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3х Ex. 2. Reduce 5a- to an improper fraction.
7 Here 5a Xy=5ay; to this add the numerator with its proper sign, viz. -- 3x ; and we shall have 5ay--3x. Hence, 5ay-3x
is the fraction required. ข
as-ya Ex. 3. Reduce x2
to an improper fraction.
Here, xo xxx ; adding the numerator qi-yo with its proper sign : It is to be recollected that the
al-y sign affixed to the fraction
means that the whole of that fraction is to be subtracted, and consequently that the sign of each term of the numerator must be changed, when it is combined with x3,
23-aa+ya hence the improper fraction required is
-a +ya Or, as
; (Art. 67),
az--Y the proposed mixed quantity 22.
y-a? put under the form x + which is reduced as Ex: 1. Thus, a Xx+y2-a2 =33+ya-a;
23 +ya hence, a to
302-a+7 Ex. 4. Reduce 5a" +
to an improper
Here, 5a2 X 2ax=10aox; adding the numerator 3x?-a+7 to this, and we have 10a3x + 3x? -a+7.
is the fraction required. 2ax
3ab tc Ex. 5. Reduce 4x2
to an improper
Here, 4x2 X 2ac=Sacr, in adding the numerator with its proper sign; the sign --prefixed to the
3ab tc fraction signifies that it is to be taken nega
2ac tively, or that the whole of that fraction is to be subtracted; and consequently that the sign of each term of the numerator must be changed when it is
8acr?---3ab combined with Sacra; hence,
is 2ac 3abtc
-3ab--f the fraction required. Or, as
2ac - 3ab
(Art. 67); hence the reason of chang2ac ing the signs of the numerator is evident.
a -802 Ex. 6. Reduce x
to an improper frac
4X-9 Ex. 10. Reduce 302
to an improper 7α
21 ax2 - 4x +9 fraction.
-5 Ex. 11, Reduce 5x
to an improper 3
13x +5 fraction.
4x4 Ex. 12. Reduce 1+2x
to an improper 5x
x + 10x +4 fraction.
To reduce an improper fraction to a whole or mixed
149. Observe which terms of the numeralor are divisible by the denominator without a remainder, the quotient will give the integral part; and put the remaining terms of the numerator, if any, over the denominator for the fractional part; then the two joined together with the proper sign between them, will give the mixed quantity required.
to a mixed quan
3 + 2ax +6 Ex. 1. Reduce
2 + 2axa Here,
=x+ 2a is the integral part, 202 b and
is the fractional part; 3
6 therefore x+2at is the mixed quantity required.
208 + x*y* +y8 Ex. 2. Reduce
to a whole quan
4 ** + xy +y* lity. Dividend.
Divisor. To+cos^+g* c+cos^ +° xo + Rya txoy4
Quotient -xoy2 + y2
24-xoya +y4 — хву? — ху* —x°у°
Here the operation is performed according to the rule (Art. 93), and the quotient 2*--raya +y4 is the whole quantity required.
Ex. 3. Reduce
to a mixed quantity.
262 Here, =a is the integral, and the frac
262 tional part; therefore ac
is the mixed quantity required.
22-a? + b Ex. 4. Reduce
to a mixed quantity. *ta b
[required. xta)x2-a2 +b(xmat the mixed quantity
xta xa tax
to Here the remainder b is placed over the denominator +a, and annexed to the quotient as in (Art. 89).
3a'b? +6ab- 2x + 2c Ex. 5. Reduce
to a mixed
3a2b2 +6ab Here
=ab+2 is the integral part, 3ab - 2x + bc 2x-2c 20-2x and
(Art. 128), 3ab
3ab is the fractional part; 2-2c
20-23 .. ab +2
or ab +-2+ is the mixed 3ab
3ab quantity required.
2iax242 +9 Ex. 6. Reduce
to a mixed quan
7а Ex. 7. Reduce 8xy-- 30x -- 66
toa mixed quan
3ax t-66 tily