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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 forths, five fifths, &c. to make a whole

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 rew unit, and that we may number these parts as weil 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 ine 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 į 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 aborc 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. į 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 1 of a barrel.

If a yard of cloth cost 3 dollars, what will of a yard cosł? What will off 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 yard can be bought for 1 dollar, and what part for 2 dolars; the answer will evidently be of a yard for 1 dollar, and for 2 dollars.

It is easy to see that any number may be divided into as inany 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, of 5, because 1 is 1 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. 70 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?

Opcration.

143 (15 Price of 9 months 135

918 months. Remainder

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

for of a month & months. Note. 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%

4 tons.
527
485

Ans. 9,

Remainder 42 dollars. After paying for 25 tons, there are 42 dollars left. Idol lar will buy o'y of a ton, and 42 dollars will buy of a ton. How many times is 324 contained in 18364 ?

Operation.
1836-4 (324
1620

563 times
2164
1914

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32

J! is contained 56 times and 220 over. I is 1 of 324, and 220 is of 324. Ans. 56. times and 22. of another time.

From the above examples, we deduce the following genesal rule for the remainder : When the division is performed, as far as it can be, if there is a remainder, in order to have the true quotient, write the remainder over the divisor in the form of a fraction, and annex it to the quotient.

XI. We observed in Art. V. that when the multiplier is 10, 100, 1000, &c. the multiplication is performed by annexing the zeros at the right of the multiplicand. In like manner when the divisor is 10, 100, 1000, &c. division may be performed by cutting off as many places from the right of the dividend as there are zeros in the divisor.

At 10 cents a pound, how many pounds of meat may be bought for 64 cents ?

The 6 which stands in tens' place shows how many times ten is contained in 60, for 60 signifies 6 tens, and the 4 shows how many the number is more than 6 tens, therefore 4 is the remainder. The operation then may be performed thus, 6.4. The answer is 64 pounds.

A man has 2347 lb. of tobacco, which he wishes to put into boxes containing 100 16. each ; how many boxes will it take?

It is evident that 100 is contained in 2300, 23 times, consequently it will take 23 boxes, and there will be 47 lbs. left, which will fill 4% of another box. The operation may be performed thus, 23.47. Answer 23%

In general if one figure be cut off from the right, the tens will be brought into the units' place, and hundreds into the tens' place, &c. If two figures be cut off, hundreds are brought into the units' place, and thousands into the tens' place, &c. And if three figures be cut off, thousands are brought into the units' place, &c. that is, the numbers will be made 10, 100, or 1000 times less than before.

Hence to divide by 10, 100, 1000, fc. cut off from the right of the dividend as many figures as there are zeros in the divisor. The remuining figures will be the quotient, and the figures cut off will be the remainder, which must be written over the divisor, and annexed to the quotient.

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XII. We observed in Art. X, that any two numbers being given, it is easy to tell what part of the one the other is. Thus :

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15, &c.

What part of 10 yards are 3 yards ? Ans. I is to of 10, and 3 is o of ten.

What part of 237 barrels is 82 barrels ? Ans. I is of 237, and 82 is of 237.

Fractions are properly parts of a unit, but by extension the term fraction is often applied to numbers larger than unity. This happens when the numerator is larger than the denominator, in which case there are more parts taken than are sufficient to make a unit. All fractions in which the numerator is equal to the denominator, as ž, B, E, are equal to unity; all in which the numerator is less than the denominator are less than unity, anu are called proper fractions; all in which the numerator is greater than the denominator, are more than unity, and are called improper fractions. Thus, 43, 4, are improper fractions.

The process of finding what part of one number another number is, is called finding their ratio.

What is the ratio of 5 bushels to 3 bushels, or of 5 to 3 ? This is the same as to say, what part of 5 is 3 ? The answer is . The ratio of 5 to 3 is .

What part of 3 is 5 ? Answer . The ratio of 3 to 5

Ansicer is.

What is the ratio of 35 yards to 17 yards. Answer ..
What is the ratio of 17 to 35 ?

To find what part of one number another is, make the number which is called the part (whether it be the larger or smaller) the numerator of a fraction, and the other number the denominator.

Also to find the ratio of one number tu another, make the number which is expressed first the denominutor, and the other the numerator.

XIII. A gentleman gave t of a collar cach to 17 pour persons ; how many dollars did it take?

It took of a dollar. But of a dollar make a dollar, consequently as many times as 5 is contained in 17, so many dollars it is. 5 is contained 3 times in 17, and 2 over

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