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Kimberley Girls’ High School
Mathematics
Grade 9
Chapter 1
The Number System
 Natural Numbers:
 Counting Numbers:
 Integers:
 Rational Numbers:
Examples:
 Irrational Numbers:
Examples:
 Real Numbers:
 Non-Real
Recurring Fraction  Common Fraction
Examples:
1.
2.
Let:
Let:
Inequalities on the Number Line
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Notation
1.
Both the above notations mean:
included and
excluded.
2. On the number line:
Included:
Excluded:
Infinity:
Examples:
1.
2.
3.
4.
5.
6.
7.
0
is any real number between
and ; where
is
Page 3 of 268
EXERCISE 1
1. Choose from the list:
1.1
Integers
1.2 Rational Numbers
1.4
Non-real Numbers
1.3 Irrational Numbers
1.5 Natural Numbers
2 Express each of the following as a decimal fraction:
2.1
2.2
a.
2.9
3.
5
8
2.6
2.3
2.4
2.7
2.8
2.10
Express each of the following as common fractions:
4.
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Represent each of the following inequalities graphically (on a number line):
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
CHAPTER 2:
INDICES




: where 2 is the coefficient, x is the base and 7 is the exponent
4
9
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


LAWS OF INDICES:
1. Multiplication:
Examples:
2. Division:
Examples:
3. Power of 0:
Examples:
4. Negative Index:
Examples:
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5. Raise to a Power:
Examples:
EXERCISE 1
1.
2.
4.
5.
7.
10.
13.
16.
19.
22.
25.
9.
14.
15.
17.
18.
20.
21.
23.
24.
26.
27.
29.
30.
32.
33.
35.
37.
39.
48.
2
12.
36.
45.
 2
2 3 
x 
11.
34.
42.
6.
8.
28.
31.
3.
38.
40.
43.
41.
44.
46.
47.
49.
50.
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51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
CHAPTER 3
SUBSTITUTION
EXERCISE 1
If
, calculate the value of:
1.
2.
4.
3.
5.
7.
6.
8.
9.
EXERCISE 2
If
, calculate the value of:
1.
2.
b)
3.
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EXERCISE 3
If
, calculate the value of:
1.
2.
3.
EXERCISE 4
1.
If a = 1; b = -1; c = 2; d = -3 and e = 0, find the values of the
following orally.
(a)
a6
(b)
a 343
(e)
b 16
(f)
b 17
(g)
18b
(h)
23b
(i)
c5
(j)
c6
(k)
13c
(l)
-16c
(m)
2abc
(n)
cde
bde
7
(p)
e5
(q)
d3
(r)
d4
6d
(t)
-d
2. If a = 3; b = -2;
(c)
17a
(o)
(s)
(d)
16a
c = 1; d = 0 and h = -4, find the values of
the following.
(a)
3b 2
(b)
(3b) 2
(c)
-b 4
(d)
(-b) 4
(e)
3b 2 a 2 c 7
(f)
2a 3 b 4 d
(g)
2b 3 - h
(h)
(2a + b)(3a - b)
(j)
-3(a - 2b) 2
(k)
4abd 2
9
(i) 2(3a - b) 2
(l) 2ab 3 + 3abc + 4a 3 bd 2
3. If x = -3; y = -2 and z = -1, find the values of the following.
4.
(-3y)2
(c)
4(-2x)2 - 3(-3y)2
(d)
x 3 + y 3 + z 3 - 3xyz
(e)
(x 2 - y 2 )(x 2 + y 2 )
(f)
x2  y2
x y
If a = 3; b = -2; c = 1; g =
(a)
(c)
5.
-3xyz + z 3
(a)
5
1
b
g
4
+ ab
g
(b)
2
and h = -4, evaluate the following.
3
(b)
(d)
1
2
h
g
bc
b
h
g
If x = -3; y = 0; z =1 and p = -2, find the values of the following.
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6.
7.
(a)
x 2 y - xy 2 + y 2 z - yz 2
(c)
4x - 3{y + 2(z + p)}
(b)
(d)
x3 - y3 + z3 - p3
xy(x + y) - pz(p + z)
If a = -2 and b = -1, find the values of the following.
(a)
a 8 b16
(d)
(a) 7  16b12
a 4  20b 6
(b)
(e)
3
(c)
3
a 5  5b 3
7a 5  (2b) 3
If n = -4; r = 3 and t = 2, find the values of the following.
(a)
n2  r 2
(b)
3
r 3t 6
EXERCISE 5
1
(Unless stated otherwise  = 3 )
7
1. The area of a circle with radius r, is given by the formula A =  r 2 .
Calculate the areas of circles with radii –
(a) 70 mm
2.
(b)
35 mm
(c)
7 mm
The area of a trapezium with parallel sides which are x mm and y mm
long and p mm apart, is given by the formula A = ½(x + y)p.
Find the area of a trapezium whose parallel sides are 34 mm and 46 mm
long, and 35 mm apart.
3.
The simple interest I that a principal P will earn in T years at R% p.a. is
Is given by the formula I =
PRT
.
100
Find the interest on R725 for 8 years at 4½% p.a.
4.
A stone is dropped. The distance, s metres, that it falls in t seconds is
given by the formula s = ½gt 2 .
Calculate the distance that it will fall in 3 seconds if g = 9,8.
5.
The total area of a cylinder with radius r and height h is given by the
formula T = 2  r(r + h). Find the total area of a cylinder of which the
diameter is 70 mm and the height 70 mm.
6.
The speed s of a motor car that travels d km in t km, is given by the
formula s =
d
.
t
Calculate the speed of a car that travels 840 km in 12 hours.
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7.
Find the value of x in the formula x =
 b  b 2  4ac
if a = 2;
2a
b = -2 and c = -1½
1
.
g
8.
T = 2
Calculate T if l = 125; g = 5 and  = 3,14
9.
Use the formula P =
1
1
1
, to find P if x = ½ and y = x
3
y
a (1  r n )
calculate s when a = 3; r = ½ and n = 2
1 r
10.
If s =
11.
In the formula s =
12.
The volume V of a circular reservoir is given by the formula V =  r 2 h,
n
[10 + 5(n - 1)], find s if n = 6
2
where r is the radius and h the height of the reservoir. What will the
volume of a reservoir with a diameter of 7 m and a height of 2 m be?
CHAPTER 4
POLYNOMIALS
, is a polynomial in descending powers of
 The degree (highest power) of the expression is:
 The numerical coefficient of the second term is:
 The numerical coefficient of
is:
 The index of the 4th term is: 2
 The constant term is: 8
EXERCISE 1
1. Given:
1.1
Simplify and write the expression in descending powers of .
1.2
Find the degree of the polynomial.
1.3
Write down the coefficient of
1.4
Find the value of the polynomial if:
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2. Given:
2.1
Simplify and write in descending powers of .
2.2
Find the constant term.
2.3
The numerical coefficient of the first term.
2.4
The index of
2.5
The value of the expression if:
in the second term.
Addition:
Subtraction:
Subtract:
Multiplication:
1.
2.
3.
Division:
Squares & Square Roots:
Examples
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
EXERCISE 2
Determine the sum of:
1.
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2.
3.
4.
5.
6.
2x 2 - 5x ;
x 2 + 4 ; - 7x + 8
7.
x 2 - 3x + 1 ;
8.
a 2 - 4a ;
9.
2b 2 - 4b + 1 ;
10.
- 2b + 5 ;
b2 + b + 4 ;
11.
x2 - 4x + 2 ;
- 2x2 - 3x - 5 ;
x2 + 7x + 3
12.
p2 - 8p - 1 ;
292 - 4p - 2 ;
- 3p2 + 10p + 3
13.
2y2 + 3y ;
- y2 + y ;
14.
a3 + 4a ;
- 2a2 + a ;
15.
x2y + xy ;
3xy - xy2 ;
- 2x2 + 7 ;
-a - 5;
- 5x - 2
2a + 7
- b2 - b ;
- 4b2 - 2b + 6
2b2 + 2b - 8
- 3y + 8
2a3 - 5a + a2
2x2 - 4xy
EXERCISE 3
1.
Subtract
2.
Subtract
3.
From
4.
From
5.
What must be added to
6.
Subtract ( a² - 4a + 2 ) from ( 2a² - a - 3 )
7.
Subtract ( x² - 3x - 5 ) from ( 2x² - 8x )
8.
From ( 5x² - 3x + 7 )
9.
Subtract ( a² - 4a - 8 ) from - 10a
10.
From ( 2b² - 4b )
11.
Subtract ( x² - 5x + 1 ) from 8
12.
Subtract 2a + 5 from ( a² - 8a )
13.
Subtract ( 2p3 - 4p - 7 ) from
14.
From 2x subtract ( 4x² - 2x )
15.
From 2 + x
subtract ( 2x² - 6x - 4 )
subtract ( - b² - 5 )
( 5p3 - 8p + 9 )
subtract ( x² + 8x - 5)
EXERCISE 4
Simplify:
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1.
2.
3.
4.
5.
6. ( x² + 2x ) + ( 3x² - 4x - 2 )
7.
( x² + 2x ) - ( 3x² - 4x - 2 )
8. ( 2b² - b - 4 ) - ( b² - 5b + 8 )
9.
( x3 + 4x² ) + ( - 2x² - 5x )
10. ( x3 + 4x² ) - ( - 2x² - 5x )
11. (p² - 4p - 5 ) + ( 7p² - p + 2 )
12. ( p² - 4p - 5 ) - ( 7p² - p + 2 )
13. ( y² + 2y ) + ( - 3y² ) - ( -y² + 4y )
14. ( 4x² - 2x - 1 ) - ( x² - 3x - 3 ) + ( 2x² + 4x )
15. ( a² - 2a - 4 ) + ( 2a² - 4a ) - ( 3a² - 7a )
16. ( b² - 2b - 8 ) - ( - 5b² + 3b ) + ( 3b² - 5 )
17. ( 3p² - 3p - 5 ) - ( 2p² - 2p - 5 ) + ( -p² - 2p )
18. ( a² - 2ab - b² ) + ( ab - 5b² ) + ( 3a² - 2b² )
19. ( 3x² - 4xy + y² ) - ( - 3x² - xy + 2y² )
20. ( - p² - p - 1 ) - ( - p² - 4p - 3 ) -
( 3p² - 5p - 7 )
21. ( 7x² - xy ) - ( - 2x² + 3xy + y² )
22. ( 2x - 5x² + 7 ) - ( - 2x² - 8x ) - ( 5 - 2x - x² )
23. ( a² - ab - 5b² ) - ( - 2a² - 3ab - b² )
24. ( - 2a - 4b ) - ( 2a - b ) + ( b - 2a ) - ( - 5a )
EXERCISE 5
Simplify:
1.
3x ( x - 2 ) + 4x
3.
2x ( x + 5 ) - 3x ( x - 1 )
5.
- 2y ( y + 2 ) + y ( y - 1 )
2.
4.
x ( x - 1 ) + 2x ( x - 3 )
a ( a - 1 ) - 2a² - 5a
6. x ( x + 2 ) - 2x ( x - 4 ) + 3x
7.
2p ( p + 7 ) - 3p ( P + 1 ) + p ( 2p + 3 )
8.
k ( k + 4 0 - 2k ( k - 4 )
9. 2 ( a - 3 ) - a ( a - 3 ) + a ( 4 - a )
10. -2b ( b - 10 ) - b ( b - 3 ) + 3b²
11. 2k ( k - 1 ) + 3k ( k + 1 ) - k
12. 5a ( a + 5 ) - 5a ( a - 5 )
13. x ( X + 1 ) = X ( X + 2 ) + X ( X + 3 )
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14. 2y² ( y² - 3y + 4 ) - 3y ( y² - 2y + 3 )
15. k ( 2k² - 3k + 4 ) + 2k ( k - 5 )
16. 7b ( b² - 2b - 1 ) - 3b² ( b - 4 )
17. 2a ( a² + 3a - 2 ) - 3a² ( a² - 4a + 1 )
18. - p² ( p3 - p ) + p3 ( 1 - p²)
19. 2a5 (5a² - a ) - a3 ( a4 - 2a3 )
20. x ( - 5 + x - 2x² ) - x + 2x ( x - 4 )
EXERCISE 6
Simplify:
1.
2x ( x - 4 )
2.
( x + 3 )5x
3.
- 3x ( x - 2 )
4.
( x - 2 ) - 3x
5.
( x - 2 )( - 3x )
7.
( y² - 2y - 3 )3y
9.
3a ( a - 4 ) + ( a - 4 )3a
6.
- 3x - ( x - 2 )
8.
- 2x ( x + 1 ) + ( x + 1 - 2x
10. 2 ( y + 3 )y - y ( y - 4 )
11. 2x ( x - 2 ) - ( x - 5 )x
12. p ( p² - 2p - 1 ) - 3 ( p - 4 )p
13. 2a - ( a + 3 ) - a
14. 2x - ( x² - x - 2 ) - 2x ( x - 1 )
15. - x ( x + 1 ) + ( x + 1 ) - x
16. a ( a + 2 ) - (2a + 1 )a
17. 2b ( b - 3 ) - ( b² - 5b ) + 2b
18. - 2x ( x + 1 ) + ( x - 2 )x
19. 2x² ( x² - 3x - 1 ) - x3 ( 2x - 5 ) + x ( 2x - 3 )
20. 3a ( a² - 2a + 1 ) - 2a ( 3a² + 4a - 3 ) - 5a² ( a + 2 )
EXERCISE 7
Simplify:
1.
4.
2.
4.
5.
6.
7.
8.
9.
10.
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11.
12.
13.
14.
15.
16.
EXERCISE 8
Simplify:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
EXERCISE 9
Simplify:
1.
2.
3.
EXERCISE 10
1.
2.
3.
4.
EXERCISE 11
1.
2.
4.
5.
6.
7.
8.
9.
10.
11.
3.
12.
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13.
14.
15.
EXERCISE 12
Simplify:
1.
-x(x – 5) + (x – 5) - x
2.
–x(x – 5) - (x - 5)x
3.
2x - (x + 2) + 2(x – 3)
4.
5xy - x(2x – y)
5.
3a(a 2 - 2a – 1) - a(a 2 - a + 1) + 4
6.
2x(x 2 - 4) - (x 2 - 3x)4x - 3(x 3 + 7x 2 - 5x)
7.
-7x(x -3) + (x – 3) - 3x(x – 7)
8.
-(ab – b) + (2ab + 5b) + b + (b – 2)a
9.
ab(a – b) + b(ab - c 2 ) - (a 2 - c 2 )b
10.
3x - (5x + 2) - 3x(5x +2)
11.
x 2 (x 3 - 2x 2 ) - 2x 2 (3x - 8x 2 ) + 10x 4
3
12. ab + a(b – 3) - (5a +2ab) + 7
13.
(2x – 3y)(-2x) - 2x(x + y) - 2xy
14.
-(p + 2)p 2 - p 2 (2p + 1) + (3p – 2)p 2 - (4p + 3) - p 2
The product of two binomials
P(x +2) = px + 2p
(Distributive law)
Now let p = x - 3, then:
(X – 3)(x + 2) = (x -3)x + (x - 3)2
= x 2 - 3x + 2x -6
= x2 - x - 6
Example 1: (x – 5)(x + 1) = (x – 5)x + (x – 5)(-3)
= x 2 - 5x - 3x + 15
= x 2 - 8x + 15
EXERCISE 13
Simplify by multiplication:
1.
x(x – 5) + 3(x – 5)
2.
(x – 5)x + (x – 5)3
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3.
p(x +3)
4.
(X – 5)(X + 3)
5.
a(x + 7)
6.
(x - 2)(x + 7)
7.
(x + 4)x + (x + 4)3
9.
(x + 9)x + (x + 9)(-2)
11.
(2x + 3)x + (2x +3)(-5)
13.
(2x + 3)(x – 5)
15.
(2x +5)(3x – 1)
16. (x – 3)(x + 2)
17. (x + 5)(x – 7)
18.
(x – 1)(x + 3)
19. (x + 3)(x – 5)
20. (x – 9)(x + 8)
21.
(x - 12)(x + 5)
22. (x – 4)(x + 3)
23. (x + 4)(x + 2)
24.
(x – 13)(x – 1)
25.
27.
(3x – 1)(2x + 1)
28. (3x + 1)(x – 5)
30
(2x – y)(x – 2y)
(x +
10.
(x + 4)(x + 3)
(x + 9)(x – 2)
12.
14.
x(2x + 3) - 5(2x + 3)
(3x – 1)(2x + 5)
(2x + 3)(x – 5)
26. (x - 2)(2x – 3)
29. (a - b)(a + 2b)
31. (2p + 3q)(2p + q)
33. (2a –b) 2
36.
8.
34. (x +
1
1
)(x 2
4
32. (2x + y)(2x – y)
35.
(2x – 1)(½x + 2)
1
2
)(x - )
3
3
EXERCISE 14
Multiply by inspection:
1.
(x +1)(x + 2)
2.
4.
(y – 5)(y – 1) 5.
(x – 7)(x + 2)
6.
7.
(a + 11)(a – 4) 8.
(b – 12)(b + 1) 9.
(x + 1)(x – 1)
10.
(x + 1)(x + 1)
(y + 3)(y + 3)
12.
(x – 2)(2x – 1)
13.
(2x + 1)(x – 5) 14.
(2x – 5)(3x + 1)
15.
(3a – 2)(2a + 3)
16.
(2p – 7q)(3p + q)
17.
(b – 10y)(2b + 3y)
19.
(2x – 7) 2
20.
(x -
11.
(a – 2)(a – 4) 3.
(b + 3)(b + 4)
(a – 8)(a + 6)
18.
(x – 5y)(2x + 3y)
1
1
)(x - )
2
4
EXERCISE 15
Squaring of a binomial
Multiply by inspection:
1.
(a +2) 2
4.
(x – 5) 2
7.
(x – 7)2
10.
(x + 12)2
2.
(x – 3) 2
5.
8.
(x – 1) 2
(y + 10)2
11.
(2x – 3)2
3. (x + 4) 2
6. (b + 3)2
9. (x – 20)2
12. (2y + 5)2
Page 17 of 268
13.
(a + b)2
(a + 2b)2
16.
(2x – 3y)2
17.
(2a – b)2
18. (3x + 2y)2
19.
(5a – 7b)2
20.
(p – 4q)2
21. (x +
22.
(
14.
15. (x – 5y)2
1 2
)
2
x
1
- )2
2
3
EXERCISE 16
Multiply:
1.
(x + 1)(x – 1)
4.
(a – 8b)(a +8b) 5.
7.
2.
(2x - 1)(2x + 1)
8.
(x – 4)(x + 4)
(p – q)(p + q)
3. (2x + 3)(2x – 3)
6. (a – b)(a + b)
(2a + 5)(3a – 5)
9.
(7y – 4)(7y + 4)
EXERCISE 17
Multiply by inspection:
1.
(a + 2)(a – 2)
4.
(x – 5)(x + 5)
7.
(x + 12)(x – 12)
2c)(ab + 2c)
13.
(x -
2. (x – 3)(x + 3)
3. (x + 4)(x – 4)
5. (x – 7)(x + 7)
6. (y – 10)(7 + 10)
8. (2a – 9b)(2a + 9b) 9. (7x – 5y)(7x + 5y) 10. (ab –
11. (4xy – 7z)(4xy + 7z) 12. (8p - 6q)(8p + 6q)
1
1
)(x + )
5
5
14. (2x - ½)(2x + ½)
EXERCISE 18
Multiply by inspection:
1.
(x + 8)2
2.
(x + 8)(x -8)
3.
(x + 8)(x – 5)
4.
((x + 8)( x + 5)
5.
(x + 5)2
6.
(x – 5)2
7.
(x + 5)(x – 5)
8.
10.
(2x – 3)2
11. (2x – 3)(5x + 2)
12. (2x – 3)(3x – 2)
13.
(5x + 2)2
14. (5X – 2)(5X + 2)
15. (5X – 2)(X + 3)
16.
(7X – 3)2
17. (7X – 3)(7X + 3)
18. (7X - 3)(2X + 3)
(x – 5)(x – 2)
9.
(x – 5)(x + 2)
EXERCISE 19
Multiply by inspection:
1.
(3a – 5b)2
2.
(5x – 1)(5x + 1)
3. (2p – 3)(2p + 1)
4.
(x – 4y)2
5.
(3x – 2y)(5x – y)
6. (7a – 5b)2
7.
(5x – 2a)2
8.
(3x – 7y)(3x + 7y)
9. (ab – c)(3ab – 5c)
Page 18 of 268
10.
(3x – 2y)(3x + 2y)
13.
(6x – 1)2
16.
(ax + by)2
19.
(2x4 -
11, (10p + 3)2
12. (7a - 9c)(a – 2c)
14. (8a – 3b)(8a + 3b)
1 2
)
3
17. (
x
+ 2y)2
2
20.
(½p -
15. (20x + 50y)2
18. (
1
1
a _ b)( a + b)
4
4
1
)(2p + 3)
3
EXERCISE 20
Multiply:
1. 2(x – 1)2
2. 3(x + 2)(x – 1)
3. -1(x + 3)2
4. -5(x – 2)(x + 2)
5. (x + 10)(x – 5)
6. -1(x – 5)(x + 5)
7. -2(x + 8)(x – 4)
8. -3(x + 2)2
9. 4(x – 1)(x + 7)
10. 2(x – 9)(x + 9)
11. 2(2x -1)2
12. 3(2x + 3)2
13. -1(2x – 5)2
14. -2(2x + 3)(2x -1) 15. -3(2x + 1)(x – 2)
18. 3(5x -1)2
16. -1(7x + 3)(7x – 3) 17. 4(2x + y)(2x + y)
19. 2(2x – 1)(2x + 1)
20. -1(x + 5)(x – 4)
22. 3(5a – 2)2
23. 6(a –b)(a + b)
25. 2a2b(a – 2b)2
26. p2(p – 2q)(p + q)
28. -5a3(a2 – 1)(a2 + 1)
29. -2a2(a – b)(a – 2b)
21. 2( 10p + 3)2
24. 2x(x – 2y)2
27. -3x2(x2 + 2)2
30. –(x – 3y)(x + 5y)
EXERCISE 21.
Simplify:
2. (x + 3)2 + (x + 3)(x – 3)
1.
(x + 2)(x + 3) + (x – 5)(x – 1)
3.
(a – 7)(a + 2) + (a + 5)2
4. (b – 1)2 + (b – 6)(b + 8)
5.
(2x + 3)(x – 2) + (2x -1)(x – 1)
6. (2x + 1)(2x – 1) + (x – 3)(2x + 4)
7.
(a – 4)(a – 5) - 2a(a – 4)
8.
(x + 5)2 + (x + 5)(x – 5) + (x + 5)(x – 7)
9.
2x(x – 3) + x(2x – 1) - 4x(x + 7)
10.
(2y – 5)2 + (y – 1)(3y + 5)
11.
2k(k – 3) + (2k + 1)(k – 3) - k(4k – 12)
12.
(a – 4)2 + (a – 3)2 + (a + 2)2 + (a + 1)2
SCIENTIFIC NOTATION
It is a convenient way to express very large or very small numbers.
Page 19 of 268
Examples:
1.
2.
3.
4.
5.
6.
EXERCISE 22
1. Write in scientific notation:
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
2 Write as ordinary numbers:
2.1
2.4
2.2
2.3
2.5
3 Simplify, without using a calculator:
3.1
3.2
3.2
3.4
1.5
Page 20 of 268
CHAPTER 5
FACTORS
COMMON FACTORS
3 is a factor of 12 and also a factor of 18, therefore 3 is a common factor of 12
and 18 but is not the biggest factor of 12 and 18. 6 is the biggest number that
can go into 12 and 18, hence 6 is the H.C.F. of 12 and 18.
EXERCISE 1
Write down the H.C.F. of each of the following:
1.
8 ; 12
2.
24 ; 32
3.
15 ; 24
4.
8 ; 18
5.
6a ; 8ab
6.
12x²y ; 18xy
7.
9a ; 12a²b
8.
7p²q ; 14pq²
9.
21x²y ; 15x²y²
10.
18a²bc ; 9a²b²c²
11.
36xy3z2 ; 54x2y4z2
12.
12a3bc5 ; 16a4b3c2
13.
ab2 ; a2b3c ; b2c3
14.
x2y3z ; xy4z2 ; x3y2
15.
16ab ; 9a2 ; 3a
16.
6ab ; 40a2b ; 24ab2
17.
6xy2 ; 12x2y2 ; 30xy3
18.
4a2 ; 18ab ; 16b2 ; 2a2b2
19.
8r2s2t ; 12rs4t2 ; 16s3t3
20. 32ab4c6 ; 24a2b2c5 ; 36a3b3c2
21.
10mn ; m2n ; 30mn2
22.
(2xyz)3 ; 4x2y2z ; 8xy3
23.
15p4q3 ; 20p3r2 ; 25p2qr
24.
108a2b ; 72ab2 ; 90a2b2
FACTORIZATION: THE HIGHEST COMMON FACTOR:
Eg.
Factorize:
Solution:
12a2b + 16ab
What is the highest common factor of 12a2b and 16ab.
The HCF is 4ab.
 12a2b + 16ab = 4ab X 3a + 4ab X 4
= 4ab ( 3a + 4 )
EXERCISE 2
Factorize:
1.
4a + 8
2.
3x - 9
3.
12ab - 6a
4.
xy + x
5.
p2 + p
6.
a²b + ab²
7.
5x² - 10xy
8.
6a + 6
9.
a3 + a²
Page 21 of 268
10.
ab² - a²b²c
11.
6r - 9s
12.
10e + 15e²
13.
9c² - 9c
14.
-4a + 2
15.
-ab + ac
16.
a3 + a² + a
17.
ab + bc + ac
18.
a²b + ab² - ab
19.
3x3 + 2x² + x
20.
-p² - 8p
21.
2x² - 4c3
22.
3a4 - 4a3
23.
8b4 - 6b5
24.
-2ab - 4a - 6b
25.
12a3bc² - 8a²b - 16a²b²
26.
24ap² - 36b²p3 - 30abp²
27.
5ef² - 5f² + 5ef
28.
-8ab + 8ac - 16a²b
29.
2x3 + 3x² + 2x + 1
30.
4g²h3 - 8g3h² - 16g4h5
EXERCISE 3
Factorize:
1.
5x² + 10x
2.
2a² + 2a
3.
5ab - 30ab²
4.
6y² - 8y
5.
14a3 + 21a² + 7a
6.
r² - r + 2
7.
12p² + 12p
8.
2a3 - 6a² + 4a
9.
b3 - 3b² - 3b
10.
a² + a(3b + a)
11.
r² + 9
12.
a² + 9a
13.
p² - 4p + 4
14.
9x² + 6y²
15.
5x² + 5xy + y²
16.
32a3b²c - 16a²bc² + 8a4c
17.
6a + 8ab - 4ac
18.
v²u - 4u² + 2vu - u
19.
22a11 - 11a10 + 33a9
20.
axy - 4ax + 4xy
THE DIFFERENCE BETWEEN TWO SQUARES.
Eg. Multiply each of the following by inspection:
(a)
(a – b)(a + b)
(b)
a – 4)(a + 4)
(c)
(x – y)(x + y)
(d)
(x – 7)(x + 7)
YOU WILL NOTE THAT EACH ANSWER IS THE DIFFERENCE BETWEEN TWO SQUARES.
NOW FACTORIZE EACH OF THE FOLLOWING.
(a)
a² - b²
Ans:
(a + b)(a – b)
(b)
x² - 9
Ans:
(x + 3)(x – 3)
(c)
16x² - 25y²
Ans:
(4x + 5y)(4x – 5y)
Page 22 of 268
EXERCISE 4
Factorize:
1.
p² - q²
2.
a² - b²
3.
a² - 1
4.
a² - 4
5.
x² - y²
6.
4a² - b²
7.
16x² - 49
8.
25b² - 1
9.
81p² - 25q²
10.
64x² - y²z²
11.
121 - 49a²
12.
100b² - 25c²
13.
9r²s² - t²
14.
81a² - b²c²
15.
1 - 25a²b4
16.
a4 - b²
17.
a²b² - 4
18.
x 8 - 16
19.
8x² - 2
20.
10a² - 40
21.
25 - 36p²
22.
x² - 121y²
23.
a²b²c² - 49d²
24.
50a²b - 2b
EXERCISE 5
Factorise completely:
1.
5a² + 20
2.
5a² - 20
3.
16x² + y²
4.
16x² - y²
5.
8p² + 2
6.
8p² - 2
7.
9xy² - 16x
8.
25a² + 50
9.
200xy - 2x3y
10.
9x² + 3x
11.
36a² - 16a²b² - 4a²c²
12.
x 20 - 1
13.
p 4 - 81
14.
3abc² - 12ab
15.
a² + 4
16.
20a²b² - 45a²c
17.
121x² + 1x
18.
b 8 - 256
19.
a8 - 1
20.
ab² - ac²
21.
a3 - a
22.
4b 5 - 16b
23.
x - 4x3
24.
b16 - 625
EXERCISE 6
Factorise completely:
1.
x(3a + 2b) + y(3a + 2b)
2.
3x(3x – 5y0 - x(3x – 5y)
2.
a(b – 3) - b(b – 3)
4.
5(c – 2d) - p(c – 2d)
5.
x²(3e + f) - x(3e + f)
6.
12y3(7g + 2h) + 9y²(7g + 2h)
7.
(2a – 3)(a + 2) + (2a + 3)(a + 2)
8.
a(3x – 2y) + (3x – 2y)
9.
(2a + 9b)(3a – b) = (3a – b)(4a + b)
10
(3a – 6b)(9a + 5b) - (3a – 6b)(6a – b)
11.
p(5t – 3) + 5t - 3
12.
p(a + 2b) - 3(a + 2b)
13.
(x + y)(x – y) - x - y
14.
(a + b)(a – b) - a + b
15.
p² - p - 3(p – 1)
16.
a(b – 3) + b(3 – b)
Page 23 of 268
17.
3(2c + 3) - p(3 + 2c)
18.
d(e – 5) + (5 – e)
19.
r(1 – 3k) - 5(3k – 1)
20.
5(p² - 3) - q(3 – p²)
21.
y(3x + 5y) - (5y + 3x)
22.
(5 – t)(3t + 1)( - (t – 5)(4t + 3)
23.
(3w – 6)(4x – 7) + (2x + 1)(6 – 3w)
24.
(5w – 1)² - (1 – 5w)
EXERCISE 7
Factorise the following trinomials:
1.
x² + 7x + 6
2.
x² - 7x + 6
3.
x² - 5x - 6
4.
x² + 5x - 6
5.
x² + 5x + 6
6.
x² - 5x + 6
7.
x² - x - 6
8.
x² + x - 6
9.
x² + 13x + 12
10.
x² - 13x + 12
11.
x² + 11x - 12
12.
x² - 11x - 12
13.
x² + 8x + 12
14.
x² - 8x + 12
15.
x² + 4x - 12
16.
x² - 4x - 12
17.
x² + 7x + 12
18.
x² - 7x + 12
19.
x² - x - 12
20.
x² + x - 12
21.
x² + 9x + 8
22.
x² - 9x + 8
23.
x² - 7x - 8
24.
x² + 7x - 8
25.
x² + 6x + 8
26.
x² - 6x + 8
27.
x² + 2x - 8
28.
x² - 2x - 8
EXERCISE 8
Factorise:
1.
x² + 4x + 3
2.
x² + 7x + 10
3.
x² + x - 2
4.
x² + 6x - 7
5.
x² + x - 6
6.
x² - 9x + 20
7.
x² + 3x - 18
8.
x² + 10x + 16
9.
x² - 4x + 3
10.
x² + 4x + 4
11.
x² - 8x + 15
12.
x² - 11x + 10
13.
x² + 5x - 14
14.
x² + 6x + 9
EXERCISE 9
Factorise:
1.
a² + 3a + 2
2.
a² - 3a + 2
3.
a² - a - 12
4.
a² + 3a - 10
5.
a² + 6a + 5
6.
a² - 9a + 8
7.
a² + 9a + 8
8.
a² + 9a - 10
9.
a² + 11a + 10
10.
a² + 9a + 14
11.
a² + 6a - 16
12.
a² + 12a + 20
13.
a² + 5a + 4
14.
a² - 7a + 6
15.
a² - 8a - 9
16.
a² - 6a + 9
Page 24 of 268
EXERCISE 10
Factorise completely:
1.
2x² - 8x + 8
2.
2x² - 4x - 6
3.
3x² - 6x - 24
4.
5x² + 10x + 15
5.
4x² - 36x + 56
6.
-2x² + 18x - 16
7.
-x² - 10x - 9
8.
-10x² - 60x - 90
9.
ax² - 5ax - 6a
10.
2x4 - 14x3 + 20x²
11.
3a3b - 12a²b + 9ab
12.
2x²y3 + 16x²y² - 40x²y
13.
-x5 - 11x4 + 12x3
14.
-10a4b - 10a3b + 120a²b
15.
p6q² - p5q² + 12p4q²
16.
a4b + 16a3b + 64a²b
17.
2a4 - 8a3 + 8a²
18.
2a4 - 8a²
19.
20x3 + 40x² + 20x
20.
-4ab3 + 24ab² + 28ab
21.
2x - 12 + 2x²
22.
10 - 4x + 2x²
23.
4xy + x3y - 5x²y
24.
2x² + 2x + 16
25.
4x² + 16
26.
x² - 17x + 30
27.
x² - 13x - 30
28.
3x² - 21x - 90
29.
2a² + 22a - 24
30.
2a² + 4a - 96
FACTORIZING TRINOMIALS WHERE THE COEFFICIENT OF YOUR SQUARE
TERM IS NOT 0NE: EXAMPLE: 5X² - 36X + 7 = (5X -1)(X – 7)
EXERCISE 11
Factorise completely:
1.
3X² + 19X + 6
2.
3X² + 11X + 6
3.
5X² + 23X + 12
4.
3X² + 35X + 15
5.
5X² - 6X + 15
6.
5X² - 27X - 18
7.
3X² - 20X + 12
8.
2X² + 13X - 18
9.
7X² + 23X + 18
10.
5X² - 32X + 12
11.
2X² - 5X + 12
12.
18X² - 21X - 4
13.
X4 - 3X² + 2
14.
X6 - 3X3 + 2
15.
2X² - 3X + 1
16.
(X + 5)² - 3(X + 5) + 2
Page 25 of 268
EXERCISE 12 (MIXED EXAMPLES)
Factorise completely:
1.
2a² - 2a
2.
3x² + 3
3.
x² + 3x + 2
4.
3x² - 3
5.
e² - 25
6.
5x² - 25
7.
a² - 4b²
8.
x² + 5x + 6
9.
3x3 + 3x² + 3x
10.
a²b - ab
11.
a²b² - 1
12.
x² - 4x + 3
13.
x² - 4x
14.
x² - 9
15.
y3 - y
16.
4x² + 11x + 6
17.
8x² - 9x - 14
18.
12x² + 2x - 10
19.
12x² + 8x - 15
20.
5x² - 21x + 18
EXERCISE 13 (MIXED EXAMPLES)
Factorise completely:
3x3 - 18x² + 24x
1.
2x² - 18
2.
4.
6x3 - 6x
5.
ax² + 3ax + 2a
6.
p3 + 4p² + 4p
7.
2x3 - 4x² - 6x
8.
2a²b - 32b
9.
2ab² - 12ab + 16a
10.
x4y + x3y - 2x3y
11.
-3a²b - 6ab - 3b
12.
16x²y4 - x²13. 32p²q4 - 2p²
14.
2x3y + 18xy
15.
-x² - 6x - 9
16.
(x – 1)² + 2(x – 1) - 3
17.
(x – 1)² - 4
18.
(x – 3)² - 6(x – 3) + 8
19.
(a – b)² + 2(a – b) + 1
3. a3 - ab²
20.
x16 - 16
Page 26 of 268
CHAPTER 6
FRACTIONS
SIMPLIFICATION OF FRACTIONS:
EXAMPLES:
1.
12 2 X 2 X 3 2 3 2
2 2

