```EXERCISE 1
WORKED EXAMPLES:
If x = 9, y = 6 and z = -3 find the value of the following.
x 9 3 1
x
9
9 3
b.
= = =1
=
= =
y 6 2
2
2y 2 ´ 6 12 4
2x 2 ´ 9 18
x -1 9 -1 8 4
1
d.
=
= =3
=
= = =1
y
6
6
y
6
6 3
3
y 6 2
y -1 6 -1 5 1
f.
= =
=
= =
x 9 3
x +1 9 + 1 10 2
z - 3 -1
z - 3 -1
=
=
h.
=
=
y 6
2
x 9
3
z -1
3- 1 - 3+- 1 - 4 - 2
2z 2´ 3
6
2
=
=
=
=
j.
=
=
=
y
6
6
6
3
x
9
9
3
a.
c.
e.
g.
i.
WRITTEN EXAMPLES:
If = 4, y = 6 and z = -2 find the value of the following fractions in their simplest form.
1.
5.
9.
13.
17.
21.
25.
29.
33.
x
y
y
x
2x + 1
y
2
x
y
3x - y
3x + y
z
x
z+4
x
z+ x
x+z
z+ x
x+z
2.
6.
10.
14.
18.
22.
26.
30.
34.
x
2y
y
3x
y
3x - 3
x
y2
2y - x
2y + x
z
y
z -1
y
x-y
y-x
z-x
x-z
3.
7.
11.
15.
19.
23.
27.
31.
35.
2x
y
3y
x
3x - 2
y -1
y-x
y+x
y+ x
y 2 - x2
3z
x
4-z
x
y+z
z+ y
x+z
y+z
4.
8.
12.
16.
20.
24.
28.
32.
36.
x -1
y
y+2
x
y +1
3x + 2
x+y
xy
y- x
y 2 - x2
2z
y
4z - 1
y
y-z
z-y
x´z
y´z
DS/GRBJMH: February 6, 2016
page 1 of 10
EXERCISE 2
WORKED EXAMPLES:
a.
b.
c.
d.
e.
g.
i.
Simplify these fractions.
10 2/ ´5 5 2
2x 2/ ´ x x
=
= = 1 and so
=
=
2y 2/ ´ y y
6 2/ ´ 3 3
3
10x 2/ ´ 5 ´ x/ 5 2
=
= =1
6x 2/ ´ 3 ´ x/ 3
3
10x 2/ ´ 5 ´ x 5 ´ x 5x
2
(which is not usually written as 1 x )
=
=
=
6
2/ ´ 3
3
3
3
10
2/ ´ 5
5
5
=
=
=
6y 2/ ´ 3 ´ y 3 ´ y 3y
4x 4/ ´ x/
10x 2/ ´ 5 ´ x 5 ´ x 5x
f.
=
=1
=
=
=
4x 4/ ´ x/
6y 2/ ´ 3 ´ y 3 ´ y 3y
2x 2/ ´ x x
2
2/ ´ 1
1
1
h.
=
=
=
=
=
6y 2/ ´ 3 ´ y 3 ´ y 3y
6 2/ ´ 3 3
ab
a/ ´ b/
1
1
ab a ´ b/ a
j.
=
=
=
2 2 =
2 =
ab
a/ ´ a ´ b ´ b/ a ´b ab
b
b ´ b/ b
WRITTEN EXAMPLES: Simplify the following fractions.
5x
2x
3y
1.
2.
3.
2y
3y
5
7x
2y
5x
5.
6.
7.
5y
7x
2
6x
4x
8y
9.
10.
11.
2y
2y
2
3x
3y
3x
13.
14.
15.
6y
12 x
9
8x
12 x
20y
17.
18.
19.
4y
4y
4
5x
5y
5x
21.
22.
23.
15 y
25x
10
8x
4x
12 y
25.
26.
27.
6y
9y
10
15x
20y
12 x
29.
30.
31.
8y
20x
16
ab
ab
a
33.
34.
35.
a
b
ab
bc
ca
a2
37.
38.
39.
ac
bc
a
3
b
a
a3
41.
42.
43.
b3
a
a2
a
b
ab
45.
46.
47.
2
2
ab
ab
a2 b
bc
a2
c 2a
49.
50.
51.
c2
ab
ca 2
4.
