Multiply 328 by 112 328 4 1312 product oy 4 9194 product by 28 4 है 36736 product by 112 It is easy to see that we may multiply by any other number in the same manner. This operation may be expressed as follows. To multiply by a composite number. Find two or more numbers, which being multiplicd together will produce the multiplier ; multiply the multiplicand by one of these numbers, and then that product by another, and so on, until you have multiplied by all the factors, into which you had divided the multiplier, and the last product will be the product required. If the multiplier be not a composite number, or if it cannot be divided into convenient factors : Find a composite number as near as possible to the multiplier, but smaller, and multiply by it according to the above rule, and then add as many times the multiplicand, as this number falls short of the multiplier. V. I have shown how to multiply any number by a single fiqure ; and when the multiplier consists of several figures, how to decompose it into such numbers as shall contain but one figure. It remains to show how to multiply by any number of figures; for the above processes will not always be found convenient. The most simple numbers consisting of more than one figure are 10, 100, 1000, &c. It will be very easy to multiply by these numbers, if we recollect that any figure written in the second place from the right signifies ten times as many as it does when it stands alone, and in the third place, one hundred times as many, and so on. If a zero be annexed at the right of a figure or any number of figures, it is evident that they will all be removed one place towards the left, and consequently become ten times as great; if two zeros be annexed they will be removed two places, and will be one hundred times as great, &c. Hence, to multiply by any ti umber consisting of 1, with any number of zeros at the 1 x 10 = 10 1 x 100 = 100 270 4200 368 x 1000 = 368000 27 X 42 x 10 = 100 = 1 VI. When the multiplier is 20, 30, 40, 200, 300, 2000, 4000, &c. These are composite numbers, of which 10, or 100, or 1000, &c. is one of the factors. Thus 20 = 2 X 10; 30 = 3 X 10; 300 = 3 X 100; &c. In the same manner 387000 = 387 X 1000. How much will 30 hogsheads of wine come to, at 87 dollars per hogshead ? Operation. 87 It appears that it is sufficient in this example to multiply by 3 and then annex a zero w the product. If the number of hogsheads had been 300, or 3000, iwo or threc zeros must have been annexed. It is plain also that, if there are zeros on the right of the multiplicand, they may be omitted until the multiplication has been performed, and then annexed to the product. VII. A man bought 26 pipes of wine, at 143 dollars a pipe ; how much did they come to ? 26 = 20+ 6. The operation may be performed thus : 143 6 858 dolls. price of pipes 143 20 2860 dolls. price of 20 pipes = 3718 dolls. price of 26 pipes 143 2860 dolls. price of 20 pipes + 858 dolls. price of 6 pipes = 3718 dolls. price of 26 pipes 143 858 dolls. price of 6 pipes + 2360 dolls. price of 20 pipes = 3718 dolls. price of 26 pipes If the wages of 1 man be 438 dollars for 1 year, what will be the wages of 234 men, at the same raje? Operution. 438 87600 uolls. wages of 200 men + 13140 do. wages of 30 men + 1752 do. wages of 4 men =102492 dolls. wages of 234 men =102492 dolls. wages of 234 men When we multiply by the 30 and the 200, we need not annex the zeros at all, if we are careful, when multiplying by the tens, to set the first figure of the product in the ten's place, and when multiplying by hundreds, to set the first figure in the hundred's place, &c. Operation. 433 234 1752 1314. 876.. 102,492 If we compare this operation with the last, we shall find that the figures stand precisely the same in the two. We may show by another process of reasoning, that when we multiply units by tens, the first figure of the product should stand in the tens' place, &c.; for units multiplied by tens ought to produce tens, and units multiplied by hundreds, ought to produce hundreds, in the same manner as tens multipiied by units produce tens. If it take 353 dollars to support a family one ycar, how many dollars will it take to support 207 such families the same time? Operation. 853 In this example I multiply first by the 7 207 units, and write the result in its proper place ; then there being no tens, multiply next by 5971 the 2 hundreds, and write the first figure of 1706 this product under the hundreds of the first product; and then add the results in the order 176571 in which they stand. The general rule therefore for multiplying by any number of figures niay be expressed thus: Multiply cach figure of the multiplicand by each figure of the multiplier separately, taking care when multiplying by units to make the fist figure of the result stand in the units' place; and when multiplying by tens, to make the first figure stand in the tens' place; and when multiplying by hundreds, to make the first figure stand in the hundreds' place, foc. and then add the several producis together. Note. It is generally the best way to set the first figure of each partial product directly under the figure by which you are multiplying. Proof. The proper proof of multiplication is by division, consequently it cannot be explained here. There is also a method of proof by casting out the nines, as it is called. But the nature of this cannot be understood, until the pupil is acquainted with division. It will be explained in its proper place. The instructer, if he chooses, may explain the use of it here. SUBTRACTION. VIII. A man having ten dollars, paid away three of them ; how many had he left ? We have seen that all numbers are formed by the successive addition of units, and that they may also be formed by adding together two or more numbers smaller than themselves, but all together containing the same number of units as the number to be formed. The number, 10 for example, may be formed by adding 3 to 7,7 + 3 = 10. It is easy to see therefore that any number may be decomposed into two or more numbers, which taken together, shall be equal to that number. Since 7+3= 10, it is evident that if 3 be taken from 10, there will remain 7. The following examples, though apparently different, all require the same operation, as will be immediately perceived. A man having 10 sheep sold 3 of them ; how many had he left? That is, if 3 be taken from 10, what number will reinain ? |