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This * rise to fractions in the same manner as in arithmetic It was shown in arithmetic, that a fraction properly expresses a quotient. Algebraic fractions are subject to precisely the same rules as fractions in arithmetic. Many of the operations are more easily performed on algebraic fractions.
In these, as in arithmetic, it must be kept in mind, that the denominator shows into how many parts a unit is divided ; and the numerator shows how many of those parts are used ; or the denominator shows into how many parts the numerator is divided.
I shall here briefly recapitulate the rules for the operations on fractions, referring the learner to the Arithmetic for a more full developement of their principles.
# of 7 is or ; for 4 of 7 is 4, and ; is 3 times as much. 4 o' a is of ; for of a is #, and # is 2 times as much. The ; part of c is: ; for ; of c is #, and ; is a times as much.
Hence, to multiply a #. by a whole number, or a whole number by a fraction, multiply the numerator of the fraction and the
whole number together, and divide by the denominator. * * Arith. Articles XW. & XVI
This cannot be done like the others, but it may be done by multiplying the denominator as in Arith. Art. XVII. For the fraction # denotes, that one is divided into as many equal parts as there are units in b, and that as many of these parts are used as there are units in a ; or that a is divided into as many equal parts as there are units in b ; hence if it be divided into twice as many parts, the parts will be only one half as large, and the fraction will have only one half the value. ,
Hence, to divide a fraction by a whole number, divide the numorator; or when that cannot be done, multiply the denominator by
the divisor. 6. Divide
10. Divide 11. Divide 12. Divide 13. Divide 14. Divide 15. Divide 16. Divide
12 a c d
3 a b.
7 b n”.
Hence, to multiply one fraction by another, multiply the numerators together for a new numerator, and the denominators together
for a new denominator.
.Arith. Art. XVII.
We have seen that a fraction may be divided by multiplying its denominator, because the parts are made smaller; on the contrary, a fraction may be multiplied by dividing its denominator, because the parts are made larger. Arith. Art. XVIII. If the denominator be divided by 2, the unit is divided into only one half as many parts ; consequently the parts must be twice as large as before. . If the denominator be divided by 5, the unit is divided into only one fifth as many parts; hence the parts must be five times as large as before, and if the same number of parts be used as at first, the value of the fraction will be five times as great, and so on.
33. Multiply #. by b. c If we divide the denominator by b, the fraction becomes
“, in which a is divided into # part as many parts; hence C © the parts, and consequently the fraction is b times as large as before.
20 m a. 38. Multiply 3 by 5