8.
12.
16.
20.
24.
28.
32.
36.
40.
44.
48.
52.
7
7y
3
3x
10
2y
3
15 x
16
4y
5
20x
18
15 y
25
30x
b
ab
b
b2
b2
b3
ab
ab2
a 3b 2c
ab2 c 3
DS/GRBJMH: February 6, 2016
page 2 of 10
EXERCISE 3
WORKED EXAMPLES:
Find these products and quotients. Simplify the answers where possible.
5 2 5 ´ 2 10
a c a ´ c ac
a.
and so ´ =
´ =
=
=
7 3 7 ´ 3 21
b d b ´ d bd
5 2 5 3 5 ´ 3 15
a c a d a ´ d ad
b.
and so ¸ = ´ =
¸ = ´ =
=
=
7 3 7 2 7 ´ 2 14
b d b c b ´ c bc
5a 2b 5 ´ a ´ 2/ ´ b/ 5a
a b a ´b/ a
c.
´ =
=
´ =
= and
6b 3c 3 6/ ´ b/ ´ 3´ c 9c
b c b/ ´c c
a a a/ c c
5a 2a 5a 3c 5 ´ a/ ´ /3 ´ c 5c
d.
¸ = ´ = and
¸
= ´
=
=
6b 3c 6b 2a 2 6/ ´ b ´ 2 ´ a/ 4b
b c b a/ b
WRITTEN EXAMPLES:
x y
x y
x z
x z
1.
2.
3.
4.
´
¸
´
¸
y z
y z
y y
y y
x z
x z
x x
x x
5.
6.
7.
8.
´
¸
´
¸
y x
y x
y z
y z
x x
x x
x y
x y
9.
10.
11.
12.
´
¸
´
¸
y y
y y
y x
y x
1
1
1
1
13. x ´
14. x ¸
15.
16.
´y
¸y
y
y
x
x
2a b
2a b
3a c
3a c
17.
18.
19.
20.
´
¸
´
¸
3b 2c
3b 2c
4b 3b
4b 3b
2a b
2a b
3a a
3a a
21.
22.
23.
24.
´
¸
´
¸
3b 6a
3b 6a
4b 6c
4b 6c
5a 2a
5a 2a
5a 3b
5a 3b
25.
26.
27.
28.
´
¸
´
¸
6b 3b
6b 3b
6b 4a
6b 4a
3a 2
3a 2
3 3b
3 3b
29.
30.
31.
32.
´
¸
´
¸
4 3a
4 3a
4b 2
4b 2
x y z
x y z
x y z
x y z
*33.
*34.
*35.
*36.
´ ´
´ ¸
¸ ´
¸ ¸
y z x
y z x
y z x
y z x
3a 2b c
3a 2b c
3a 2b c
3a 2b c
*37.
*38.
*39.
*40.
´ ´
´ ¸
¸ ´
¸ ¸
4b 3c 2a
4b 3c 2a
4b 3c 2a
4b 3c 2a
EXERCISE 4
WORKED EXAMPLES:
5 1 5 +1 6
a b a+b
5 1 5 +1 6
and so
and
+ =
=
+ =
+ =
=
7 7
7
7
7 7
7
c c
c
c
5 1 5 -1 4
a b a -b
5 1 5 -1 4
b.
and so
and
- =
=
- =
- =
=
7 7
7
7
7 7
7
c c
c
c
5a a 5a + a 6a 3a
c.
+ =
=
=
8 8
8
8
4
5a a 5a - a 4a a
d.
- =
=
=
8 8
8
8 2
a b a+b
a b a-b
In general,
and
+ =
- =
c c
c
c c
c
a.
DS/GRBJMH: February 6, 2016
page 3 of 10
WRITTEN EXAMPLES:
x y
x y
x y
x y
1.
2.
3.
4.
+
+
2 2
3 3
4 4
5 5
x y z
x y z
x y z
x y z
5.
6.
7.
8.
+ +
+ - +
- 2 2 2
3 3 3
4 4 4
5 5 5
3 2
3 2
4 1
4 1
9.
10.
11.
12.
+
+
y y
y y
x x
x x
3 1
3 1
4 2
4 2
13.
14.
15.
16.
+
+
y y
y y
x x
x x
4 3 2 1
4 3 2 1
4 3 2 1
4 3 2 1
17.
