This question is the same as the following: In 18 bushels, how many “ of a bushel ? That is, how many 7ths of a bushel ? In 1 bushel there are 7, consequently, in 18 bushels there are 18 times 7 sevenths ; that is, 146, and more make 14°. Answer 129 persons. Reduce 284} to an improper fraction. That is, in 2837 how many 25. Since there are is in 1, in 28 there must be 28 times as many. 28 times 25 are 700, and 17 more are 717. Ans. 717 Hence to reduce a whole number to an improper fraction with a given denominator, or a mixed number to an improper · fraction : multiply the whole number by the denominator, and if it is a mired number add the numerator of the fraction, and write the result over the denominator. XV. A man hired 7 labourers for 1 day and gave them of a dollar apiece ; how many dollars did he pay the whole ? If we suppose each doliar to be divided into 5 equal parts, it would take 3 parts to pay 1 2 men, &c. and 7 times 3 or 21 parts, that is, of a dollar to pay the whole. ? of a dollar are 41 dollars. Ans. 43 dollars. A man bought 13 bushels of grain, at of a dollar a bushcl; how many dollars did it come to ? of a dollar are 5 shillings. 13 bushels at 5 shillings a bushel, would come to 65 shillings, which is 10 dollars and 5 shillings. In the first form, 13 times of a dollar are 6 of a dollar ; that is 104 dollars, as before. A man found by experience, that one day with another, his horse would consume jî of a bushel of oats in a day; how many bushels would he consume in 5 weeks or 35 days? If we suppose each bushel to be divided into 37 equal parts, he would consume 13 parts each day. In 35 days he would consume 35 times 13 parts, which is 455 parts. But the parts are 37ths, therefore it is a = 12 bushels. 35 13 105 35 44 - 1237 455 This process is called multiplying a fraction by a whole number. Multiply 1:43 by 48. The fraction 2013 signifies that 1 is divided into 1372 equal parts, and that 253 of those parts are used. To multiply it by 48, is to take 48 times as many parts, that is, to multiply the numerator 253 by 48. 253 134 = = Q1168 12144 The product of 253 by 48 is 12144 ; this written over the denominator is 123142.. which being reduced is 81391. Ans. To multiply a fraction then, is to multiply the number of parts used; hence the rule : multiply the numerator and write the product over the denominator. Note. It is generally most convenient, when the numerator becomes larger than the denominator, to reduce the fraction to a whole or mixed number. It is sometimes necessary to multiply a mixed number. Bought 13 tons of iron, at 97/4 dollars a ton ; what did it come to ? In this example the whole number and the fraction must be multiplied separately. 13 times 97 are 1261. 13 times 34 are 187, equal to 43*; this added to 1261 makes 1265 dollars. Ans. 1261 182 1261 +44 = 126544 dolls. Hence, to multiply a mixed number : multiply the whole pumber and the fraction separately; then reduce the fraction to a whole or mixed number, and add it to the product of the whole number. XVI. We have seen that single things may be divided into parts, and that numbers may be divided into as many parts as they contain units ; that is, 4 may be divided into 4 parts, 7 into 7 parts, &c. It now remains to be shown, how every number may be divided into any number of equal parts. If 3 yards of cloth cost 12 dollars, what is that a yard ? It is evident that the price of 3 yards must be divided into 3 equal parts, in order to have the price of 1 yard. That is, of 12 must be found. We know by the table of Pythagoras, that 3 times 4 are 12, therefore { of 12, or 4 dollars is the price of 1 yard. If 5 yards of cloth cost 45 dollars, what is that a yard ? 1 yard will cost } of 45 dollars. 5 times 9 are 45, therefore 9 is į of 45, or the price of 1 yard. The two last examples are similar to the first example Art. 9. If we take 1 dollar for each yard, it will be 5 dollars, then one for each yard again, it will be 5 more, and so on, until the whole is divided. The question, therefore, is to see how many times 5 is contained in 45, and the result will be a number that is contained 5 times in 45. 5 is contained 9 times, therefore 9 is contained 5 times in 45. This is evident also from Art. III. When a number, therefore, is to be divided into parts, it is done by division. The number to be divided is the dividend, the number of parts the divisora, and the quotient is one of the parts. A man owned a share in a bank worth 136 dollars, and sold } of it; how many dollars did he sell it for? 136 (2 Ans. 68 dollars. 2 is contained 68 times in 136, therefore 2 times 68 are 136, consequently 68 is į of 136. A ticket drew a prize of 2,845 dollars, of which A owned }; what was his share? 2845 (5 Ans. 569 dollars. Since 5 is contained 569 times in 2,845, 5 times 569 ars equal to 2,845, therefore 569 is į of 2,845. Division may be explained, as taking a part of a number. In the above example I say, 1 of 28(00) is 5(00) and 3(00) over. Then supposing 3 at the left of 4, I say, 1 of 34(0) is 6(0) and 4(0) over. Then of 45 is 9. Writing all together it makes 569, as before. The same explanation will apply when the divisor is a large number. Bought 43 tons of iron for 4,171 dollars ; how much was it a ton ? 1 ton is as part of 43 tons, therefore the price of 1 ton will be as part of the price of 43 tons. 4171 (43 97 dollars. Two men A and B traded in company and gained 456 dollars, of which A was to have and B }; what was the share of each? The name of the fraction shows how to perform this example of 456 signifies that 456 must be divided into 8 equal parts, and 5 of the parts taken. of 456 is 57, 5 times 57 are 285, and 3 times 57 are 171. A's share 285, and B's 171 dollars. 456 (8 57 57 5 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. oy of 1224 is 18. 47 times are 18 are 846. Ans. 846 dollars. To find any fractional part of a number, divide the number by the denominator of the fraction, and multiply the quotient by the numerator. A man bought 5 yards of cloth for 28 dollars ; what was that a yard? } of 25 is 5, and } 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 16. 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. 853 (7 |