r ? - - 152 Applications of U.  - - 180 Denominate Fractions, - • - - - 182 Compound Rules, - -  is the science of numbers. 3« What is a single thing called ? A Unit, or One. 3. If another is put with it^what? Two. 4. If another, and another, etc., what ? Three, four, five, six, etc. 5« What is number ? Number is a unit, or a collection of units. 6. When a nmnber is not applied to any object, what is it called? An Abstract Number. 7. When it is applied to some object, what ? A Concrete Number. Give examples. §. When nnmbers express nnits of the same Mnd, as, 3 . apples and 4 apples, 5 and 7, etc., what are thej called ? Like Numbers. 9. When they express units of different kinds, as, 4 books ** and 6 pencils, what ? Unlike Numbers. NOTATION. iO. What is Notation ? Notation is the art of expressing numbers by figures, letters, and other numeral characters. 11* What are the two principal methods in use? The Arabic and the Roman, ARABIC NOTATION. 1. What is the Arabic Notation, and why so called 1 The Arabic Notation is the method of express- ing numbers ^j figures; and is so called because it was introduce into Europe from Arabia. 3, How manj figures does it employ, and what called ? leti, called — 1, 2, 3, 4, 5, 6, 7, 8, 9, o. One, two, three, four, five, six, seven, eight, nine, naught. 3. What are the first nine called, and why ? The first nine are called significant figures, because each of them always expressera number. They are also called digits, from digitus, a finger, because the ancients used to" reckon upon their fingers. What is the last cabled, and why? The last is called naught, because when standing alone it has no value, and when connected with significant figures, it denotes the absence of the orders in whose place it stands. It is also called zero, or cipher. Note. — The pupil should learn to distinguish and write the Arabic figures with readiness before proceeding further. How is each of the first nine numbers expressed? By a single figure. What are these numbers called? Units of the first order, or simply units. NOTATION. What is the greatest number expressed by one figure? Nine. How is ten expressed? Ten is expressed by writing i in the second places with a cipher on the right; as, lo. What is the i called, standing in the second place? A unit of the second order. Explain and write each of the numbers from ten to twenty. Eleven is composed of one ten and one unity and is expressed by writing i in the second place to denote the ten, and i in the first or right hand place to denote the unit; as, 1 1. Twelve is composed of one ten and two units, and is expressed by writing i in the second place, and 2 in the first; as, 12. Thirteen is composed of one ten and three units, and is expressed by writing i in the second place, and 3 in the first; as, 13, etc. Twenty is two tens, and is expressed by placing 2 in the second place, and o in the first; as, 20. Explain in like manner, and write each of the numbers from ten to twenty. numbers from twenty to thirty. 3. Prom thirty to forty. Prom forty to fifty. 4. Prom fifty to sixty. From sixty to seventy ; and so on to one hundred. 10. What is the greatest number that can be expressed by tiDO figures? Ninety-nine. !!• How is a hundred expressed? A hundred is expressed by writing i in the third place, with two ciphers on the right ; as, 100. 12. What is the i called, standing iu the third place? A unit of the third order. 10 NOTATIOJS^. 5. Explain and write each of the numbers from one hundred to one hundred and ten. One hundred and one equals one hundred, no tens, and one unit, and is expressed by writing i in the third place, o in the seqond, and i in the first ; as, loi. One hundred and two is expressed by 102 ; one hun dred and three by 103, etc. 6. Explain and write each number from one hundred and ten to one hundred and twenty. One hundred and ten is composed of one hundred, one ten, and no units, and is expressed by writing i in the third place, i in the second, and o in W\q first j as, 1 10. One hundred and eleven by 1 1 1 ; one hundred and twelve by 112, etc. 7. Explain in like manner, and write the numbers from one hundred and twenty to one hundred and ' thirty. 