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 C may be 2 2 302-a+7 Ex. 4. Reduce 5a" + to an improper 2ar fraction. Here, 5a2 X 2ax=10aox; adding the numerator 3x?-a+7 to this, and we have 10a3x + 3x? -a+7. 10a3+3r---a+7 Hence, is the fraction required. 2ax 3ab tc Ex. 5. Reduce 4x2 to an improper 2ac fraction. 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 =t 2ac 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 -C 4X-9 Ex. 10. Reduce 302 to an improper 7α 21 ax2 - 4x +9 fraction. Ans. 7а -5 Ex. 11, Reduce 5x to an improper 3 13x +5 fraction. Ans. 3 4x4 Ex. 12. Reduce 1+2x to an improper 5x x + 10x +4 fraction. Ans. 5x CASE JI. To reduce an improper fraction to a whole or mixed quantity. RULE. 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 tity. 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. x2 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° 4 8 Here the operation is performed according to the rule (Art. 93), and the quotient 2*--raya +y4 is the whole quantity required. ax -262 X ax 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 -ax-a? a 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 3ab quantity. 3a2b2 +6ab Here =ab+2 is the integral part, 3ab - 2x + bc 2x-2c 20-2x and :to (Art. 128), 3ab 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 429 tity. Ans. 3x 7а Ex. 7. Reduce 8xy-- 30x -- 66 toa mixed quan 3ax t-66 tily Ans. 2y 4x? a 7α 4x2 |