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in the quotient, then multiplying and subtracting as before, I find a remainder 1284. I bring down the next figure of the dividend, which gives 12847 for the next partial dividend. I find that the divisor is not contained in this at all. I put 0 in the quotient, so that the other figures may stand in their proper places, when the division is completed. Then I bring down the next figure of the dividend, which gives for a partial dividend, 128474. The divisor is contained twice in this. Multiplying and subtracting as before, I find no remainder. The division therefore is completed.

Proof. It was observed in the commencement of this Art. that division is proved by multiplying the divisor by the quotient. This is always done during the operation. In the last example, the divisor was first multiplied by 6 (6000,) and then by 8 (800,) and then by 2; we have only to add these numbers together in the order they stand in, and if the work is right, this sum will be the dividend. The asterisms show the numbers to be added.

From the above examples we derive the following general rule for division : Place the divisor at the right of the dividend, separate them by a mark, and draw a line under the divisor, to separate it from the quotient. Take as many

figures on the left of the dividend as are necessary to contain the divisor once or morc. Seek how many times the first figure of the divisor is contained in the first, or two first figures of these, then increasing the first figure of the divisor by one, scek how many times that is contained in the same figure or figures. Take the figure contained within these limits, which appears the most probable, and multiply the two left hand figures of the divisor by it; if that is not suficient to determine, multiply the third, and so on. When the first figure of the quotient is discovered, multiply the divisor by it, and subtract the product from the partial dividend. Then write the next figure of the dividend by the side of the remainder. This is the next partial dividend. Seek as before how many times the divisor is contained in this, and place the result in the quotient, at the right of the other quotient figure, then multiply

and subtract, as before ; and so on, until all the figures of the dividend have been used. If it happers that any partial dividend is not so large as the divisor, a zero must be put in the quotient, and the next figure of the dividend written at the right of the partial dividend.

Note. If the remainder at any time should exceed the

divisor, the quotient figure must be increased, and the multiplication and subtraction must be performed again. If the product of the divisor, by any quotient figure, should be larger than the partial dividend, the quotient figure must be diminished.

Short Division.

When the divisor is a small number, the operation of division may be much abridged, by performing the multiplication and subtraction in the mind without writing the results. In this case it is usual to write the quotient under the dividend. This method is called short division.

A man purchased a quantity of flour for 3045 dollars, at 7 dollars a barrel. How

many

barrels were there? Long Division.

Short Division.
3045 (7

3045 (7
28
435

435
24
21

35
35

In short division, I say 7 into 30, 4 times; I write 4 underneath ; then I say 4 times 7 are 28, which taken from 30 leaves 2. I suppose the 2 written at the left of 4, which makes 24; then 7 into 24, 3 times, writing 3 underneath, I say 3 times 7 are 21, which taken from 24 leaves 3. I şuppose the 3 written at the left of 5, which makes 35; then 7 in 35, 5 times exactly ; I write 5 underneath, and the division is completed.

If the work in the short and long be compared together, they will be found to be exactly alike, except in the short it is not written down.

X. How many yards of cloth, at 6 dollars a yard, may be bought for 45 dollars ?

42 dollars will buy 7 yards, and 48 dollars will buy 8 yards. 45 dollars then will buy more than 7 yards and less than 8 yards, that is, 7 yards and a part of another yard. As cases like this may frequently occur, it is necessary to know what this part is, and how to distinguish one part from another.

When any thing, or any number is divided into two equal parts, one of the parts is called the half of the thing or number. When the thing or number is divided into three equal parts, one of the parts is called one third of the thing or number ; when it is divided into four equal parts, the parts are called fourths ; when into five equal parts, fifths, &c. That is the parts always take their names from the number of parts into which the thing or number is divided. It is evident that whatever be the number of parts into which the thing or number is divided, it will take all the parts to make the whole thing or number. That is, it will take two halves, three thirds, four fourths, five fifths, &c. to make a whole one. It is also evident, that the more parts a thing or number is divided into, the smaller the parts will be. That is, halves are larger than thirds, thirds are larger than fourths, and fourths are larger than fifths, &c.