8. Write the numbers from one hundred and thirty to one hundred and fiftjTT From one hundred and fifty to two hundred. 9. Write two hundred. ' Three hundred. Four hun- dred. Five hundred. Write the following numbers in figures : 10. One hundred and twenty-three. 1 1. Two hundred and thirty-seven. 12. Three hundred and forty-five. 13. Four hundred and ten. 14. Six hundred and seven. 15. Five hundred and sixty-three. 16. Six hundred and five. 17. Seven hundred and thirty. 18. Six hundred and seventy-five. 19. Eight hundred and forty-three. 20. Nine hundred and ninety-nine. NOTATION. 11 13. What is the largest number that can be expressed by three figures ? Nine hundred and ninety-nine. Note. — ^The preceding exercises should be repeated and sup- plemented \>j dictation, until the class become perfectly familiar \?ith writing numbers less than a thousand. 14* How are numbers larger than 999 expressed ? Bj forming other orders, called thousands, tens of thou- sands, hundreds of thousands, millions, etc., and assigning to each succeeding order ten timss the value of the pre- ceding. 15. What is the general law by which the orders of units increase? Tfiey increase from right to left by the scale of ten ; that is, Ten simple units make one ten; Ten tens make one hundred; Ten hundreds make one thousand; and, universally, ten of any lower order make one of the next higher. 16. What places do the different orders occupy ? Simple units occu})y the right hand place ; Tens, the second place ; Hundreds, the third place ; Tfiousands, the fourth place ; Tens of thousands, the fifth place ; Hundreds of thousands, the sixth place ; Millions, the seventh place, etc. ; the order of units corresponding with the place which the figure occupies. IT. What is the effect of moving a figure from right to left, or from left to right. Its value is increased tenfold for every place it is moved from right to left ; and is diminished tenfold for evory place it is moved from left to right. What are the different values of a figure called? The simple and local values. What is the simple value of a figure? The simple value of a figure is the number of units it expresses when it stands alone. The local value of a figure is the number of units it expresses when connected with other figures, and is determined by the place it occupies, counting from the right. What is the rule for expressing numbers by figures? Begin at the left hand, and write the figures of the given orders in their successive places toward the right. If any intermediate orders are omitted, supply their places with ciphers. Write the following numbers in figures: One thousand, three hundred, and sixty. Five thousand, seven hundred, and thirty-five. Seven thousand, three hundred, and sixty. Twenty-six thousand and seventy-five. Thirty-seven thousand, one hundred, and six. Ninety-five thousand and seventeen. Ninety-five thousand and seventeen. Three hundred and forty-eight thousand and two hundred. Four hundred and forty-five thousand, six hundred, and thirty. Seven hundred and fifty-five thousand, two hundred, and three. Eight hundred and fifty-five thousand, two hundred, and three. Eight hundred and fifty-five thousand, two hundred, and three. Nine hundred and thirty-eight thousand and sixty-eight. NOTATION. 13 ROMAN NQTATION. 22. What is the Roman Notation; and why so called? The Mofnan Notation is the method of expressing D ambers by letters; and is so called because it was employed by the Romans. 