 X X  1X 1X 
30 2 X 3 X 5 2 3 5
5 5
2.
5ab
5 XaXb
5b
a
a
a


X
 1X

10bc 2 X 5 XbXc 5b 2c
2c 2c
EXERCISE 1
Simplify:
2.
16x ² y 3
12xy
3.
9abc²
12a ²bc
8 pq
10 p ²
5.
22x ² y 3
11x 3 y 5
6.
2rs ²
8r ² s 4
5a 7 b 5 c 4
10a 3b 3 c 3
8.
18e 3 fg ²
9e² fg
1.
10ab²
15a ²b
4.
7.
32x ² y 3 z
64x 3 y 3 z ²
9.
10.
12kmx²
20k ² mx
11.
3ab².2a ²b
12ab 3
12.
36u ²v 3 .uv²
27u 3 v 4
13.
7 abx²
14ax.4bx
14.
4a ² p.9bp²
6ab.15 p 4
15.
48x ² y 3 .36x 3 y ²
27 xy 3 .32x 5 y 4
16.
x12
x14
17.
5a ²b15
15a 5 b16
18.
x 3 y.xy 3
x 4 y ².x ² y ²
EXERCISE 2
Simplify:
1.
ab  a
ab
2.
7 x  14
7x
3.
5a  a ²
5a
4.
x²  x
x²
5.
2 x²  2 x
2x
6.
3a ²b  6ab
6ab
7.
10a  5
5
8.
2 p²  4 p
p²  4
9.
5 x ² y 3  10 xy²
5x 3 y 3
10.
9abc  12ab²
6abc²
11.
14 pqr  21p ² qr
49 p ² q ²
12.
12u ²v 3  8u 3 v 3
6u ²v 4
Page 27 of 268
13.
3a ²b  3ab²
3a ²  3b²
14.
2 x ²  2 xy
4 x²  4 y ²
15.
x²  2 x  1
x²  1
16.
x ²  7 x  12
3 x  12
17.
x²  5x  6
3x  6
18.
a ²  2a  15
a ²  25
19.
a²  a  2
a²  a
20.
2 p²  4 p
p²  4
21.
x²  5x  6
x²  9
22.
6 x 4  8x 3  2 x²
2x
23.
a ²  b²
pa  pb
24.
a ²b  2ab²
a 3  4ab²
25.
3a 3  6a ²b  3ab²
4a 3b  8a ²b²  4ab3
27.
p²  9
p ²  p  12
28.
xy  x
y 1
29.
d e
ed
30.
x ²  xy
x ²  xy
31.
x ²  xy
xy  x ²
32.
ab
ba
33.
2m  n
2n  4m
34.
3r  2 p
8 p  12r
35.
mn(2  p)
21m( p  2)
36.
2 ( x  4)
4( 4  x )
37.
8a ²  8b ²
a4  b4
38.
x 1
( x  1)²
39.
8m 4  32x 4
m²  2 x ²
40.
22 p  44x
9 p ²  36x ²
41.
x²  y ²
x y
42.
( x  3 y )²
3y  x
43.
16x ²  25
5  4x
44.
9m  3n
n  3m
45.
( x  3)( x  2)
4  2x
46.
3 x ²(2a  b)
x(b  2a )
47.
8a 3 ( x  y )²
4a ²( x  y )
26.
9a 3b  18a ²b²
3a 3  12ab²
EXERCISE 3
Simplify:
1.
15a -2
2.
-6a -2
3.
-5 + a0
4.
6
a²
5.
3
- 2x -3
x3
6.
5
x y 3
7.
6 p 2
 3 p  2 m 1
8.
 6 y 2
x 1
9.
14m² n 2
 7mn
10.
2ab
 4ac
c
11.
2a 9b
X
3
a
12.
2 x 5 y 3 4by²
b
X
X 4
b²
x²
8y
13.
x ² xy x 3
X

y
z ² zy
14.
a ² bc b²c
 X
bc² a ²
a²
15.
a ²  bc b ²c 
 X

bc²  a ²
a² 
2
Page 28 of 268
16.
a ² bc b²c
X

bc² a ² a ²
17.
a ² bc b ² c


bc² a ² a ²
18.
a ²  bc b ² c 
 

bc²  a ² a ² 
EXERCISE 4
Simplify:
1.
(a²b + 3ab) ÷ a²b
2.
x ² y ( y  2)
 ( x ²  3 x)
y
3.
4a ²  12a
4
X
8
a3
4.
x3
 ( x  3x)
2
5.
2 x²  2 x
1

x²  x
x 1
6.
x
4 x²

x  1 x²  x
7.
a ²  ab
a

ab
a 1
8.
2b
4b

b  7b² b²  49
9.
x
x²  4 x

x  2 x²  4
10.
x  5 5x²  5x
X
5
x ²  25
11.
x²  1 x²  2 x
X
3x  6
x 1
12.
x
x²  9
X
x  3 x²  3x
13.
x
4

x  2 x²  2 x
14.
2 x  6 6 x ²  18x
X
3x
x²  9
15.
x
x²  4
X
x  2 x²  x
16.
2 x ²  2 x 2 x ²  10 x
X 3
x ²  25
4 x  4 x²
17.
3x  21 x
X
x ²  49 3
18.
x 4  81x ² x ²  9 x