When a thing or number is divided into parts, any number of the parts may be used. When a thing is divided into three parts, we may use one of the parts or two of them. When it is divided into four parts, we may use one, two, or three of them, and so on. Indeed it is plain, that, when any thing is divided into parts, each part becomes a new unit, and that we may number these parts as well as the things themselves before they were divided.

Hence we say one third, two thirds, one fourth, two fourths, three fourths, one fifth, two fifths, three fifths, &c.

These parts of one are called fractions, or broken numbers. They may be expressed by figures as well as whole numbers; but it requires two numbers to express them, one to show into how many parts the thing or number is to be divided (that is, how large the parts are, and how many it takes to make the whole one); and the other, to show how many of these parts are used. It is evident that these numbers must always be written in such a manner, that we may know what each of them is intended to represent. It is agreed to write the numbers one above the other, with a line between them. The number below the line shows into

how many parts the thing or number is divided, and the number above the line shows how many of the parts are used. Thus şof an orange signifies, that the orange is divided into three equal parts, and that two of the parts or pieces are used. of a yard of cloth, signifies that the yard is supposed to be divided into five equal parts, and that three of these parts are used. The number below the line is called the denominator, because it gives the denomination or name to the fraction, as halves, thirds, fourths, &c. and the number above the line is called the numerator, because it shows how many parts are used.

We have applied this division to a single thing, but it often happens that we have a number of things which we consider as a bunch or collection, and cf which we wish to take parts, as we do of a single thing. In fact it frequently happens that one case gives rise to the other, so that both kinds of division happen in the same question.

If a barrel of cider cost 2 dollars, what will of a barrel cost ?

To answer this question, it is evident the number two must be divided into two equal parts, which is very easily done. of 2 is 1.

Again, it may be asked, if a barrel of cider cost 2 dollars, what part of a barrel will one dollar buy ?

This question is the reverse of the other. But we have just scen that 1 is į of 2, and this enables us to answer the question. It will buy į of a barrel.

If a yard of cloth cost 3 dollars, what will of a yard cost ? What will f of a yard cost ?

If 3 dollars be divided into 3 equal parts, one of the parts will be 1, and two of the parts will be 2. Hence ; of a yard will cost 1 dollar, and will cost 2 dollars.

If this question be reversed, and it be asked, what part of a yard can be bought for 1 dollar, and what part for 2 dollars; the answer will evidently be of a yard fór 1 dollar, and for 2 dollars.

It is easy to see that any number may be divided into as many parts as it contains units, and that the number of units used will be so many of the parts of that number. Hence if

it be asked, what part of 5, 3 is, we say, 1 of 5, because 1 is } of 5, and 3 is three times as much.

We can now answer the question proposed above, viz. How many yards of cloth, at 6 dollars a yard, may be bought for 45 dollars ?

42 dollars will buy 7 yards, and the other 3 dollars will buy of a yard. Ans. 78 yards, which is read 7 yards and

of a yard. A man hired a labourer for 15 dollars a month ; at the end of the time agreed upon, he paid him 143 dollars. How many months did he work?

Operation.

143 (15 Price of 9 months 135

97 months. Remainder

8 The wages of 9 months is 135 dollars, which subtracted from 143, leaves 8 dollars. Now 1 dollar will pay for i's of a month, consequently 8 dollars will pay for 18 of a month. Ans. 9 months.

Notc. The number which remains after division, as 8 in this example, is called the remainder.

At 97 dollars a ton, how many tons of iron may be bought for 2467 dollars ?

Operation.
2467 (97
194

25% tons.
527
485

Remainder 42 dollars.
After paying for 25 tons, there are 42 dollars left. I dol-
lar will buy sy of a ton, and 42 dollars will buy of a ton.
How many times is 324 contained in 18364?

Operation.
18364 (324
1620

5631. times.
2164
1944

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