ON THE MULTIPLICATION AND DIVISION' OF FRACTIONS.
To multiply a fraction by any quantity, smiltiply the intmerator by that quantity, and retain the denominator.
Thus, Ix c= a c. For if the quantity to be divided be c times as great as be fore, and the divisor the same, the quo tient must be c times as great.
The product tti two fractions is found by tmdtiplying the numerators togetherfor a ne-w numerator, and the denominators for a ne-is denominator.
Let <-1- and be the two fractions; then b d a c ac a X --- For if = b d b a multiplying the equal quantities Tand by b, a =b x ; in the same manner, c dy; therefore a c = 6 11 x y; dividing- these equal quantities, a c and b d y, by b d, C a c we have y--= X b d b d To divide a fraction by any quantity, multiply the denominator by that quantity, anti retain the numerator.
The fraction divided by c, is ne.
b c a a c a 'cause c' a cth part of this is ; b c the quantity to be divided, being a cth part of what it was before, and the divisor the saint..
The result is the same, whether t de. nominator is multiplied by the quautity, or the numerator divided by it.
I.et thp fraction be I-% ; Utile dencmi.
nator be multiplied by c, it becomes ific t e quantity- which arises from the division of the numerator by c.
To divide one fraction by another, invert the numerator and demminator of the divi sor, caul proceed as in imatiptication.
a c Let F and be the two fractions, then a cadad T x =re For if as x, and cc-, =_-. y, then ci= b x, ancl d y ; also, a d=bdx,andbc= ad bdx x rz c d y; therefore Fyy-= The rule for multiplying the powers of the same quantity will hold, when one or both of the indices are negative.
Thus, am X an = arnn; for am x 1 am = x -=-_ -=-. ; in. the same an on 1 manner, x3 x x-5 = = = X5 .10 _kgain, X ; b, cause 1 1 1 a'-'n X fr"== X =-- =a-m " arn an cim+a If m==72, Lm x a-m =am-rn =.aa ; also, tOn x a-m = 1 ; therefore a0= 1 ; tan according to the notation adopted.
The rule for dhiding any power of a quantity by any other power of the same quantity holds, whether those powers are positive or negative.
1 Thus, arn-i-a-n= am = am X a" an =on+n.
1 1 a" Again, a-m-χ-n-n a-T.±-a3= 07-n =- anz+n.
lience it appears, that a quantity maybe t.ransferred from the numerator of a frac tion to the denominator, and the contrary, by changing the sig-n of its index. Thus, x an_ tins and am _am x a-n bp bp (in' an bp bp