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37. Multiply 12 by f Ans. 84.
38. Multiply 15 by T6r. Ans. 8T2T.
39. A merchant owning fa of a ship sells -fa of his share to A. What part is that of the whole ship?
40. Multiply 3fa by 104. Ans. 39£f.
41. Multiply § of 7£ by f of llf. Ans. 49|§f.
42. Multiply \ of 9 by § of 17. Ans. 26^.
43. Multiply \ of 8T3a by 4 of 9f Ans. 25?VV
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.
I. Multiply 71 by 9. 2. Multiply 12 by 3f.
I X 9 = V- 5t f of 12 - 9
7X9 =63 12 X 3 = 36
681 Ans. 45 Ans.
3. Multiply 8f by 7. Ans. 604,
4. Multiply 17 by 3£.
5. Multiply 13 by 8f Ans. 1094.
6. Multiply 37 by 13^. Ans. 5074$.
7. Multiply llf by 8. Ans. 94f.
8. What cost 7T6T1b. of beef at 5 cents per pound?
9. What cost 23^bbl. of flour at $ 6 per barrel?
Ans. $ 1414.
10. What cost 8gyd. of cloth at $ 5 per yard? Ans. $ 41 fa.
II. What cost 9 barrels of vinegar at $ 6$ per barrel?
12. What cost 12 cords of wood at $ 6.374 per cord?
13. What cost llcwt. of sugar at $ 9f per cwt.?
14. What cost 4| bushels of rye at $ 1.75 per bushel?
15. What cost 7 tons of hay at $ 11$ per ton?
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 $7,624 per cwt.?
19. What cost 40 tons of coal at $ 8.37£ 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 is that fraction inverted, or since X T9o"= 1
239. To divide when the divisor or dividend, or both, are fractions.
Ex. 1. Divide ^ by 7. Ans. T2T.
First Operation. It is evident that the fraction If is di
J-f -5- 7 = -^y 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.
Second Operation. It is evident the fraction is also di
^-f -=- 7 = y1^- = T2T Ans. vided by 7 by multiplying its denominator 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 if by T%. Ans. 2.
Operation. Since the fractional units of the two
^fr -r- Ttj- = 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; 12-5-6 = 2,Ans 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 f by f. Ans. Iff.
First Operation. Having reduced the frac
\ -5- §, = ff ,+, § £ = Ans. tions to a common denominator, 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
Second Operation. in the second oper
f-7-f = ^X § = f T = l£i -^ns 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 fo a common denominator, and divides the numerator of the dividend by that of the divisor, will be seen, if we consider that the divisor, |, 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, ^ 8 = 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, * 8 = if = l^f' 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.