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times as much. 15 times = 645 = 113. Ans. 114! bushels.
If 13 bbls. of beef cost 957 dollars, what will 25 bbls. cost ?
1 bbl. will cost iof 957 dollars, and 25 bbls. will cost 4 of it. To find this, it is best to multiply first by 25, and then divide by 13. For is of 95} is the same as its of 25 times 953.
Operation. 951 x 25 = 23967. 23963 (13
41 = V. Ans. 184,6 dolls. In this example I divide 23967 by 13. I obtain a quotient 184, and a remainder 47, which is equal to . Then 3. divided by 13, gives 3.5, which I annex to the quotient, and the division is completed.
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 barret, how many barrels of cider may be bought for 8 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=*, and 8} = 42. As many times as are contained in 43, that is, as many times as 15 are contained in 43, so many barrels may be bought.
Expressing the division 4 = 213. 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.
times is 2 contained in 5 ? 2= = ; 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 i lb. of sugar cost i of a dollar, what will 4 lb. 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 ž, for j = 1, and I of g = .
If 1 lb. of figs costs of a dollar, what will 7 lb. cost ?
Dividing the denominator by 7, the fraction becomes : Now it is evident that fourths are 7 times as large as twentyeighths, because if fourths be divided into 7 parts, the parts will be twenty-eighths. Ans. dolls.
Or multiplying the numerator, 7 times it is it. 28, and
so that the answers are the same. Therefore, to multiply a fraction, divide the denominator, when it can be done without a remainder.
Two ways have now been found to multiply fractions, and two ways to divide them. To multiply a fraction The numerator, Art. 15. To divide a fraction The denominator, Art. 17. To divide a fraction
The numerator, Art. 17. Té multiply a fraction @ The denominator, Art. 18
XIX. We observed a remarkable circurastance in the last article, viz. that 1 = and 1=i}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 hat he left ?
į and cannot be added together, because the parts are of different values. Their sum must be more than ž, and less than ı 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 į = 4 = , &c. And s== }, &c. It appears, therefore, that they may both be changed to sixths.
Ž, which added together make & He had sold and had left.
A man sold of a barrel of flour at one time, and at another, how much did he sell in the whole ?
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 žš, and will make is, and the two added together make 4}, that is, 16 Ans. 17 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 alıke, is called reducing them to a common denominator.
In order to reduce pounds to shillings, we muliiply 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 I of the former. If the denominator be multip'ied 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 irito 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 multiplied 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. Ilence 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 de nominator, 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 is. 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 35. This is called reducing fractions to lower terms. Hence
To reduce a fraction to lower terms, divide both the nume
rator 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 4 pieces of cloth, the first contained 239 yards; the second 2812 ; the third 3715 ; 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 denominator. But before reducing them to a common denominator, I observe that some of them may be reduced to lower terms, which will render it much easier to find the common denominator. In the numerator and denominator may both be divided by 2, and it becomes i may be reduced to s, and i to š. I find also that halves may be reduced to iourths, therefore I have only to find the common denominafor of the three first fractions, and the fourth can be reduced to the same.
Multiplying the denominators together 3 X 4 X 5 = 60. The common denominator is 60. Now 3 is multiplied by 4 and by 5 to make 60, therefore, the numerator of į must be multiplied by 4 and by 5, or, which is the same thing, by 23, which makes 40, ? 6o. In 1, the four is multiplied by 3 and 5 to make 60, therefore these are the numbers by which the numerator 3 must be multiplied. = 46. In the fraction ž, the 5 is multiplied by 3 and 4 to make 60, therefore these are the numbers by which the numerator 1 must be multiplied. =3 = 20 These results may be verified, by taking s, i, and of 60. It will be seen that of 60 is 20, the product of 4 and 5; 1 of 60 is 15, the product of 3 and 5; and of 60 is 12, the product of 3 and 4 Now the numbers may be added as follows: 235 233 2376
Ans. 10770 yards.
127 I add together the fractions, which make 13 = 27. I write the fraction õõ, and add the 2 whole ones with the others.