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Ans. 84. 37. Multiply 12 by 4. 38. Multiply 15 by ft.
Ans. 811: 39. A merchant owning of a ship sells t of his share to A. What part is that of the whole ship? 40. Multiply 37 by 104.
Ans. 3943. 41. Multiply of 71 by of 114.
Ans. 49133. 42. Multiply 4 of 9 by of 17.
Ans. 2635 43. Multiply 4 of 81% by of 9
Ans. 252415 236. When one of the factors is a whole number, and the other a mixed number, we may
Multiply the fractional part and the whole number separately, and add together the products.
EXAMPLES. 1. Multiply 75 by 9.
2. Multiply 12 by 31.
of 12 = 9
12 x 3 36
Ans. $ 1411. 10. What cost 8Žyd. of cloth at $ 5 per yard ? Ans. $ 41%. 11. What cost 9 barrels of vinegar at $ 6% per barrel ?
Ans. $ 573. 12. What cost 12 cords of wood at $ 6.374 per cord ?
Ans. $ 76.50. 13. What cost 11cwt. of sugar at $ 93 per cwt. ? 14. What cost 43 bushels of rye at $ 1.75 per bushel ?
Ans. $ 7.65ą. 15. What cost 7 tons of hay at $ 11% per ton ?
Ans. $ 834
16. What cost 9 dozen of adzes at $ 10% per dozen ? 17. What cost 5 tons of timber at $3$ per ton?
Ans. $ 15. 18. What cost 15cwt. of rice at $ 7.621 per cwt. ?
Ans. $ 114.371. 19. What cost 40 tons of coal at $8.374 per ton ?
Ans. $ 335.
DIVISION OF COMMON FRACTIONS.
237. DIVISION of Fractions is the process of dividing when the divisor or dividend, or both, are fractional numbers.
NOTE. — If the divisor is less than 1, the quotient arising from the division will be as many times the dividend as the divisor is contained times in 1. Therefore, the quotient arising from dividing a whole or mixed number by a proper fraction will always be larger than the dividend.
238. The reciprocal of a fraction is the number resulting from taking its numerator as denominator, and its denominator as numerator, since any two numbers, whose product is 1, are the reciprocals of each other. Thus, the reciprocal of La is that fraction inverted, or 1, since in Xo=1.
239. To divide when the divisor or dividend, or both, are fractions. Ex. 1. Divide 14 by 7.
It is evident that the fraction 14 is di#:7= Ans. vided by 7 by dividing its numerator by 7,
since the size of the parts, as denoted by the denominator, remains the same, while the number of parts taken is only as large as before.
It is evident the fraction is also di14 = Ans. vided by 7 by multiplying its denomi
nator by 7, since the number of parts taken, as denoted by the numerator, remains the same, while the size of the parts is only as large as before. Therefore,
Dividing the numerator or multiplying the denominator of a fraction by any number divides the fraction by that number (Art. 217). 2. Divide 13 by 15
Ans. 2. Since the fractional units of the two - 2 Ans. fractions are of the same kind, it is
evident that 12 thirteenths contain 6
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 numoers, by dividing the numerator of the dividend by the numerator of the divisor. 3. Divide 4 by š.
Having reduced the frac= 141 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 141.
In the second oper$_$= 4 X 24 = 141 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, f, is an expression denoting that 3 is to be divided by 8. Now, regarding 3 as a whole number, we divide the fraction # by it, by multiplying the denominator; thus, * x.8 t. 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, , is } as large as it should be, and must be multiplied by 8; thus, 붉
= 141, 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 method. 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.
Ans. Ans. 14
6. Divide 11 by 18. 7. Divide f by } 8. Divide 18 by 11. 9. Divide by . 10. Divide i by. 11. Divide is by 28. 12. Divide 1 by 27. 13. Divide is by 128. 14. Divide 17 by 98. 15. Divide if by 19. 16. Divide & by 167. 17. Divide ng by 49. 18. Divide I'} by 15. 19. Divide 27 by it. 20. Divide 128 by Ps. 21. Divide 98 by 14. 22. Divide 19 by 24. 23. Divide 167 by få. 24. Divide 49 by 26. 25. Divide 15 by is. 26. Divide zi by it: 27. Divide 4 by 30. 28. Divide å by s. 29. Divide it by Ir.
Ans. gło Ans. Tot Ans. 4 Ans. TÓCZ Ans. IST Ans. 25
Ans. 960. Ans. 151 Ý Ans. 31 Ans. 2003
Ans. 4. Ans. 374.
Ans. A Ans. 116 Ans. 365
30. Divide by 14.
Ans. 1118 36. Divide 17 | by 28.11.
Ans. His 37. Divide 1611 by 143.
Ans. 111211 38. Divide 11 of 4 by of li. 39. Divide of 73 by 1 of 174. 40. Divide of 15 by 15 of 22.
41. Bought of a coal-mine for $3675, and having sold 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 daugh
Ans. $ 15844.
Ans. 14 Ans. 35%.
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 together the results for the quotient required.
Ex. 1. Divide 273 by 6.
6) 27} 4, rem. 33; 3} = 1; 44x6 = }&= $; 4+= 43, Ans.
Ans. $ 419. 7. Divide $ 96ş among 11 persons.
Ans. $ 8. 8. What is of 16711 cwt. of iron ? Ans. 2041 cwt.
9. Divide z of a prize, valued at $ 1723, equally between 12 seamen.