50. How much will 69 cords of wood cost at $4.50 a cord ? 51. If a railroad engineer travels 794 miles a week, how far will he travel in 48 weeks ? 52. Mr. Moore sold 1,324 acres of land at $193 an acre : how much did he receive ? 53. How much will it cost to build 734 miles of railroad at $48,356 per mile ? 54. How much pension money do 679 soldiers receive, if each receives $2,495 ? 55. What is the cost of 185 city lots at $3,125 each ? 56. A butcher bought 48 sheep at $7 a head, and 32 barrels of beef at $17 a barrel : how much more did he pay for the beef than for the sheep ? 57. What is the cost of 3,150 acres of prairie land at $38 an acre, and 2,542 acres of woodland at $46 an acre ? 58. Mr. Robinson's yearly income is $7,250, and his average daily expenditure $7: how much can he save in a year of 365 days ? 59. A certain field contains 4,238 rows of corn ; each row contains 3,102 hills, and each hill has 4 stalks : how many stalks of corn are in the field ? 60. A builder used y loads of brick a day for 139 days. If each load contained 1,244 bricks, how many bricks did he use ? 61. Since a day contains 24 hours and each hour contains 60 minutes and each minute 60 seconds, how many seconds are there in 19 days? 62. A certain street in a city contains 48 houses, each house 24 windows, and each window 16 panes : how many window panes are in the 48 houses ? 63. Illustrate by an original problem the multiplication of one number by another. ART. 62.—To Multiply when the Multiplier is a Composite Number. 1. Multiply 342 by 18. Process. Analysis.—Since 18 is composed of the factors 342 3 and 6, it is evident that if the multiplicand 6 be multiplied by either of them and that product 2052 by the remaining factor, the result is the same as 3 that obtained by multiplying by 18. 6156 2. Multiply 643 by 16. 7. Multiply 678 by 48. 3. Multiply 584 by 22. 8. Multiply 7654 by 64. 4. Multiply 963 by 24. 9. Multiply 3087 by 72. 5. Multiply 4765 by 50. 10. Multiply 6009 by 81. 6. Multiply 7649 by 32. 11. Multiply 908070 by 144. ART. 63.—To Multiply by 10, 100, 1000, etc. 1. Multiply 34 by 10; by 100; by 1000. It is evident that when any number is multiplied by 10, it is moved one order to the left; when multiplied by 100, it is moved two orders to the left, and when multiplied by 1000, it is moved three orders to the left. 34 x 10 = 340; 34 x 100 = 3400; 34 x 1000 = 34000. Therefore, To multiply any number by 10, 100, 1000, etc., annex as many ciphers to the number as there are ciphers in the multiplier. 2. Multiply 368 by 10; by 100; by 1000. Division. Progressive Oral and Written Drills. SUGGESTION.—Division is the reverse of multiplication, and is the process of finding how many times one number is contained in another. Take the multiples of 2, and ask, How many 2's in 4? 12? 16? 6? 8? 14? 10? 18? Illustrate division by separating objects into groups of 2. If odd numbers are taken, there will always be one over, which may be written over the divisor, giving rise to the fractional form f. REMARK.—The dividend may contain any figure, but the divisor must not be larger than the digit which is the object of study. What is į of 4? 6? 8 ? 10 ? 12 ? 14 ? 16 ? 18 ? SUGGESTION.—Count by 3 to 27. Ask, How many 3's in 9? 18? 12? 6? 21 ? 15? 24 ? 27 ? Begin at 1 and count by 3's. The numbers thus obtained will contain a certain number of 3's with one over, which is } of 3. How many 3's in 7 ? 13? 19 ? 25? 16 ? 28? 10 ? 22 ? Begin at 2 and count by 3's. These numbers will all have 2 over, which is # of 3. How many 3's in 8? 14? 20 ? 26 ? 17 ? 29? 11? 23? SLATE AND BLACKBOARD EXERCISES. 3)345963 3)6481582 3)6378457 3)4579864 REMARK.—When there is 1 or 2 over with the last division, write what remains over the divisor. What is $ of 6 ? 9 ? 12 ? 15 ? 18 ? 21 ? 24 ? 27? What is of 6 ? 9 ? 12 ? 15 ? 18 ? 21 ? 24 ? 27 ? NOTE.—The process of finding the fractional parts of multiples, is an excellent review exercise in division and multiplication. SUGGESTION.—Count by 4's up to 36. Ask, How many 4's in 16? 24 ? 20 ? 32? 12? 36 ? 8? 28? Illustrate by groups of 4. Begin with 1 and count by 4's up to 37. These numbers give a certain number of 4's and 1 over, which is of 4. Begin with 2 and count by 4's up to 38. These numbers contain a certain number of 4's and 2 over, which is 4 of 4. Begin with 3 and count by 4's up to 39. These numbers contain a certain number of 4's, and 3 over, which is 4 of 4. SLATE AND BLACKBOARD EXERCISES. 4)4845286476 4)8976842676 4)6880462416 REMARK.—Dictate additional examples. Find f and of the above numbers. SUGGESTION.-Begin with 0, 1, 2, 3, 4, and count by 5. Tell how many 5's and how many over there are in each number. SLATE AND BLACKBOARD EXERCISES. 5)4056849686 5)8402868405 5)6847967245 What is of 10 ? 30 ? 45 ? 20 ? 35 ? 15 ? 40 ? 25 ? Find , and of the above numbers. SUGGESTION.—After beginning with 0, 1, 2, 3, 4 or 5, and counting by 6's, let pupils take such columns as the following and tell how many 6's and how many over there are in each number. Thus: În 14 there are two 6's and 2 over, and so on. SLATE AND BLACKBOARD EXERCISES. 6)7264816743 6)8463948765 6)4376798268 REMARK.—Dictate additional examples. What is f of 18 ? 36 ? 54 ? 42 ? 24 ? 12 ? 30 ? 48 ? e Find , 4, 4 and 4 of the above numbers. SUGGESTION.—Begin with 0, 1, 2, 3, 4, 5, or 6, and count by 7's up to 69. After the pupil can do this with facility, let him tell how many 7's and how many over there are in each of the following numbers: REMARK.—This table contains all numbers from 0 to 70. What is 4 of 14 ? 21 ? 28 ? 35 ? 42 ? 49 ? Find 4, 5, 4, and of the above numbers. SLATE AND BLACKBOARD EXERCISES. 7)8964068976 7)8604964843 7)8774879543 REMARK.—Dictate additional examples. SUGGESTION.—Begin with 0, 1, 2, 3, 4, 5, 6, or 7, and count by 8's .p to 79. This exercise is not only a review of addition, but prepares the pupil for the analytic process of division. |