18.
19.
20.
+ + +
+ - - + - - +
y y y y
y y y y
x x x x
x x x x
x x
x x x
y y
y y y
21.
22.
23.
24.
+
+ +
+
+ +
2 2
3 3 3
4 4
6 6 6
3x x
3x x
3y y
3y y
25.
26.
27.
28.
+
+
2 2
2 2
4 4
4 4
5x 3x
5x 3x
5y y
5y y
29.
30.
31.
32.
+
+
4
4
4
4
6 6
6 6
7x 5x
7x 5x
5y y
5y y
33.
34.
35.
36.
+
+
8
8
8
8
8 8
8 8
7
5
7
5
5
1
5
1
37.
38.
39.
40.
+
+
8y 8y
8y 8y
8x 8x
8x 8x
5
1
5
3
4
2
7
5
41.
42.
43.
44.
+
+
3y 3y
6y 6y
2x 2x
4x 4x
11x 7x 5x x
11x 7x 5x x
11y 7y 5y y
45.
46.
47.
+
+
+
+
+
12 12 12 12
12 12 12 12
12 12 12 12
11y 7y 5y y
11
7
5
1
11
7
5
1
48.
49.
50.
- +
+
+
+
+
12 12 12 12
12x 12x 12x 12x
12x 12x 12x 12x
11
7
5
1
11
7
5
1
51.
52.
+
+
12y 12y 12y 12y
12y 12y 12y 12y
x y
x y
x y z
x y z
53.
54.
55.
56.
+
+ - +
z z
z z
a a a
a a a
EXERCISE 5
WORKED EXAMPLES:
a.
c.
3x x 6x x 6x + x 7x
+ =
+ =
=
2 4 4 4
4
4
3x x 9x 2x 9x + 2x 11x
+ =
+
=
=
4 6 12 12
12
12
b.
d.
4y 5y 8y 5y 8y -5y 3y y
=
=
=
=
3
6
6
6
6
6 2
3x 2y 9x 4y 9x - 4y
=
=
2
3
6
6
6
Note: Always choose your common denominator to be the lowest one possible.
DS/GRBJMH: February 6, 2016
page 4 of 10
WRITTEN EXAMPLES:
x x
x x
3y y
3y y
1.
2.
3.
4.
+
+
2 4
2 4
4 2
4 2
x x
x x
5y y
5y y
5.
6.
7.
8.
+
+
2 6
2 6
6 2
6 2
x x
2x x
5y 2y
5y y
9.
10.
11.
12.
+
+
3 6
3 6
6
3
6 3
x x
3x x
5y 3y
5y y
13.
14.
15.
16.
+
+
4 6
4 6
6
4
6 4
x y
x y
2x y
2x y
17.
18.
19.
20.
+
+
2 3
2 3
3 2
3 2
x y
2x y
3x y
3x 2y
21.
22.
23.
24.
+
+
3 4
3 4
4 3
4
3
5x 2x x
5x 2x x
5x 2x x
5x 2x x
25.
26.
27.
28.
+
+
+
- +
- 6
3 2
6
3 2
6
3 2
6
3 2
5x 2y z
5x 3y 2z
5x 3y z
3x 2y z
29.
30.
31.
32.
+
+
+
- +
- 6
3 2
6
4
3
6
4 2
4
3 2
EXERCISE 6
WORKED EXAMPLES:
3
1
3
2 3+2 5
5
1 10 1 10 -1 9
3
a.
b.
+
=
+
=
=
=
=
=
=
3y 6y 6y 6y
6y
6y 2y
4x 2x 4x 4x
4x
4x
c.
3
5
9
10 9 + 10 19
+
=
+
=
=
4y 6y 12y 12y
12y 12y
d.
2 4 2y 12x 2y - 12x
- =
=
3x y 3xy 3xy
3xy
Note: Always choose your common denominator to be the lowest one possible.
WRITTEN EXAMPLES:
3
1
3
1
5
3
5
3
1.
2.
3.
4.
+
+
2y 4 y
2y 4y
2x 4 x
2x 4x
3
1
3
1
3
5
3
5
5.
6.
7.
8.
+
+
2y 6y
2y 6y
2x 6x
2x 6x
4
1
4
5
7
4
11 4
9.
