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17. Multiply 893254:25 by 999 by subtraction, and prove by adding complement.

In a similar manner the process of multiplying by 49, 29, or any other two figures ending in 9, may be abridged by using 50, 30, &c., as multiplier, and at the same time subtracting the multiplicand once, or adding its complement. Thus,

18. Multiply 7236408 by 69; that is, by 70 with the addition at the same time of the complement of the multiplicand, and prove by multiplying by 69. 7236408

70

499312152=70—1 time by adding the complement of the

multiplicand while performing the multiplication. 19. Multiply 1423746 by 49 [50 and complement], and

prove by 49.

20. Multiply 653492 by 399, as above [by 400 and complement], and prove.

21. Multiply 28656 by 29, as above, and prove. 22. Multiply 67345 by 79, as above, and prove.

23. Multiply 6248 by 2499 by division and addition of complement separately, and prove.

24. Multiply 4588-24 by 499 by division and addition of complement separately, and prove.

25. Multiply 2346 by 8; that is, by supposing the multiplicand to be multiplied by 10, and adding double its complement. For example:

8.2346

18768

Suggestive Questions. If the multiplicand were multiplied by 10, what would be the right hand figure? If to that we add twice the complement of 6, what would it then be? What would the next figure of the product be if twice the complement of 4 were added to the 6? The next figure of the product + twice the complement of 3? and so on, dropping 2 for twice the complement of the rank next above the highest figure of the complement.

26. Multiply 45827 by 38 [by 40 and twice the complemont], and prove by dividing by 38.

27. Multiply 372-65 by 98 [100 and twice the complement], and prove by dividing by 98.

28. Multiply 2536 by 248 by division, adding two complements afterwards, and prove. 29. Multiply 3847 by 78 [80—2], and prove by division.

3. Multiplication partially or wholly by Addition. 30. Multiply 62305496 by 11 by addition, and prove.

Suggestive Questions.—If a number has only been taken 10 times, when it ought to have been taken 11 times, how many times too few has it been taken? How shall the error be rectified ?

Then 62305496 =10 times by position,
more. 62305496= 1 time,

makes 685360456=11 times.

The intelligent pupil will readily perceive that the above depends on a similar principle to that shown in example 12, and that the second line is wholly superfluous, being merely inserted here to show the principle. Of course, in the following exercises, he will omit the superfluity. In fact, it would be preferable that he should even omit the first line also, and write the product by 11 simply by inspection of the book.

31. Multiply 52643889 by 11 by addition, and prove by division.

32. Multiply 846-25 by 11, and prove by division. 33. Multiply 2345421 by 22, and prove by division.

This, and the 17 succeeding exercises, can be performed without writing any figures on the slate except the product.

34. Multiply 2862-75 by 33 by addition and multiplication, and prove by division.

35. Multiply 243692801 by 44 by addition, &c., and prove. 36. Multiply 8210432 by 55 as above. 37. Multiply 560438 by 66 as above. 38. Multiply 2439-004 by 77 as above. 39. Multiply .008 by 8-8 as above. 40. Multiply 3926 by 111 as above. 41. Multiply 62-25 by 111 as above. 42. Multiply 426832 by 222 as above. 43. Multiply 364852 by 444 as above. 44. Multiply 125896 by 555 as above.

45. Multiply 28-96 by 333 as above. 46. Multiply ·006 by 6-66 as above.

47. Multiply 1236 by 442 as above, and the use of the double complement.

48. Multiply 7216 by 886 as in last exercise.

49. Multiply 18425 by 774 by addition and use of treble complement.

50. Multiply 3215 by 996 as in last exercise.
51. Multiply 7326489 by 4829 by addition.
1.

7326489

4829

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14652978 multiplied by 10 by position. 29305956

1000 by position. 58611912

6 100 by position. 65938401

35379615381 Total product.

52. Multiply 6248351 by 3624 by addition, and prove by multiplication.

53. Multiply 1324658 by 18532 as above. 54. Multiply 9024368 by 2936 as above.

4. Multiplication by Resolution. 55. Multiply 73250147 by 64328.

73250147

64328

No. 1. 586001176 multiplied by

8 No. 2. 2344004704 No. 1 by 4 and by 10 by pos. 320 No. 3. 4688009408 No. 2 by 2 and by 1000 by pos. 64000

4712035456216 Product by 61328.

64328

56. Multiply 16348792 by 8567. This may be done in two ways : 1st, by 7, by 8=56:7=8; or 2d, by 7=56:8=7. Prove by division.

57. Multiply 27364895 by 64816 two ways: 1st, by 8 and by 8=64:4=16; 2d, by 6 and by 8=48-3. Prove by division.

58. Multiply 39460582 by 72836 [8.9=722]. Prove by division.

59. Multiply 82964523 by 27315 [3, 9, 5). Prove by division.

60. Multiply 21879030 by 28442 by resolution. Prove by division.

5. Division by Multiplication. 61. Divide 3265 by 5 by multiplication, and prove by division. 3265

5)3265 2

653 Proof. 653-0

62. Divide 8244 by 5 as above.
63. Divide 91.23 by 5 as above.
64. Divide 26345 by 5 as above.
65. Divide 3276885 by 25 as above.

3276885 25)3276885
4

131075-4 Proof.
131075.40

66. Divide the following numbers severally by 25 by multiplication, and prove by division : 9258, 326725, 8396289.

67. Divide 62845936 by 125 by multiplication, and prove by division. 62845936

125)62845936 8

5027674488 Proof. 502767-488

68. Divide the following numbers severally by 125 by multiplication, and prove by division: 74263485, 29632652, 81297124.

69. Why is it that in dividing integers by 5, the number of decimal places cannot exceed one; in dividing by 25, two; by 125, three?

70. What is the greatest number of decimal places possible in dividing integers by 2? by 4? 8? 16 ? 32 ?

CHAPTER III.

THE SHORTENED PROCESSES OF INCREASE AND DECREASE APPLIED TO COMMON FRACTIONS AND DENOMINATE

FRACTIONS.

DEFINITIONS.

THERE are three kinds of fractions : Decimal, Common, and Denominate Fractions. All these, as well as integers, have two names or values, namely, the primary, or simple, or absolute value; and the secondary value. The primary name originates alike in all. It is the same with that of the character or characters by which it is represented. The secondary name is derived as follows:

1. In integers and decimal fractions from their horizontal position ; that is, from their distance to the right or left of the place of units. (See Chap. I., p. 112, 1. 33). Thus, in the number

44'44, the primary name of each of the figures is the same, namely, four ; but their secondary names are different; that of the first being ty (or tens); of the second, units ; of the third, tenths ; of the fourth, hundredths.

2. In common fractions, the secondary name is written under the number of the fraction in figures. Thus, in , the 4 represents the primary, or simple, and the 5 the secondary name and value of the fraction, which, therefore, is called four fifths. The chief difference between decimal and common fractions is, that in the former, the integer can be divided only by some power of ten, and hence we can only have tenths, hundredths, &c.; whereas, in common fractions, any number whatever may be used as the divisor. Thus, we have not only for (four tenths), ito (four hundredths), but may use as divisor 5, 6, 18, 356, and so on without end, making , 4, 4s, 56, &c. From this definition it results, that a common fraction may be changed into an equivalent decimal fraction by performing the division

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