23. How many, and what letters are used ? Seven capitals, viz,: I, V, X, L, C, D. and M. 2<i« What does each of these letters express ? The letter I expresses one; Y,five; X, ten; L, fifty ; C, one hundred; D, five hundred; and M, one thou^ sand. TABLE. I denotes one. XXTV denotes twenty-four. II « two. XXV « twenty-five. m €t three. XXVI ti twenty-six. IV it four. • XXVII tt twenty-seven. V n five. XXVIU It twenty-eight. VI tt six. XXIX . " twenty-nine. vn seven. -XXX a thirty. VIU it eight XL a forty. IX tt nine. L ti fifty. X *t ten. LX it sixty. XI tt eleven. LXX tt seventy. XTI a twelve. LXXX a eighty. xin It thirteen. XC ti ninety. XIV tt fourteen. C it one hundred. XV it fifteen. CC it two hundred. XVI tt sixteen. CCC it three hundred. XV 11 tt seventeen. CCCC ti four hundred. XVIII tt eighteen. D tt five hundred. XTX it nineteen. DC it six hundred. XX a twenty. DCC it seven hundred. XXT tt twenty-one. DCCC, tl eight hundred. XXII u twenty-two. DCCCC It nine hnndred. xxin it twenty-three. M tt one thousand. MDGCCLXXII, one thousand, eight hundred, and seventy-two. 14 NUMEKATION. NoTER — I. Repeating a letter repeats its value. Thus, 1 de- notes one; II, two; III, Hiree; X, tea ; XX, twenty ^ etc. 2. Placing a letter of less yalue 6e/ore one of greater value, diminisJiea the value of the greater by that of the less; placing the less after the greater increases the value of the greater by that of the less. Thus, I denotes one, and V five ; but IV VAfour, and VI siac. 3. Placing a horizontal line over a letter increases its value at $10 times. Thus, I denotes a thousand; X, ten thousand; C, a hundred thousand; M, a million. 4. Four was formerly denoted by IIII; nine, by Villi; forty, by XXXX; and ninety, by LXXXX. Express the following numbers by letters: I. 17. 7. 48. 13. 98. 19. 564. 2. 26. 8. 53. 14. III. 20. 896. 3- 24. 9. 67. 15. 109. 21. 1116. 4. 37-. 10. 78. 16. 114. 22. 1320. 5. 29. II. 84. 17. 118. 23- 1536. 6. 44. 12. 89. 18- 377- 24. 1876. NUMERATION. 25. What is Numeration? Numeration is the art of reading numbers expressed by figures, letters, and other numeral characters. 36. How read numbers expressed by figures? Divide them into periods of three figures each, counting from the right. Beginning at the left hand, read the periods in succession, and add the name to each, except the last. 27. Why is the name of the last period omitted? Because the right hand period always denotes simple units, therefore its name need not be mentioned. 1 NUMERATION. 15 Repeat the Numeration Table, beginning with units.  a m § 1 I Si CD o OB 98. 5 7 9 s 1 2 3» Period V. Trillions. Period IV. Billions. Period m. Millions. Period n. Thousands. Period L Units. The figures in the Table are thus read: 298 trillions, 570 billions, 923 millions, 874 thousand, two hundred and sixty-seven. Note. — This method of reading figures is commonly ascribed to the French Numeration. Others ascribe it to the Italians, and thence call it the Italian Method. Copy and read the following: I. 107. 13- 10315- 25- 3423201. 2. 260. 14. 12065. 26. 2300400, 3- 315- 15- 24308. 27. 5408025. 4. 809. 16. 13020. 28. 45320265. 5. 1020. 17. 20460. 29. 63205308. 6. 2405. 18. 35007- 30. 265310275. 7. 3200. 19. 40800. 31- 8045007. 8. 5007. 20. 85408. 32. 925604300. 9. 6080. 21. 115326. 33- 4260345809. 10. 7650. 22. 208065. 34. 61204400000. II. 8075. 23. 400304. 35- 160240030045. 12. 9364. 24. 803025. 36. 407008300416. 16 NUMERATION. Copy and read the following: I. m. 2. VL 3. IV. 4. vm. 5. VII. 6. IX. 7. xm. 8. xxxn. 9. XXVIII. 10. XLII. 21. CIV. 22. CVIII. 23. CXII. 24. CXII. 24. CXII. 25. CXII. 26. CXII. 27. CXII. 28. CXII. 29. CXII. 30. CXII. 31. CXII. 32. CXII. 33. CXII. 34. CXII. 35. CXII. 36. CXII. 37. CXII. 38. CXII. 39. CXII. 40. CXII. 41. CXII. 42. CXII. 43. CXII. 44. CXII. 45. CXII. 46. CXII. 47. CXII. 48. CXII. 49. CXII. 50. CXII. 51. CXII. 52. CXII. 53. IX. 25. CXL. 26. CXLIX. 