10.
11.
12.
+
+
6y 3y
6y 3y
3x 6x
3x 6x
3
1
3
1
5
1
5
1
13.
14.
15.
16.
+
+
6y 4 y
6y 4y
4x 6x
4x 6x
1
1
4
1
1
1
5
2
17.
18.
19.
20.
+
+
2x 3x
3x 2x
3x 4 x
4x 3x
1 1
3 2
3 4
2 3
21.
22.
23.
24.
+
+
x y
x y
2x y
x 4y
5
4
4
5
2
5
5
2
25.
26.
27.
28.
+
+
2x 3y
3x 2y
3x 4 y
4x 3y
5
2
1
5
2
1
5
2
1
5
2
1
29.
30.
31.
32.
+
+
+
- +
- 6x 3x 2x
6x 3x 2x
6x 3x 2x
6x 3x 2x
5
2
1
5
3
2
5
3
1
3
2
1
*33.
*34.
*35.
*36.
+
+
+
+
6x 3y 2z
6x 4 y 3z
6x 4y 2z
4x 3y 2z
DS/GRBJMH: February 6, 2016
page 5 of 10
EXERCISE 7
ORAL EXAMPLES:
1
1. 2 ´ = ?
2
1
´3=?
3
4 ´? =1
2.
19 ´ 2 ´
28 ´
1
=?
2
3. Given that 25 x 4 = 100
1
´3=?
3
then 25 x 4 x 2 = ?
1
and 25 x 4 x 5 = ?
37 x 4 x ? = 37
1
1
2
46 x 5 x ? = 46
and 25 x 4 x 5 = ?
5 x?=1
2
2
64 x 3 x ? = 64
4. Given that 125 x 8 = 1000
3 x?=1
4
4
73 x 5 x ? = 73
then 125 x 8 x 3 = ?
5 x?=1
3
3
1
82 x 4 x ? = 82
and 125 x 8 x 4 = ?
4 x?=1
2
2
3
91 x 5 x ? = 91
and 125 x 8 x 4 = ?
5 x?=1
5. Hence we can multiply one side of an equation by a number provided we also multiply the
WORKED EXAMPLES:
Solve these equations for x.
3
1
a.
b.
c.
4x =
4x =
4x = 3
2
2
1
1
1
1 3
Þ ´ 4x = ´ 3
Þ ´ 4x = ´
1
1 -1
4
4
4
4 2
Þ ´ 4x = ´
4
4 2
1 4x 1 3
1 4x 1 3
1 4x 1 -1
i.e. ´
i.e. ´
i.e. ´
= ´
= ´
= ´
4 1 4 1
4 1 4 2
4 1 4 2
Þ x=
d.
g.
3
4
x = 3
4
x
Þ 4 ´ = 4´- 3
4
4 x
i.e. ´ = 4´- 3
1 4
Þ x= 12
5x
=3
6
6 5x 6 3
Þ ´ = ´
5 6 5 1
18
5
3
i.e. x = 3
5
i.e. x =
Þ x=
e.
-
1
8
x 1
=
4 2
x
1
Þ4´ =4´
4
2
4 x 4 1
i.e. ´ = ´
1 4 1 2
Þ x=
f.
Þ x=2
5x -1
=
6
2
h.
Þ
6 5x 6 -1
´
= ´
5 6 5 2
6
i.e. x =
10
3
i.e. x =
5
i.
3
8
x -3
=
4 2
x
3
Þ4´ =4´
4
2
4 x 4 -3
i.e. ´ = ´
1 4 1 2
Þ x= 6
5x 3
=
6 2
6 5x 6 3
Þ ´ = ´
5 6 5 2
18
10
4
i.e. x = 1
5
i.e. x =
DS/GRBJMH: February 6, 2016
page 6 of 10
WRITTTEN EXAMPLES:
Solve these equations for x (x Q).
1. 2x =1
2. 2x=- 3
3. 2x = 5
3
4
5
5. 2x =
6. 2x =
7. 2x =
4
3
6
x
x x
9.
10.
11.
=1
= 3
=5
2
2
2
x 4
x 3
x -6
13.
14.
15.
=
=
=
2 3
2 4
2 5
17. 3x =1
18. 3x=- 2
19. 3x = 4
1
3
4
21. 3x =
22. 3x =
23. 3x =
2
4
5
x
x x
25.
