Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 1, A-AME

6io
ALGEBRA.
Ex. i. Let *+ -L, and *-
, „ «*(o+*)=a,*+«*‘ ,,, l new numerators,
be reducert to ^ai x*^(a—Ar)~a3—^x—ax^-^-x* ^ v
improper fractions.
Firft x -j — —, the anfvver.
^ a a
a2— x2 x2-—a1 + ** _ ix1—a*
a—;eVaa.»)=:o*—x2 the common denominator.
' ‘ ‘ * -a2x+x*
— 1—r- amt = ,—
-x
ax a2x
Hence =
i—x a2'—x
a*—x1
And x-
x
X2
, Anf.
Ex. 2. Reduce a—*+ —to an improper fradtion.
a-\-x
x* ( a -j-^) ( a—x^> x*
0 ^ ' o-J-a; a-j^x’
54. Pros. IV. To Reduce an improper Traci ion to
a whole or mixed number.
Rule. Divide the numerator by the denominator for
the integral part, and place the remainder, if any,
over the denominator, and it will he the mixed
quantity required.
ax-\-a2
Ex. 1. Reduce
to a whole or mixed quantity.
d x
56. Pros. VI. To Add or Subtratt Fra&ions.
Rule. Reduce the fractions to a common denominator,
and add or fubtradl their numerators, and the fum
or difference placed over the common denominator,
is the fum or remainder required.
Ex. 1. Add together^, ^ ant^
a adf
"b~W
c bcf
d~ bdf
e bde
ace adf+bcf+bde
Hence ^ +/_ ^
Ex. 2. From — — - fubtract
the fum required
axJrE.=:a\ fL the anfwer required.
x x
_ , ax4-2x2 x2—y* , . .
Ex. 2. Reduce —-7 , alfo -——, to whole or nnx-
a-{-x
x
a
a + »
a2 + 2ax + x2
a2ax
a
A—y
ed quantities.
_ axTJX2 , x2' . r
Firft ~ x A the aniwer.
a-\-x tf+tf
And — =A;4-y a whole quantity, which is the
x—y
anfwer.
55. Pros. V. To Reduce TraBions of different De¬
nominators to others of the fame value which Jhall
have a common Denominator.
Rule. Multiply each numerator feparately into all the
denominators except its own for the new numera¬
tors, and all the denominators together for the com¬
mon denominator.
Ex. 1. Reduce^, 7 and ^ to fradtions of equal value
b d J
which have a common denominator.
aXdy.f=adf‘)
cy.b yf— cbf >■ New' numerators.
eybydz=. ebd J
bydyf—bdf Common denominator.
„ , a adf c cbf e ebd
Hence we find rf_^and^_^, where
the new fractions have a common denominator, as was
required.
Hence
a-\-x a2-\-ax
a-\-x a 2ax-\-x2
a a-\.x a2A-ox
Ex. 3. Add together —^ and —
A?-f-2
x x— 3 8 a? i 6 -j- 6 x -J- 12 x-
44"
-60
^ , aX T
Ex. 2. Reduce and
-x"
3 ■ 4 ' 2 24
If ft be required to add or fubtract mixed
12
quantities, they may either be reduced to the form of
fraftions by prob. 3. and then added or fubtra£ted, or
elfe thefe operations may be performed firft on the inte¬
ger quantities, and afterwards, on the fractions.
57. Prob. VII. To Multiply Fra&ions.
Rule. ' Multiply the numerators of the fractions for the
numerator of the product, and the denominators ftr
the denominator of the product.
b d
Ex. 1. Multiply - by
- X — the product required.
a c ac
Ex. 2. Multiply by —-j->
aJ-b a—b a2—b2
—-— X —7- — 7— 5 the product.
c d cd
If it be required to multiply an integer by a frac¬
tion, the integer may be confidered as having unity
• mi /■ *\ 3^ ® 3</
for a denominator. Thus (<7-f-Af) X —
a—x a-{-x
value and having a common denominator.
to fraaions of equal ^ ^d+^dx
[Mixed