27. CXLIX. 28. CXLIX. 29. CXLIX. 30. CXLIX. 31. CXLIX. 31. CXLIX. 32. CXLIX. 33. CXLIX. 34. CXLIX. 35. CXLIX. 36. CXLIX. 37. CXLIX. 38. CXLIX. 39. CXLIX. 40. CXLIX. 41. CXLIX. 42. CXLIX. 43. CXLIX. 44. CXLIX. 45. CXLIX. 46. CXLIX. 47. CXLIX. 48. CXLIX. 49. CXLIX. 50. CXLIX. 51. CXLIX. 52. CXLIX. 53. CXLIX. 54. CXLIX. 55. CXLIX. 56. CXLIX. 57. CXLIX. 58. CXLIX. 59. CXLIX. 60. CXLIX. 61. CXLIX. 62. CXLIX. 63. CXLIX. 63. CXLIX. 64. CXLIX. 65. CXLIX. 66. CXLIX. 67. CXLIX. 68. CXLIX. 69. CXLIX. 70. CXLIX. 71. CXLIX. 72. CXLIX. 72. CXLIX. 72. CXLIX. 73. CXLIX. 74. CXLIX. 74. CXLIX. 75. CXLIX. 76. CXLIX. 76. CXLIX. 77.  Process of adding Two digits together. If young, let them illustrate the examples by counters or not marks. 1. If you have a apple, and I give you a apple more, how many apples will you have? "One apple and a apple more are 2 apples." 2. If you have 2 cents, and you find 1 more, how many cents will you have? "Two cents and 1 cent more are 3 cents." 3. How many are 3 marbles and 2 marbles? 4. Show this by your fingers. 5. Sarah has 3 red roses, and 3 white ones: how many roses has she of both kinds? Show it. 6. If an orange costs 5 cents, and a lemon 4 cents, how much will both cost? Show it. 7. William earned 6 cents in the morning, and 4 in the afternoon: how much did he earn in both? 8. Sanford obtained 6 credit marks, and his sister 6: how many did both obtain? Show it. 9. If I gather 6 quarts of cherries, and buy 8 quarts, how many quarts shall I have? Show it. Joseph picked 5 quarts of blackberries, and his brother 7 quarts: how many quarts did both pick? If I pay 8 dollars for a barrel of flour, and 7 dollars for a ton of coal, what shall I pay for both? If I picked 12 peaches from one tree, and 3 from another: how many did he pick from both? George picked 12 peaches from one tree, and 3 from another: how many did he pick from both? 2 and 3 and 4 and 5 and I are 3 I are 4 I are 5 I are 6 2 " 4 2 " S 2 " 6 2 " 7 3 " 5 3 « 6 3 " 7 3 " 8 4 « 6 4 " 7 4 " 8 4 " 9 5 " 7 5 « 8 • 5 " 9 S ** 10 6 " 8 6 ^' 9 6 " 10 6 « II 7 " 9 7 « 10 7 « II 7 « 12 8 " 10 8 « II 8 " 12 8 " 13 9 " II 9 " 12 9 " 13 9 " 14 lO " 12 10 " 13 10 *^ 14 10 " 15 6 and 7 and 8 and 9 and I are 7 I are 8 I are 9 I are 10 2 « 8 2 " 9 2 " 10 2 " II 3 " 9 3 " 10 3 " II 3 " 12 4 " 10 4 " II 4 " 12 4 " 13 5 « II 5 '* 12 5 " 13 S " 14 6 <c 12 6 " 13 6 " 14 6 « 15 7 " 13 7 " If 7 " 15 7 « 16 8 " 14 8 " 15 8 " 16 8 " 17 9 " ^5 9 « 16 9 " 17 9 " 18 lO " 16 10 " 17 10 " 18 10 « 19 9 1. Show by counters or unit marks how many 2 added to 3 will make. 2. Show how many 2 added to 4 will make. 3. Show how many 2 added to 5 will make. Show how many 3 added to 4 will make. Show how many 3 added to 5 will make. Show how many 3 added to 5 will make, etc. Particular care should be taken to see that beginners understand how the Addition Table is constructed; that they fully comprehend the results of adding two parts together, before they are required to commit them to memory. ADDITION. DEFINITIONS. 1. What is Addition? Addition is uniting two or more numbers in one. 3. What is the number obtained by addition called? The Sum or Amount Note. — The sum or amount contains as many urvUs or ones as all the numbers added. 3. When the numbers to be added are the same denomination, what is the operation called? Simple Addition. 4. How is Addition denoted? By a perpendicular cross called plus ( +), placed between the numbers to be added. Thus, 5 + 3 shows that 5 and 3 are to be added together, and is read, "5 plus 3," "5 and 3," or "5 added to 3." NOTE. — The term plus is a Latin word, signifying more, or added to, 5. How is the equality between two numbers or sets of numbers denoted? By two short paralleltines, called the sign of equality (=). The expression 5 + 3 = 8, shows that 5 increased by 3 equals 8, and is read, "5 plus 3 equal 8," or the sum of "5 plus 3 equals 8." Read the following expressions: 7 + 2 = 9; 6 + 4 = 7 + 3; 17 + 3 + 5 = 12 + 7 + 6; 21 + 8 + 9 = 12 + 20 + 6; 30 + 3 + 20 = 40 + 13. MENTAL EXERCISES. To Teaches.