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: 143 858 dolls. price of 6 pipes 143 20 2860 dolls. price of 20 pipes +858 dolls. price of 6 pipes =3718 dolls. price of 26 pipes The operation may be performed more simply thus; 143 26 858 dolls. price of 6 pipes +2860 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 rate? Operation. 234 87600 dolls. wages of 200 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. 438 234 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 multiplied by units produce tens. If it take 853 dollars to support a family one year, how many dollars will it take to support 207 such families the same time? Operation. 853 207 5971 1706 176571 In this example I multiply first by the 7 units, and write the result in its proper place; then there being no tens, I multiply next by the 2 hundreds, and write the first figure of this product under the hundreds of the first product; and then add the results in the order in which they stand. The general rule therefore for multiplying by any number of figures may be expressed thus: Multiply each figure of the multiplicand by each figure of the multiplier separately, taking 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, &c. and then add the several products 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 re main ? A man gave 3 dollars to one son, and 10 to another; how 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 hat, 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 7 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. Ox 47 dollars. Cow 23 dollars. It will be best to perform this 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 unIder each other. The smaller number is usually written under the larger. Since units are to be taken from units, and tens from tens, it will be best to write units under units, tens under tens, &c. as in addition. It is also most convenient, and, in fact, frequently necessary, to begin with the units as in addition and multiplication. Operation. Ox 47 dollars. Cow 23 dollars. I say first 3 from 7, and there will remain 4. Then 2 (tens) from 4 (tens) and there will remain 2 (tens), 24 difference. and the whole remainder is 24. A man having 62 sheep in his flack, sold 17 of them; how many had he then? Operation. He had 62 sheep In this example a difficulty immediately presents itself, if we attempt to perform the operation as before; for Had left 45 sheep. we cannot take 7 from 2. We can, however, take 7 from 62, and there remains 55; and 10 from 55, and there remains 45, which is the answer. The same operation may be performed in another way, which is generally more convenient. I first observe, that 62 is the same as 50 and 12; and 17 is the same as 10 and 7. They may be written thus: 6250+12 17=10+ 7 That is, I take one ten from the six tens, and write it with the two units. But the 17 I separate simply into units 4540+ 5 and tens as they stand. Now I can take 7 from 12, and there remains 5. Then 10 from 50, and there remains 40, and these put together make 45.* This separation may be made in the mind as well as to write it down. Operation. 62 Here I suppose 1 ten taken from the 6 tens, 17 and written with the 2, which makes 12. I say 7 from 12, 5 remains, then setting down the 5, I say, 1 ten from 5 tens, or simply 1 from 5, and there remains 4 (tens), which written down shows the remainder, 45. 45 The taking of the ten out of 6 tens and joining it with the 2 units, is called borrowing ten. Let the pupil perform a large number of examples by separating them in this way, when he first commences subtraction. |