6's, 58 and five, 63-carry 6; 66 166535 8's, 79 and seven, 86-carry 8; The number in italics is in each case the number in the multiplicand just to the right of the one multiplied. To multiply by 31, 41, 91, it is best to write the multiplier over the multiplicand, and use the multiplicand itself as the partial product from the digit 1 in the multiplier. For example: 81 the multiplier. product by 1. product by 8 (tens). 7814773927 The product can be obtained in one line, as in multiplying by 19, but there is greater risk of error in the mental working. Such examples as 84 × 76 afford interesting and useful mental practice. Multiplying crosswise and summing the products, 76 tens; multiplying the units, 2 tens 4 units; multiplying the two tens, 56 hundreds; hence 63 hundreds, 8 tens, and 4 units-i. e., 6384. 2. Practice in finding the squares of numbers is very useful. The rule for finding the square of the sum of two numbers and the difference between the squares of two numbers may be readily arrived at. For example, multiply 85 by 85: 80 + 5 80 + 5 80+ 5 X 80 + 5 × 80+ 5′ 80+10 × 80 +5o = 7225 This may be illustrated by intuitions, symbolising units by dots. Let the following indicate the square of 7 (5+2). We see at once that to make up the whole square there is (i) the square of 5, (ii) 5 taken twice, and (ii) the square of 2-that is, the square of the first number, twice the product of the first by the second, and the square of the second number. It will be readily seen that the difference of the two squares (7-5) is twelve times two; but twelve is the sum of the numbers and two their difference. Does this hold for other numbers? The pupil will be greatly interested in discovering for himself the general principle: the difference of the squares of two numbers is equal to the sum of the numbers multiplied by their difference. If, in the figure, he compares the square of 3 with the square of 4, of 4 with that of 5, he will see that the square of any of these numbers is got from the square of the next lower by simply adding the sum of the numbers to the square of the lower. The square of 3 is 9; the square of 4 is 9 (= 3') + (3+4); the square of 5 is 16+ (4+5) ; and the square of 7 is 36 +(6+7). The pupil will deduce for himself that, given the square of any number, the square of the next consecutive number is obtained by adding the sum of the numbers to the given square. All these principles, and many others, may be made the basis of exercises equally interesting and useful in mental arithmetic: Square of 95; multiply 95 by 105, (100 — 5) (100+5); 295 by 305, (300+5) (300 — 5); the square of 250; the square of 251, etc. 3. The making-up method in subtraction may be conveniently used when the product of one number by another has to be taken from a given quantity. From 89713 take 8 times 8793. The work is done as follows: 89713 Eight 3's, 24 and nine = 33—carry 3; 9's, 75 and six 81-carry 8; 8 19369 7's, 64 and three = 67-carry 6 The numbers in italics indicate the remainder, 19369. In this example we multiply by 7, and, observing that 28 is 4 times 7, we multiply the first line of the product by 4, getting the second line; then the multiplicand by 3, taking care, of course, to put the product in the thousands' place. We may often take advantage of this method by breaking the order of finding the partial products. Thus, if the product of 567392 by 218126 is required, we may use the former as multiplier, and work thus: We notice that 56 is 8 times 7, and that 392 is 7 times 56. Begin, therefore, with 7. Multiplying this product by 8, we have the second line of partial products; and, finally, multiplying this second line by 7, we get the third line of partial products. Or we might have used 218126 for multiplier, observing that 9 times 2 are 18, and 7 times 18 is 126; thus: 567392 218126 113 478 4 102 13 0 5 6 71491 392 1 2 3,7 6 2,9 4 7,3 9 2 Multiplying first by 2 (hundred thousand); multiplying this product by 9; multiplying this second partial product by 7, taking care as to the proper placing of the products, we have the complete product. 5. Another plan that affords a good exercise in mental additions, and subsequently proves useful, is the method of finding a product of two factors in a single line. To multiply, e. g., 487 by 563, write the multiplier, with the digits in inverted order, on the lower edge of a slip of paper, thus, 365. Place the paper over the multiplicand so that the units (3) shall be just over the units of the multiplicand. The artifice consists in moving the slip of paper along the multiplicand, figure by figure, till the last digit (5) of the inverted multiplier is over the last digit of the multiplicand, and taking the product of any pair, or the sum of products of any pairs, of numbers that may be in column. Thus: Three 7's, 21-one and carry 2; 365 487 365 487 365 487 365 487 365 487 Three 8's, 26; six 7's, 68-eight and carry 6; Three 4's, 18; six 8's, seven 5's; 101 one and carry 10; Six 4's, 34; five 8's; 74-four and carry 7; Five 4's and 7 carried-twenty-seven. The numbers in italics, taken in order, are the product, 274181. Proofs of Multiplication.—(1) By repeating the operation with the factors interchanged. (2) The product divided by either factor should give the other factor. (3) By casting the nines out of the multiplier and the multiplicand, then multiplying these remainders together and casting the nines out of their product; the remainder thus obtained should equal the remainder from casting the nines out of the product of multiplier and multiplicand. For example, test the following by casting out the nines: |
