EXAMPLE  S.
1.  Reduce  a?  to  a  fraction  whose  denominator  is  a,     Ans,  —.
2.  Reduce  2az  to  a  fraction  whose  denominator  is  z^.         ^

Ans.  ..
22  •

3.  Reduce  a:+y  to  a  fraction  whose  denominator  is  x — y.
Ans.  ^2izt

4.  Reduce  m — n  to  a  fraction  whose  denominator  is  a(m — ny.
Ans.  "(."'-^y
a(m — ny
Art.  12s.  To  convert  a  fraction  to  an  equivalent  one,  having
a  given  denominator.
Rule.  —  Divide  the  given  denominator  by  the  denominator  of  the
given  fraction,  and  multiply  both  terms  by  the  quotient.
Remark.  —  This  rule  is  perfectly  general,  but  it  is  never  applied  except
when  the  required  denominator  is  a  multiple  of  the  given  one.  In
other  cases  it  would  produce  a  complex  fraction.  Thus,  if  it  were
required  to  reduce  §  to  an  equivalent  fraction  with  a  denominator  5,  the
numerator  of  the  new  fraction  would  be  3J.
ALGEBRAIC  FRACTIONS.  69

EXAMPLES.
1 .  Convert  f  to  an  equivalent  fraction,  having  for  its  denomi-
nator, 49.  Ans.  |-i.
2.  Convert  -  and  _  to  equivalent  fractions  having  the  denomi-

.02^^  A       3ac2  45c

nator  9c^.  Ans. ,  —

9c2    9c2-

3.  Convert  —L-  and  -iZl  to  equivalent  fractions  having  the
denominator  a^ — b^,  Ans.    o ^2  '  ^2 ^72  •

Case  V.  —  Addition  and  subtraction  of  fractions.
Art.  129.  It  is  self-evident  that  two  algebraic  fractions,  like
two  arithmetical  fractions,  must  have  a  common  denominator^,
before  we  can  find  either  their  sum  or  their  difference.
1.  Let  it  be  required  to  find  the  value  of  ?,  -,  and  _.
7  d   d  d

Let  -=m,  _=w,  and  _=r.
d  d  d

Then  a=mdf  b=nd,  and  c=rd ;
and     a-\-b-{-c=md'-\'nd-\-rd ;
or,      a-\-b-\-c=(m-{-n-\-r)d ;

hence  ^IIlJX?=m+w+r.
