umy number consisting of 1, with any number of zeros at the right of it it is sufficient to annex the zeros to the multiplicand. VI. When the multiplier is 20, 30, 40, 200, 300, 2000, 1000, &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 = 337 X 1000. How much will 30 hogsheads of wine come to, at 87 dollars per hogshead ? Operation. 87 261 dolls. price of 3 hhds. 2010 dolls. price of 30 hhds. More simply thus 87 30 2610 dolls. price of 30 hhds. It appears that it is sufficient in this example to multiply by 3 and then annex a zero lo the product. If the number of hogsheads had been 300, or 3000, two 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 2860 dolls. price of 20 pipes + 858 dolls. price of 6 pipes = 3718 dolls. price of 26 pipes 143 2860 doils. price of 20 pipes + 858 doll:. 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 pipeg If the wages of I man be 433 dollars for 1 year, what will be the wages of 234 men, at the same raje ? Operation. 438 Or thus 433 1752 dolls. wages of 4 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. 438 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 hy 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 multiplied by units produce tens. If it take 353 dollars to support a family one yerır, horo 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, I multiply next hy 5971 the 2 hundreds, and write the first figure of 1706 this product under tha 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 nay be expressed thus : Multiply each figure of the multiplicand by each figure of the multiplier separately, tak ing care when multiplying by units to make the first 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. easy to 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 fuccessive 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 see therefore that any number may be decomposed into two or inore 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 remain ? 1 A man gave 3 dollars to one son, and 10 to another ; hoid much more did he give to the one than to the other ? That is, how much greater is the number 10 than the number 3 ? A man owing 10 dollars, paid 3 dollars at one time, and the rest at another; how much did he pay the last time ? That is, how much must be added to 3 to make 10 ? From Boston to Dedham it is 10 miles, and from Boston to Roxbury it is only 3 miles; what is the difference in the two distances from Boston ? A boy divided 10 apples between two other boys ; to one he gave 3, how many did he give to the other ? That is, if 10 be divided into two parts so that one of the parts may be 3, what will the other part be? It is evident that the above five questions are all answered by taking 3 from 10, and finding the difference. This operation is called subtraction. It is the reverse of addition, Addition puts numbers together, subtraction separates a number into two parts. A man paid 29 dollars for a coat and 7 dollars for a kat, how much more did he pay for his coat than for his hat ? In this example we have to take the 7 from the 29; we know from addition, that 7 and 2 are 9, and consequently that 22 and 7 are 29; it is evident therefore that if a be taken from 29 the remainder will be 22. A man bought an ox for 47 dollars ; to pay for it he gave a cow worth 23 dollars, and the rest in money ; how much money did he pay ? Operation. It will be best to perform thi: example by parts. It is plain that we must take the twenty from the forty, and the three from the seven ; that is, the tens from the tens, and the units from the units. I take twenty from forty, and there remains twenty. I then take three from seven, and there remains four, and the whole remainder is twenty-four. Ans. 24 dollars. It is generally most convenient to write the numbers under each other. The smaller number is usually written under the larger. Since units are to be taken from units, and tens from .ens, it will be best to write units under units, |