plier be only a part of 1, the product will be only a part of the multiplicand, 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. 07 For in the last example but one, in which 43 was multiplied by 25, 25 and 43 were multiplied together, and the product written over the denominator 63, thus 8. The same would have been done, if 25 had been multiplied by 43. In the last example also, 138 was multiplied by 3. The result would have been the same if had been multiplied by 138. 63 63 > This may be proved directly. It is required to find 25 of 43. of 1 is 25, of 43 must be 43 times as much, that is, 43 times 2, or 87 = 17. So also of 1 is 3. of 138 must be 138 times as much, that is, 138 times, or 4824. 5 4 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? of 3 parts is 1 part. Ans. of a dollar. of a barrel of flour equally among 4 did he give them apiece? of 12 parts is 3 parts. Ans. of a 17 are 3 parts. A man divided 12 families; how much are 12 parts. 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. A boy wishes to divide of an orange equally between two other boys; how much must he give them apiece? If he had 3 oranges to divide, he might give them 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, 11⁄2, 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 and of 4. 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 is . Each will have and of an orange. Or he may cut each of the three parts into two equal parts, and give of each part to each boy, then each will have 3 parts, that is 8. Therefore of 3 is 3. Ans. . te÷22 3 A man divided of a barrel of flour equally between 2 labourers; what part of the whole barrel did he give to each? To answer this question it is necessary to find 1⁄2 of 1. If the whole barrel be divided first into 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, of } is o He gave them 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 is, and of is 7 times as much, that is, Ans.. Or since, and of 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 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 frac tion, if the numerator cannot be divided exactly by the divisor, multiply the denominator by the divisor. A man divided 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 T. 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 12: A man owned 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, of 18. of much, that is, must be 4 times as Ans. of the whole loss. Or it may be said, that since he owned of the ship, he must sustain of the loss, that is, of of =2 of, and is 7 times as much, that is, 16, as Thof, and 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 denominator, and multiplied by multiplying its numerator. We have seen in the above example, that when two fractions are to be multiplied, either of them may be made multiplier, without affecting the result. Therefore, to take a fraction of a fraction, that is, to multiply one fraction by another, multiply the denenominators together for a new denominator, and the numerators for a new numerator. dollars will buy 53 bushels of rye, how much will 1 dollar buy? How much will 15 dollars buy? If 43 1 dollar will buy of 53 bushels. In order to find of it, 5 must be changed to eighths. 53. of 48 = 23. 1 dollar will buy of a bushel. 15 dollars will buy 15 36 times as much. 15 times = 6,45 11. Ans. 1138 bushels. If 13 bbls. of beef cost 957 dollars, what will 25 bbls. cost? 1 bbl. will cost of 957 dollars, and 25 bbls. will cost 2 of it. To find this, it is best to multiply first by divide by 13. For is the same as 25, and then 1 of of 25 times 957. 56 52 184,34 47 39. Ans. 18432 dolls. In this example I divide 23967 by 13. I obtain a quotient 184, and a remainder 47, which is equal to 39. Then 39 divided by 13, gives 3, 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. 35, 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 83 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 §! 4 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 1 lb. of sugar cost 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 Dividing the denominator by 7, the fraction becomes 3. 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 is . But 7 , and 23, 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. |