26.
27.
=1
= 2
=4
3
3
3
x 1
x 4
x -4
29.
30.
31.
=
=
=
3 2
3 3
3 5
33. 6x =1
34. 6x=- 2
35. 6x = 4
2
3
3
37. 6x =
38. 6x =
39. 6x =
3
4
2
x
x x
41.
42.
43.
=1
= 2
=4
6
6
6
x 2
x 3
x -3
45.
46.
47.
=
=
=
6 3
6 4
6 2
2x
3x 2x 1
49.
50.
51.
=6
=6
=
3
2
3 4
2x 4
2x 5
3x - 4
53.
54.
55.
=
=
=
3 5
3 6
2
5
3x
4x 3x 1
57.
58.
59.
=6
=6
=
4
3
4 2
3x 5
3x 2
4x - 5
61.
62.
63.
=
=
=
4 6
4 5
3
6
6x
6x 3
5x -1
65.
66.
67.
=4
=
=
5
5 4
6
3
-
4.
2x= 7
8.
2x =
12.
16.
20.
24.
28.
32.
36.
40.
44.
48.
52.
56.
60.
64.
68.
-
6
5
x = 7
2
x -5
=
2 6
3x= 5
6
3x =
5
x = 5
3
x -6
=
3 5
6x= 3
4
6x =
3
x = 3
6
x 4
=
6 3
3x - 1
=
2
4
3x
5
=
2
6
4x - 1
=
3 2
4x - 2
=
3
5
5x
3
=
6
4
DS/GRBJMH: February 6, 2016
page 7 of 10
REVISION EXERCISE
If x = 8 and y = -6 find the value of the following fractions in their simplest form.
y
y
x+y
x-y
1.
2.
3.
4.
y
x+y
x
3x
2y
y+2
x
y2
5.
6.
7.
8.
y-2
x
x
x
Simplify the following fractions.
12 x
8x
6
2x
9.
10.
11.
12.
6y
12 y
2y
8
3x
6y
15 x
4
13.
14.
15.
16.
8y
4x
3
10 x
10 x
xy
20
y
17.
18.
19.
20.
15 y
xy
20
x
2
2
x y
y
xy
x y
21.
22.
23.
24.
2
2
xy
xy
x 2y 2
x
Find the following sums, differences, products and quotients. Simplify your answers where possible.
1 x
1 x
x
x
25.
26.
27. y ´
28. x ¸
´
¸
x y
y y
y
y
3x 2y
3x 9x
3x 12z
3x 12z
29.
30.
31.
32.
´
¸
´
¸
4y 3z
4y 4z
4y 5x
4y 5y
5x 3x
5x 3x
5
3
5
3
33.
34.
35.
36.
+
+
4y 4 y
4y 4y
4
4
4
4
7x 5x 3x x
7x 5x 3x x
7y 5y 3y y
7y 5y 3y y
37.
38.
39.
36.
+
+
+
+
+
- +
8
8
8 8
8
8
8 8
8
8
8 8
8
8
8 8
7
5
3
1
7
5
3
1
7
5
3
1
7
5
3
1
41.
42.
43.
44.
+
+
+
+
- +
8y 8y 8y 8y
8y 8y 8y 8y
8x 8x 8x 8x
8x 8x 8x 8x
3x x
4x 5x
3y y
2x y
45.
46.
47.
48.
+
+
2 6
3
6
4 6
3 4
3
5
4
5
3 4
2
1
49.
50.
51.
52.
+
+
2x y
3x 4y
2x 4 x
3x 6x
5x 3x 2x
5
3
2
5y 3y 2y
5
3
2
53.
54.
55.
56.
+
+
+
+
- +
+
6y 4y 3y
6
4
3
6x 4 x 3x
6
4
3
Solve these equations for x.
5x 2
x -1
6x - 2
57. 4x = 6
58.
59.
60.
=
=
=
6 3
4 6
5
3
DS/GRBJMH: February 6, 2016
page 8 of 10
EXTRA EXERCISE
1.
x x2 xy x2
y , xy , y2 , y2
Which ones are equal to each other?
Which one is undefined if x = 0?
What can you say about y in all them?
Consider the following fractions
a.
b.
c.