2x
6
19.
x²  2 x x²  4

x²  2 x x  2
20.
2 x 3  32x
x
X
x²  x
x²  4 x
21.
x ²  3x
x ²  3x  2
X
x²  5x  6
x²  4 x
22.
x²  x
x²  4
X
x²  x  2 x²  3x  2
23.
2 x²  8x
x²  4
X
x²  6 x  8 x²  2 x
24.
x²  2 x  1
x5
X
x ²  6 x  5 x ²  3x  2
25.
x
x ²  3x
x²  9

X
x  2 x²  4
x²  2 x
26.
x ²  10x  21
3x ²  6 x
X
2 x ²  14x
x²  5x  6
27.
x(2 x  3)( x  2) 2 x ²  3x

( x  2)( x  1)
x²  1
28.
4a ²  12a
1
X
a
a²  9
29.
(2 x  10) ( x  2)( x  5) (2 x  4)
X

( x ²  25)
5
25
30.
(a  3)(a  2) (a  3)(a  1)
(a ²  4)


a²
(a ²  a)
( 2a ²  4a )
3
Page 29 of 268
31.
(3 y ²  12 y )
(2 y ²  32)
(4 y  16)

X
( y  1)
( y  2)( y  1) ( y ²  2 y )
32.
( x ²  2 x)( x  4) ( x  3)( x  1) (4 x ²  36)
X

( x  2)( x  1)
( 2 x  6)
( x 3  4 x ²)
33.

xy  2 x  y ²  2 y
xy  x

X
xy  2 x  y ²
x( y  2)( y  1) 
EXERCISE 5
Write each of the following as a single fraction:
1.
1 1

2 3
2.
2 3

3 4
3.
2 x 3x

3
4
4.
3 4

a² a
5.
2 3

a ab
6.
2
5

3b 4b
7.
x y

y x
8.
2 3

x x²
9.
k 3k

6 8
10.
1
2
3


ab² ab a ²b
11.
2 3
 5
x² x
12.
3p 5p 7 p


4
8 16
13.
5a 2b 3


b²c ac bc
14.
a5 a4 3


3
2
4
15.
a  2 3a  5

5
15
16.
b2
+1
5
17.
3 -
18.
d 1 d 3
 
12
8 6
c2
5
EXERCISE 6
Find the L.C.M. of each of the following:
1.
12 ; 18
2.
6 ; 10
3.
32 ; 12
4.
a²b ; b²
5.
2xy² ; 6x3y
6.
3a²b3 ; 9a3b ; 12ab²
7.
b²c ; ac ; bc²
8.
4xy ; 8x² ; 16y² 9.
5pq3r4 ; 15p²q²r ; 20p3
10.
18a3b²c ; 24a²b4c5
11.
16r²s ; 8rst² ; 12s²t3
12.
9xy ; 8x² ; 6xy3
13.
10a ; 15a² ; 6ab
14.
4pq3 ; 12p² ; 10q
15.
6a²bc ; 8abc ; 12ab²
EXERCISE 7
Write each of the following as a single fraction and simplify if possible:
1.
5x x

12 18
2.
a 5a

16 24
Page 30 of 268
3.
b 5b

36 24
4.
3
1 2


4x 6x x
5.
1+
3
2

x 3x ²
6.
1
2a
3


a ²bc b ² c ac
7.
3
1
3


4 x ² 6 xy 8 y ²
8.
2
3
5


3 p ² q 8 pq² 12 pq
9.
2
3 4

 5
a ²b ab a
10.
2 x  1 x  1 3x ²  x  1


x
2x
x²
11.
a  1 a  3 4  a²


3a ²
4a
6a ²
12.
y  3 2 2  y  6y3


4 y² 3 y
6y3
13.
b  2 2b ²  3 3


5b
10b ²
2b
14.
a²  1 a²  2 a  5


8a
4a
16
15.
a 1 a  2 a  3


a
2a
3a
16.
b2 b3

2
b²
b
17.
x2 x
 3
x²
2
18.
a 2a 3  b


b b²
2b
19.
2x  3 4x  2 x  1


x
3x
6x
20.
y  2 2 y  5 y²  2 y  1


y²
3y
4 y²
21.
2y 1
1
3y
22.
2x  3 x  4

3x
x
23.
2a  b 5a  2b a  2b


4
6
8
24.
2 m  n 4 n  5m
1


3m² n
2mn ²
4mn
25.
y  4 2 y 1 3y  2


3
2
4
26.
x  3 4x  5 x  3


2x
5x
4x
EXERCISE 8 (Revision)
1.
Determine the HCF and LCM of each of the following.
(a)
16a²b; 8ab²c; 4ac
(b)
6xy; 12x²; 8xy²
(b)
x²  6 x  8
x4
2.
Simplify:
(a)
x²  4
x2
3.
Simplify:
(a)
5  x 5  3x 1


2
6x
3
(b)
2x  3 4x  1 1


3x
2x
6
(c)
a ²  4a a 3  2a ²

a ²  16 2a ²  8a
(d)
3
1
2


4 x 6 x 3x
(b)
3a ²b 3 2ab² 6a 3b 4
X 3 
c
c²
c
4.
Simplify:
(a)
3a 3  3x
2 x²  2 x
5.
Simplify:
(a)
6+
1
3
(b)
6X
1
3
(c)
a+
1
b
4x  8
4x
(c)
(d)
aX
1
b
Page 31 of 268
6.
Simplify:
(a)
2 x  4 x²  x  2
X
x²  4
2 x²  2 x
(b)
a4  b4
a b
7.
Simplify:
(a)
 1  1 
1  1     x ²  1
x 
x

(b)
x y
yx
8.
Simplify:
(a)
4 x ²  6 xy 6 xy  4 y ²

2x
2y
(b)
1-
9.
Simplify:
(a)
2a  2b
b²  a ²
(b)
ab
ba
X
2b  2a  3a  3b
(c)
2 x²  8 y ²
 ( x  2 y )
6 xy  3x ²
(d)
5a 6b 16a ²  b ²


2b 3a
6ab
(e)
y  2 3 2 y²  3


5y
2y
10 y ²
(f)
2a  3 3 4  a  5a ²


3a ²
4a
6a 3
10.
The expression
(a)
x ²  3x  18
x6
x 1
x
is given.
For which values of x is the expression
x ²  3x  18
x6
not defined?
(b)
Simplify the expression.
(c)
Calculate the value of the expression if:
(i)
x = -5
(ii)
x = 3.53
m4 2
m 1
 (m  1) 
2
3
4
11.
Simplify:
(a)
12.
Simplify:
p ²  16
p²  2 p  8
X
( p  4)²
p²  4
(b)
a  ab 2a

b  b²
b
(b)
x  1 2 x  1 3x  2


2
6
3
CHAPTER 7
EQUATIONS
Remove all brackets and fractions. Then simplify your sides by adding all the like terms.
EXERCISE 1
Solve the following equations:
1.
2a
6
3
2.
x 3
 5
2 2
3.
3x – 2 = 13
4.
2x – 3 = 3 - x
5.
2(x + 1) = 2(2 - x) + 6
6.
2(x – 7) + 3x = 2(x – 4)
7.
2(2x – 1) = x + 1
8.
-5x = 10
Page 32 of 268
9.
-12 + x = 3
10.
-12x = 3
11.
3x
2
4
12.
2
1
x
5
10
13.

2x 1

5 10
14.
 4x 2

3
9
15.
4 - 2x = 5
16.
2x - 1 = 3x
17.
x + 4 = 2x + 5
18.
7x + 9 = 4x
19.
6x + 2 = 2x - 2
20.
1
x 3 1
2
EXERCISE 2
Solve the following equations:
1.
3x - 6 = x - 4
2.
5x - 3 = 2x + 9
3.
3x + 8 = x + 7
4.
7x + 5 = 3x - 7
5.
5x + 7 = 3(3 - x)
6.
5(x + 5) = 2(x - 3) - 5
7.
2(2x + 5) = -2(1 + x)
8.
4(3x - 8) = 5(x - 3) - 3
9.
3(x - 4) = 2(x - 2) + 2 - x
10.
7(x + 3) + 3x + 1 = 2(x - 13)
11.
2(3a + 1) = 7 - 4a
12.
6(p – 1) - p = -2(P + 1) + p – 1
13.
2(2y - 1) - 3y = 4 - y
15.
4(x - 1) + 2(2x - 3) - 2 = 15 - x + 3(3 - x)
16.
7(x + 2) + 2 = -5(x - 6) + 2(5 - 3x)
17.
2(x + 3) + x + 7 = 2(2x + 9) + 5(4 - x) + 3
18.
8(x - 1) + 2(3x - 2) = -7(x + 2) + 3(x - 6) - 2x
14.
2(b + 2) - 4 = 2(b + 1) - b + 2
EXERCISE 3
Solve the following equations:
1.
(x - 2)(x - 3) + 2(x - 2) = (x + 4)² - 3x + 10
2.
(x + 3)(x + 4) - 2x(x + 4) = -2x - (x - 2)² + 4
3.
(x - 1)(x + 1) + (x - 2)² = 2(x + 3)(x - 2) + 3(x - 4)
4.
(x - 2)(x + 7) + (x - 5)(x - 3) = (2x - 7)² + 2x(11 - x)
5.
(x - 5)² = (x - 4)²
6.
2(x - 3)(x + 1) - (x + 4)(x - 2) = x(x - 7)
7.
(2x - 3)(x + 3) - (2x + 1)² = 4x - 2(x + 1)² - 5
8.
(2x - 3)(2x + 1) + 2x(x - 5) = 3x(2x - 5)
9.
(x + 3)(x + 4) + (x - 2)² = x(x - 5) + (x - 1)²
10.
(x - 1)² - (x - 2)² - (x - 3)² = -(x - 4)² + 4x
Page 33 of 268
EQUATIONS WITH FRACTIONS.
Whenyou solve equations with fractions, eliminate all fractions by multiplying both sides of the equation by the Lowest Common Denominator
(L.C.D) of the fractions.
Eg.
3x x
5x
 6
4 3
2

 3x x

 5x 
12    6   12 
 4 3

 2 
12 X
In this case the L.C.D. of 4, 3, 1 and 2 is 12.
3x
x
5x
+ 12 X + 12 X 6 = 12 X
3
2
4
9x + 4x + 72 = 30 x
13 x - 30 x
= - 72
- 17x
= - 72
x =
 72
=
17
4
4
17
EXERCISE 4
Solve the following equations:
1.
3x
= x - 1
2
3.
a -
5.
y 1 y 1
 

3 2
6
2
6.
x+
7.
b b
 4
3 2
8.
2 y 3y 1


3
4
2
9.
5 x 3x
1

1
2
4
2
11.
x
5
 x  x2
6
2
1
a
= -2 +
2
2
2.
4.
P +
10.
12.
4x
1
= x 3
3
3
p
= 1 +
4
2
1 x x 3
  
2 4 4 4
3a
1
a  2
5
2
3x
- 4 = x + 1
2
EXERCISE 5
Solve the following equations:
1.
x2
x2
2
2.
a–1-
a a 1

3
3
Page 34 of 268
3.
x
4x  7
 2( x  1) 
2
2
2
4.
y
4  3y
2 y
7
3
5.
a  2 a 1 3

 a2
4
2
4
6.
2 x  3 x  1 3x  1


2
3
2
7.
y3 y2 y

 1
4
8
2
8.
2x x  2
3x  2

 3  1
3
2
6
9.
2 y 1 y  2 3y  8


 y2
5
2
10
10.
x x9 x
x

    x 1
5
3
6
2
11.
x2
x 1
x
3
6
12.
3a  4 7a  5 a  2


4
8
2
13.
2a  3 a  9

0
5
6
14.
b  1 b 1
2b

 1
3
2
3
15.
7(3 x  1) 3(2 x  3)
1

8
6
2
3
16.
x  2 2x  5
1

 x 1
8
6
3
17.
3x  5 3x  2

20
6
4
18.
3
5
2
(r  2)  (r  3)   0
8
12
3
19.
5x  2 x
3x  5
  x
3
5
15
20.
2( x  1) 3(2 x  1)

 2
3
2
Zero denominators.
If an equation has a variable in the denominator, you must check your
answers to make sure that the denominators of the original equation
are not zero. If for example, you solve the following equation:
x 1 1  3  x
 
3x
9
9x
Multiply both sides by 9x: 3(x - 1) - x = -3 + x
3x – 3 - x = -3 + x
2x - 3
= -3 + x
2x – x
= -3 + 3
X
= 0
But if you substitute 0 into the original equation, you get a 0
denominator which is undefined. Therefore this equation has no
real solution.
Page 35 of 268
EXERCISE 6
Solve the following equations:
1.
1
3 4
x
2.
2 3
 1
x 4
3.
15 2
 1
2x 3
4.
2 4 2
 
x 3 3x
5.
3
1 9
2 
y
2 2y
6.
3
7.
2
1
1 
3x
x
8.
5 1 3
 
p 3 4
9.
1 1 2 x


2 3x
6x
10.
11.
2 3

7
a 2a
12.
1 1 1 1
3
  

2 y 3 2y 2y
13.
5 2
x3

 4
4 3x
12x
14.
5
15.
1
2

x 1 x
16.
1
2
3


x  1 x  1 ( x  1)( x  1)
2 3

4
x 2x
1 2
5
 
y 3 6y
8 2( a  4)

a
a
EXERCISE 7
1.
Find m if 3 is a root of the equation:
2  x 2x  m 5


1
3
4
12
(a root of an equation is a value of x that makes the equation true)
2  x a  2x x  3


3
4
5
2.
Find a if -3 is a root of:
3.
Find m if
4.
show that 4 is a root of
5.
show that 5 is a root of
6.
Show that -3 is a root of
7.
See if
8.
Check whether 2 is a root of
1
is a root of
5
2( x  1) 11( x  2)

1
5
m
2
1
( x  1)  (3 x  2)  3  2 x
3
2
1
1
1
1
( x  3)  x  1  (2 x  4)  1
4
2
3
2
5(3 – x) -
1
(5 – x) = 2(4 – 3x)
2
1
1
1
1
is a root of 4(x + 1) - (3x + 1 ) + 5
2
3
2
2
3x² - 5x + 6 = 0
Page 36 of 268
9.
Check whether -4 is a root of
1
1
1
(9 - 2x) = 1 (7x – 18)
2
2
10
EXERCISE 8
In this exercise, if what is given is an equation, solve it;
if what is given is an expression, simplify it.
1.
5 x  1 x  1 3x  1 x  1



6
2
4
12
2.
1
1 1
(9  2 x)  1  (7 x  18)
2
2 10
3.
3
1
1
x  (4  x)  2  3( x  2)
2
3
3
4.
2 x  3 3x  2 7 x  2


5
4
20
5.
3x  1 1 2 x  3
 
0
4
3
12
6.
3x  1 1 2 x  3
 
4
3
12
7.
2 x  3 3x  5 5 x  3 7 x  5



3
5
6
10
8.
2 x  1 3x  2 x


3
6
6
9.
1  x 2( x  2) 3(1  2 x) 31  32x



2
3
4
12
10.
2 x  1 3x  2

3
6
11.
2 x  1 3x  2 x

 1
3
6
6
12.
2 x  1 3x  2 x


3
6
2
EXERCISE 9 (REVISION)
Solve each of the following equations:
1.
2(x + 3) = 4(x - 2) = -3(x + 1) + 10(x - 3)
2.
7(x - 2) + 3x = 4(2x - 4)
3.
(x - 1)² - (x + 2)² = 2(x - 1) -2(x + 2)
4.
(x + 3)(x - 1) + 2(x - 2)² = 2x(x - 3) + (x + 1)(x - 2)
5.
1
1
1 1
2
1 a  a  2  a  a 1
2
3
4 4
3
6.
2a
1
 a 1
3
2
7.
x x2 5 4 x

