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figures on the left of the dividend ; I find 7 times, and 7 is contained 6 times. The limits are 6 and 7. 7 times 6 are 42, and 42 from 43 leaves 1, which I
supposo placed by the side of 6; this makes 16. But 4, the second figure of the divisor, is not contained 7 times in 16, therefore 6 will the first figure of the quotient.
It is easy to see that this must be 6000, when the division is completed; because there being five figures in the divisor, and the first figure of the divisor being larger than the first figure of the dividend, we are obliged to take the six first figures of the dividend for the first partial dividend; and the dividend containing nine figures, the right hand figure of this partial dividend, is in the thousands' place. I write 6 in the quotient, and multiply the divisor by it, and write the result inder the dividend, so that the first figure on the right hand may stand under the sixth figure of the dividend, counted from the left, or under the place of thousands. This product, subtracted from the dividend as it stands, leaves a remainder 51518; by the side of this I bring down the next figure of the dividend, which is 0, and the second partial dividend is 515180. Trying as before with the 6, and then with the 4, into the first figures of this partial dividend, I find the divisor is contained in it 8 (800) times. 1 write 8 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 pui 0 in the quotient, so that the other figures inay 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-t 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 more. Seek how
many times the first fiyure of the divisor is cortained in the first, or two first figures of these, then increasing the first figure of the divisor by one, seek 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 sufficient 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 dividerid by the side of the remainder. This is the next partial dividend. Seek as before how many times the divisor is contuined 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 happens 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.
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.
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 suppose 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 dollarc 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 bay 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 10 distinguish one part from another.
When any thing, or any number is divided into tiro 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 in 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. li 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 of 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 of 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 į 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 i dollar buy?
This question is the reverse of the other. But we have just seen that 1 is 1 of two, 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 į of a yard cost ? What will 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