- £1) equal as many pounds as there are times 20 in 130, which are 6 times, with a remainder of 10. Hence, 130 s. = £6 10 s. 7 times £22 = £154, and £6 added £160. Hence, the product is £160 10 s. 11 d. 2 qr. NOTE. — When any denomination to be multiplied is very near a unit of the next higher, the work may frequently be much shortened by considering it a unit of that higher denomination, and subtracting for its deficiency in value. For instance, in the example above: since 18 s. = £1 2 s., 7 times 18 s. must equal £7 — 14 s., or £6 6 s., to which adding 4 s., (from the previous product,) we have £6 10 s., as before. What is the product 50. Of £29 8 s. 11 d. 1 qr. multiplied by 9 ? 51. Of 37 T. 19 cwt. 2 qr. 24 lb. 11 oz. 7 dr. multiplied by 3 ? 52. Of 273 bu. 1 pk. 5 qt. 1 pt. multiplied by 2 ? 53. Of 9 lb. 8 oz. 13 dwt. 22 gr. multiplied by 7 ? 54. Of 28 da. 17 h. 27 m. 58 sec. multiplied by 6 ? 55. Of 47 lb 83 73 23 18 gr. multiplied by 4 ? 56. Of 238 gal. 1 qt. 1 pt. 3 gi. multiplied by 9 ? 57. Of 674 lb. 4 oz. 19 dwt. 20 gr. multiplied by 8? 58. Of 23 lb. 4 oz. 16 dwt. 22 gr. multiplied by 8 ? 59. Of 13 owt. 2 qr. 17 lb. 13 oz. 9 dr. multiplied by 6? 60. Of 6 T. 18 cwt. 1 qr. 24 lb. 2 oz. 1 dr. multiplied by 7 ? 61. Of 9 bu. 3 pk. 7 qt. multiplied by 9 ? 62. Of 9 gal. 2 qt. 1 pt. multiplied by 5 ? 63. Of 8 w. 1 da. 23 h. 59 m. 56 sec. multiplied by 7 ? * 64. Of £8 19 s. 11 d. 3 qr. multiplied by 8 ? 65. Of 9 T. 19 cwt. 3 qr. 24 lb. 14 oz. multiplied by 7 ? 66. Of 7 lb. 11 oz. 19 dwt. 21 gr. multiplied by 4? 67. Of 483 yd. 3 qr. 1 na. multiplied by 9 ? 68. Of 4978 bu. 3 pk. 5 qt. multiplied by 5 ? 69. Of 37 lb. 11 oz. 19 dwt. 23 gr. ultiplied by 6 ? 70 Of £5871 18 s. 4 d. 1 qr. multiplied by 2? * In performing this example, much labor may be saved by observing that the multiplicand is only 4 seconds less than 8 w. 2 da. Similar things can frequently be applied, as in several of the subsequent examples. 80. Multiplication by Factors. (a.) It often happens that a number used as a multiplier is the product of two or more factors. In such cases it is sometimes convenient to resort to processes similar to those explained below. Note. - In writing the work here, as in several other places through. out the book, we have used letters for convenience of indicating to the eye the operations which have been performed, and the relations which the numbers bear to each other. For instance, in the first four given belove," a= 743,” means that the letter "a" stands for 743. “2 Xa= 1486,” means that two times the number “a," i. c., 2 times 743, = the number represented by “b,” which is 1486. “6 X b= 12 X a,' means that 6 times the number“b” (i. e., 6 times 1486) equals 12 times the number “a,” (i. e., 12 times 743.) The student will observe that the letter which in one form stands for one number, may in another form stand for a different number. Thus, in the first form “b” is used to represent 1486, while in the second it is used to represent 2972. How many are 12 times 743 ? 2 Xc= 12 X a = 8916 3 xc= 12 X a = 8916 Explanations. First Method. — Since 12 = 6 times 2, 12 times a number must be equal to 6 times 2 times the number. Second Method. — Since 12 = 3 times 4, 12 times a number must be equal to 3 times 4 times the number. Third Method. -- Since 12 = 2 times 2 times 3, 12 times a number must be equal to 2 times 2 times 3 times the number. Fourth Method. - Since 12 : 3 times 2 times 2, 12 times a number must equal 3 times 2 times 2 times the number. (6.) Solve the following examples in a similar manner. 4. Of 6427 x 42 ? 5. Of 4.379 X 64? 3. Of 8764 X 36 ? 6. Of 2976.4 X 28 ? 