 
2
6
3
2
8.
4 1
1 1
 1 
3x 5
3 5x
9.
(y - 2)(y + 3) + (y - 4)(y + 5) = 2(y - 4)²
10.
4
1
1
1
1 
3
3p
4 12 p
2
12.
1
1
1
(2 x  5)  (4 x  3)  2  1
3
6
6
11.
1
1 1
(2 x  9)  1  (7 x  18)  0
2
2 10
EXERCISE 10
Write the following in algebraic language:
1.
Twice the product of x and y.
2.
The sum of the squares of x and y.
Page 37 of 268
3.
The square of the some of x and y.
4.
x is 2 more than y. (Write this in two ways)
5.
x is twice y.
6.
The sum of x and y exceeds the difference of x and y by 2.
7.
The sum of x and y is twice the difference between x and y.
8.
If x is divided by y the result is m.
9.
The sum of the square of x and y is equal to 41.
10.
The square of the sum of x and y is 8 more than the sum of the
square of x and y.
11.
Half of the sum of x and y is equal to 10.
12.
Five times x decreased by 4 is equal to y.
13.
The difference between x and y is 5 (where x > y)
14.
The difference between x and y is 10 (where y > x)
15.
The square root of x is greater than the square of y by 10.
EXERCISE 11
Translate the following word sentences into mathematical equations and solve them where
possible:
1.
x is the same number as y.
2.
A certain number increased by five equals thirty.
3.
Five times a certain number equals thirty.
4.
Three time a number decreased by two equals nineteen.
5.
The sum of two consecutive integers equals thirteen.
6.
One natural number is twice another natural number and their difference is equal to
twenty.
7.
The sum of three consecutive natural numbers is equal to fourteen.
8.
The product of two consecutive even integers is forty-eight.
9.
The sum of two consecutive odd numbers is equal to twenty-four.
10.
If a certain number is increased by seven it will be equal to twice that same number
decreased by three.
11.
The sum of a and b is equal to c.
12.
The product of a and b is equal to c.
13.
The difference between a and b is equal to c.
14.
The sum of twice a and c is equal to b decreased by five.
Page 38 of 268
15.
a exceeds b by 4.
16.
The product of a and b is greater than c by twenty-two.
17.
The sum of two numbers equals 10. Twice the one number plus 5 will equal 9.
18.
The difference between two natural numbers equals 11. Twice the
smaller number plus 3 will also equal 11.
EXERCISE 12
Solve the following problems by finding an appropriate equation first:
1.
If you multiply a certain number by 5, and then subtract 3, the answer is 17. Find the
number.
2.
Find 2 numbers which have a sum of 60, and a difference of 36.
3.
A number increased by three times the number, is 28. Find the number.
4.
The length of a rectangle is 10 cm longer than its breath. The perimeter is 48 cm.
Determine the area of the rectangle.
5.
The denominator of a fraction is 1 more than the numerator. If 4 is added to the
numerator and 6 to the denominator, the new fraction will be equal to the original
fraction. Determine the original fraction.
6.
The perimeter of a rectangle is 30 cm. If the breath is doubled (length unchanged)
it becomes a square. Determine the dimensions of the rectangle.
7.
A salesman know he has 36 tables in his store-room, some of which have 3 legs and
other 4 legs. He discovers by counting that there are 124 legs. How many tables
with 4 legs does he have?
8.
The joint ages of A and B is 82 years. In 6 years time A will be
twice as old as B was 4 years ago. Determine their ages.
9.
The ages of three children in our family total 32 years. I am twice as
old as my sister, and my brother is 2 years older than I. How old am I?
10.
A tank is 2/3 full of water. If 35 litres are tapped off, the tank is 1/6
full. Determine the volume of water in a full tank.
11.
If one side of a square is extended by 8 m and the adjacent side is
Is reduced by 6 m, a rectangle is obtained which has a perimeter of
Page 39 of 268
100 m. Determine the area of the original square.
12.
There were 400 coins in the container after a street collection. If
there were only 5c- and 10c-coins, with a total value of R27,50,
how many 5c-coins and how many 10c-coins were there?
13.
A man may fire 30 shots at a target. Each time he hits the target, he
gets 25 c, but for every miss, he has to pay in 15c. If he makes a profit of R3,50,
haw many times did he hit the target?
14.
In a test, consisting of multiple choice questions, one mark is awarded
for each correct answer and two marks are deducted for every wrong
answer.
If a pupil answers 30 questions and obtains 21 marks, how
many correct answers were there?
15.
I bought 25 CD’s . Some were R50, others R80 each. Altogether I
paid R 1430. How many of each CD did I buy?
16.
A CD costs R3 more than last year. I have to pay R496 for 12 CD’s
at the current price, after being given a discount of R80. What did a
CD cost last year?
17. A bottle contains 120 ml medicine. A second bottle has twice the
amount. Determine how much of the first bottle’s contents must be
added to the second bottle, so that the second bottle will have 3
times as much as the first?
18.
A man has to complete a journey of 520 k, in 6 hours. Having travelled
for 4 hours, he realises that he will have to increase his speed by 20
km/h in order to complete the journey on time. What was his original
speed?
19.
A cyclist rides against the wind for 36 km at a speed of 12 km/h. On his
return journey with the wind behind him, his speed is 18 km/h. What
was his average speed?
Remember: Average speed = Total distance ÷ Total time
20.
120 People attended a concert. Entrance fees were R12 per adult
and R8 per child. If the concert raised R1280, how many adult how
many children attended the concert?
Page 40 of 268
EXERCISE 13
1.
I think of a number, treble it and subtract 12. My answer is 60. Find the number.
2.
The result of adding 30 to a certain number is the same as multiplying
that number by 5. what is the number?
3.
Two numbers differ by 4 and the sum of twice the greater number
and three times the smaller number is 23. Find the numbers.
4.
The sum of two numbers is eleven and four times the smaller number is
half of three times the bigger number. Find the numbers.
5.
The sum of
3
1
of a number and
of the number is 2 more than the number. Find
4
3
the number.
6.
A man is seven times as old as his son. In five years time he will be four
times as old as his son. How old are they now?
7.
John has four times as many marbles as Peter. If he gives Peter 8 marbles, he will have
twice as many marbles as peter. Hoe many did each one have originally?
8.
The equal sides of an isosceles triangle are twice as long as the third side.
If the perimeter is 350 mm, find the lengths of the sides.
9.
In a right angled triangle, the other two angles differ by 32º. Find the
size of each.
10.
In a leaque competition, 3 points are awarded for a win, 2 points for a draw and
no points for losing. At the end of a season a team had played 24 games and lost
7 of them, scoring 45 points. How many games did it win?
11.
A man cycles from P to Q at 20 km/h and does the return trip in
48 minutes less by cycling 4 km/h faster. How far is it from P to Q?
12.
Find tree consecutive numbers so that three times the sum of the first two exceeds
twice the sum of the second two by 13.
13.
A man leaves
7
1
of his money to his elder son and
to his
12
3
younger son. The rest which is R 800 he leaves to charity. How
much did he leave?
14.
A man had six hours in which to do a journey of 520 km. After four
Page 41 of 268
hours of travelling he realised he would have to increase his speed by 20 km/h to
complete his journey in time. What was his original speed?
15.
John travels from A to B at 80 km/h and back at 100 km/h. peter
does the same double journey at 90 km/h and he takes 24 minutes
less for the journey. How far is it from A to B?
16.
In a multiple choice examination one mark is given for each correct answer, and
two marks are deducted for every wrong answer. A boy answers 30 questions and
gets 21 marks. How many correct answers did he have?
17.
I walk at 4 km/h and run at 6 km/h. I can save 2½ minutes by
running to the station instead of walking. How far is the station?
18.
If one tap can fill a tank in two hours and another can fill it in
three hours, how long will the two together take?
EXERCISE 14
1.
Alan has 12 rand more than Paul.
(a)
If Paul has x Rand, How many does Alan have?
(b)
Together they have 83 Rand. Formulate an equation in x to
determine x.
(c)
2.
How many Rand do they each have?
A rectangle is x mm wide and three times as long.
(a)
What is the length in terms of x?
(b)
If the perimeter of the rectangle is 400 mm, write down an
equation in x to determine the width of the rectangle?
(c)
3.
Hence calculate the dimensions of the rectangle.
Alison and Heather share R 123 between them so that Alison has
R 15 more than Heather. How many Rand does Alison get?
4.
A rectangular plot of ground is 5 m longer than it is wide. If the
perimeter of the plot is 42 m, determine the dimensions of the plot.
5.
The sum of an odd number and the next odd number is 92. Write
down an equation to determine the two numbers, and then solve the equation.
6.
Tank A contains 240 l of water and tank B contains 50 l.
(a)
If k litre of water are added to each tank, how much water will each tank
contain?
Page 42 of 268
(b)
Write down an equation in k, if tank A now contains twice
as much water as tank B.
(c)
7.
Determine k.
Andrew, Adrian and Justin together have R 865. If Andrew has R 83
more than Adrian, and Adrian has R 52 more than Justin, How much
money does Justin have?
8.
The sum of two consecutive odd numbers is 72. Determine the
smaller number.
9.
One integer exceeds another integer by 18. Determine the two
numbers if their sum is 84.
10.
A dealer buys 120 pens. x of them cost R 3 each, the others cost
R 4 each.
(a)
Write down, in terms of x, how many of the more expensive
pens he buys.
(b)
If the pens cost him R 400, write down an equation to
determine x.
(c)
11.
How many of each type does he buy?
Bottle A contains four times the amount of liquid in bottle B. If 50 ml
is poured from bottle B into bottle A, then bottle A will contain five
times the amount of liquid in bottle B. How much liquid did each
bottle contain at the beginning?
12.
If the sum of five consecutive odd integers is 845, determine the smallest integer.
13.
A bottle contains 120 ml of medicine and a second bottle contains twice as much.
How much must be poured from the first bottle
into the second if the latter is to contain three times as much as the first?
14.
The price of an article increased by 25% is equal to R 100. What
was the original price. (Let the original price of the article be x)
15.
1
One rational number exceeds another by 6 . The two numbers have a sum of
3
1
12 . Find the smaller number.
4
Page 43 of 268
16.
A dealer buys 200 digital watches. The cheaper watches costs R24 each and the more
expensive watches cost R36 each. If the watches cost him R 5 760, how many of each
type does he buys?
17.
Two numbers differ by 17; their sum is 45. Find the numbers.
18.
Peter and Steven scored 36 points between them during the rugby season. If Peter
scored twice as many points as Steven, how many did Steven score?
19.
I have 20 coins, some of them are 10 c coins, the rest 5 c coins. I find that the total
value is R1,20. How many of each coin do I have?
20.
A man takes 26 minutes to cover 3 km walking part of the way at
6 km/h and running the rest at 10 km/h. How far does he walk?
21.
Joe has to travel 8 km from home to school. After riding part of the way at 15 km/h, his
bike breaks down and so he walks the remaining distance at 8 km/h. He covers the
whole distance in 53 minutes. How far does he walk?
22.
a) If a framer ploughs a field with a tractor in 4 hours, what fraction of a
the field is ploughed in 1 hour?
b) If the farmer uses his new tractor, he can plough the same field in 3
hours. What fraction of the field does he plough in 1 hour using his
new tractor?
c) How long will it take to plough the field using both tractors?
23.
Peter is twice as old as Paul. Ten years ago Peter was three times as old
as Paul. How old is Peter?
24.
Two towns, A and B, are 195 km apart. A plane leaves town A
travelling at a speed of x km/h toward B. Another plane leaves town
B at exactly the same time, travelling 60 km/h faster than the other
plane, towards town A.
The planes meet after ¾ of an hour.
Determine the speed of each plane.
25.
A bricklayer builds the walls of a certain house in 20 days working by
himself. Another bricklayer can do the same job in 5 days less. If they
were to work together, how many days would it take to build the walls?
Page 44 of 268
CHAPTER 8
THE STRAIGHT LINE GRAPH
EXERCISE 1
1.
10
K
L
G
M
P
O
T
S
U
V
9
K
J
H
P
O
N
M
L
Q
A
8
A
J
L
D
J
F
E
M
T
Y
7
A
D
C
M
I
H
O
Q
P
X
6
P
L
C
M
N
I
D
O
Z
Y
5
H
J
I
G
E
N
O
P
L
K
4
Y
L
L
O
U
N
D
P
R
S
3
A
D
O
N
W
U
T
C
T
C
Page 45 of 268
2
L
B
E
F
G
V
S
O
K
K
1
N
D
C
H
J
M
R
P
M
T
1
2
3
4
5
6
7
8
9
10
Y
XX
Use the above grid to decode the message below:
Message:
( 3 ; 10 ); ( 5 ; 9 ); ( 7 ; 7 ); ( 7 ; 6 ); ( 3 ; 4 ); ( 5 ; 4 ); ( 8 ; 3 );
( 10 ; 2 )
(Clue: two words)
2.
Decipher the secret message in the following grid:
Y
X
D
P
M
N
K
5
Y
T
S
T
O
P
E
S
T
E
4
R
F
E
N
M
Q
S
R
A
O
3
T
M
S
N
W
T
D
C
S
A
2
M
L
A
D
A
S
T
K
L
M
1
I
V
I
J
S
-5
-4
-3
-2
-1
1
2
3
4
5
M
N
P
E
P
-1
S
F
T
U
N
A
O
R
R
N
-2
L
F
L
T
F
T
S
C
-3
M
M
M
O
D
N
L
V
A
K
-4
H
N
A
T
V
P
E
D
C
B
-5
P
Q
S
N
W
P
E
X
Page 46 of 268
Y
Message: ( -5 ; 5 ); ( -4 ; 4 ); ( -4 ; 3 ); ( -3 ; 2 ); ( -2 ; 3 ); ( 1 ; 4 ); ( 2 ; 5 );
( 3 ; 4 ); ( 3 ; 3 ); ( 5 ; 3 ); ( 5 ; 2 ); ( 5 ; 1 ); ( -5 ; -2 ); ( -4 ; -3 ) ;
( -3 ; -2 ); ( -2 ; -1 ) ( -1 ; -2 ); ( -1 ; -3 ); ( 1 ; -4 ); ( 2 ; -3 );
( 3 ; -4 ); ( 4 ; -5 )
Clue: four words.
THE CARTESIAN PLANE:
An ordered pair is a pair of numbers or letters in which the order matters. If we say the time is
12:10 it means it is “10 minutes past 12” and not “ 12 minutes past 10”.
Ordered pairs are made up of a first component and a second component. The first
component is always the x- coordinate or the coordinate on the horizontal axis. The second
component is always the y-coordinate or the coordinate on the vertical axis.
EXERCISE 2
1.
Use squared paper and plot the following points. Join the points in alphabetical order
and then check your drawing with a friend.
A( 8 ; 1 ); B( 4 ; 1 ); C( 4 ; 2 ); D( 7 ; 2 ); E( 3 ; 5 ); F( 3 ; 1 ); G( 2 ; 1 ); H( 2 ; 7 ); I( - 3
; 2 ); J( 1 ; 2 ); K( 1 ; 1 ); L( -4 ; 1 ); M( -3 ; -1 ); N( 7 ; -1 ); A( 8 ; 1 ).
2.
Use squared paper and plot the following points. Join the points in alphabetical order
and then check your drawing with a friend.
A( -6 ; -3 ); B( -3 ; -1 ); C( 2 ; 1 ); D( 5 ; -1 ); E( 5 ; -4 ); F( 6 ; 0 ); G( 3 ; 3 ); H( 4 ; 4 );
I( 5 ; 4 ); J( 3 ; 5 ); K( 2 ; 4 ); L( 0 ; 6 ); M( -5 ; 5 ); N( -1 ; 5 ); O(0 ; 3 ); P( -3 ; 0 ); Q( 7 ; -1 ); R( - 4 ; -1 ); A( -6 ; -3 ).
3.
Read off the coordinates of the points A to L in the following drawing:
Page 47 of 268
4.
Copy the sketch below and identify each of the following:
(a)
The x-axis
(b)
The y-axis
(c)
The coordinates of P
(d)
The origin
(e)
The coordinates of the origin
(f)
The y-coordinate of P
(g)
The x-coordinate of P
(h)
The Cartesian plane
(i)
The four quadrants
y
.P(3;2)
O
5.
X
In the diagram below, P is the point ( 3; 2 ), PQ  x-axis and
PR  y-axis.
(a) Determine the coordinates of Q.
(b) Determine the coordinates of R.
(c) What is the length of PQ ?
(d) Determine the coordinates of S if PQ = QS.
(e) Determine the coordinates of T if TR = RP.
Page 48 of 268
Y
T
R
P
O
Q
X
S
6.
Form the coordinates of a point S if the x-coordinate is a and the
y-coordinate is b.
7.
If the y-coordinate of a point P is d > 0 and the x-coordinate is
c < o, determine in which quadrant point P will lie.
8.
A point S lies on the x-axis. Determine the value of the
y-coordinate of S.
9.
A point Q lies on the y-axis. Determine the value of the x –
coordinate of Q
10.
A point P lies on the x-axis and on the y-axis. Determine the value
of the ordinate(x-coordinate) and the abscissa(y-coordinate) of P.
VARIATION:
EXERCISE 3
1.
R 100 can be won in a crossword competition. The prize is to be shared by those
who complete the crossword correctly.
(a) If 2 people send in correct entries, how much money will they
each receive ?
(b) If 5 people send in correct entries, how much money will they
each receive ?
(c) A total of 10 correct entries are sent in. What is the prize money
per person ?
(d) What will the effect of more and more people sending in correct
entries be ?
(e) What is the cause of the prize money per person getting smaller and
smaller ?
(f) Suppose nobody sent in a correct entry for three weeks and each
week the prize money were doubled. How much money would
Page 49 of 268
then be available ?
(g) What is the cause of the prize money being increased ?
(h) What is the effect of nobody winning the prize ?
We say that the amount of money won depends on the number of correct entries sent in each
week. Also we say that the amount of money won varies inversely according to the number of
winners. In other words, the more winners there are, the less money any person will win.
2.
It costs R 5 per day for a cat to stay at the kennels.
(a) What will it cost to have the cat stay at the kennels for:
(i) 2 days ? ( a weekend )
(ii) 7 days ( a week )
(iii) 30 days ? ( a month )
(b) What is the effect of leaving the cat at the kennels for a long time ?
(c) What is the cause of spending a great deal of money for the cat to
stay at the kennels ?
We say that the amount of money spent depends on the length of time the stays at the
kennels.
We say that the amount of money spent varies directly according to the number of
days the cat stays at the kennels, i.e. the more days the cat stays, the more money is
spent.
If we represent direct proportion graphically, it will always be a straight line graph.
The formula for the cost of the cat staying at the kennels is y = 5x where y is the cost
and x is the number of days. This can be represented in a table as follows:
X
0
1
2
3
4
5
6
7
8
9
10
Y
0
5
10
15
20
25
30
35
40
45
50
Represented graphically it will look like this.
Page 50 of 268
y
50
40
Cost in Rands
30
20
10
2
4
6
8
10
x
No of days
Please note that the independent variables ( in this case the number of days), will always be
your x-values and represented on the x-axis or horizontal axis. The dependant variables (in
this case the amount to be paid), will always be represented on the y-axis or the vertical axis).
EXERCISE 4 (Revision)
1.
(a) Determine the coordinates of each point in the figure below:
(b)
State in which quadrant each point lie.
(c)
List the points which have a positive y-coordinate.
(d)
List the points which have a positive x-coordinate.
Page 51 of 268
(e)
List the points which have a positive x-coordinate and a negative ycoordinate.
2.
In which quadrant(s) of the Cartesian plane:
(a) Is the x-coordinate positive ?
(b) Is the y-coordinate negative ?
(c) Are both the x-coordinate and the y-coordinate positive ?
(d) Are both the y-coordinate and the x-coordinate negative ?
(e) Are the y-coordinate and the x-coordinate of opposite signs ?
(f) Are the y-coordinate and the x-coordinate of the same sign ?
(g) Is the x-coordinate negative and the y-coordinate positive ?
3.
From the figure below, determine:
Y
(a) The y-coordinate of P and S.
(b) The x-coordinate of Q and P.
(c) The coordinates of Q.
(d) The coordinates of the origin.
4.
The relationship between the approximate distance at which lightning strikes from you,
and the time between the flash and the thunder, is given in the table below:
Page 52 of 268
Time (in seconds)
1
2
3
4
5
Distance (in metres)
400
800
1200
1600
2000
(a) Draw a graph, letting the x-axis represent time and the y-axis represent distance. Let
1 unit on the y-axis represent 200 metres.
(b) How far away does lightning strike if the time is 0 seconds ?
(c) How far does lightning strike if the time is 3,5 seconds ?
(d) Try to write a formula for the relationship. Let d be the distance
and t be the time.
DRAWING THE STRAIGHT LINE GRAPH:
We have three methods to draw the straight line graph:
Method 1: The table method:
Consider F = { ( 2 ; 4 ); ( 3 ; 6 ); ( 4 ; 8 ) }.
We can say F is the set of ordered pairs ( x ; y ) where y is double x and x is 2, 3
or 4.
Algebraically we can write this as: F = { ( x ; y ): y = 2x, x ε { 2 ; 3 ; 4 }}
We have found a pattern which we can express algebraically, relating x to y.
The pattern shows up very clearly in the Cartesian plane, because the three points are in
a straight line.
If we extend the possible values of x to the Real numbers we get:
Page 53 of 268
G = { (x ; y): y = 2x ; x ε R }. We are not able to list all the possible ordered pairs
that are elements of the set G, but we can make a table in which we select certain values
for x ε R and calculate the corresponding values of y.
X
-1
0
1
Y
-2
0
2
If x = -1 the corresponding y value will then be: y = 2 X ( -1) = -2.
Once you have calculated all the corresponding y-values for the
selected x-values, the points must be plotted. Since x ε R, the graph of
G will be a continuous line, therefore the points plotted must be joined.
This method is called the table-method for drawing a graph, i.e. draw up
a table by selecting certain x-values, work out the corresponding yvalues and then plotting the points.
NB. If we use the table method to draw the straight line graph, we must
First write the equation of the graph in standard form, i.e. in the form
y = mx + c.
Example 1:
Sketch y = 2x + 1; x ε { -1 ; 0 ; 1 }.
Solution: Since x ε { -1 ; 0 ; 1 } the table must be of the following form:
X
-1
0
1
Y
-1
1
3
We may only plot the three points with ordered pairs ( -1 ; -1 ), ( 0 ; 1 )
and (1 ; 3 ).
Page 54 of 268
y
.(1;3)
.(0;1)
O
x
.
(-1;-1)
Note the points all lie in a straight line, but you are not allowed to join
them, since the graph is only defined for x = -1, 0 and 1 and not for any
other x-values.
Example 2:
Sketch y = 2x + 1; x ε R.
Solution: Since x ε R, we can choose any values of x for the table, and
calculate the corresponding values of y.
Let x = -2, x = 0 and x = 2. Then y = 2(-2) + 1 = -3, y = 2(0) + 1 = 1
and y = 2(2) + 1 = 5. Plot these points and draw a continuous line
through the points, because x ε R.
From the graph we can read off, for example, that when y = 7,
then x = 3.
EXERCISE 5
Page 55 of 268
Use the table method to sketch the following graphs. Pay careful attention to whether you may
draw a continuous line through the points or not.
1.
y = -x + 2; x ε R.
2.
3.
y =
2x
+ 1; x ε [ 3 ; 6 ; 9 } 4.
3
y =
2x
+ 1; x ε R
3
5.
y =
 3x
; x ε R.
2
y =
 3x
; x ε { -4 ; -2 ; 0 ; 2 ; 4 }
2
6.
y = -x + 2; x ε { -2 ; 0 ; 2 }
You should have noticed that in the previous exercise the points that you have plotted all lie in
a straight line, whether you were allowed to draw the line or not.
We say that y = mx + c, x ε R is the standard equation of the straight line, no matter what
the values of m and c are.
EXERCISE 6
1.
H = { ( x ; y ): y = 3x; x ε { 1 ; 0 ; -1 }}
(a)
List the ordered pairs of H.
(b)
Plot H in the Cartesian Plane. May you draw a continuous line
through the points you have plotted ? Give a reason for your
answer.
2.
G = { ( x ; y ): y = 3x; x ε R}
(a)
Copy and complete the following table:
X
0
1
2
Y
3.
4.
(b)
Plot the points you have determined in the above table.
(c)
May you join the points ? Complete the graph.
(d)
Use the graph to determine the value of x when y = -3.
(e)
Use your calculator to determine the value of y if x = 0,725.
(f)
Check your answer to (e) on your graph.
F = { ( x ; y ): y = x + 1; x ε { -2 ; 1 ; 2 }}
(a)
List the ordered pairs of F.
(b)
Sketch the graph of F in the Cartesian plane.
A = { ( x ; y ): y = x + 1}
Note: If no restrictions are given for x, then we assume that x ε R.
Page 56 of 268
(a)
Copy and complete the following table:
X
-2
0
1
Y
5.
(b)
Sketch the graphs of y = x + 1 in the Cartesian plane.
(c)
From your sketch determine the value of x if y = 7.
(a)
Sketch the graph of y = x and y = -x on the same set of
axes.
(b)
6.
What do you notice about the angle between the two graphs ?
B = { ( c ; y ) : y = -2x ; x ε R }
(a)
(b)
7.
Copy and complete the following table:
x
-2
y
4
-1
0
1
2
-2
Sketch the graph of B.
y = -3x + 1 defines C.
(a)
Copy and complete the following table:
X
-2
0
2
Y
(b)
Sketch the graph of C.
(c)
Show on your graph where you would read off the value of x
when y = 5,26.
(d)
8.
Use your calculator to determine x when y = 5,26.
y = -x - 1
(a)
Draw up a table to determine three ordered pairs that satisfy
the above equation.
(b)
9.
Hence sketch the graph of y = -x - 1.
F = { ( x ; y ): y = 3 }
(a)
Use the table given below to sketch the graph of y = 3.
X
1
2
3
Page 57 of 268
Y
(b)
10.
3
3
3
The graph of y = 3 is parallel to one of the axes. Which one ?
H = { ( x ; y ) : x = -2 }
(a) Use the table below to sketch the graph of H.
X
-2
-2
-2
Y
2
5
7
(b)
The graph of x = -2 is parallel to one of the axes. Which one ?
11.
(a) On the same set of axes sketch the graphs of y = 0 and x = 0.
(b) The lines have special names. What are they called ?
12.
(a) Determine the coordinates of A, B, C and D in the graph below.
( b) Try to establish an equation for the graph.
13.
(a)
Determine the coordinates of A, B, C, D, E and F.
(b)
Determine the equation for the graph below.
Page 58 of 268
14.
15.
(a)
Determine the coordinates of A, B and C in figures (i) and (ii).
(b)
Determine the equation that defines each of the graphs below.
(c)
Determine the length of OB in each case.
(a)
Determine the coordinates of A, B C and D and the
coordinates of E, the point of intersection.
(b)
16.
Determine the lengths of AD and BC.
(a) Copy and complete the following table, for y = 2x - 1.
(i.e. Determine the value of y if x = 0 and the value of x if y = 0)
X
Y
(b)
Hence sketch the graph of y = 2x - 1.
0
0
Page 59 of 268
17.
Copy and complete the following table in order to sketch the graph of
y
x
2
2
X
0
Y
18.
0
Determine the value of x if y = 0, and the value of y when x = 0, in
order to sketch the graph of y = -3x + 2.
19.
By determining the intercepts on the x-axis and the y-axis, sketch the
graph of y + x = 2.
Method 2 for drawing the straight line graph: The dual-intercept method:
If we use this method it is not necessary to write the equation in standard form first.
In this method we determine the intercepts on the x-axis and on the y-axis. It is useful to
determine the coordinates of a third point to check that the other two points are correct.
Example: Use the dual-intercept method to sketch the graph of y + 2x = 3.
Solution: If y = 0: 0 + 2x = 3
 x =
y
3
 Intercept on x-axis is A(1 12 : 0)
2
3
If x = 0: y + 2(0) = 3
 y + 0 = 3
 y = 3
O
1
2
1
x
 Intercept on y-axis is B(0 ; 3)
Method 3 for drawing the straight line graph: The gradient y-intercept method.
For this method the equation of the line must be written in standard form:
i.e. y = mx + c. m is called the gradient and tells you what the direction and slope of the
graph is. c is called the y-intercept and tells you where your graph will cut the y-axis.
Example: Use the gradient y-intercept method to draw the graph of:
3y - 4x + 6 = 0
Solution: First write in standard form:
3y = 4x - 6
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y =
4
x - 2. Now you know that the y-intercept is -2 and that the
3
gradient is positive
4
. Plot the point ( 0 ; -2 ) on the y-axis. Since
3
the gradient is always the
increase/ in / y
increase/ in / x
you move from -2 on the
y-axis 3 units horizontally and then 4 units vertically.
y
4
O
-2
x
3
.
EXERCISE 7
1.
(a) On the same set of axes sketch the graphs of the following relations.
Use any of the three methods that you prefer.
(i)
(iv)
y = 2x
(ii)
y - 2x = 1
y = 2x - 1
(v)
(iii)
y = 2x - 2
y - 2x - 3 = 0
(b) (i) What is the same in each graph ?
(ii) To which number do you attribute this?
(c) (i) What is different in each graph ?
(ii) To which number do you attribute this ?
2.
On the same set of axes draw the graphs of: y = 2x - 2 and
1
y = - x + 1.
2
What do you notice about the angle between the two lines ?
Note: If M1XM2 = -1, then the two graphs are perpendicular to each
other. In this case: 2 X -
1
2
perpendicular to each other.
= -1,
 The two lines are
Page 61 of 268
3.
Rewrite each of the following pairs of equations in the form y = mx + c and
determine whether the pairs of lines will be parallel, perpendicular or neither. (Do not
sketch the graphs).
4.
(a)
y - x = 2;
y = 3 + x
(b)
Y - 2x - 4 = 0 ;
y - 5 = 2x
(c)
2y - 3x = 6 ;
(d)
y -
(e)
2y = x + 4 ;
(a)
Sketch the following lines, all on the same set of axes:
x
= 7;
2
3y - 2x = 6
y = 6 +
x
2
4y = 2x + 8
(i)
y = 2
(ii)
y = 
(iv)
x = 6
(v)
x = -2
3
2
(iii)
(vi)
x =
y = 5
5
2
(b)
What do you notice about the graphs in questions (i) to (iii) ?
(c)
What do you notice about the graphs in questions (iv) to (vi) ?
Two lines: y = m1x + c1 and y = m2x + c2 are parallel if m1 = m2.
Two lines: y = m1x + c1 and y = m2x + c2 are perpendicular if
m1Xm2 = -1
Lines of the form y = c, i.e. y = (0)x + c, are all parallel to the x-axis, and hence
parallel to each other.
Lines of the form x = c, are all parallel to the y-axis, and hence parallel to each other.
Example 1:
Determine whether the following pairs of lines are parallel,
perpendicular or neither:
Solutions:
(a)
y = 2x + 2 ;
y - 2x = 3
(b)
y = 2x + 2 ;
2y + x = 4
(c)
y = 2x + 2 ;
y = -2x + 3
In each case rewrite the equation in the form y = mx + c.
(a)
y = 2x + 2 ;
y = 2x + 3
m1 = 2 and m2 = 2 the two lines will be parallel.
(b)
y = 2x + 2 ;
y = 
1
x = 2
2
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m1 = 2 and m2 = 
1
2
 m1 X m2 = -1
 The two lines will be
perpendicular.
(c)
y = 2x + 2 ;
y = -2x + 3
m1 = 2 and m2 = -2. Since m1 ≠ m2 , the lines are not parallel. Since
m1 X m2 = 2 X -2 = -4 ≠ -1 , the lines are not perpendicular.
Example 2:
Solutions:
Determine a if:
(a)
(a)
y = 4x + 2 is parallel to y = ax + 7.
(b)
2y - 6x = 7 is parallel to y = ax - 2.
m1 = 4 and m2 = a.
For parallel lines, m1 = m2, so a = 4
(b)
First write 2y - 6x = 7 in standard form.
i.e. 2y = 6x + 7
 y = 3x +
7
2
For parallel lines, m1 = m2, so a = 3
Example 3:
Determine which of the following pairs of lines will be parallel or
perpendicular to each other:
Solutions:
(a) x = 2 ;
y = 2
(b) x = 3 ;
x = 2
(c) y = 3 ;
y = 2
(a) Since x = 2 is parallel to the y-axis and y = 2 is parallel to
the x-axis, the lines will be perpendicular to each other.
(b)
Since x = 3 and x = 2 are both parallel to the y-axis,
they will be parallel to each other.
(c)
Since y = 3 and y = 2 are both parallel to the x-axis,
they will be parallel to each other.
EXERCISE 8
1. For each pair of lines:
(a) Rewrite each equation in the form y = mx + c.
(b) Determine whether the lines will be parallel, perpendicular or neither.
(c) Sketch the graphs of the lines, using any of the three methods that
Page 63 of 268
you prefer.
(i)
2y - x = 6 ;
(ii)
2.
3.
y - 4 =
x - y = 7;
x
2
y - x = 6
(iii)
x + y = 2;
x - y = 2
(iv)
2x + 3y - 6 = 0 ;
3x - 2y + 4 = 0
(v)
100x + 50y = 700 ;
x +
(vi)
x + 2y = 4 ;
-x + 2y = 4
(vii)
x + 2y = 4 ;
x - 2y = 4
(viii)
x + 2y = 4 ;
y + 2x = 4
y
= 7
2
Determine m if:
(a)
y = 3x + 7 is parallel to y - mx + 6
(b)
y - 4x = 2 is perpendicular to y = mx + 8
(c)
2y + x = 10 is parallel to y = mx +
(d)
2y - 3x - 6 = 0 is perpendicular to y = mx - 7
(e)
3y
+ 8x - 2 = 0 is: (i) parallel (ii) perpendicular to y = mx
4
1
2
Determine whether the following pairs of lines are parallel,
perpendicular or neither:
4.
3
x + 2
2
(a)
2y + 3x = 2 ;
y = 
(b)
2y + 3x = 2 ;
y =
3
x + 2
2
(c)
2y + 3x = 2 ;
y =
2x
+ 2
3
(d)
2y + 3x = 2 ;
y = 
(e)
y = 7;
y = -2
(f)
x = 7;
x = -2
(g)
y = 7;
x = -2
(h)
y = 0;
x = 0
2x
+ 2
3
(a) On the same set of axes sketch the graphs for the following
equations. Use any method you prefer.
Page 64 of 268
(i) y = x + 2
(ii) y = -x + 2
(iv) y = -3x + 2
(v) y -
(iii) y = -2x + 2
x
= 2
2
(b) (i) What is the same in each graph ?
(ii) To which number do you attribute this ?
(c) (i) What is different in each graph ?
(ii) To which number do you attribute this ?
5.
Determine where each of the following graphs will intersect the y-axis
by determining the value of y when x = 0.
(a) y = 
3x
- 2
2
(d) y = x - 2
6.
(b) y + 2 = x
(c) y + x + 2 = 0
(e) 0 = y - x + 2
Rewrite each of the following in the form y = mx + c and then
determine where each one will cut the y-axis by determining the
value of y when x = 0.
(a) y + x = 3 = 0 (b) y - 3 = 0 (c) y = 3 + 2x
(d) 3 = y - 2x
(e) 3y + 6 + x = 0 (f) -8y + 24 + 2x = 0
Example: (a) Determine the equation of the line y = mx + c if m =
1
and the y2
intercept is -3.
(b) Determine the equation of the line y = mx + c that is parallel to
y =
7x
+ 7 and that cuts the y-axis at -2.
2
Solutions: (a) m =
 y =
1
2
and c = -3. Substitute into y = mx + c.
1
x - 3
2
(b) Since the new line cuts the y-axis at -2, c = -2, and because
the new line is parallel to y =
7x
7
+ 7 , m = . Substitute
2
2
into y = mx + c and the equation of the new line will be
y =
7x
- 2.
2
EXERCISE 9
1. (a) Rewrite each equation in the form y = mx + c.
Page 65 of 268
(b) Hence determine the gradient and the y-intercept of each graph.
(c) Draw a sketch graph of each one on a different set of axes for each
one.
(i) x + y - 2 = 0
(ii) x - y - 2 = 0
(iv) y + x + 2 = 0
(v) y + 2 = 0
(iii) y - x + 2 = 0
(vi) x + 2 = 0
(vii) 2x + 2y - 2 = 0 (viii) 2x - 2y - 2 = 0
2.
(a) Rewrite each equation in the form y = mx + c.
(b) Hence determine the gradient and the y-intercept of each graph.
(c) Draw a sketch graph of each one on a different set of axes for
each one.
(i) 3y + 5x = 15
(ii) 5y + 3x = 15
(iv) 3y - 5x = 15
(v) 3x - 5y = 15
(vi) 5x - 3y = 15
(vii) 3y + 5x + 15 = 0
(viii) 5y + 3x + 15 = 0
(ix) 3x - 5y + 15 = 0
(x) 5x - 3y + 15 = 0
0
(xiii) 3y - 15 = 0
(iii) 5y - 3x = 15
(xi) 5x + 15 = 0
(xii) 5x - 15 =
(xiv) 3y + 15 = 0
3. (a) Rewrite each equation in the form y = mx + c.
(b) Hence determine the gradient and the y-intercept of each graph.
(c) Draw a sketch graph of each one on a different set of axes for
each one.
(i) 4y + 3x = 12
(ii) 3y + 4x = 12
(iv) 3y - 4x = 12
4.
5.
(v) 3x - 4y = 12
(iii) 4y - 3x = 12
(vi) 4x - 3y = 12
(vii) 3y + 4x + 12 = 0
(viii) 4y + 3x + 12 = 0
(ix) 3x - 4y + 12 = 0
(x) 4x - 3y + 12 = 0
Determine the equation of each of the following lines given that:
(a) m = -1 ; c = 2 (b) m =
3
; c = 0 (c) m = 0 ; c = 5
4
(d) m = -7 ; c = 2 (e) m =
5
5
; c =
(f) m = 0 ; c = 0
2
2
In each case determine the equation of the line that is :
(i)
parallel
(ii) perpendicular
to the given line and that cuts the y-axis at the point indicated:
(a)
y = 2x + 7 ;
y-intercept is 2
(b)
y = 7 - 2x ;
y-intercept is -7
Page 66 of 268
6.
(c)
y = 7x + 2 ;
y-intercept is -2
(d)
y = 2 - 7x ;
(e)
7y + x = 2 ;
y-intercept is 7
(f)
y = x;
y-intercept is 0
y-intercept is 7
Sketch the graphs of the following pairs of linear relations, each pair on
a separate set of axes:
7.
(a)
y = -2x + 1 ;
y =
1
x + 1
2
(b)
y =
3x
+ 2;
2
y =
 2x
+ 4
3
(c)
y = -x + 1 ;
(d)
y =
y = x + 1
x
- 2;
3
y = 3x + 2
Make y the subject of the following equations:
(a) 3x - 2y = 6
8.
(b) 4x + 3y = 6
(c) x = 2 -
3y
4
Arrange the equations in standard form:
(a) 2y - 3x = 4
(b) x - 2y = 0
(d)
3 y
x
4
(e)
2
1
yx
3
2
(g)
y x 1