— The Mental and Slate Exercises are intended to be combined in each recitation. Hence, no more examples should be assigned to a lesson, than the class can thoroughly master. I. Henry gave 4 cents for an orange, 3 cents for a pear, and 2 cents for an apple: how many cents did he give for all? Analysis. — 5 cents and 3 cents are 7 cents, and 2 are 9 cents. Therefore, he gave 9 cents for all. 30 ADDITION. John paid 5 cents for a writing-book, and 2 cents for a pen: how many cents did he pay for both? 5 and what make 7? An orange costs 4 cents, a pear 3 cents, and a peach 2 cents, what will all three cost? George gave 4 apricots to one of his sisters, 3 to another, and 5 to another: how many did he give to all? If you pick up 4 apples under one tree, 3 under another, and 5 under another, how many apples will you have? A man paid 3 dollars for a cane, 5 dollars for an umbrella, and 4 dollars for a hat: how much did he pay for all? How many are 5 cents, and 3 cents, and 2 cents, and 1 cent? Sarah paid 9 dollars for a hat, and 4 dollars for a parasol: what did she pay for both? Count by two rapidly to 60. Thus, two, four, six, eight, ten, twelve, etc. Count by threes in like manner to 60. Slate exercises. Columns of Single Figures. I. Write and add 2, 4, 5, 3, 6, and 1, upward and downward four times. Explanation. — Write the numbers in a column, and draw a line under it. First. Beginning at the bottom, add upward; as, one, seven, ten, fifteen, nineteen, twenty-one. Set 21 under the column. Second. Begin at the top, and Add downward; as, two, six, eleven, etc. 21 Ans, ADDITIOK. 21 Copy and add the following upward and downward orally till rapidity is attained : -■) (30 (4.) (5.) (6.) (7.) (8.) (9.) 2 3 4 5 2 3 4 5 4 2 3 2 I 2 I 3 3 4 5 3 5 4 3 2 I 2 I 2 2 3  3 3 3 2 2 3 3 4 4 3 4 3 I 2 2 I 2 4 5 4 5 5 4 5 3 S 4 %* Care should be taken to write figures with neatness and BTmmetry, and set them in perpendicular columns. MENTAL EXERCISES. 1. There are 5 ducks in one pond, 6 in another, and 7 in another : how many are there in all the ponds ? 2. A man picked 8 quarts of blackberries, and his son 6 quarts : how many quaiis did both pick ? 3. "William has 9 chickens in one coop, and 7 in another : how many chickens has he in both ? 4. 7 and what make 16 ? 9 from 16 leaves what? 5. How many are 7, and 6, and 5, and 4, and 8 ? 6. K there are 9 pears in one dish, and 8 in another, how many are there in both dishes ? 7. 8 and what make 17 ? 9 from 17 leaves what ? 8. Henry solved 6 examples, George 9, and Samuel 8 : how many did they all solve ? 9. Lucy gave her teacher 7 roses, Julia 9, and Hattie 10 : how many roses had the teacher ? 10. Count by fives to 100, with rapidity. 11. Count by sixes to 100, in like manner. 22 ADDITION. SLATE EXERCISES. Copy and add the following as before : (I.) (2.) (3.) (4.) (5.) (6.) (7-) (8.) 6 5 4 7 8 7 4 9 3 4 5 4 6 5 7 5 S 2 6 7 2 8 9 4 4 5 7 3 8 3 I 9 2 3 4 5 4 4 3 6 5 4 7 7 8 8 9 9 4 6 3 4 5 6 7 8 6 4 7 8 9 5 6 4 3 7 6 9 4 8 4 3 7 5 3 5 6 7 8 9 MENTAL EXERCISES. 1. Henry had 5 cents, and earne.d 6 more: how many cents had he then ? 5 and what make 11? 2. William picked 7 quarts of cherries, and his brother 6 quarts : how many quarts did both pick ? 3. 5 and what make 12 ? 6 and what make 13 ? 4. A farmer had 8 cows, and bought 6 more : how many had he then ? 5. If you pay 7 dollars for a velocipede, and 8 dollars for an overcoat, what will both cost you ? 6. 6 and what make 14 ? 7 and what make 15 ? 7. If Helen pays 8 dollars for a music-box, and 9 dol- lars for a fur cape, what will both cost her ? 