zty

or from

1
1
4.246

7x12 4. From take or 5.0 + take 2. + 1-Y x+y? 7

13 13(4x—6)—7172—12) 52x—784493+84 3x+6

and 7x13591

91

91 52-20=31; -. 3x+

3x+6 3X

Answer.

91
a+b a- -6

10x49 5. From

3—5 take

take a+B

8

7 (4+5)(a+b=a+2ab +6 a®+2ab+B2—a*+2ab477 4ab (a−b)(a—b)=a_2ab +62

a2b2 (a+b)(a+b=a12 as required. (104c-937=70-63 70x463—24x+40 466-23

is the (3.1--58=24.0-40

56

56 8x7=7856 fraction required, or answer. 2x+? 54-6

11x410 6. From 2x +

take a
or 3x +

take
8
21

15
3x45
7

42x+147 40-48 By art. 42, ex. 9, the fractions are

and

-; 168

168 40x448, 42x+147 therefore 2.1 +

82x+99 (

= (2-1) +
168
168

168 45x75

772270 By art. 42, the fractions are

and
105

105
774–70
452–75

32x+5
--(22+ -) = x+ Ans.
105
105

105
+2y 12x 3х
take

take by

be bto 39%

4x 118y-1 2a.ca 35

8

2x +

; then

axt

ax

7. From 15y take 1724, OF

or

Answers,

Ext 35

or

8. Take

or

с

> or 2an

+

с

с

+35

2a

2

X3

За
5a 2a 3a to

26 8

from 6a, or 4

4 3 6 2a +6 4a+8

36

2a 9. Take from

from 4a+ 9.

5 The fractions reduced to a common denominator. 2x 5x 2x X7 , 5x X5 14.x 25%

39.x + +

Ans. 7 5 5X7 5X7 35

35
Зr 2aXx 3x X5 2ax 15x 15x+2ax
Here +

ti
ti

Ans. 5 2aX5 2a X5 10a

10a 10a 12x 8x

6x 13. Here

tä tä

+ 2

Ans. 4 24

+24+

24 12 4x -2 20x , 7(242) 202+72—14 272—14 + +

Ans. 7 5 35 35

35

35 Here these fractions may be written thus ; 2a + 3atat 2x 8x 2x 8x 18x 40x

22x
=6a+
hat

=6a- Ans. 5 9 5 9 45 45

45 Here the fractions reduced to a common denominator are 3a-x-3ax? 5a?

5a*_10az+522 and

-; therefore the sum is 5a5ax '5a_5ax

ба” Бах
3a*x3ax+100
-10ax+52%

3a_3ax' +52 2a+

=a+2+

Ans. 5a_5ax

5a_5ax

57_102 62_9. 5%-10% The fractions reduced are

and

; ..5x+
15x
15%

15% 63-9 52–16x+9 +42- =9x+ =

Ans.
152
15x

2x-3 But if the fractions were 5x, and 4.x

the final result 3

5x
2x-3 5x+10x 6x9
=
=9x+

=9x+ 3 5x

15% 15x
5x+10x46x+9 5x +4x+9
=9x+

Ans.
15x
152

8a The fractions reduced to a common denominator are and

1222 3ax+622

8a Зах+6х? 8a+3ax+6x2
–5x+

Ans. 12x2

1222

122*

MULTIPLICATION OF ALGEBRAIC FRACTIONS.

or

or

2

by

or

or

by

or

by

by

52. Rule. Multiply all the numerators together for the numerator of the product, and all the denominators together for its denominator ; the former placed over the latter will give the product required. 2x За Зr

5x 2x 32
1. Multiply zby
5

by
2 2 38

7
21х3х 6x2 322 50X3x 1522 2.x X 32 622
5X2a 10a 5' 2X36 26

7(241) 72–79 325x 7а

3x? 152-30 2. Multiply

by 14 2x 32 5—10 2x 3az—52 3x4(15—30) 3x* x3 9m2 9x

Ans. 4_6

22(5x-10) 2x 2x 2
4.2

3.x 5x 2x 322 3. Multiply

by or 2x,

7 5

or 7x,

7? 4

5

2a (7—2)4x 4x482 2x X3x X 5x 3020 152

Ans. 5X7 35

1X4X7 28
7X2x X 3x*7*XXX 3.x*
”

2124 Lastly,

Ans.
1x5X2a 1x5xa ба
2x 4x4

X-1

2+1 by or 3x,

by atx

2a

atx
6
6

6
or 5ato by 3am
3x

52
a2-x2 2+1
6. Multiply at by or 3.x,

by

2a atb a a_62 7. Multiply

and at

together.
atb' axta
a2 a2x

al
at
Х

-Xa ato axta

or

14

za

a 8. Multiply at by X- + 2a 4a?