4
3
(h)
2( x  3)
5
y
(c) 5(y - 1) = 3x
(f)
y 3 1
 x
4
2
SIMULTANEOUS EQUATIONS:
Example:
(a)
Draw the graphs of x + 2 = 0 and x - y = 3 on the same set of
axes and determine the point that represents the point of intersection
of the two graphs.
(b)
Check your answer in (a) by solving the two equations x + 2 = 0 and
x - y = 3 simultaneously.
Solution: (a) If x + 2 = 0
then x = -2
(i)
If x - y = 3 then -y = -x + 3
y
 y = x - 3 (ii)
Page 67 of 268
-2
3
O
x
-3
(-2 ; -5)
The point of intersection of the two graphs is ( -2 ; -5 )
(b) We solve the equation with one unknown and then substitute into
the equation with two unknowns.
If x + 2 = 0
x - y = 3
then x = -2
(i)
(ii)
Substitute (i) into (ii):
 -2 - y = 3
 y = -5
-y = 5
 point of intersection ( - 2 ; -5 )
EXERCISE 10
In each of the following cases:
(a) Solve the systems of equations simultaneously for x and y.
(b) Draw the graphs defined by the equations on the same system of axes.
(c) Check geometrically that the point of intersection of the two graphs is the
same as the solution in (a).
1. y + x = 4 ; y = 2
3.
2.
y + 2x = 3 ; x = 1
X = -2 ; y - x = 6
4.
2y - x - 4 = 0 ; x + 2 = 0
5. 2y - 10 = x ; x + 4 = 0
6.
2x + y - 5 = 0 ; y + 1 = 0
7. x - y - 6 = 0 ; x + 2 = 0 8.
x - y - 4 = 0; y - 4 = 0
9. x = 4 ; 4y + x = 12
3x + y = 5 ; y + 4 = 0
10.
EXERCISE 11
1.
Solve the following algebraically:
(a)
x = 3; x + y = 4
(b)
2x = 4 ; x + 2y - 4 = 0
Page 68 of 268
(c)
y = 10 ; x + y = 6
(e)
3(a + 1) = -3 + a ; 2a - b = 1
(g)
2(x - 4) = x - 2 ; x - 2y + 4 = 0
(h)
2x + 1 = -5 ; y - x = 2
(f)
y = -3 ; x - y = 12
5(a - 1) = a - 3 ; 2a - b + 3 = 0
(i)
y + 5 = 0 ; 2y + 12 = x
(j)
2.
(d)
3(x - 2) = x - 10 ; 2y - 3x = 4
(k)
3x + 2 = x ; x + 2y = 4
(l)
y - 4 = 0; y = x
(m)
x = 5 - y ; 2(x + 2) = x + 3
(n)
y = 2x + 3 ; x + y = -3
(o)
4x + 2y = 20 ; y + 2 = 0
Determine the following algebraically:
(a)
{ (x ; y) : x + y = 12 }  { (x ; y) : x + 6 = 0 }
(b)
{ (x ; y) : x - 2y - 8 = 0 }  { (x ; y) : y + 10 = 0 }
(c)
{ (x ; y) : 3 x + 4y = 6 }  { (x ; y) : 2y = 6 }
EXERCISE 12
Write down the equations of the following graphs:
1.
2.
y
2
-3
4.
1
1
O
x
O
3
x
2
x
O
-2
y
5.
y
y
6.
2
2
O
X
O
y
3.
y
O
x
1
6
x
-2
7.
y
8.
y
y
9.
1 12
3
x
O
-2
4
x
-2
O
x
Page 69 of 268
O
y
10.
11.
1 12
3
13.
x
-1
x
y
5
15.
O 45˚
x
y
16.
x
y
135˚
x
O
O
x
-1 12
14.
O
4
O
Y
1
12.
O
-3
O
y
y
x
-2
17.
y
2
18.
y
2
3
6
O
x
x
O
2
3
19.
y
20.
y
B
1 12
-1
O
X
O
-1
A
-3
D
x
Page 70 of 268
C
21.
y
22.
.
3
O
y
(1;2)
x
x
O
.
-1
EXERCISE 13
Find the equations of the following graphs:
1.
y
-3
2.
O
y
1 12
x
-2
3.
y
4.
O
x
x
O
-2
y
3
2
-2
O
x
-3
5.
y
6.
y
3
1
O
7.
1½
2
y
3
O
x
8.
x
y
2
(4;-4)
Page 71 of 268
2
x
O
3
x
O
-2,4
9.
y
y
10.
3
3
-2
 1 12
11.
y
2
x
O
x
O
12.
y
3
-3
O
x
O
x
-4
13.
y
14.
1
y
B
1 12
2
D
-3
x
O
-4
x
O
A
-1
C
C
15.
y
16.
B
5
O
y
4
x
-3
O
A
2
D
y=3x+6
17.
y
18.
B
B
Page 72 of 268
4x+3y-6=0
O
x
O
-2
A
A
EXERCISE 14
Find the equations of the following graphs:
1.
2.
y
y
45˚
O
3. y
-3
135˚
O
x
x
3
4.
C
y
A
5
4
-1
x
O
x
O
3
B
5. y
D
4
6.
y
A
O
D
x
.
-3
(1;-3)
B
O
5
-5
C
7.
8.
y
O
2
x
y
5
x
Page 73 of 268
O
-10
x
-5
9. y
10.
.
y
(-3;3)
6
O
x
O
11.
-2
y
12.
6
O
y

