Or thus 438 234 1752 dolls. wages of 4 men =102492 dolls. wages of 234 men When we multiply by the 30 and the 200, we need not amex 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 Bat 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 853 dollars to support a family one ycar, hovo 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 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 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, foc. 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 unis 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 remain ? That is, 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 ? it 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. 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 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, 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. I say first 3 from 7, and there will Cow 23 dollars. 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 flock, sold 17 of them ; how many had he then ? Operation. He had 62 sheep In this example a difficulty immedi Sold 17 sheep ately presents itself, if we attempt to perform the operation as before; for Had left 45 sheep we cannot take 7 from 2. We can, liowever, 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 : 62 = 50 + 12 That is, I take one ten from the six 17 = 10 + 7 tens, and write it with the two units. But the 17 I separate simply into units 45 = 40 + 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 8 tens, 17 and written with the 2, which makes 12. I say 7 from 12, 5 remains, then setting down the 5, I 45 say, 1 ten from 5 tens, or simply 1 from 5, and there remains 4 (tens), which written down shows the remainder, 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. Sir Isaac Newton was born in the year 1642, and he died in 1727; how old was he at the time of his decease ? It is evident that the difference between these two numbers must give his age. Operation. 1600 + 40 +2=1642 Ans. 80+5= 85 years old. . In this example I take 2 from 7 and there remains 5, which I write down. But since I cannot take 4 (tens) from 2 (tens,) I borrow 1 (hundred) or 10 tens from the 7 (hundreds,) which joined with 2 (tens) makes 12 (tens,) then 4 (tens) from 12 (tens) there remains 8 (tens,) which I write down. Then 6 (hundreds) from 6 (hundreds) there remains nothing. Also 1 (thousand) from 1 (thousand) nothing remains. The answer is 85 years. A man bought a quantity of flour for 15,265 dollars, and sold it again for 23,007 dollars, how much did he gain by the bargain ? Operation. 0, and it does not immediately appear where 2 I shall borrow the 10 (tens,) since there is nothing in the hundreds' place. This will be evident, how ever, if I decompose the numbers into parts. Operation. 7,000 + 700 + 40 +2= 7,742 The 23,000 is equal to 10,000 and 13,000; this last is equal to 12,000 and 1,000 ; and 1,000 is equal to 900 and 100. Now I take 5 from 7, and there remains 2 ; 60 from 100, or 6 tens from 10 tens, and there remains 40, or 4 tens; 2 hundreds from 9 hundreds, and there remains 7 hundreds; 5 thousands from 12 thousands, and there remains 7 thousands; and I ten-thousand from 1 ten-thousand, and nothing remains. The answer is 7,742 dollars. This example may be performed in the same manner as |