456 (8 57 3 57 B's share 171 dollars. A's share 285 dollars. A man bought 68 barrels of pork for 1224 dollars, and sold 47 barrels, at the same rate that he gave for it. How much did the 47 barrels come to ? To answer this question it is necessary to find the price of 1 barrel, and then of 47. 1 barrel costs as of 1224 dollars, and 47 barrels cost of it. o of 1224 is 18. times are 18 are 846. Ans. 846 dollars. To find any fractional part of a number, divide the nunber by the denominator of the fraction, and multiply the quotient by ihe numerator. A man bought 5 yards of cloth for 28 dollars; what was that a yard? of 25 is 5, and į of 30 is 6. of 28 then must be between 5 and 6. Cases of this kind frequently occur, in which a number cannot be divided into exactly the number of parts proposed, except by taking fractions. But it may easily be done by fractions. of 25 dollars is 5 dollars. It now remains to find } of 3 dollars. Suppose each dollar divided into 5 equal parts, and take 1 part from each. That will be 3 parts or of a dollar. Ans. 5} dollars. of a dollar is of 100 cents, which is 60 cents. Ans. $5.60. A man had 853 lb. of butter, which he wished to divide into 7 equal parts ; how many pounds would there be in each part? of 847 lb. is 121 lb. Then suppose each of the 6 remaining pounds to be divided into 7 equal parts, and take 1 part from each ; that will be 6 parts, or of a pound. Ans. 1214 A man having travelled 47 days, found that he had travel led 1800 miles ; how many miles had he travelled in a day on an average? How many miles would he travel in 53 days, at that rate ? In one day he travelled dy of 1800 miles, and in 53 days he would travel of it. 37 of 1800 is 38, and 14 over. of 1 is a't, as of 14 is 14 times as much, that is, 11. In one day he travelled 3844 miles. In 53 days he would travel 53 times 3814 miles. 1800 (47 38 53 141 53 114 212 190 53 14 14 742 + 2014 1527 4 = 1541 Ans. 202937 miles in 53 days. Hence to divide a number into parts; divide it by the number of parts required, and if there te a remainder, make it the numerator of a fraction, of which the divisor is the denominator. N. B. This rule is substantially the same as the rule in Art. X. When one part is found, any number of the parts may be found by multiplication. It was shown in Art. X. that, in a fraction, the denominator shows into how many parts 1 is supposed to be divided, and that the Lumerator shows how many of the parts are used. It will appear from the following examples, that the numerator is a dividend, and the denominator à 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 ex pressed, though it cannot be performed. A gentleman wishes to divide 23 barrels of flour equaily 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 only 1 barrel to divide, he 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, H of a barrel. Hence it appears that 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 fi of one time. If 10 lbs. of copper cost 3 dollars, what is it per 16. ? Here 3 must be divided by 10.1 of 1 is to, and i'y of 8 must be Ans. io 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 og of 43 dolls., and to find the price of 25 galls . is to find of 43 dolls. obs of 1 is 65, of of 43 is 43 times as much, that is, **. is 25 times as much as o's, that is, 25 times . 25 times and are 1835 177 dolls. Ans. If 5 tons of hay cost 138 dolls. what cost 3 tons ? 3 tons will cost of 138 dolls. This may be done as follows. } of 138 is 27, and 3 times 27}, are 824 dolls. Ans. Or, Expressing the division, instead of performing it, of 138 is 138 of 138 are 3 times *38, that is, *}* = 82 dolls. as before. Note. of 138 by the above rule is 27}. But the same result will be obtained, if we say, of 138 is ®, for are equal to 277. The process in this Art. is called multiplying a whole num ber by a fraction. Multiplication strictly speaking is repeating the number a certain number of times, but by exten sion, 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 multi 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. For in the last example but one, in which 43 was multiplied by , 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 had been multiplied by 138. This may be proved directly. It is required to find any of 43. of 1 is, of 43 must be 43 times as much, that is, 43 times, or 137 1775. So also of 1 is , of 138 must be 138 times as much, that is, 138 times, or 44 = 82%. Hence to multiply a fraction by a whole number, or a rnhole number by a fraction ; multiply the whole number and the numeruter 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. I of a dollar. A man divided ik of a barrel of flour equally among 4 families ; how much did he give them apiece? 1 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 folllowing example. A boy wishes to divide of an orange equally between two other boys; how much must he give them apiece ? urt of mbers mnade same the 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 lj orange. Or he night cut them all into two equal parts each, which would make six parts, and give 3 parts to each, that is, 1į, 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 į of .. 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 į of is 5. Each will have 1 and ğ 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 s. Therefore į of is š. Ans. š. 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 ž of }, 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 to 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 į is z', and of } is 7 times as much, that is, Ans. 27 Or since == , and of 11 =1 In the three Jast 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 |