x
-3
13.
x
1
2
x
O
-2
y
14.
y
135˚
O
1
4
x
O
-2
-1
15.
y
(-1;3)
x
16.
.
y
5
O
O
x
10
x
Page 74 of 268
17.
y
18.
y
1
3
-4
2
x
O
x
O
-2
EXERCISE 15
Determine the equations of the following graphs:
1.
y
2.
y
3
3
O
x
-2
-3
3.
y
A
x
O
4.
B
y
D
C
2
1
O
-2
-1
x
x
O
-1
E
-3
F
-2
B
D
C
5.
A
C
y
6.
4
y
B
2
3
O
-1
x
6
9
2
O
x
Page 75 of 268
D
A
7.
y
8.
6
O
y
1
4
x
O
-2
x
4
2
9.
y
10.
y
O
-2
.
(5;5)
3
x
x
O
-3
EXAMPLE:
Determine the equation of the line passing through the following points:
X
-3
-2
0
1
2
3
Y
-1
0
2
3
4
6
Solution:
m =
y = x + c
-
y  2  (3)  2  3 1


 1
x
0  (1)
0 1 1
Now substitute any of the points , say the point (-3 ; -1)
 -1 + 3 = c
I = -3 + c
 c = 2
 Equation: y = x + 2
EXERCISE 16
Find the equations of the lines passing through the following points:
1.
X
-3
-2
-1
1
2
3
y
-9
-7
-5
-1
1
3
Page 76 of 268
2.
X
-3
-2
-1
0
1
2
3
y
10
7
4
1
-2
-5
-8
3.
X
-3
-2
-1
1
5
y
-11
-9
-7
-3
1
X
-10
5
20
y
50
5
-40
4.
5.
X
0
1
-1
1/
y
-2
2
-6
0
-1/2
2
-4
EXERCISE 17
1. Determine whether the points: (a)
on the graph of
( -2 ; 1 )
(b)
lie
3y - 2x = 7.
2. Test whether the following points lie on the graph of
(a)
(4;6)
( -2 ; -7 )
(b)
(
1
2
;-
1
2
5x - y = -3:
)
3. If the point ( k ; -5 ) lies on the line 2x + 4y = 8 find k.
4. If the point ( -3 ; -2 ) lies on the graph of 3y - 2k = -2x, find the value
of k and the equation of the graph in standard form.
5. Will the point ( -2 ; -3 ) lie on the graph of y = -2x + 1 ?
6. Will the graph defined by y - 2x = 3 12 pass through the point ( 2 ; - 12 )?
7. If ( 3 ; a ) and ( b ; 2 ) lie on the line y = 2x + 6, find a and b.
8. If y = 4x + 2 is parallel to y = ax + 7, find a.
9. Determine b if y = bx - 1 is (a) parallel
(b) perpendicular to
3x - 2y = 4.
10. Determine the value of k, if ( 2 ; 1 ) lies on the line with equation
kx + y = 2.
11. Given: 3x - 5y - 15 = 0. Find:
(a) the y-intercept
(b) the x-intercept.
EXERCISE 18
Page 77 of 268
1.
Determine the equation of the line:
(a)
parallel to y - 3x = -2 and passes through ( 0 ; 1 ).
(b)
perpendicular to the line 2x + 3y - 6 = 0 and passing
through the origin.
(c)
2.
parallel to the y-axis, and passing through ( 5 ; 1 ).
Determine the equation of the line which:
(a)
cuts the y-axis at 2 and has a gradient of -1.
(b)
passes through the point ( 3 ; 2 ) and cuts the y-axis at -2.
(c)
has a gradient of 3 and passes through the point ( -1 ; 2 )
.
3.
AA
y
Use the sketch and find:
2
(a)
The equation of AB.
(b)
The values of a and b if
1
O
A is the point (-2;a) and B
B
the point (b ; -6)
4.
Show algebraically which of the following points lie on the graph
with equation
(a) ( -1 ; -1 )
5.
.
x
x - 4y = 3
(b) ( 3 ; 0 )
(c) ( 7 ; 1 )
(d) ( 6 ; 2 )
(e) ( 1 ; - 12 )
(f) ( -7 : 2 12 )
If the point ( 2 ; 1 ) lies on the graph with the following equation,
find the value of k:
(a) 2x - y = k
6.
(a)
(b) kx + y = 2
(c) 3x - ky = 1
Solve the following pairs of simultaneous equations
algebraically and
(b)
Draw graphs of each set of equations and illustrate your
answer graphically:
(i) 2x - 5y = 4
(ii) 3x + 4y = 6
y = -3
(iv) x + y = 8
2x - 3 = 1
(v) y = 2x - 1
2y - 3 = 8
7.
2y + 5 = 1
(iii) y =
2
3
x - 1
2x - 5 = 1
(vi) 3x - 5y = -7
y=2
Find the equation of the graph:
(a)
Which is parallel to y =
3
2
x - 3 and passes through the origin.
Page 78 of 268
(b)
Which is parallel to y =
(c)
Which is parallel to y =
3
2
3
2
x - 3 and cuts the y-axis at 2.
x - 3 and passes through the
point ( -2 ; -1 ).
8.
(a)
Solve the following pairs of simultaneous equations
algebraically and
(b)
Draw graphs of each set of equations and illustrate your
answer graphically:
(i) y + x = 4; y = 2
(ii) x = -2 ; y - x = 6
(iii) y + 2x = 3 ; x = 1
(iv) 2y –x – 4 = 0 ; x + 2 = 0
(v) 2y – 10 = x ; x + 4
(vi) 2x - y – 5 = 0 ; y + 1 = 0
(vii) x – y – 6 = 0 ; x + 2 = 0
(viii) x – y – 4 = 0 ; y – 4 = 0
(ix) x = 4 ; 4y + x = 12
9.
(x) 3x + y = 5 ; y + 4 = 0
Solve the following simultaneously:
(a) x = 3 ; x + y = 4
(b) 2x = 4 ; x + 2y - 4 = 0
(c) y = 10 ; x + y = 6
(d) 2x + 1 = -5 ; y - x = 2
(e) 3(a + 1) = -3 + a ; 2a – b = 1
(f) y = -3 ; x – y = 12
(g) 2(x – 4) = x – 2 ; x – 2y + 4 = 0
(h) 5(a – 1) = a – 3 ; 2a – b + 3 = 0
(i)
y + 5 = 0 ; 2y + 12 = x
(k) 3x + 2 = x ; x + 2y = 4
(j)
3(x – 2) = x – 10 ; 2y – 3x = 4
(l) y – 4 = 0 ; y = x
(m) x = 5 – y ; 2(x+ 2) = x + 3 (n) y = 2x + 3 ; x + y = -3
(o) 4x + 2y = 20 ; y + 2 = 0
10.
Determine the following algebraically:
(a) {(x ; y) : x + y = 12}  {(x ; y) : x + 6 = 0}
(b) {(x ; y) : x – 2y – 8 = 0}  {9x ; y) : y + 10 = 0}
(c) {(x ; y) : 3x + 4y = 6}  {(x ; y) : 2y = 6}
EXERCISE 19
1. Find the equation of the graph:
(a)
Which is perpendicular to y =  34 x + 2 and passes through the origin.
Page 79 of 268
(b) Which is perpendicular to y =  34 x + 2 and passes through the
point ( 4 ;-2 ).
(c) Which passes through the point ( 2 ; 1 ) and cuts the y-axis at 3.
(d) Which cuts the y-axis at -1 and has a gradient of
3
4
.
(e) Passes through the origin and the point ( 3 ; -2 )
(f) Has a gradient of  1 12 and cuts the y-axis at 2.
2.
Find the equation of the straight line graph which cuts the x-axis and
the y-axis respectively at:
3.
(a)
-3 and 2
(d)
-2 and
2 12
(b)
2 and 2
(c)
1 12 and 3
(e)
3 and -2
(f)
2 12 and  1 12
Find the equations defining the following:
(a) The line parallel to the y-axis and passing through 2 on the x-axis.
(b) The set of all points with x-co-ordinate of -2.
(c) The x-axis.
(d) The line perpendicular to the y-axis at 3.
(e) The y-axis.
(f) The line parallel to the y-axis and passing through -3 on the x-axis.
(g) The set of all points with y-coordinate of 3.
(h) Gradient = 2,
y-intercept = -1.
(i)
passing through the origin.
Gradient = -3,
(j) Parallel to the x-axis and passing through -1 on the y-axis.
(k) Perpendicular to the x-axis, cutting it at x = 2.
(l) Parallel to 3x - 2y + 2 = 0 and passing through the origin.
(m) Perpendicular to y = x + 1 and passing through ( 0 ; 2 ).
(n) Perpendicular to y = 2x - 1 and passing through ( 2 ; 2 ).
(o) Parallel to 2x - 2y - 5 = 0 and passing through ( 1 ; -2 )
(p) Perpendicular to 3x + 2y - 6 = 0 and passing through ( 1 ; 2 ).
EXERCISE 20
1.
y
A sketch of line CA is given.
(a) Find the equation of CA if A is
the point ( 4 ; -4 )
C
D(0;2)
G 1
Page 80 of 268
(b) Find the length of OB.
(c) If the y-coordinate of D is 2,
find the length of CD.
(d) If the x-coordinate of E is 3,
A
y
Given the line x + 2y - 2 = 0.
D
(a) Find the slope of the line.
(b) Find the length of (i) OA
A
(ii) OB.
(c) If the x-coordinate of C is -2, find
B
O
C
x
E
the length of CD.
F
(d) The y-coordinate of E is -1, find the
Length of EF.
y
3.
(a) Find the equations of the lines
AD and CE.
B
A
(b) Find the length of AB if the
E
-3
y-coordinate of A is 2.
(c) Find the length of CD if the
x-coordinate of C is 1.
C
O
x
-4 F
(d) Find the length of EF.
D
4.
x
F
find the length of EF.
2.
E(3;0)
B
O
Determine whether the following pairs of lines are parallel,
perpendicular or neither. Show all your work.
(a) 2x + 3y = 2 and y =
3x
+ 2
2
(b) 2y + 3x = 2 and y = 
(c) 2y + 3x = 2 and y =
(d) 2y + 3x = 2 and 
(e) y = 7 and y = -2
3
x+ 2
2
2x
+ 2
3
2
x + 2
3
(f) x = 7 and x = -2
Page 81 of 268
(g) y = 7 and x = -2
5.
(h) y = 0 and x = 0
Determine b if:
(a) y = 2x + 3 is parallel to y = bx + 3
(b) y = 2x + 3 is perpendicular to y = bx + 3
(c) y = x + 2 is parallel to y = bx + 2
(d) y = x + 2 is perpendicular to y = bx + 2
6.
Determine whether the gradient of each of the following lines is
positive, negative, zero or undefined.
(a) y = 2x + 3
(d)
(b) y + 3x - 1 = 0 (c) 2y - 3x - 2 = 0
y x 1
  0
2 3 6
(e) 2y - 8 = 0
(f) 3x = 6
EXERCISE 21
1.
Determine a if y = -3x + 2 is
(a) parallel to y = ax + 7
(b) perpendicular to y = ax + 7
2.
Find the value of b if
y x
  1 is :
3 2
(a) parallel to y = bx + 4
3.
(b) perpendicular to y = bx - 4
Find the equation of the line that is parallel to y =
x
+ 2 and that
2
cuts the y-axis at 2.
4.
find the equation of the line that is perpendicular to y =
3x
+ 2
2
and that cuts the y-axis at -2.
5.
Use any method to sketch the following lines:
(a) 2x + 4y = 8
(e) x = 0
6.
(b) x = 8
(f) 5y + 3x + 10 = 0
(c) y = -2
(g) 2x = 6y = 12
(a) On the same system of axes sketch the graphs of y = 2x and
2y = -x + 5.
(b) From your graph determine the coordinates of the point of
Intersection of the two lines.
2. Sketch the graph of y + 2x = 3. Showing clearly where your
readings are taken, determine approximately from your graph:
(a) The value of x if y is 3.
(b) The value of y if x is 0.
(d) y = 0
Page 82 of 268
(c) The value of y if x is 2.
(d)
8.
(d) The value of x if y is 7.
The value of y if x is -1.
Sketch y = mx + c if:
9.
(a) m > 0, c > 0
(b) m = 0, c > 0
(c) m < 0, c > 0
(d) m < 0, c < 0
(e) m = 0, c < 0
(f) m > 0, c < 0
(g) m > 0, c = 0
(h) m = 0, c = 0
(i) m < 0, c = 0
Determine whether m an c are positive, negative, zero or
undefined in each of the following:
(a)
Y
(b)
y
(c)
y
O
x
O
(d)
y
(e)
O
x
O
y
(f)
x
y
x
O
xO
EXERCISE 22
1.
Determine whether the following pairs of lines will be parallel,
perpendicular or neither:
2.
(a)
3x + 3y = 3 ; 4x + 4y = 4 ;
(b)
(c)
x - -y ; x = y + 1
x
x
= -2y ;
= 3y - 1
2
3
(e)
x =7; y = 7
(d)
x = y; x = y + 1
There is a relationship between the number of chirps a cricket
makes in a minute and the temperature, so it is possible to use the
cricket as a thermometer. A formula for the relation is t =
n
+ 40
4
where t is the temperature in degrees Fahrenheit and n is the number
of cricket chirps in one minute.
(a) Copy and complete the following table:
N
40
T
50
60
80
100
120
140
75
x
Page 83 of 268
(b) Draw a graph letting the x-axis represent the number of cricket
chirps per minute and the y-axis represent the temperature.
(c) If you hear 170 chirps, what is the approximate temperature ?
(d)
3.
At what temperature do crickets stop chirping ?
Determine whether the gradient of each of the following is positive,
negative, zero or undefined:
(a)
(b)
Y
y
O
x
O
x
(c)
(d)
Y
y
O x
x
O
(e)
y
(f)
x
O
4.
5.
y
O
Draw each pair of the following graphs on a different set of axes:
(a)
y = 2x + 3 ; y = 
(c)
y = 
3
x + 2;
4
1
x - 1 (b)
2
y =
y = -2 12 x ; y =
2
5
x - 4
4
x + 1
3
Draw rough sketches of the graphs with the following equations:
x
Page 84 of 268
6.
y = -x + 3
(b)
y = 2x + 5
(d)
y = -2 12
(e)
2x + 5y = -4 (f) 2x + 3y + 12 = 0
(g)
x + y = 0
(h)
x = -4
(j)
x y

