46. Checks by Interchanging Factors or by Dividing Product. Multiplication may be checked by interchanging multiplier and multiplicand. The most certain check, however, is to divide the product by one of the factors. be the other factor. The quotient must 47. Casting out 9's. Example. Multiply 493 by 78. 493 7 78 6 42 6 3944 3451 38454 - 6 To check by casting out 9's (see page 14), first find the excesses left after casting out 9's in 493 and 78, which are 7 and 6. Then find the excess after casting out 9's in the product of 7 and 6, which is 6. The excess after casting out 9's in the product 38,454 is also 6. Hence the multiplication checks. As in addi tion, the casting out of 9's is not an absolute test. Rule: The excess of 9's in the product is equal to the excess of 9's in the product of the excesses. WRITTEN EXERCISES Multiply each of the following and check by casting out 9's: SHORT CUTS 48. Means for Finding Short Cuts. The most useful short cuts in arithmetic involve doing part of the work mentally, thus saving writing. Familiarity with short cuts materially increases one's speed in computing. Cross multiplication as shown in § 49 is the most important in multiplication and should be thoroughly mastered. It is almost useless to learn a set of rules for short cuts. The short cuts must be learned by constant practice and if possible devised by the learner himself. The examples given on this and the following pages suggest what can be done in this direction. Speed is by no means the only consideration in favor of short cuts. The appearance and compactness of the work is equally important. In this respect any method which enables the computer to write the product horizontally as in 268 X 63 16884 is vastly superior to the ordinary method. = 49. Cross Multiplication. - By a little study of the multiplication of two numbers like 74 by 46 we can find a method which will enable us to multiply mentally two such numbers, using pencil and paper only to put down the product. Example 1. Multiply 74 by 46. 74 46 24 42 16 28 3404 6 ones X 4 ones = 24 ones. 4 tens X 4 ones = 16 tens. 4 tens X 7 tens = - 28 hundreds. = Add = 24, = This work can all be done mentally. We simply say 6×4 2 to carry. 6X7 42 and 4 X 4 = 16, which with the 2 carried gives 60. Carry 6. 4 X 7 28. To this add the 6 carried, obtaining 34. Example 2. Multiply 268 by 63. = 24, carry 2. 3 × 6 = 268 × 63 = 16884. = = 18, 6 X 8 48, 218 +48 6, 6 × 6 = 36, 6+ 6 + 36 48, carry 4. 6 × 2 Hence the product is 16884. = The facts to be noticed in connection with cross multiplication are: Example 3. Multiply 2397 by 183. 2397 X 183 = 438651. 1 x 2 = 2, 2 + 2 = 4. = 16 = (The numbers carried may be jotted down.) After a little practice this work can all be done mentally, the results only being put down from right to left as the successive figures are obtained. Cross multiplication is the most useful of devices for shortening multiplication. It should be mastered perfectly. The method of multiplication given in § 49 is used, for instance, in extending bills, since the numbers can be multiplied horizontally, only the final results being put down. 50. Multiplying by 11. Example. Multiply 3946 by 11. 3946 X 11 3946 43406 By arranging the work as shown here, we save writing one line of figures. The work can be shortened still further if we notice that the first figure at the right in the multiplicand is the first figure in the product and that the remaining figures are obtained as follows: 6 + 4 = 10, 1 to carry. 4+ 9 + 1 = 14, 1 to carry. 9 + 3 + 1 = 13, 1 to carry. 1 + 3 = 4. Example. Multiply 1947204 by 11. 1947204 X 11 21419244 In the manner just described the product is written down at once. With a little practice products of this sort can be written down very rapidly. WRITTEN EXERCISES Write down the product of each of the following: 52. Multiplying by 25, 50, 75. Example 1. Multiply 4938 by 25. Since 25 is one fourth of a hundred, we may multiply a number by 25 by multiplying it by 100 and then dividing by 4. The result can be written down at once by annexing the zeros mentally and then dividing by 4. In this case also the result may be written at once. 53. Multiplying by Numbers like 104, 501, etc. Example 1. Multiply 2047 by 104. 2047 X 104 8188 212888 To multiply by 104, multiply by 100 and by 4 and add the products. Notice how this is accomplished by putting the figure 8 in the second place to the right of 7. This has the effect of putting 7 into hundreds' place, the 4 in thousands' place, and so on, thus multiplying the whole multiplicand by 100. Example 2. Multiply 19274 by 301. 19274 X 301 57822 5801474 Explain how 19274 is multiplied by 300 in this example. Example 3. Multiply 837 by 299. 299 = 300 1. 837 X 299 251100 250263 Hence multiply by 300 and then subtract 1 × 837. To save writing we place 300 × 837 251100 below 837 and then subtract the upper from the lower number. = 3748 X 8 29984 X 7 209888 First multiply by 8 and then the product by 7. This saves writing one line of figures and adding two numbers. WRITTEN EXERCISES Multiply, using an appropriate short cut in each case: |