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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.
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
numerator or denominator, or in both. Thus, - and - are
242i To reduce complex to simple fractions.
Ex. 1. Reduce ~ to a simple fraction. Ans. ^-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).
2. Reduce —- to a simple fraction. Ans. ^ = 4£.
Operation. We reduce the nu
7 f . , . . merator, 7, and the
l|=r==tAf = .% — **, -*ns. denominator, If, to
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.
3. Reduce A to a simple fraction. Ans.
243. Complex fractions, after being reduced to simple ones, may be added, subtracted, multiplied, and divided, according to the respective rules for simple fractions.
1. Add I of f of 28|| to 3^|. Ans. 6^.
2. Add i, 2 J, —, and _» together. Ans. 8J|}*H§*
3. What is the difference between — - and \-?
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
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
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