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thirteenths as many time, as 6 is contained in 12; 12 = 6 = 2, Ang Therefore,
When the fractions have a common denominator, the division can be performed as in whole numbers, by dividing the numerator of the dividend by the numerator of the divisor. 3. Divide 4 by g
Having reduced the frac4
1i Ans. tions to a common denomi
nator, we divide the numerator 32 of the dividend by the numerator 21 of the divisor, as in working the last example, and obtain as the required result 111.
In the second oper4f= 4 X = 21 = 111 Ans. ation, we invert the
divisor, and then proceed as in multiplication of fractions (Art. 235). The reason of this process, which in effect reduces the fractions to a common denominator, and divides the numerator of the dividend by that of the divisor, will be seen, if we consider that the divisor, g, is an expression denoting that 3 is to be divided by 8. Now regarding 3 as a whole number, we divide the fraction 4 by it, by multiplying the denominator; thus, & 1. But the divisor 3 is 8 times as large as it ought to be, since it was to be divided by 8, as seen in the original fraction; then the quotient, ft, is į as large as it should be, and must be multiplied by 8; thus, * 8 111, the answer, as before. By this operation we have multiplied the dividend by the reciprocal of the divisor, the denominator of the dividend having been multiplied by the numerator of the divisor, and the numerator of the dividend by the denominator of the divisor. Therefore,
Dividing by a fraction is the same as multiplying by its reciprocal.
When either divisor or dividend is not a fraction, it may be changed to a fractional form, and the division performed by the last methoả. Hence the general
RULE. Invert the divisor, and then proceed as in multiplication of fractions.
NOTE 1. When either divisor or dividend is a whole or mixed number, or a compound fraction, it must be reduced to the form of a simple fraction before dividing.
NOTE 2. — Factors common to both numerator and denominator should be cancelled.
NOTE 3. When the given fractions have a common denominator, the answer may be obtained by dividing the numerator of the dividend by that of the divisor. Also, if the fractions have numerators alike, the answer may be obtained by dividing the denominator of the divisor by that of the dividend. NOTE 4. —
When the numerator of the divisor will exactly divide the numerator of the dividend, and the denominator of the divisor exactly divide the denominator of the dividend, the division can be effected in that way.
6. Divide 11 by 18. 7. Divide by . 8. Divide 18 by 1t: 9. Divide by 10. Divide i by . 11. Divide 1 by 28. 12. Divide it by 27. 13. Divide is by 128. 14. Divide 17 by 98. 15. Divide by 19. 16. Divide & by 167. 17. Divide if by 49. 18. Divide its by 15. 19. Divide 27 by 14. 20. Divide 128 by 15. 21. Divide 98 by 17. 22. Divide 19 by it. 23. Divide 167 by tế. 24. Divide 49 by 19. 25. Divide 15 by 1s. 26. Divide ji by it. 27. Divide %7 by 31. 28. Divided by s. 29. Divide if by TT
Ans. 960. Ans. 151 ir Ans. 31 14 Ans. 2003
Ans. lh Ans. 11085
30. Divides by 14. 31. Divide tf by I's.
Ans. 114. 32. Divide zas by 74. 33. Divide i by 165.
tot 34. Divide 114 by 4. 35. Divide 214 by 184.
Ans. 1118 36. Divide 17 31 by 2816.
Ans. 4940 37. Divide 161, by 143.
Ans. 111441 38. Divide 11 of by of . 39. Divide 8 of 73 by of 174. 40. Divide of 15 by 15 of 22.
41. Bought of a coal-mine for $3675, and having sold 5 of my share, I gave of the remainder to a charitable society, 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. $ 15845.
Ans. 14. Ans. 9.
240. When the dividend is a mixed number, and the divi. sor 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 together the results for the quotient required.
Ex. 1. Divide 273 by 6.
6 ) 27
4, rem. 33; 3= 16; 18 = }&= $; 4+ = 4$, Ans. 2. Divide 293 by 9.
Ans. 343. 3. Divide 14.} by 7.
Ans. 213 6. Divide $ 37 among 9 men.
Ans. $ 45%. 7. Divide $ 96ş among
Ans. $ 81 8. What is s of 167 1 cwt. of iron ? Ans. 2041 cwt.
9. Divide / 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 $ 1073 ?
Ans. $ 5.65s. 11. If 15 pounds of raisins can be obtained for $ 34, what will 1 pound cost ?
Ans. $ 0.214 12. If 12 quarts of wine cost $ 3.75), what will a quart cost?
13. If $ 19 will buy 37514 acres of land, how much can be bought for $1?
Ans. 1938 acres.
REDUCTION OF COMPLEX FRACTIONS.
241. A COMPLEX fraction is one having a fraction in its
3 21 numerator or denominator, or in both. Thus, and
} 을 complex fractions.
242. To reduce complex to simple fractions.
Ex. 1. Reduce
to a simple fraction.
Since the numerator of a fraction
is the dividend, and the denominator s x
the divisor (Art. 216), we divide .
the numerator, , by the denomi: nator, s, as in division of fractions (Art. 239).
7 2. Reduce to a simple fraction.
We reduce the nu7
merator, 7, and the Í X 21 1
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 denomie
Or, multiply by the least common multiple of the denominators,
Ans. 64 2 /
16. Reduce to a simple fraction.
Ans. 3. 123 17. If 7 were to be the denominator of the fraction whose
numerator is what would be its value ?
114 18. If y is the numerator of the fraction whose denominator
what is its value ?