XV. A man hired 7 laborers for 1 day, and gave them of a dollar apiece; how many dollars did he pay the whole ? If we suppose each dollar to be divided into 5 equal parts, it would take 3 parts to pay 1 man, 6 parts to pay 2 men, &c. and 7 times 3 or 21 parts, that is, 1 of a dollar to pay the whole. 24 of a dollar are 45 dollars. Ans. 41 dollars. A man bought 13 bushels of grain, at of a dollar a Bushel ; how many dollars did it come to Z of a dollar are 5 shillings 13 busbeis 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 one of a dollar; that is 10dollars, as before. A man found by experience, that one day with another, his horse would consume j} of a bushel of oats in a day; how many bushels would he consume in 5 weeks or 35 daus ? 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 154 = 1211 35 105 35 = 1237 455 This process is called multiplying a fraction by a whole number, Multiply it by 48. The fraction 107 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. The product of 253 by 48 is 12144; this written over the denominator is 114*, which being reduced, is 8114. 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 971 dollars a ton ; what did it come to ? In this example the number and the fraction must be multiplied separately. 13 times 97 are 1261. 13 times 34 are 57, equal to 1024; this added to 1261 makes 1271}dollars. Ans. Operation. 97 14 13 13 1261 182 1261 + 10% = 1271}} dolls. Hence, to multiply a mixed number : multiply the whole number and the fraction separately; then reduce the fraction to a whole or mixed number, and add it te 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 member 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 I yard. That is, į 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? i 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. IX. 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 divisor, 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 236, therefore 2 times 68 are 136, consequently 68 is 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 are equal to 2,845, therefore 569 is 1 of 2,845. Division may be explained, as taking a part of a number. In the above example 1 say, } of 28:00) is 5(00) and 3100) over. Then supposing 3 at the left of 4, I say, } 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 a 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. 57 3 B's share 171 dollars. 456 (8 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 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 of of gave for it. 1224 dollars, and 47 barrels cost of it. o of 1224 is 18. 47 times 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 1 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 1 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 lbs. of butter, which he wished to divide into 7 equal parts ; how many pounds would there be in each part? 1 of 847 lbs. is 121 lbs. 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 (7 121.4 lbs. Ans. A man having travelled 47 days, found that he had travelled 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 as of 1800 miles, and in 53 days he would travel 19 of it. At of 1800 is 38, and 14 over. at of 1 is 17, at of 14 is 14 times as inucb, that is, 14. In one day he travelled 38 14 miles. In 53 days he would travel 53 times 3817 miles. |