PROOF. First Method. 6 2 1 less than 3 12 the former mul tiplier. 6 multiplicand. 18 The product the You may take 1 from the multiplier, and multiply by the remainder. Thus, one taken from three, the multiplier, leaves 2. Multiply the 6 by the 2, which makes 12, and add the multiplicand to the product; and if the sum be equal to the first product, the work is right. Thus 6, the multiplicand, added to 12, makes 18. The principle of this method of proving multiplication is very plain; for it is perfectly evident, that two times six, and one time six, make three times six. Second Method. same as that of the other operation, and is therefore right. 6 multiplicand. 3 multiplier. 18 product. 6 6 You may prove Multiplication by Substraction. You have learned that Substraction is the reverse of Addition; and you have just been told that Multiplication is a short way of doing Addition; and, therefore, Substraction may 12 be used to prove Multiplication. Thus, in the present example, the product is 18; that is, 6 three times expressed; consequently, if 6 be taken from 18 three times, it will reduce it to nothing; as it is taking away 6, the multiplicand, as many times as it has been repeated by the 3, the multiplier; and, therefore, the work is right. 6 6 0 2. If you should pay 4 cents for one top; how many cents, at the same rate, must you pay for 36 tops? Ans. 144 cents. EXPLANATIONS. You must write the figures down according to the first example; that is, place the larger 36 multiplicand. 4 multiplier. sum for the multiplicand, the 144 product. smaller for the multiplier, and draw a line underneath. You must then say, 4 times 6 are twentyfour, and write down the figure 4 only; bearing or carrying, in mind, the twenty, or two tens, until you have the product of the 3 multiplied by the 4, and then add or carry it to that product: thus, 4 times 3 are twelve, and remembering that these are twelve TENS, you must call the twenty, which you have carried, two, that is, two tens; and say, 4 times 3 are twelve, and two make 14, that is, fourteen tens; and you must write them down in their proper place, that is, on the left hand of the figure 4, which will be the product, 144, the product required. In this manner are any sums, or any number of figures, to be multiplied by a single figure; that is, by any figure from one to ten; and, indeed, you may go on to twelve, obtaining the product of eleven or twelve times any line of figures whatever, in a manner equally simple; that is, beginning to multiply the first figure on the right hand, and, so proceeding, taking each figure in its turn through all the figures of the multiplicand, or sum to be multiplied; so that millions are as easily multiplied as are hundreds or tens. Take the following examplo. 3. Multiply 5274139 by 4. EXPLANATIONS. 5274139 4 Beginning with the multiplier at the right hand figure of the multiplicand, you must say, 4 times 9 are thirty-six, that is, thirty-six units. You must set 21096556 down the 6 in the place of units, and carry the three tens to the place of tens: thus, 4 times 3 are twelve tens, and the three tens carried make fifteen tens, that is, one hundred, and five tens. You must set down the 5 in the place of tens, and carry the one hundred to the place of hundreds: thus, 4 times 1 are four hundreds, and the one hundred carried makes five hundreds, which 5 is, of course, to be set down in the place of hundreds, leaving nothing to be carried to the place of thousands. You must then begin anew, and say, 4 times 4 are sixteen, that is, sixteen thousands. You must set down the 6 in the place of thousands, and carry the one to the place of tens of thousands thus, 4 times 7 are twenty-eight, and the one carried makes twenty-nine tens of thousands, that is, two hundreds of thousands, and nine tens of thousands. You must set down the 9 in the place of tens of thousands, and carry the two hundreds of thousands to the place of hundreds of thousands: thus, 4 times 2 are eight hundreds of thousands, and the two carried make ten hundreds of thousands, that is, one million. You must set down a cipher in the place of hundreds of thousands, and carry the one to the place of millions: thus, 4 times 5 are twenty, and the one carried makes 21; which sum, as the process is now finished, is to be set down. Thus you will have, as the product of your mul tiplicand and multiplier, the sum of twenty-one millions, ninety-six thousand, five hundred and fiftysix; that is, 4 times the 9 units, 4 times the 3 tens, 4 times the 1 hundred, 4 times the 4 thousands, 4 times the 7 tens of thousands, 4 times the 2 hundreds of thousands, and 4 times the 5 millions; and, consequently, 4 times the 5274139. In this easy manner you can multiply any line of figures whatever, by any number from 1 up to 12, and bring out the product of it in a single line. 4. Multiply 653412 by 12. EXPLANATIONS. 653412 12 Beginning, as before, with the multiplier at the right hand figure of the multiplicand, you must say, 12 times 2 are twenty-four, set down 4 and carry two; and 7840944 proceeding to the next figure, you must say, 12 times 1 are twelve, and the two carried make fourteen, set down 4 and carry one; and say, 12 times 4 are fortyeight, and the one carried makes forty-nine, set down 9 and carry four; and say, 12 times 3 are thirty-six, and the four carried make forty, set down a cipher and carry four; and say 12 times 5 are sixty, and the four carried make sixty-four, set down the 4 and carry six; and say, 12 times 6 are seventy-two, and the six carried make 78; which sum, as the process is now finished, is to be set down. By paying particular attention to the preceding EXPLANATIONS, you will be able to work any sum in Multiplication; that is, any sum in which the multiplier does not exceed 12; for you must remember that the object of these EXPLANATIONS is to explain to you the principles upon which each rule is founded. 16. If you pay 2 cents for one apple; how many cents must you pay for 24 apples? Ans. 48 cents. 17. If you pay 25 cents for one arithmetick; how many cents must you pay for 3 arithmeticks? Ans. 75 cents. 18. If you pay 7 dollars for one barrel of flour; how many dollars must you pay for 125 barrels? Ans. 875 dollars. 19. If you pay 8 cents for one pound of pork; how many cents must you pay for 375 pounds? Ans. 3000 cents. 20. A merchant bought 225 lemons at 5 cents apiece; how many cents did he pay for the whole? Ans. 1125 cents. |