7. Of 2377 T. 17 cwt. 2 qr. 19 lb. 6 o 11 dr. x 63 ? 8. Of 27 lb 83 63 23 17 gr. X 45 ? 9. Of 19 w. 5 d. 17 h. 38 m. 29 sec. X 36 ? 10. Of £28 13 s. 10 d. 2 qr. X 35 ? 11. Of 48 lb. 10 oz. 16 dwt. 19 gr. X 24? 12. Of 837 bu. 3 pk. 6 qt. 1 pt. X 18? (c.) The most useful application of the foregoing principle is made when the multiplier is some multiple of 10. 13. What is the product of 8746 X 400 ? Solution. — Since 400 = 4 times 100, 400 times a number must equal 4 times 100 times the number; to obtain which we have only to remove the point two places towards the right, (24) and multiply by 4. Hence, 8746 400 3498400 14. What is the product of 9.7487 X 7000 ? Solution. Since 7000 7 times 1000, 7000 times a number must equal 7 tim 000 tim the number; to obtain which we have only to multiply by 7, and remove the point three places towards the right Hence. 9.7487 7000 68240.9 (d.) In like manner, to multiply by 60, we may multiply by 6, and remove the point one place towards the right; to multiply by 9000000, we may multiply by 9, and remove the point six places towards the right. In any case, all places left vacant between the number and the point must be filled with zeros. (See 15.) What is the product 15. Of 874379 X 20 ? 19. Of 627.34 X 80 ? 16. Of 27.9863 X 5000 ? 20. Of 9137.6 x 30000 ? 17. Of 714.26 X 90000 ? 21. Of 84273 X 60 ? 18. Of 62.794 X 40 ? 22. Of 7643 x 7000000 ? 81. When both Factors are large Numbers. (a.) We can find the product of two numbers by multiplying one of them by the parts into which we choose to separate the other, and then adding the products thus obtained together. Illustration. - 8 times 7 = 6 times 7 + 2 times 7 = 3 times 7 + 5 times 7 = 7 times 7 + once 7 = 4 times 7 + 4 times 7 = 56. (6.) This principle, and the one illustrated in article 80, are ordinarily employed when the multiplier contains several denominations. Illustrations. We usually get 83 times a number, by adding together 80 times the number and 3 times the number. We get 647 times a number by adding together 600 times the number, 40 times the number, and 7 times the number. We get 8009 times a number by adding together 8000 times the number and 9 times the number. (c.). It can make no difference which part of a number we multiply by first, provided we multiply by all its parts; yet for the sake of uniformity it may be well generally to begin with the lowest denomination. 1. Suppose that we are required to find the product of 5794 X 78? Explanation. We may find the product by 8 in the usual manner. To find the product by 70, we have only to multiply by 7, and remove the point one place towards the right, or, which is the same thing, the figures one place towards the left. Adding these results together will give 78 times the number. WRITTEN WORK. a = 5794 = Multiplicand. 78 = Multiplier. 70 X a=c= 405580 = Product by 70. b+c= 78 X a = 451932 = Product by 78. (d.) Since the product of the multiplication by the units is sufficient to fix the place of the figures in the subsequent products, the zero at the right of the second product need not have been written. The product would then stand, 4345 = Product by 5. 6083 70. 65175 = 66 75. (e.) Examples in which the multiplier consists of more than two figures are performed in a similar way. 2. What is the product of 780.69 X 20850 ? WRITTEN WORK. a = 780.69 = Multiplicand. 20850 = Multiplier. a X 50 = b = 39034.50 = Product by 50. a X 800 == 624552. = Product by 800. a X 20000 =d=156138 = Product by 20000. b+c+d= = 16277386.50 = Product by 20850. NOTE. Thus: - In practice, only the necessary figures should be written 780.69 39034.50 16277386.50 |