8. How many are 9 dollars, and 6 dollars, and 7 dol- lars? 9. How many are 15 rods, 6 rods, and 8 rods ? 10. Count by sevens to 100, with rapidity. 11. Count by eights to 100, in like manner. ADDITION. 23 SLATE EXERCISES. Copy and add the following upward and downward as before : (I.) (2.) (3.) (4.) (5.) (6.) (7.) (8 4 3 2 9 5 6 8 9 3 4 3 8 3 8 8 8 6 2 S 4 8 9 5 4 8 3 7 3 4 7 6 7 I I 6 I 2 4 7 5 2 6 I 7 3 5 3 6 7 7 3 6 9 3 S 7 3 2 4 5 6 7 9 8 5 7 6 8 7 6 5 7 8 6 7 4 6 7 8 9 • Write in columns, and add the following : 9. 3 pounds, 4 pounds, 5 pounds, 2 pounds, 7 pounds, and I pound. 10. 6 yards, 5 yards, 3 yards, 4 yards, 2 yards, 8 yards, and 7 yards. MENTAL EXERCISES. 1. How many are 3 and 10 ? 13 and 10 ? 23 and 10 ? S^ and 10 ? 43 and 10 ? 53 and 10 ? 63 and 10 ? 73 and 10 ? S^ and 10 ? 93 and 10 ? 2. 7 and 10 ? 17 and 10 ? 27 and 10 ? 37 and 10 ? 47 and 10? 57 and 10? 67 and 10? 77 and 10? 87 and 10 ? 97 and 10? 3. 3 and 4 ? 13 and 4 ? 23 and 4 ? 33 and 4 ? 53 and 4 ? 43 and 4 ? 6^ and 4 ? 83 and 4 ? 93 and 4 ? 4. 5 and 3 ? 15 and 3 ? 25 and 3 ? 45 a^^d 3 ? 35 and 3 ? 55 and 3 ? 65 and 3 ? 85 and 3 ? 75 and 3 ? 24 ADDITION. 5. 17 and 5? 37 and 5? 27 and 5? 57 and 5?  5 hundred, and 2 are 7 hundred. Set the 7 under the hundreds, because it is hundreds. Note.— In practice, it is better simply to pronounce the results; as, two, five, nine, etc. Copy and add the following in like manner: (2.) (3.) (4.) (5.) (6.) (7.) (8.)  make 7 units, or 7 tens? Why? MENTAL EXERCISES. 1. A certain school had 40 girls and 30 boys in attendance; how many pupils did it contain? Akaltsis. — 40 is equal to 4 tens, and 30 is equal to 3 tens. Now 4 tens and 3 tens are 7 tens, or 70. The school, therefore, contained 70 pupils. 2. 50 is how many tens? 40? 60? 70? 80? 100? 3. 5 tens are how many? 7 tens? 6 tens? 8 tens? 8 tens? 10 tens? 4. How many are 3 tens and 5 tens? 6 tens and 4 tens? 5 tens and 7 tens? 5. How many are 20 and 30? 50 and 80? 6. How many are 70 and 50? 80 and 90? 2 20 ADDITION. 7. In a certain grove there are 89 sugar-maples and 46 elms: how many trees are there in the grove? Analysis. — 89 is 8 tens and 9 units, and 46 is 4 tens and 6 units. Now 8 tens and 4 tens are 12 tens, or 120; and 9 units and 6 units are 15 units, which, added to 120, make 135. Therefore, there are 135 trees in the grove. Note. — When numbers to be added mentally are large, it is advisable to separate them into the units, tens, etc., of which they are composed, and begin to add with the highest order. 8. How many are 34 and 43? 26 and 51? 9. How many are 45 and 62? 71 and 46? 10. A man paid 68 dollars for a cow, and 55 dollars for a colt: what did he pay for both? SLATE EXERCISES. When the Sum of a Column is 10, or More. I. A farmer had 3 flocks of sheep; one of 325, another 436, the other 541: how many sheep had he? Analysis. — We write units under units, tens Operation. Under tens, etc., and begin at the right hand as before. Thus, 1 unit and 6 units are 7 units, 436 S. and 5 are 12 units, or 1 ten and 2 units. We set the 2 units under the column added, because it is the same order as this column, and add the 1 ten -^' ^S^^ S. to the next column, because it is the same order as that column. Now 1 ten added to 4 tens makes 5 tens, and 3 are 8 tens, and 2 are 10 tens, or 1 hundred and 2 tens. We set the o, or right hand figure, under the column added, because there are no units of this order, and add the 1 hundred to the next column, because it is the same order as that column. Adding the 1 hundred to the next column, the sum is 13 hundred, or 1 thousand and 3 hundred. This being the last column, we set down the whole sum, putting the 3 under the column added, because it is the same order as this column, and the 1 in the next, or thousands place, because it is thousands. Therefore, etc. ADDITION. 