2x 4x2

4a8+2ax— 4.28—2axtas By art. 42, ex. 7, last set,

-) XC 4a

4x2 4.'—80*x+16a%282—50*x+20*x+-2ax8 + 8ax_420 °z, Ans.

. 160 m

6a

9. Multiply 5

by Fc

by Zc

20

ato

or

a

a

с

4a

За 462 3a Зас 400
gr by or Х
4
За

6 b
ab 3a? 20?_262 4a? +262
10. Multiply 2a+2

by or

- by together.
b „Зbc
2a+1

2a
206 3ab

бас 11. Multiply 3a and

by

by 2a+b

2b

Ans., 15ax. 4a 6a 24a? 8a2 За 462

За

4ab

Зас
Х

Х Х 3

150
50 4 За

b

3c 3283

Ans.
6
12ac3a6 8_4a2b2_404 12a_3

Ans.
2bc
3abc+38c 2a

+6

x 5c

or

-, or

or

DIVISION OF FRACTIONS IN ALGEBRA.

or

1. Divide $by * 30 x+

•l

53. RULE. Invert the divisor, and then proceed as in Multiplication.

EXAMPLES. 29 + 422

9y:—37 3y!

" or by 5, or

by 7

7' 2+1

2x by 6

3 2?_9 2+3 (249)4 4.2-12

4x2

4.22 Ans., or

• 5= X 5 4 (2+3)5 5.

7

7 1 4x2 9y2—37 Зу” •9° — Зу 7 Зу—1 Ans.

X.

Ans., 52 7

7 Зу?
c+1 2x
3 6X2x

4.2
X 9y

За 7x 2. Divide by

by
1.

5
5 5x 5

9y?

342 Х

Ans. 5

x-x2 12
9y"

3
5y—10 154-30

172 Х

or 2y 3y2 2y

2y

5 172 Х

C

2

2

x2+xy

or 2

2d2_4a6+202 by

,4

or

by

or

by

3

2ax+22

24-64 22+bx 3. Divide by ΟΙ

by 03-23

?—26+B2

X-6 (2a+x)2 2ax+22

2a+a Х

Ans. c

c2texta X4_6 xtbx 204_74 X-6 Again,

x x2 _ _ 2bat 2

(x-6)

22+bx 12? +6?) (22—62) (2—6) (2+0) (x2-04)

2° +62

Ans. (1-6) (3-6) (+6) X3 (22—62) XX

x2y

5a4_564 6a2+5ab 4. Divide by X-Y (24)

4a-46 (x+xy) (22—2xyty) (2+xy) (r-y) ?+xy=x(x+y) (-y) (x4)

x-Y x+xy+xy' +yo 50% 564 6a? + 5ab (5a — 56) x (4a - 46)

" 2 - 4ab+ 20 4a - 46 (2a — 4ab-204) (6a? + 5ab (6X2) (at_84) 5.2.(a+a'btab +63 10(a'+a’b+ab?+68 (2-6) (6a' +5ab) 1X (64+5ab)

6a2+5ab 3x 11 622

3x+1

4x 4x 5. Divide by

by 3x, or 4 12 5

9 3 22_1 3r 11 Зr 12 9.3

3.2 6x? 1 2x X

х

A. 4 12 4 11 11 5 1 5 За

5
4x 3a
2a-6 5ac

5a4_564 6. Divile by

by

or 5

by 56 4cd

6d 202—4ab+26* 60° +5ab 40-45 бас (2a-6) 6d ба —30 Х

Ans, 4cd 6d 4cd

бас

10ac? 5a4_564 4a-46 5a-7)

2 x G

X 24?_4ab+262 6a +5ab

ab

6a? +-5ab 5(e2 ta’b+ab? +B) 2

10(a +apb+ab2+28) xt

A. 1

6a2 +5ab 7x 3

3cx 2axt x2 7. Divide by

by

by
8cd 4d 2x
x2 - a2

X4__64

22+bx 8. Divide

by tax xta 2?_2bx+62 X-6 8+1 42 (3x+1) 3

1

4.2 Х

or 9 3

9
4.2

122
2x-1

3 3 12

4x

3a 4x 50 462
X
or

Х

Ans. (2+1)

2x-1 5 50 5 За