= 2
3 4
(k)
y = 0,6x + 1,5
AM
(b)
(c)
(i)
Write down the equation of:
(a)
A
2 12 y = 3x + 5
(l)
2,4x + y = 7,2
y
BM
3
Calculate:
7.
1
y =  x - 1
3
(a)
(b)
The coordinates of point M
(c)
The equation of OM
4
O
B
-3
x
M
On graph paper using a scale 2 cm : 1 unit on both axes draw the
graphs of:
A = {(x;y) : y =
2
x + 2 }
3
B = { ( x ; y ) : 3x + 2y = 6 }
C = { ( x ; y ) : y = -3 }
D = ={ ( x ; y ) : y = 
(a)
3
x - 3 }
2
What kind of figure is enclosed between the four lines ?
Could you deduce this without drawing the graphs ? Explain.
(b)
Join the point of intersection of graphs B and C to the origin
and write down the equation of this line.
EXERCISE 23
1.
In each of the following find:
(a)
the equation of AM
(b)
the equation of BM
(c)
the coordinates of M
(a)
y
(b)
y
B
M
2
B
A
1
O
-3
1
M
3
2
x
O
Page 85 of 268
A
(c)
(d)
y
y
A

1
2
1
x
O
M
-2
-1
1
O
x
A
M
B
-2
B
(e)
(f)
Y
y
M
M
B
x
O
-3
2
.
O 45˚
B
(g)
x
A(3;-2)
A
A
(h)
y
y
4
3
M
2
2
O
-2
x
B
O
4
B
2.
M
A
-1
Find algebraically the equation of the straight line graph which:
(a) cuts the y-axis at 2 and the x-axis at -3
(b) has a gradient of 
3
and goes through ( -1 ; 2 )
5
(c) goes through ( 0 ; 2 ) and ( 3 ; -2 )
(d) is parallel to the y-axis and goes through ( -1 12 ; -3 )
(e) is parallel to the x-axis and goes through ( -2 ; 5 )
x
Page 86 of 268
(f) is parallel to (a) and goes through ( 0 ; -1 )
(g) is perpendicular to (a) and goes through ( -1 ; -2 )
3.
4.
Sketch y = mx + c if :
(a) m > 0 ; c < 0
(b) m < 0 ; c = 0
(c) m = 0 ; c < 0
(d) m < 0 ; c > 0
(e) m > 0 ; c > 0
(f) m = 0 ; c > 0
(a) Solve the following pairs of equations simultaneously.
(b) Draw graphs of each pair of equations and show the point of
intersection on your graph.
(i) y = -x + 9; y = x - 1
(ii) y = 2x - 2 ; y = -x + 1
(iii) y = 3x - 5 ; y = x + 1
(iv) y = -x + 2 ; y = 2x + 5
(v) y = -x + 1 ; y = -2x - 3
(vi) y = 7 - 2x ; y = x + 2
CHAPTER 9
RATIO AND PROPORTION
The meaning of ratio:
The masses of two boys, John (60 kg) and Peter (48 kg), can be compared.
We say that John’s mass is to Peter’s mass as 60 kg is to 48 kg or as 60:48 or as
15:12 or as 5:4.
NB: Quantities may be compaired only when they are of the same kind.
8.1
The simplification of ratios
The value of a ratio does not change if its terms are multiplied or divided by
the same number.
Examples:
Simplify:
Solution:
(a)
3,6 : 5,4
(b)
5 2 5
: :
8 3 6
(c)
1 2
1 :
4 3
Page 87 of 268
36
2
3,6
3,6 X 10
=
=
=
3
54
5,4
5,4 X 10
(a)
3,6 : 5,4 =
(b)
5 2 5
15 16 20
: : =
:
:
= 15 : 16 : 20
8 3 6
24 24 24
(c)
1
5 3 3
1 2
1 :
= 4  X 
2 4 5 4
4 3
1
3
1
EXERCISE 1
Simplify the following ratios:
1.
12 : 15
2.
14 : 35
3.
4.
63 : 42
5.
108 : 84
7.
½:5
8.
3:
10.
3 7
:
4 8
11.
13.
5,1 : 6,8
44 : 77
6.
1
3
70 : 105
9.
1 3
;
3 4
3
:6
8
12.
1,5 : 4,5
14.
2,1 : 2,8
15.
0,55 : 0,121
16.
0,1 : 0,03
17.
1 3
1 :
4 4
19.
3 5 1
: :1
8 16 4
20.
2
1
4
5
5
8
2
18.
5
1 6
:3 :7
14 7 7
One quantity as a ratio of another quantity
To find the ratio of one to another, you must express both quantities in the
same unit.
Examples:
What is the ratio of: (a)
R 1,25 : 75 c
(b)
4,5 l : 18 000ml
Answers:
(a)
R1,25 125c 5


75c
75c 3
(b)
4,5l
4,5l 9 1 1

 X

18000ml 18l 2 18 4
NB: The answer is not
5
cents
3
EXERCISE 2
Simplify the ratios:
1.
R 1,45 : 29 c
2.
6,6 m : 880 mm
3.
20 l : 0,16 kl
Page 88 of 268
4.
2 h : 20 min
5.
3,2 km : 600 m
7.
8,4 kg : 189 g 8.
2,25 l : 75 cm3 9.
10.
1,35 m² : 3000 cm²
8.3
Comparing ratios
6.
2,5 m² : 0,000 2 km²
2,8 t : 140 kg
Examples: Determine which of the following ratios is bigger:
(a)
4 : 9 or 3 : 8
(b)
50 mm : 1,8 m or 5 cm² ; 0,18 m²
Answers:
(a)
4 32
3 27


and
NB: bring the fractions to the same denominator.
9 72
8 72

(b)
4 3

9 8
50mm
50mm
5
1
5cm ²
5cm ²
1





and
1,8m 1800mm 180 36
0,18m² 1800cm ² 360

50mm
5cm ²
>
1,8m
0,18m²
EXERCISE 3
1.
2.
8.4
Which ratio in each of the following pairs is the larger?
(a)
16 : 20 or 21 : 25
(b)
9 : 10 or 21 : 23
(c)
1 3
3 1
2 :1
or 4 : 3 (d)
8 8
3 4
3,8 : 1,4 or 25,5 : 10,5
(e)
4 min : 1 h 10 min or 350 mm : 4 m
Two squares have sides of 30 mm and 90 mm respectively. Compare
(a)
their sides
(b)
their areas
Increasing or decreasing ratios
Examples:
1.
The price of paper increases from 132 c/kg to 165 c/kg. Calculate
the price of a book which originally cost R 8,80.
Solution:
165 5

132 4
 New price of book =
 New price of book =
2.
5
X R 8,80 = R 11
4
(a)
Increase 12 in the ratio 3 : 4.
(b)
Decrease 12 in the ratio 4 : 3.
5
of old price
4
Page 89 of 268
Solution:
(a)
Increased number =
12 4
X  16 . Multiply by fraction > 1
1 3
(b)
Decreased number =
12 3
X  9.
1
4
Multiply by fraction < 1
EXERCISE 4
1.
The enrolment in a school is 120 pupils. The enrolment increases in the
ratio 5 : 6. How many pupils are there now?
2.
A number of pupils out of a class of 36 left school and enrolment
consequently decreased in the ratio 12 : 11. Determine how many
pupils:
3.
4.
(a)
remained in the class
(b)
left the school.
A man’s monthly salary of R 1 200 was increased in the ratio 21 : 22.
(a)
What was his monthly salary after the increase?
(b)
By How much did his salary increase per month?
The price of an article costing R 60 was increased in the ratio 4 ; 5 and
then again in the ratio 5 : 7. What was the final price of the article?
8.5
Proportional division
Examples;
1
A sum of R 144 is divided among three boys A, B and C in the ratio
3 : 4 : 5. How much does each receive?
Solution
3 + 4 + 5 = 12
A receives
3
of R 144 = R 36
12
Check the answer by addition
B receives
4
of R 144 = R 48
12
R 36 + R 48 + R 60 = R 144
C receives
5
of R 144 = R 60
12
2.
Divide 1025 oranges among A, B and C so that
A’s part : B’s part = 3 : 4 and B’s part : C’s part = 3 : 5
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Solution
Ratio: A : B : C
3 : 4
3 : 5
9
:
12
12 : 20
i.e.
9 : 12 : 20
Now 9 + 12 + 20 = 41 parts
A receives
9
of 1025 = 225 oranges
41
B receives
12
of 1025 = 300 oranges
41
C receives
20
of 1025 = 500 oranges
41
3.
Complete:
Check the answer by addition
2 ? 14 ? 1
 


3 9 ? 36 ?
Solution
2 6 14 24
1
 


3 9 21 36 1½
EXERCISE 5
1.
Two partners A and B invest R 150 000 and R 200 000 respectively
in a business. How must a profit of R 7000 be divided between them?
1 1 1
: :
2 4 5
2.
Divide R 9 400 in the ratio
3.
Divide R 413 among A, B and C so that A’s share : C’s share is
as 3 ; 4 and B’s : C’s is as 6 : 5.
4.
A certain sum of money is divided among A, B and C in the ratio
10 : 4 : 3. If B and C together receive R 665, what sum is divided
among the three?
5.
Divide R 280 among four boys and two girls so that each girl receives
R 20 more than each boy.
Page 91 of 268
6.
The sides of a right-angled triangle are in the ratio 3 : 4 : 5. If the
longest side is 115 mm, how long are the other sides?
3 ? 15 ?



4 12 ? 2
(a)
8 16 ?


15 ? 45
7.
Complete:
(b)
8.
A chemical compound contains 25 % potassium, 45 % aluminium,
10 % sulphur and 20 % oxygen. What mass of each element is
present in 200 kg of the compound?
Direct proportion and the proportional constant:
A proportion is a statement that two ratios are equal. Look at the following example:
Perimeter (P)
60 mm
32 mm
28 mm
Side (S)
15 mm
8 mm
7 mm
The ratio of perimeter : side is 4 : 1 so that we can calculate the length of
a side of a square if the perimeter is given, and vice versa. If the perimeter
is doubled, the side of the square is also doubled. In this case
If
P
is a constant.
S
y
P
= k ( or
= k as in the example above), then k is called the proportional constant,
x
S
which is 4 in the example. If y and x are not like quantities
y
is called a rate.
x
Example 1.
If 8 kg of tea costs R 79,20 we can tabulate this result as follows:
Cost in rand
79,20
39,60
9,90
Kilograms of tea
8
4
1
In this example the rate =
This is read as
Example 2
R79,20 R39,60 R9,90