27 10. When the sum of a column exceeds 9, what do you do? Write the units figure under the column added, and carry the tens to the next column. 11. What do you do with the last column? Set down the whole sum. 13. What is meant by carrying the tens?  Unit Marks. Analysis. — 325 = 3 hundred + 2 tens + 5 units; 436 = 4 hundred + 3 tens + 6 units; and 541 = 5 hundred + 4 tens + 1 unit. Now, to represent the first number, we place three counters in the column of hundreds, two in the column of tens, and five in the column of units. The other numbers are represented in a similar manner. Beginning at the right hand, we find there are 12 counters in units column. With drawing ten of them, we will be left, which we put under the column added. Now 10 units make 1 ten; hence, to represent the ten units withdrawn, we put a single counter (T.), denoting a unit of the 2nd order, in the column of tens. Again, adding the column of tens, we find there are ten counters, and withdrawing ten tens from this number, no tens are left. To express the absence of tens, we put a cipher in tens place. But ten tens make 1 hundred; hence, to represent the ten tens withdrawn, we put a single counter (H.), denoting a unit of the 3rd order, in the column of hundreds. Finally, in the hundreds column, there are 13 counters, and withdrawing ten of them leaves three. We therefore put three counters in the column of hundreds, and to represent the ten withdrawn, we put a single counter (M.), determining a unit of the 4th order, in the column of thousands. The amount is 1 thousand, 3 hundred, or tens, and 2 units. Hence, carrying the tens is simply taking a part from a lower order and adding it to the next higher, which can no more affect the amount of money a man has, if he changes 10 cents for a dime. ADDITION. (2.) (3.) (4.) (5.) (5.) (6.) (6.) 4683 3605 5072 7304 8097 3427 7036 3816 6og4 3845 5032 5783 7301 3752 5«2 13. The preceding principles may be summed up in the following GENERAL RULE. 1. Place the numbers one under another, units under units, etc., and beginning at the right, add each column separately, II. If the sum of a column does not exceed nine, write it under the column added. If the sum exceeds nine, write the units figure under the column, and carry the tens to the next higher order. Finally, set down the whole sum of the last column. Proof. — Begin at the top and add each column downward. If the two results agree, the work is right. Note. — This method of proof depends upon the principle that reversing the order of the figures will be likely to detect any error that may have occurred in the operation. The  840 965 937 85 37 6 4 685 967 4 28 382 129 49 132 394 41 845 732 563 825 985 29 11. Herbert read 235 pages one day, 264 the next, and 362 the next: how many pages did he read in all? 12. What is the sum of 2362 days, 375 days, and 27 days? 13. If a yoke of oxen cost 250 dollars, and a cart 119 dollars, what will both cost? 14. A man paid 67 dollars for his coat, 16 for his vest, 23 for his pants, and 13 dollars for his boots: what did he pay for his suit? 15. A farmer has four flocks of sheep, one flock containing 256 sheep, the second 320, the third 195, and the fourth 168: how many sheep had he? 16. What is the sum of five hundred and sixty-one, two hundred and seven, and nine hundred and fourteen? 17. What is the sum of twelve thousand and twelve, six thousand and two, and ninety-five hundred? 18. A man paid 2250 dollars for his farm, 1600 dollars for stock, and had 168 dollars left: how much money had he at first? 19. Washington was born in the year 1732, and lived 67 years: what year did he die? 20. A. man paid 6 270 dollars for a horse and sold it for 1565 dollars more than he paid for it: how much did he get for it? 21. In what year will a person born in 1865, be 21 years old? 