ALGEBRA. 
Fractions are changed to others of equal value 
with a common denominator, by multiplying each 
numerator by every denominator except its own, 
for the new numerator ; and all the denomina- 
tors together for the common denominator . 
ace 
be the proposed fractions; then 
and 
Thus, - 
, , — , reduced to 
m y m z 
ay % 
mxy % m x y z 
mon denominator, are 
crv 
m x y z 
ON THE ADDITION AND SUBTRACTION OF 
FRACTIONS. 
Jf the fractions to be added have a common 
denominator, their sum is found by adding the 
numerators together and retaining the common 
denominator. Thus, 
a . c a-i-c 
b~*~b 
If the fractions have not a common deno- 
isninator they must be transformed to others 
of the same val ue, which have a common de- 
nominator, and then the addition may take 
place as before. 
/ 
Ex. 2. 
Ex. 3, 
« i £ , !_1 ad-\-hc 
b ' d b d* b d 0 d 
a -j -h 
-b 2 
Ex. 4. a + j=ZS + t=lL±l. Here 
a is considered as a fraction whose denomi- 
nator is unity. 
are fractions of the same 
edb e 
*— r>— ? the numerator and deno- 
bdf f 
a df c,h f e db 
bdf’ oaf’ u df’ 
value w ith the former, having the common 
denominator bdf. For —j ; '-J- — 
bdf b’ bdf 
e 
d 
minator of each fraction having been multi- 
plied by the same quantity, viz. the product 
of the denominators of all the other fractions. 
When the denominators of the proposed 
fractions are hot prime to each other, find 
their greatest common measure; multiply 
both the numerator and denominator of each 
fraction, by the denominators of all the rest, 
divided respectively by their greatest com- 
mon measure ; and the fractions will be re- 
duced to a common denominator in lower 
terms than they would have been by proceed- 
ing according to the former rule. 
If two fractions have a common denominator, 
their difference is found by taking the difference 
of the numerators and retaining the common dec 
nominator. Thus, 
If they have not a common denominator, 
they must be transformed to others of the 
same value, which have a common denomi- 
nator, and then the subtraction may take 
place as above. 
Ex. 2. - — - — — 
,_d 
b d 
h r 
b d " 
Ex. 3. a — — — 
r rt 
b 
a h 
0 
a d — b c 
b <i 
ah — c d 
Ex. 4. £ 
r-i-d ac — ad fic-J-irf 
a c — a d ■ 
c — d bc- 
— he — h d 
-b d 
- b d ... 
- b d 
The sign of b d is negative, because every 
part of the latter fraction is to be taken from 
the former. 
ON THE MULTIPLICATION AND DIVISION OF 
FRACTIONS. 
/ 
To multiply a fraction by any quantity, mul- 
tiply the numerator by that quantity and retain 
the denominator. 
Thus, - X c = . 
For if the quantity to 
b b ' 
be divided be c times as great as before, and 
the divisor the same, the quotient must be c 
times as great. 
The product of two fractions is found by 
multiplying the numerators together or a new 
numerator, and the denominators for a new de- 
nominator. 
Let _ and _ be the two fractions ; then t 
b d i 
X 3 = = 1 C -=y, by 
multiplying the equal quantities and x, by 
l,az=bx; in the same manner, c = dy; 
therefore, a c~b d x y ; dividing these equal 
quantities, ac and bdxy, by bd, we have 
ac a c ■ ’ 
M= x y=- b *d- 
