out affecting the result. Therefore, to take a fraction of a fraction, that is, to multiply one fraction by another, multiply the denominators together for a new denominator, and the numerators for a new numerator.X

If 7 dollars will buy 53 bushels of rye, how much will 1 dollar buy? How much will 15 dollars buy?

43

7

I dollar will buy of 53 bushels. In order to find of it, 5 must be changed to eighths. 53 43. of 4343. 1 dollar will buy 3 of a bushel. 15 dollars will buy 15 times as much. 15 times = 451128. Ans. 113 bushels.

8

6 3

56

=

If 13 bbls. of beef cost 957 dollars, what will 25 bbls. cost?

1

1 bbl. will cost of 957 dollars, and 25 bbls. will of it. To find this, it is best to multiply first by 25, and then divide by 13. For of 957 is the same

cost

In this example I divide 23967 by 13. I obtain a quotient 184, and a remainder 47, which is equal to 39. Then divided by 13, gives, which I annex to the quotient, and the division is completed.

39

The examples hitherto employed to illustrate the division of fractions, have been such as to require the division of the fractions into parts. It has been shown (Art. XVI.) that the division of whole numbers is performed in the same manner, whether it be required to divide the number into parts, or to find how many times one number is contained in another. It will now be shown that the same is true with regard to fractions.

At 3 dollars a barrel, how many barrels of cider may be bought for 83 dollars?

The numbers must be reduced to fifths, for the same reason that they must be reduced to pence, if one of the numbers were given in shillings and pence.

3=15, and 83 = 43. As many times as 15 are contained in 43, that is, as many times as 15 are contained in 43, so many barrels may be bought.

Expressing the division 43213. Ans. 213 barrels. This result agrees with the manner explained above. For 8 was reduced to fifths, and the denominator 15 was formed by multiplying the denominator 5 by the divisor 3.

How many times is 2 contained in §?

2=

14; 14 is contained in 5, of one time. The same result may be produced by the other method.

XVIII. We have seen that a fraction may be divided by multiplying its denominator, because the parts are made smaller. On the contrary, a fraction may be multiplied by dividing the denominator, because the parts will be made larger. If the denominator be divided by 2, for instance, the denominator being rendered only half as large, the unit will be divided into only one half as many parts, consequently, the parts will be twice as large as before. If the denominator be divided by 3, the unit will be divided into only one third as many parts, consequently the parts will be three times as large as before, and if the same number of these parts be taken, the value of the fraction will be three times as great, and so on.

If 1 lb. of sugar cost of a dollar, what will 4 lbs. cost? If the denominator 8 be divided by 4, the fraction becomes; that is, the dollar, instead of being divided into 8 parts, is divided into only 2 parts. It is evident that halves are 4 times as large as eighths, because if each half be divided into 4 parts, the parts will be eighths. Ans. doll.

If it be done by multiplying the numerator, the answer is, which is the same as, for = 1, and 1⁄2 of

If 1 lb. of figs cost of a dollar, what will 7 lbs. cost? Dividing the denominator by 7, the fraction becomes

Now it is evident that fourths are 7 times as large as twenty-eighths, because if fourths be divided into 7 parts, the parts will be twenty-eights. Ans dolls. Or multiplying the numerator, 7 times is. But 1=278, and 2, so that the answers are the same. Therefore, to multiply a fraction, divide the denominator, when it can be done without a remainder.

21 289

28

Two ways have now been found to multiply fractions, and two ways to divide them.

the last article, viz. that

and 2 This will

=

be found very important in what follows. A man having a cask of wine, sold

of it at one time, and of it at another; how much had he left?

and cannot be added together, because the parts are of different values. Their sum must be more than , and less than or 1. If we have dollars and crowns to add together, we reduce them both to pence. Let us see if these fractions cannot be reduced both to the same denomination. Now = 2 = 3 = 4, &c. And == }, &c. It appears, therefore, that they may both be changed to sixths. & and, which added together make. He had sold and had left. A man sold of a barrel of flour at one time, at another; how much did he sell in the whole?

3

and

Fifths and sevenths are different parts, but if a thing be first divided into 5 equal parts, and then those parts each into 7 equal parts, the parts will be thirty-fifths. Also if the thing be divided first into 7 equal parts, and then those parts each into 5 equal parts, the parts will be thirty-fifths. Therefore, the parts will be alike. But in dividing them thus, will make 3, and 4 will make

20

, and the two added together make, that is, 158. Ans. 1

13

barrel.

When the denominators of two or more fractions are alike, they are said to have a common denominator. And the process by which they are made alike, is called reducing them to a common denominator.

In order to reduce pounds to shillings, we multiply by 20, and to reduce guineas to shillings, we multiply by 28. In like manner to reduce two or more fractions to a common denominator, it is necessary to find what denomination they may be reduced to, and what number the parts of each must be multiplied by, to reduce them to that denomination.

If the denominator of a fraction be multiplied by 2, it is the same as if each of the parts were divided into 2 equal parts, therefore it will take 2 parts of the latter kind to make 1 of the former. If the denominator be multiplied by 3, it is the same as if the parts were divided each into 3 equal parts, and it will take 3 parts of the latter kind, to make 1 of the former. Indeed, whatever number the denominator be multiplied by, it is the same as if the parts were each divided into so many equal parts, and it will take so many parts of the latter kind to make 1 of the former. Therefore, to find what the parts must be multiplied by, it is necessary to find what the denominator must be inultiplied by to produce the denominator required.

The common denominator then, (which must be found first) must be a number of which the denominators of all the fractions to be reduced, are factors. We shall always find such a number, by multiplying the denominators together. Hence if there are only two fractions, the denominators being multiplied together for the common denominator, the parts of one fraction must be multiplied by the denominator of the other. If there be more than two fractions, since by multiplying all the denominators together, the denominator of each will be multiplied by all the others, the parts in each fraction, that is, the numerators, must be multiplied by the denominators of the other fractions.

In the above example to reduce and to a common denominator, 7 times 5 are 35; 7 is the number by which the first denominator 5 must be multiplied to produce 35, and consequently the number by which the numerator 3 must be multiplied. 5 is the number, by which 7, the second denominator, must be multiplied to produce 35, and consequently the number by which the numerator 4 must be multiplied.

N. B. It appears from the above reasoning, that if both the numerator and denominator of any fraction be multiplied by the same number, the value of the fraction will remain the same. It will follow also from this, that if both numerator and denominator can be divided by the same number, without a remainder, the value of the fraction will not be altered. In fact, if the numerator be divided by any number, as 3 for example, it is taking

of the number of parts; then if the denominator be divided by 3, these parts will be made 3 times as large as before, consequently the value will be the same as at first. This enables us frequently, when a fraction is expressed with large numbers, to reduce it, and express it with much smaller numbers, which often saves a great deal of labour in the operations.

Take for example 1. Dividing the numerator by 5, we take of the parts, then dividing the denominator by 5, the parts are made 5 times as large, and the fraction becomes, the same value as 15. This is called reducing fractions to lower terms. Hence

To reduce a fraction to lower terms, divide both the numerator and denominator by any number that will divide them both without a remainder.

Note. This gives rise to a question, how to find the divisors of numbers. These may frequently be found by trial. The question will be examined hereafter.

A man bought four pieces of cloth, the first contained 23 yards; the second 288; the third 373; and the fourth 17. How many yards in the whole?

The fractional parts of these numbers cannot be added together until they are reduced to a common de