70. Supplement and Complement Methods of Multiplication. The supplement of a number is the difference between it and the next smaller number which has zeros in all places except the last one to the left. For definition of complement see §30. Thus the supplement of 108 is 8, while the complement of 97 is 3. Example 1. Multiply 94 by 97. = 94 X 97 9118. The two figures to the right are the product of the complements of the figures in ones' places in the factors. That is, 6X 3 = 18. To find the rest of the product subtract from either factor the complement of the other. That is, 94 - 3 = 91 or 97 Example 2. Multiply 95 by 98. - 6 = 91. = 10. 95 2 = 93. The product 95 X 98 = 9310. We say, 2 X 5 9310 may therefore be written down at once. This method holds only when both factors are between 90 and 100. Example 3. Multiply 103 by 108. = 24. 103 X 108 = 11124. For the first two figures in the product take the product of the figures in units' places in the factors. That is, 3 X 8 Then add to either factor the supplement of the other. Thus, 103 + 8 The product 11124 may therefore be written down at once. Example 4. Multiply 104 by 109. = 111. 71. Multiplying by Numbers like 624 or 189. —A special device is convenient in the case of a multiplier such that a part of it is a multiple of another part. (In 624, 24 = 4 X6, and in 189, 18 = 2 × 9.) Example 1. Multiply 396 by 624. 22338 44676 469098 First multiply by 9 and then this product by 2 (20) and add. WRITTEN EXERCISES In the manner just described find the following products: 72. Products of Numbers Ending in 5. - A special rule may be found for multiplying two such numbers as 85 and 45. (1) If the sum of the digits to the left of the 5's is even (as in 85 and 45, where this sum is 8 + 4 = 12), take the product of these digits and add one half the sum of the digits. To this sum annex 25. In simple examples the products may be written down at once. (2) If the sum of the digits to the left of the 5's is odd (as in 145 = 75, where this sum is 14 +7 21), take the product of these digits and add one half their sum, neglecting the . To this sum annex 75. DRILL EXERCISES IN MULTIPLICATION Find the products of the following by ordinary multiplication. Check by casting out 9's. Find the products of the following in the manner shown in § 62 or Find the product of the following as shown in §§ 65-67. Find the following products as shown in § 68. Find the following products as shown in § 69. Find the following products as shown in §§ 63, 64. CHAPTER V DIVISION 73. Uses of Division. — In business division is used less frequently than addition and multiplication. Nevertheless, it occurs so often that any candidate for a business position should be able to divide with reasonable speed and accuracy. 74. Definition of Division. - Division is the process of finding one of two numbers when their product and the other number are given. 75. Dividend, Divisor, Quotient. -The given product is called the dividend, the other given number is the divisor, and the number to be found is called the quotient. The sign means that the number preceding it is to be divided by the number following it. Thus 72 8 is read " 72 divided by 8." To find the missing number in 6 X ? 48 is a problem in division. 48 is the dividend, 6 the divisor, and the number to be found is the quotient. = 76. Other Definitions of Division.- Division is also defined as the process of finding how many times one number is contained in another. 782 146 636 146 490 146 344 146 198 146 52 From this point of view division may be regarded as a short cut for subtracting a number a certain number of times. Thus 782 may be divided by 146 by repeated subtraction in the manner shown in the margin. When the remainder becomes less than 146, the process ends. We thus find that 146 is contained 5 times in 782 with 52 as a remainder. This method of dividing is required occasionally in certain Civil Service examinations. As late as the only method of division in general use. year 1000 A. D. this was the See page 6. |