2.
In this question consider any 2 natural numbers (e.g. 4 and 3)
a. i. Find the sum of the numbers and hence find the reciprocal of their sum.
ii. Find the reciprocal of each number and hence find the sum of their reciprocals.
iii. (True or False) The reciprocal of their sum is equal to the sum of their reciprocals.
b. i. Find the product of the numbers and hence find the reciprocal of their product.
ii. Find the reciprocal of each number and hence find the product of their reciprocals.
iii. (True or False) The reciprocal of their product is equal to the product of their reciprocals.
c. Rewrite the statements in (a) (iii) and (b) (iii) which are true in terms of x and y (where x and y are
natural numbers).
d. Is this statement in (c) true if x and y are integers?
3.
Copy down and check the first four statements. Copy down and complete the next four statements.
1
a. If x ´ (1+1) = y then y ´ (1- ) = x
2
1
1
If x ´ (1+ ) = y then y ´ (1- ) = x
2
3
1
1
If x ´ (1+ ) = y then y ´ (1- ) = x
3
4
1
1
If x ´ (1+ ) = y then y ´ (1- ) = x
4
5
1
If x ´ (1+ ) = y then
5
1
If x ´ (1+ ) = y then
6
1
If x ´ (1+ ) = y then
10
1
If x ´ (1+ ) = y then
n
b.
Prove the last statement in (a).
c.
Copy down and complete this statement.
1
If x ´ (1- ) = y then y ´ (1+ ?) = x
n
DS/GRBJMH: February 6, 2016
page 9 of 10
4.
a.
c.
d.
e.
5.
Copy down and check these statements.
1
1
If x ´ (1+ ) = y then y ´ (1- ) = x
5
6
2
2
If x ´ (1+ ) = y then y ´ (1- ) = x
5
7
3
3
If x ´ (1+ ) = y then y ´ (1- ) = x
5
8
4
4
If x ´ (1+ ) = y then y ´ (1- ) = x
5
9
Copy down and complete this statement.
m
If x ´ (1+ ) = y then y ´ (1- ?) = x
n
Prove the statement in (c).
b.
Copy down and complete these statements.
1
If x ´ (1+ ) = y then y ´ (1- ?) = x
n
2
If x ´ (1+ ) = y then y ´ (1- ?) = x
n
3
If x ´ (1+ ) = y then y ´ (1- ?) = x
n
4
If x ´ (1+ ) = y then y ´ (1- ?) = x
n
Copy down and complete this statement.
m
If x ´ (1- ) = y then y ´ (1+ ?) = x
n
Copy down and check the first four statements. Copy down and complete the next four statements.
a.
b.
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
= 1-2
1
+ (2 )2
1
= 1 - (2 )2
1
1
+ (2 )2 + (2 )3
1
1
1
+ ( )2 + ( )3 + ( )4
1
= 1 - (2 )3
+ (1 )2
2
+ (1 )2
2
+ (1 )2
2
+ (1 )2
2
=
2
+ (1 )2
3
+ (1 )2
3
+ (1 )2
3
+ (1 )2
3
+ (1 )2
3
+ (1 )2
3
+ (1 )2
3
2
+ (1 )3
2
+ (1 )3
2
+ (1 )3
2
+ (1 )3
2
+ (1 )3
3
+ (1 )3
3
+ (1 )3
3
+ (1 )3
3
+ (1 )3
3
+ (1 )3
3
+ ( 1 )4
2
+ (1 )4
2
+ ( 1 )4
2
+ ( 1 )4
2
+ ( 1 )4
3
+ ( 1 )4
3
+ ( 1 )4
3
+ ( 1 )4
3
+ ( 1 )4
3
2
+ (1 )5
2
+ (1 )5 + (1 )6
2
2
+ ... + (1 )10
2
+ ... + (1 )n
2
1
= 1 - (2 )4
=
=
=
= 1 -1
2 2
= 1 -1
2 2
= 1 -1
2 2
= 1 -1
2 2
+ (1 )5
3
+ (1 )5 + (1 )6
3
3
1
10
+ ... + ( )
3
+ ... + (1 )n
3
x1
3
x (1 )2
3
x (1 )3
3
x (1 )4
3
=
=
=
=
DS/GRBJMH: February 6, 2016
page 10 of 10
```