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Ans. 202937 miles in 53 days. Hence to divide a number into parts; divide it by the number of parts required, and if there be a remainder, make it the numerator of a fraction, of which the divisor is the denominator.x

N. B. This rule is substantially the same as the rule it Art. X.

When one part is found, any number of the parts may be found by multiplication.

In was shown in Art. X. that, in a fraction, the denominator shows into how many parts i is supposed to be divided, and that the numerator shows how many of the parts are used. It will appear from the following examples, that the numerator is a dividend, and the denominator a divisor, and that the fraction expresses a quotient. The denominator shows into how many parts the numerator is to be divided. In this manner division may be expressed without being actually performed. If the fraction be multiplied or divided, the quotient will also be multiplied or divided. Hence division may be first expressed, and the necessary operations performed on the quotient, and the operation of division itself omitted, until the last, which is often more convenient. Also, when the divisor is larger than the dividend, division may be expressed, though it cannot be performed.

A gentleman wishes to divide 23 barrels of flour equally among 57 families ; how much must he give them apiece ?

In this example, the divisor 57 is greater than the dividend 23. If he had had only 1 barrel to divide, he

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could give them only z' of a barrel apiece; but since he had 23 barrels, he can give each 23 times as much, that is, of a barrel.

Hence it appears that it rightly expresses the quotient of 23 by 57.

If it be asked, how many times is 57 contained in 23? It is not contained one time, but y of one time.

If 10 lbs. of copper cost 3 dollars, what is it per lb.?

Here 3 must be divided by 10. to of 1 is o, and 1o of 3 must be i Ans. 7 of a dollar, that is, 30 cents.

At 43 dollars per hhd., what would be the price of 25 galls. of gin?

25 galls. are of a hogshead. To find the price of 1 gallon is to find it's of 43 dolls., and to find the price of 25 galls. is to find 's of 43 dolls. is of 1'is ofof of 43 is 43 times as much, that is, it is 25 times as much as o's, that is, 25 times i . 25 times are

= 17 dolls. Ans. If 5 tons of hay cost 138 dolls. what cost 3 tôns ? 3 tons will cost of 138 dolls. This may be done as follows. of 138 is 27}, and 3 times 273, are 827 dolls. Ans. Or,

Expressing the division, instead of performing it, } of 138 is 148.of 138 are 3 times 138, that is, *1 = 82 dolls. as before.

Note. 1 of 138 by the above rule is 27. But the same result will be obtained, if we say { of 138 is 138, for 138 are equal to 27.

The process in this Art. is called multiplying a whole number" by a fraction. Multiplication strictly speaking is repeating the number a certain number of times, but by extension, it is made to apply to this operation. The definition of multiplication, in its most extensive sense, is to take one number, as many times as one is contained in another number. Therefore if the multiplier be greater than 1, the product will be greater than the multiplicand; but if the multiplier be only a part of 1, the product will be only a part of the multiplicand.

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It was observed in Art. III. that when two whole numbers are to be multiplied together, either of them may be made the multiplier, without affecting the result. In the same manner, to multiply a whole number by a fraction, is the same as to multiply a fraction by a whole number.

For in the last example but one, in wbich 13 was multiplied by 5, 25 and 43 were multiplied together, and the product written over the denominator 63, thus 1:35. The same would have been done, if it had been multiplied by 43.

In the last example also, 138 was multiplied by s. The result would have been the same if i had been multiplied by 138.

This may be proved directly. It is required to find it of 43. f of 1 is ''}, 'of 43 must be 43 times as much, that is, 43 times des, or 1:35 = 174. So also of 1 is }, },of 138 must be 138 times as much, that is, 138 times , or *** = 82.

Hence to multiply a fraction by a whole number, or a whole number by a fraction ; multiply the whole number and the numerator of the fraction together, and write the product over the denominator of the fraction.

XVII. If 3 yards of cloth cost of a dollar, what is that a yard?

are 3 parts. } of 3 parts is 1 part. Ans. of a dollar.

