Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

That

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? 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 ope ration is called subtraction. It is the reverse of addition. Addition puts numbers together, subtraction separates & 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 hatꞌ

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 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,

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 flock, sold 17 of them; how many had he then ?

Operation.
He had 62 sheep
Sold 17 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 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 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.*

4540+ 5

This separation may be made in the mind as well as to write it down.

Operation. 62

17

Here I suppose 1 ten taken from the & tens, 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 soparating 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.

Ans.

Operation.

1600 + 120 + 7 = 1727
1600+ 40+2=1642

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.

23,007 Here I take 5 from 7 and there remains 15,265 2; but it is impossible to take 6 (tens) from

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.

10,000+12,000+ 900 + 100+7=23,007
10,000+ 5,000+200+ 60+5=15,265
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 1 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

the others, without separating it into parts except in the mind.

I say 5 from 7, there remains 2: then borrowing 10 (which must in fact come from the 3 (thousand), I say, 6 (tens) from 10 (tens) there remains 4 (tens ;) then I borrow ten again, but since I have already used one of these, I say, 2 (hundreds) from 9 (hundreds) there ren.ains 7 (hundreds ;) then I borrow ten again, and having borrowed one out of the 3 (thousand,) I say, 5 (thousand) from 12 (thousand) there remains 7 (thousand ;) then I (ten-thousand) from 1 (tenthousand) nothing remains. The answer is 7,742 as before.

The general rule for subtraction may be expressed thus: The less number is always to be subtracted from the larger. Begin at the right hand and take successively each figure of the less number from the corresponding figure of the larger number, that is, units from units, tens from tens, &c. If it happens that any figure of the less number cannot be taken from the corresponding figure of the larger, borrow ten and join it with the figure from which the subtraction is to be made, and then subtract; before the next figure is subtracted take care to diminish by one the figure from which the subtraction is to be made.

N. B. When two or more zeros intervene in the number from which the subtraction is to be made, all, except the first, must be called 9s in subtracting, that is, after having borrowed ten, it must be diminished by one, on account of the ten which was borrowed before.

Note. It is usual to write the smaller number under the greater, so that units may stand under units, and tens under tens, &c.

Proof. A man bought an ox and a cow for 73 dollars, and the price of the cow was 25 dollars; what was the price of the ox?

The price of the ox is evidently what remains after taking 25 from 73.

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

It appears that the ox cost 48 dollars. If the cow cost 25 dollars, and the ox 48 doliars, it is evident that 25 and 48 added together must make 73 dollars, what they both cost.

Hence to prove subtraction, add the remainder and the smaller number together, and if the work is right their sum will be equal to the larger number.

Another method. If the ox cost 48 dollars, this number Laken from 73, the price of both, must leave the price of the cow, that is, 25. Hence subtract the remainder from the larger number, and if the work is right, this last remainder will be equal to the smaller number.

Proof of addition. It is evident from what we have seen of subtraction, that when two numbers have been added together, if one of these numbers be subtracted from the sum, the remainder, if the work be right, must be equal to the other number. This will readily be seen by recurring to the last example. In the same manner if more than two numbers have been added together, and from the sum all the rumbers but one, be subtracted, the remainder must be equal to that one.

DIVISION.

IX. A boy having 32 apples wished to divide them equally among 8 of his companions; how many must he give them apiece?

If the boy were not accustomed to calculating, he would probably divide them, by giving one to each of the boys, and then another, and so on. But to give them one apiece would take 8 apples, and one apiece again would take 8 more, and so on. The question then is, to see how many times 8 may be taken from 32; or, which is the same thing, to see how many times 8 is contained in 32. It is contained four times. Ans. 4 each.

A boy having 32 apples was able to give 8 to each of his companions. How mary companions had he?

This question, though different from the other, we perceive, is to be performed exactly like it. That is, it is the question to see how many times 8 is contained in 32. We take away 8 for one boy, and then 8 for another, and so on.

A man having 54 cents, laid them all cut for oranges, at 6 cents apiece. How many did he buy?

1

« ΠροηγούμενηΣυνέχεια »