36. Divide 17ft by 28^. Ans. ff£g. 37. Divide 161ft by 14f. Ans. llT|fT. 38. Divide ft of \ by \ of ft. Ans. l\. 39. Divide | of 7ft by ft of 17f Ans. fff. 40. Divide ft of 15 by ft of 22. 41. Bought \ of a coal-mine for $3675, and having sold $ of my share, I gave \ of the remainder to a charitable societ}-. and divided the residue among 7 poor persons; what was the share of each? Ans. $ 50 for each poor person. 42. Of an estate valued at $5000, the widow receives the oldest son § of the remainder; the residue is equally divided among 7 daughters; what is the share of each daughter? Ans. $ 158||. 240. When the dividend is a mixed number, and the divisor a whole number, we may Divide the integral part of the mixed number as in division of whole numbers, and the remainder divide as in Art. 239; and add tot/ether the results for the quotient required. Ex. 1. Divide 27f by 6. Ans. 4f. OPERATION. 6)27f 4, rem. 8f; S| - JJ; J£ x „ = ft = f; 4+ f ~ 4f, Ans. 2. Divide 29$ by 9. Ans. 3f§. 3. Divide 14£ by 7. Ans. 2ft. 4. Divide 13J by 8. 5. Divide 14| by 6. Ans. 6. Divide $ 37f among 9 men. Ans. $ 4£j}. 7. Divide $96| among 11 persons. Ans. $8§|. 8. What is £ of 167ft cwt. of iron? Ans. 20f£ cwt. 9. Divide J of a prize, valued at $ 1723, equally between 12 seamen. 10. What will a barrel of flour cost, if 19 barrels can be purchased for $ 107f? Ans. $ 5.65 11. If 15 pounds of raisins can be obtained for $3^, what •will 1 pound cost? Ans. $ 0.2 l£f 12. If 12 quarts of wine cost $3.75J, what will a quart cost? 13. If $ 19 will buy 375-}£ acres of land, how much can be bought for $1? Ans. 19§g£ acres. REDUCTION OF COMPLEX FRACTIONS. 241 i A Complex fraction is one having a fraction in its 3 24 numerator or denominator, or in both. Thus, - and - are if complex fractions. 242i To reduce complex to simple fractions. 2 Ex. 1. Reduce ~ to a simple fraction. Ans. ^-f. f Operation. Since the numerator of a fraction § 16 A 's dividend, and the denominator T = S X f = if, Ans. the divisor (Art. 216), we divide . "the numerator, by the denomi nator, as in division of fractions (Art. 239). 7 2. Reduce —- to a simple fraction. Ans. ^ = 4£. * 3 Operation. We reduce the nu 7 f . , . . merator, 7, and the l|=r==tAf = .% — **, -*ns. denominator, If, to improper fractions, and then proceed as in Ex. 1. Hence, to reduce complex to simple fractions, Consider the denominator as a divisor, and the numerator as a dividend, and proceed as in division of fractions (Art. 239). Note. — Another and often a ready method of reducing a complex fraction is to multiply both its terms by the least common multiple of their denomi> nators. Examples. 3. Reduce A to a simple fraction. Ans. OPERATION. 4=^=±><^=&=2-v,ax« 243. Complex fractions, after being reduced to simple ones, may be added, subtracted, multiplied, and divided, according to the respective rules for simple fractions. Examples. 1. Add I of f of 28|| to 3^|. Ans. 6^. 45 47* 2. Add i, 2 J, —, and _» together. Ans. 8J|}*H§* 495 34a 3. What is the difference between — - and \-? 97 145^ Ans. ^ftWbV 4. What is the continued product of the following numbers: 37i' 98** 27 and ~128' 5. Divide £ of 7£ by ± of 11^. Ans. 6. Divide f of 91 by & of 87. Ans. ffljf. MISCELLANEOUS EXAMPLES IN MULTIPLICATION AND DIVISION OF FRACTIONS. 1. At 2f bushels to an acre, how many bushels of wheat will be required to sow 7^- acres? Ans. 17f> 2. Bought 84 bushels of apples for $ 4.68$-; what did they cost per bushel? Ans. $ 0.57^. 3. Bought a bale of cloth for $ 96f; I dispose of it for | of the cost, and by so doing I lose $ 2 on a yard; required the number of yards in the bale. Ans. 18T§ ?yd. 4. If a dividend be 18 J times £ and a quotient 6£ times what was the divisor? 5. By what number must l| be multiplied, that the product o shall just equal 1? Ans. #. 6. Bought a horse and chaise for $ 250, and paid for the harness T7T of what I paid for the horse. The chaise cost \£ \he value of the horse. What was the price of each? Ans. Horse, $130£$; chaise, $119£§; harness, $832-|3. 7. S. Walker has engaged to work at yearly wages of $ 200 and a suit of clothes. At the end of 9 months, falling sick, and being unable to labor longer, he receives the suit of clothes and $ 144, as the amount justly due. What was the cost of the clothes? Ans. $24. 8. What will be the result if | of £ of 3£ be multiplied by £ of itself, and the product divided by £? 9. Bought 13\ acres of land at $25£ per acre, and paid for it in wheat at $ 2% per bushel. How many bushels did it require? Ans. 137^-f-f bushels. 10. How long will it take a man to travel 553 miles, provided he travels 3£ miles per hour, and 9J hours per day? 11. If $T9,y per cord is paid E. Holmes for sawing into three pieces wood that is 4 feet long, how much more should he receive per cord for sawing into pieces of the same length wood that is 8 feet long? Ans. $ 0.22£. 12. A steamboat leaves New Orleans, January 1st, bound up the river to a place distant 2317^ miles. Her forward motion is at the rate of 9 J miles per hour for 16f hours each day, and she lies at anchor in the night for fear of running upon a snag. But having lost her anchor on the fifth day, she each succeeding night drifts backward, at the rate of 2 miles per hour. On what day of January will she reach her point of destination? Ans. 15th day. A PROPOSED NUMERATOR, OR DENOMINATOR. 244. To reduce one fraction to another of equal value, having a proposed numerator, or denominator. Ex. 1. Reduce f to an equivalent fraction having 4 for a 4 numerator. Ans. H Operation. The proposed numerator, 4, is such a 4 part of the given numerator as 4 divided It" X $ by 5, or A. Now, as the numerator pro — = _ Ans. posed is only £ as large as the given 4 5f numerator, in order that the value of the 1} X 7 two fra(.tions be the same, the denomi nator of the proposed fraction should be only £ as large as the denominator of the given fraction. Taking £ of the given denominator, 7, we obtain |