A man divided 14 of a barrel of flour equally among 4 families ; how much did he give them apiece ?

14 are 12 parts. 1 of 12 parts is 3 parts. Ans. of a barrel each.

This process is dividing a fraction by a whole number. A fraction is a certain number of parts. It is evident that any

number of these parts may be divided into parcels, as well as the same number of whole ones.

The numerator shows how many parts are used; therefore to divide a fraction, divide the numerator.

But it generally happens that the numerator cannot be exactly divided by the number, as in the following example.

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A boy wishes to divide of an orange cqually between two other boys; how much must he give them apiece?

If he had 3 oranges to divide, he might give theme 1 apiece, and then divide the other into two equal parts, and give one part to each, and each would have 1 orange. Or he might cut them all into two equal parts each, which would make six parts, and give 3 parts to each, that is, i=1, as before. But according to the question, he has or 3 pieces, consequently he may give 1 piece to each, and then cut the other into two equal parts, and give 1 part to each, then each will have 1 and 1 of 1. But if a thing be cut into four equal parts, and then each part into two equal parts, the whole will be cut into 8 equal parts or eighths; consequently of 1 is š. Each will have and of an orange.

Or he may cut each of the three parts into two equal parts, and give į of each part to each boy, then each will have 3 parts, that is s. Therefore į ofis g. Ans. .

A man divided } of a barrel of four equally between 2 laborers ; what part of the whole barrel did he give to each ?

To answer this question it is necessary to find į of . If the whole barrel be divided first in 5 equal parts or fifths, and then each of these parts into 2 equal parts, the whole will be divided into 10 equal parts. Therefore, I of } is jó. He gave them 75 of a barrel apiece.

A man owning of a share in a bank, sold of his part; what part of the whole share did he sell ?

If a share be first divided into 8 equal parts, and then each part into 3 equal parts, the whole share will be divided into 24 equal parts. Therefore of j is za and of is 7 times as much, that is r. Ans. at:

Or since j = = 1, and į of 31

In the three last examples the division is performed by multiplying the denominator. In general, if the denominator of a fraction be multiplied by 2, the unit will be divided into twice as many parts, consequently the parts will be only one half as large as before, and if the same number of the small parts be taken, as was taken of the large the value of the fraction will be one half as

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much. If the denominator be multiplied by three, each part will be divided into three parts, and the same number of the parts being taken, the fraction will be one third of the value of the first. Finally, if the denominator be multiplied by any number, the parts will be so many times smaller. Therefore, to divide a fraction, if the numerator cannot be divided exactly by the divisor, multiply the denominator by the divisor.

A man divided 16 of a hogshead of wine into 7 equal parts, in order to put it into 7 vessels ; what part of the whole hogshead did each vessel contain ?

The answer, according to the above rule, is tír. The propriety of the answer may be seen in this manner. Suppose each 16th to be divided into 7 equal parts, the parts will be 112ths. From each of the take one of the parts, and you will have 5 parts, that is in.

A man owned its of a ship's cargo; but in a gale the captain was obliged to throw overboard goods to the amount of of the whole cargo. What part of the loss must this man sustain ?

It is evident that he must lose of his share, that is,

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must be 4 times as much, that is, o. Ans. 1 of the whole loss. .

Or it may be said, that since he owned te of the ship, he must sustain is of the loss, that is, is off. of 1 / 2 공 = Tás: i't of = 1 it, and it is 7 times as much, that is,

as before. This process is multiplying one fraction by another, and is similar to multiplying a whole number by a fraction, Art. XVI. If the process be examined it will be found that the denominators were multiplied together for a new denominator, and the numerators for a new numerator. In fact to take a fraction of any number, is to divide the number by the denominator, and to multiply the quotient by the numerator. But a fraction is divided by multiplying its denommator, and multiplied by multiplying its numerator.

We bave seen in the above example, that when two fractions are to be multiplied, either of them may be made multiplier, with

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