.
8kg
4 Kg
1kg
9,90rand
= 9,90 rand per kilogram = R 9,90/kg.
1kg
Page 92 of 268
If
P
= 4, Where P is the perimeter of a square and S is the length of
S
Its side, find the perimeter of the square if the side measures 14 mm.
Solution:
P = 4s = 4 X 14 = 56 m.
Example 3
If 8 kg of tea cost R 79,20, what will 12 kg cost?
Method 1.
8 kg of tea cost R 79,20
 12 kg of tea cost R 79,20 X
12
(12 kg of tea cost more than 8 kg)
8
= R 118,80
Method 2.
The rate is
R79,20
= R 9.90/kg
8kg
 12 kg of tea cost R 9,90 X 12 = R 118,80
EXERCISE 6
1.
For each of the following tables:
(i)
Show that the variables remain in direct proportion.
(ii)
Calculate the constant of proportionality for the ratio indicated.
(a)
x
3
5
7
9
11
y
6
10
14
18
22
for
x:y
(b)
x
15
45
135
210
270
y
5
15
45
70
90
for
y;x
(c)
x
2,5
3,5
11,5
15
20
y
5
7
23
30
40
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for
x:y
(d)
x
3
4
6
9
21
y
3,6
4,8
7,2
10,8
25,2
for
y:x
(e)
x
3,5
4,5
5,5
7
9,4
y
5,25
6,75
8,25
10,5
14,1
for
2.
x:y
In each of the following examples:
(i)
Calculate the constant of proportionality.
(ii)
Use the constant of proportionality to find values for the unknowns.
(a)
X
10
15
20
25
30
35
40
Y
20
30
40
50
a
b
c
(b)
X
12
16
20
24
28
32
Y
3
4
5
a
b
c
(c)
No of books
2
5
10
12
32
Cost in Rand
18,50
46,25
a
b
c
Rand
2,50
4
8
a
82,25
Dollars
1
1,6
b
18
c
(d)
(e)
Page 94 of 268
3.
x
25,5
a
5,1
125
d
y
5,1
25,2
b
c
2c
If it is known that x and y are in direct proportion, and that y and z are also in direct
proportion, complete the table for a, b, c, d, e and f.
X
2
3,5
c
e
Y
3
a
13,5
f
Z
7,5
b
d
45
EXERCISE 7
1. if 7 pencils cost 56c, what will 18 pencils cost?
2. For 18 kg of tea one pays R 158,50. How much will 12 kg cost?
3. How far will a motor car travel on 9 litre of petrol if it travels 42 km on 5 litre.
4. Three eights of a sum of money is R 3,54. Calculate:
(a)
1
8
(b)
7
8
(c)
The full amount.
5. Between them 15 horses consume 9 bales of Lucerne per day. How
many bales will 20 horses consume per day?
6. Working together, 11 workmen can pack 341 trays of fruit a day. How
many trays will 15 workmen pack in a day?
7. Two fifths of the learners of a school are boys. If there are 152 boys,
how many girls and how many learners are there at the school in all?
8. (a) The maximum speed of a car is 180km/h. Calculate its speed in
metres per second.
(b) If the speed is 30 m/s, calculate the speed in kilometres per hour.
EXERCISE 8
1.
if 45 articles cost R 270, how many can be bought for R 120?
2.
If it takes 4 workers 12 days to complete a building project, how long
will it take the men to complete 5 such projects if we assume that they
work at the same rate?
3.
Mrs Bester finds that to cater for 45 rugby players after their match costs
R 67,50. How much will she need to spend to cater for 60 rugby players ?
4.
Alan finds that he spends R 20 a month on petrol for his motor cycle, to
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cover 500 km. what will he have to spend per month if he has to travel
750 km per month.
5. To go 216 km my car uses 24 litre of petrol.
(a) How far will it go on 45 litre?
(b) How many litres will it use for 342 km?
6. 15 books cost R262.
(a) Find the cost of 8 books.
(b) How many books can I buy for R 210?
7. 32 litre of petrol cost R 33, 60.
(a) what will 42 litre cost?
(b) How many litres can one buy for R 15,25?
8. 2 dozen eggs cost R 3,84.
(a) How much will 9 eggs cost?
(b) How many eggs can one buy for 48c?
9. (a) If 3,5 m of dress material cost R 25,20 how many metres of
material can I buy for R9?
(b) How much will 10 m cost?
10. A book of 210 pages is 1,2 cm thick ( not including covers ). How
many pages would it have if the thickness were increased to 1,8 cm?
11. A vertical stick of length 2,5 m casts a 3,2 m shadow on the ground. What
is the height of a flagstaff of which the length of the shadow is 19,2 m?
12. (a) If I can travel 261 km in three hours, how long will it take to travel 435 km?
(b) How far can I go in 1 h 20 min if the speed remains constant.
13. To lay a floor of area 12 m² I use 5 pockets of cement. How many pockets will I need for a
floor of area 18 m² ?
14. In 5 working days a boy ears R87,50.
(a)
What should he earn in 12 working days?
(b)
How many days must he work to earn R 350 ?
15. if the radius of a circle is 3,5 cm, the circumference is 22 cm ( to the
nearest cm ). What is the circumference of a circle of which the
radius measures 8,4 cm ?
16. A speed of 108 km/h is equivalent to a speed of 30 m/s.
17.
(a)
What is a speed of 60 km/h in m/s ?
(b)
What is 900 m/s in km/h.
If the American dollar is worth 760 cents, and a haircut in the U.S.A.
costs 3 dollars, how much will this be in South African money ?
18.
a clock loses 3 minutes and 5 seconds in 4 days. How much time
Page 96 of 268
will it lose in a week ?
19.
A clock loses 4½ minutes in 1 day and 10 hours. If it is correct at 03:00
on Monday, what time will it show at 17:00 on the following Sunday ?
Give your answer correct to the nearest minute.
20.
If 16 boys eat 48 buns in 25 minutes, how long will it take 30 boys to
eat 54 buns, it they eat at the same rate ?
21.
The length of line segment PQ is 15 cm. Point R divides PQ in the
ratio of 1 : 4. Find the lengths of PR and RQ.
22.
If the lengths of the sides of the triangles below are in proportion,
find the length of BC.
A
6
D
4
2
E
B
23.
3
4
F
C
Points p and Q lie on sides AB and AC of ∆ ABC respectively.
Side AB is 12 cm in length and side AC is 15 cm in length. If
P and Q divide sides AB and AC in the ratio 2 : 1, find the lengths
of PB and AQ.
A
P
B
Q
C
The graphical representation of a direct proportion.
Example:
Page 97 of 268
Answer the following questions from the graph below. It displays the motion
of an object.
(a) Is the speed constant over the whole 9 seconds? Explain.
(b) Calculate the average speed for the sections: (i) AB (ii) CD
(c) What does the section BC suggest with regard to the motion
of the object?
(d) What was the average speed of the object over the 9 seconds?
INDIRECT PROPORTION
Table A
The distance which a car covered at constant speed.
Distance in km(at constant speed)
70
140
Time taken in hours
1
2
175
210
......
1
2
3
......
2
Table B
The time men took to reap corn on a farm.
Number of men reaping corn
100
80
60
40
20
.....
Number of hours
24
30
40
60
120
.....
In table A, the ratio of each column is always 70;1. Therefore we can say that the distance
covered is directly proportional to the time taken. If the one quantity increases, so does the
other.
Page 98 of 268
In table B we see that 100X24 = 80X30 = 60X40 = 40X60 = 20X120, which we call the
constant product. We say that the two quantities are inversely proportional to the other.
If the one quantity increases, the other decreases.
EXERCISE 9
The following questions are examples of direct proportion, inverse proportion or neither. Say
which, and give the constant ratio (if the sets are directly proportional) or the constant product
(if the sets are inversely proportional).
1.
A bookshelf is filled with books of the same thickness, but the number of books depends
on their thickness.
Thickness in mm
10
15
20
30
40
50
12
Number of books
60
40
30
20
15
12
50
What is the length of the bookcase?
2.
3.
The distance which a car travels in a given time depends on the average speed at which it travels:
Speed in km/hour
50
65
75
80
60
Distance in 5 hours (km)
250
325
375
400
300
The time a car takes to travel a certain distance depends on its average speed:
Speed in km/hour
40
50
60
75
80
Time in hour to go 480 km
12
9,6
8
6,4
6
Number of books
3
5
2
6
10
Mass in kg
1
1,3
0,5
8
3
4.
5.
1st number
2nd number
6.
1
9
8
2
4
6
9
1
7
3
0
10
The number of pieces of equal length which can be cut from a metre of wire depends on
the length of each piece.
Length of each piece in mm
Number of pieces
7.
3
7
5
200
62,5
16
20
50
From a piece of wire 400 mm long I can make various rectangles.
250
4
125
8
Page 99 of 268
Length of one side in mm
Length of other side in mm
60
140
150
50
40
160
130
70
100
100
SOLVING PROBLEMS INVOLVING INVERSE PROPORTION:
Example 1:
If the numbers in the second set are inversely proportional to the numbers in the second set, find
p and q.
3
24
12
P
Q
4
Method 1:
Inverse proportion therefore the products of the numbers in the pairs must be equal, hence:
P X 12 = 24 X 3 therefore p = 6
And q X 4 = 24 X 3 therefore q = 18
Method 2:
3 has increased by a factor of 4. To get p, we must divide 24 by 4,
24
i.e. p =
= 6
4
24
Similarly, since 4 =
, q = 3 X 6 = 18
6
Example 2: A group of 6 boys on a camping trip to the Drakensberg have an 8-day supply of
food. At the last moment 2 of the boys drop out but leave their food for the others. How many
days can the remaining 4 boys spend at the camp?
Method:
Set up the ordered pairs:
Number of boys
6
4
Number of days
8
X
Ask yourself whether it is direct or inverse proportion. Since the number of boys has decreased
the food should last longer-therefore it is inverse proportion.
Therefore x X 4 = 8 X 6
 x = 12
Example 3: A certain distance is covered in 4 hours at 63 km/h. In what time will the same
distance be covered at 84 km/h?
Method 1:
At 63 km/h the distance is covered in 4 hours
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4 63
X
= 3 hours
1 84
The greater the speed, the sooner the distance is covered (i.e. less time is required).
63
Therefore multiply by a proper fraction, i.e.
where the numerator < denominator.
84
At 84 km/h the distance is covered in
Method 2:
Inverse proportion therefore constant product is 63 X 4 = 252
Therefore 84 X x = 252
252
= 3 hours
 x =
84
EXERCISE 10
I1.
If 16 men complete a job in 12 days, how many men can complete the
job in 8 days?
2.
If 111 articles at 20 c each can be bought for a certain sum of money, how many
articles at 30 c each can be bought for the same sum?
3.
4.
A motorist covers a distance in 2h15 min at 59,2 km/h.
(a)
How long will he take to cover the distance at 74 km/h ?
(b)
At what speed must he travel to cover the distance in 1
1
hours ?
2
If the variables are inversely proportional to each other, find the values of the unknown
variables:
5.
(a)
X
5
R
25
Y
10
12,5
T
If 6 men can do a job in 15 day, how long will 9 men take to do
the same job?
(b)
6.
How many men would be needed if the available time is 45 days?
6 men can lay 2000 bricks in a day. How many bricks can 9 men lay in a
day?
7.
8.
If the variables are in direct proportion, find u and v:
21
35
V
24
U
4
From a piece of timber I can cut 12 pieces each 180 mm long. How many pieces of length
240 mm could I cut from the same length of timber?
Page 101 of 268
9.
A party of 10 men in Antarctica has food to last them 6 months. They are joined by
2 men from another base. How long will the food last them now?
10.
Mr Smith has enough money to buy 4 cinema tickets at R 3 each. His party is joined by 2
friends and he finds that now he has exactly the right amount of money to buy 6
cheaper seats. What do the cheaper seats cost?
11.
A ferry can carry 6 cars each of mass 1 200 kg. How many cars of mass 720 kg
could it carry instead ?
12.
Jim could travel to Pretoria in 6 hours at an average speed of 95 km/h but now,
with speed and petrol restrictions, he finds that he can do the same trip in 7
1
hours.
2
What is his new average speed ?
13.
A party of 8 men on an expedition require 5 000 kg of supplies to last them one
year. What mass of supplies would be needed for 10 men ?
14.
(a)
On Peter’s tape recorder a reel of tape will record for 11 minutes at a speed of
9,5 cm/s. For how long will it record at a speed of 4,75 cm/s ?
(b)
Peter computes that he has 66 m of tape on the reel which plays for 11 minutes
at the faster speed. How many metre are left after 8 minutes ?
15.
A jumbo jet flies from Oliver Tambo airport to London in 15 hours at an average
speed of 1 000 km/h. If the time is reduced to 12 hours, what must the average
speed be ?
EXERCISE 11 ( Mixed examples)
1.
6 Packets of chips cost 9 cents. What should I pay for 15 packets of the
same type ?
2.
A casual labourer is paid R 81 for 6 days work. How mush should he earn in 22 days at
the same rate ?
3.
My car used on average 40 litres of petrol to go 300 km. How far should I be able to
go on a full tank of 52 litres ?
4.
If I pay 42 francs for my lunch in Paris and the rate of exchange is 5,6 francs = R 1, what
did my meal cost me in South African money ?
5.
If a car travels at a steady speed of 80 km/h:
(a)
How far will it go in 2
1
hours ?
4
Page 102 of 268
(b)
6.
How long will it take to travel 220 km ?
If a athlete using a flying start does the 200 m sprint in 20 seconds, what is his speed in
km/h ?
7.
8.
9.
(a)
If x pencils cost 30 cents, what will 1 pencil cost ?
(b)
If x pencils cost 30 cents, what will y pencils cost ?
(c)
If p pencils cost q cents, what will t pencils cost ?
(a)
If a car is travelling at 60 km/h, what is the speed in m/s ?
(b)
If a car is travelling at x km/h, what is the speed in m/s ?
(c)
Write down a formula for changing km/h to m/s.
A motorist covers a journey in 5 hours at an average speed of 80 km/h. If he wishes to
do the same journey in 4 hours, at what average speed should he travel ?
10.
I have enough money to buy 8 packets of chips at 21 cents each. If the price increases to
24 cent per packet, how many can i buy now ?
11.
At an athletics stadium it usually takes an hour for the spectators to get in if 6 turnstiles
are used. How long will it take for them to get in if 3 more turnstiles are opened ?
12.
An eight-man life-raft is equipped with iron rations to last 8 men for 30 days, but
there are only 5 survivors on the raft. How long should the rations last ?
13.
Each week i spend all my pocket money on y packets of mint-chewies at x cents
per packet. This week I had to pay z cents per packet. How many packets did I
get ?
14.
A car travels 384 km in 6 hours.
(a)
What is the average speed in km/h ?
(b)
At the same rate, how far should it go in 8 hours ?
(c)
At the same rate, how long should it take to cover 480 km?
15.
My watch loses 3 minutes in 18 hours. How much will it lose in 24 hours ?
16.
I can buy 20 litres of petrol at R 6 per litre. If the price increases to R 8 per litre,
how many litres can I get for the same amount ?
17.
A vertical pole of 2 metres long casts a shadow 1,5 metres long. At the same time
of the day a tree casts a shadow of 12 metres. Calculate the height of the tree.
18.
If 3 pumps can empty a storage dam in 16 days, how long will it take 4 similar
pumps to empty the dam ?
19.
A journey by car takes 2 hours at an average speed of 72 km/h. How long will the
journey take at an average speed of 64 km/h ?
Page 103 of 268
20.
A map shows the distance between towns A and B as 48 mm and between B and
C as 32 mm. Town A and B are known to be 120 km apart..
(a)
What is the actual distance between town B and C ?
(b)
What is the scale of the map ?
C
B
21.
A
(a)
If x books cost y cents, what is the cost of z books ?
(b)
If a car travelling at k km/h completes a journey in h hours
(i) how long will it take for the same journey travelling at m km/h?
(ii) at what speed will I have to travel to cover the same journey in
p hours ?
22.
Working 6 hours a day, 14 men finish a job in 5 days. How many days
would 15 men take to finish the same job working 7 hours per day ?
EXERCISE 12
The following questions are examples of direct proportion, inverse proportion or neither.
Say which, and give the constant ratio (if the sets are directly proportional) or the
constant product (if the sets are inversely proportional
1.
(a)
x
3
6
9
12
18
y
8
16
24
36
45
x
10
30
60
100
200
y
5
15
30
50
100
(b)
(c)
Page 104 of 268
x
3
6
9
12
15
y
8
16
24
32
40
x
7,5
15
21
36
144
y
2,5
5
7
12
48
x
1
2
3
4
5
y
6
7
8
9
10
(d)
(e)
2.
Calculate the constants of proportionality for the following examples (all direct
proportion). Calculate the proportionality constant for
(i) x : y and
(ii) y : x in each case:
(a)
x
3
12
100
y
9
36
300
x
12
21
99
y
4
7
33
x
2,5
12
20,5
y
7,5
36
61,5
(b)
(c)
3.
Handkerchiefs are sold at 3 for R 3,90. What will 13 cost ?
4.
At the local nursery 12 daffodil bulbs cost R 6,12. What will 63 bulbs cost ?
5.
The Exchange rate on a particular day is given as follows ( in Madrid, Spain ):
45,2 Pesetas
(a)
How many Pesetas will I receive for: (i) R 12
(ii)
R 25,50
1 Rand =
Page 105 of 268
(b)
How many Rand will I receive for:
(i) 1 627,2 Pesetas
6.
(ii)
2 820,48 Pesetas
How many French Francs will I receive in exchange for R 24 if R 10
buy 25 Francs?
7.
If your 50 cc motor cycle goes 280 km on 7 litre of petrol, how far can you expect it to go
on a full tank of 18,5 litres ?
8.
Draw a graph to make conversions from Rand to Lira. The Exchange Tate quoted at the
bank in Rome when you cash your cheques is:
1 Rand = 600 Lira.
(a)
Read off, from your graph, what the cost of goods will be ( in Rand), when you
spend
(b)
9.
Allow for a maximum of 12 000 Lira.
(i) 2 700 Lira (ii) 4950 Lira
What would R 7,50 amount to in Lira ?
The following questions are examples of direct proportion, inverse proportion or neither.
Say which, and give the constant ratio (if the sets are directly proportional) or the
constant product (if the sets are inversely proportional:
(a)
The time a certain amount of food lasts depends on the number of people there
are to eat it:
Number of people
20
40
12
15
30
Number of days food lasts
24
12
40
32
16
(b)
(c)
1st number
2,4
0,5
6
1,2
2nd number
1
4,8
0,8
2
The quantity of sweets I can buy for R 3 depends on the cost of the sweets per
kilogram:
(d)
Cost per kg ( in cents)
50
120
80
100
150
Number of kg for R 3
6
2,5
3,75
3
2
The time taken to cut a field of sugar-cane depends on the number of men
employed to do the work:
10.
Number of men
6
9
12
18
36
Number of days
12
8
6
4
2
If a car travels at 120 km/h, calculate:
(a)
How far does the car travel in 10 minutes?
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11.
(b)
What distance is covered in 1 hour and 15 minutes?
(c)
After how long will it reach the 184 km mark ?
We are given that y  x and z  y. Find the values of a, b, c, d, e and f.
X
3
7
20
D
e
Y
9
21
A
c
0,5
Z
4,5
10,5
B
112,5
f
EXERCISE 13
Use cross-multiplication to solve for x:
1.
5 x

6 12
2.
x 3

8 4
3.
4.
1 8

10 x
5.
5 : x = 4 : 8 6.
3 : x = 4,5 : 6
7.
x : 5 = 3 : 10
8.
4 : 7 = 12 : x
9.
2x 8

3 9
10.
x 2

5 15
11.
4
5
12.
5 3

x 2
13.
x2 4

3
5
14.
x x2

5
3
15.
x2 x4

5
3
x
1
2

5 15

x 9
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CHAPTER 10
LINES AND TRIANGLES
Revision of parallel and intersecting lines:
Two or more straight lines, which are always the same distance apart, are
called parallel lines.
When two or more parallel lines are cut by a transversal, the corresponding
angles are equal. (  a = corresponding  b )
a
b
When two or more parallel lines are cut by a transversal, the alternate
angles are equal. (  c = alternate  d )
c
d
When two or more parallel lines are cut by a transversal, the co-interior
angles add up to 180˚. ( They are supplementary)  e +  d = 180˚.
e
f
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When two lines intersect, the pairs of vertically opposite angles are equal.
 g =h
g
h
EXERCISE 1
1. Calculate the values of these variables: (Give reasons for all your steps
2.
Find the values of the unknowns:
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3. Calculate the values of the unknowns:
4. Calculate the values of the unknowns:
5. Proof by calculation that PQR is a straight line:
6. Calculate the values of x that will make P, Q and R collinear.
(a)
(b)
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EXERCISE 2
1.
Calculate the values of x and y in each of the following figures:
2. Determine the value of x and y in each of the following figures:
3. Calculate the value of x in each of the following figures: ( You will have to make a
construction ).
(a)
A
B (b) A
B
125º
50º
E x
4
P
E
R
4x
x
160º
C
(c) Q
D
Prove that  P =  R.
C
3x
5x
D
C
D
5 In  PQR,  P =  Q = x and RT || QP, QR is
produced to S. Prove that RT bisects  PRS.
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THE ISOSCELES TRIANGLE:
(a)
If a triangle has two sides equal, the angles opposite the equal sides are also
equal. The two equal angles are called the base angles and the remaining angle is
called the vertical angle.
A
B
C
If AB = AC then
B  C
(b) If a triangle has two angles equal, the sides opposite the equal angles are also
equal, so the triangle is isosceles.
P
Q
If
x
x
Q = R
R
then PQ = PR
THE EQUILATERAL TRIANGLE:
The angles of an equilateral triangle are all 60º.
EXERCISE 3
1.
Complete the following statements:
(a) If the vertical angle of an isosceles triangle is 48º each base angle is .........º.
(b) If one base angle of an isosceles triangle is 64⁰ the vertical angle is ..........º.
(c) If one angle of an isosceles triangle is 50⁰, the other two angles are .........º
and .............º or ..............º and ..............º.
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2.
Find the value of x in each of the following. Give reasons for all your steps.
DF = EF = EG
AB = BC = AD
What kind of ∆ is ACD?
3.
AB = BC. Prove that AC = CD.
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4.
Prove that ΔADB is isosceles.
5.
BD = DC = DA. Find the size of ACB in degrees.
6.
AB = BC and AD = AC. Find in terms of x:
(a) ADC
(b) C
(c) DAC
(d) Hence find the value of x in
degrees.
7.
BE = ED. Prove that ΔAEO is isosceles. (Hint: Let B = x )
8.
Prove that S is the midpoint of PQ.
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9.
EC = BC. Prove that BE bisects ABD.
EXERCISE 4
1.
In ΔABC, AB = AC. D is a point on AB so that CD = BC and ACD = 12º.
Calculate
2.
A .
In ΔABC, A = 30º, C = 45º and BD  AC. E is a point on AB so
that AE = ED.
(a) Calculate
ABC and ABD .
(b) Prove by calculating angles that ΔDEB is equilateral.
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3.
In ΔABC, AB = AC and AB produced to D so that BD = BC. If
AC = CD and A = x, complete the following in terms of x:
D = .................
[ AC = CD]
(b) BCD = ..............
[BC = BD]
(c) ABC = .............
[Ext. Angle of ΔDBC]
(a)
(d) ACB = .............[AB = AC]
(e) Hence calculate x in degrees.
4.
ABC is an equilateral triangle with BA produced to D so that
AD = BA. Prove that (a) ACD = 30º and (b) DCB = 90º.
5.
BAC of ΔABC is obtuse. AD bisects BAC with D on BC. If
AD = AC and B = 15º, calculate BAC and C.
(Hint: Let BAD = x and find ADC and
6.
C in terms of x)
D is a point on BC of ΔABC so that AD = BD = AC, and
BAD  DAC. Calculate the angles of ΔABC and hence show that
ΔABC is isosceles.
(Hint: Let B = x and find the angles of ΔADC in terms of x)
7.
Find the values of x and y in the diagram:
A
81
B⁰
C
y
3x
6x
F
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D
8.
E
Find the value of a:
H
2a
G
9.
2a
J
K
Find the value of b:
L
32º
⁰
P
48º
M
b
O
N
EXERCISE 5
1.
Determine the sizes of the angles marked with small letters. Give
reasons:
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2.
Determine the sizes of the angles marked with small letters. Give
reasons:
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The theorem of Pythagoras and its converse:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares
on the other two sides.
If C = 90º then
A
c2 = a2 + b2
b
C
c
a
B
Exercise
In ΔABC, B  90 . Calculate the third side in each of the following
cases:
1. a = 3 cm, b = 4 cm
2. a = 8 cm, c = 15 cm
4. a = 8 cm, b = 17 cm 5. b = 13 cm, c = 5 cm
3. a = 7 cm, c = 24 cm
6. b = 41cm, c = 40 cm
7. a = 10 cm, c = 24 cm 8. a = 9 cm, b = 15 cm
Find the value of x in each of the following:
C
3.
In ΔABC, C  90 . Calculate the length of the third side. If your answer
is an irrational number, leave it in the square root form. (We call this the
surd form).
(a) a = 1 cm, b = 2 cm
(b) a = b = 2 cm
(c) b = 3 cm, c = 4 cm
(d) a = 5 cm, b = 6 cm
(e) a = b and c = 6 cm.
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4.
Find the area of each of the following triangles:
5.
Find the length of the diagonal of:
(a) A rectangle 10 cm by 8 cm
(b) a square with side 10 cm.
The converse of Pythagoras:
If the square on the longest side of a triangle is equal to the sum of the squares of the other two
sides, the angle opposite the longest side is a right angle.
A
If AB2 = AC2 + BC2 then C  90.
C
B
If the square of the longest side of a triangle is less than the sum of the squares of the other
two sides, the angle opposite the longest side is acute and the triangle is acute-angled.
A
If AB2 < AC2 + BC2
B
then C  90
C
If the square of the longest side of a triangle is greater than the sum of the squares of the other
two sides, the angle opposite the longest side is obtuse and it is an obtuse-angled triangle.
A
If AB2 > AC2 + BC2 then C  90.
B
C
Exercise:
1.
What kind of triangle is ΔABC in each of the following cases:
(a) a = 5 cm, b = 12 cm, c = 13 cm
(b) a = 17 cm, b = 15 cm, c= 8 cm
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(c) a = 7 cm, b = 8 cm, c = 9 cm
(d) a = 15 cm, b= 20 cm, c = 25 cm
(e) a = 12 cm, b = 9 cm, c = 7 cm
(f) a = 16 cm, b = 20 cm, c = 12 cm
(g) a = 4 cm, b = 4,1 cm, c = 0,9 cm
(h) a = 8 cm = b, c = 4 cm
(i) a = 10 cm, b = 10 cm, c = 13 cm
(j) a = 9 cm, b = 9 2 cm, c = 9 cm
2.
Prove that ACD  90 in each of the following:
3.
Find the length of median AD.
4.
Find the area of ΔADC.
5.
Find the length of BC and hence the area of quad ABCD.
A
44 cm
B
D
60 cm
80 cm
C
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6.
Find AD and hence the area of ΔABC.
7.
Find the area of quad ABCD.
8.
ABCD is a rectangle. Find the length of BC.
9.
Calculate: (a) the length of BC
(b) The length of AB
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CHAPTER 11
QUADRILATERALS
Sum of angles of a polygon.
In any polygon an angle such as 1 is known as an interior angle. If a side of the polygon is
extended, an exterior angle, such as E is formed.
Activity to derive the sum of the interior angles of a polygon:
(a) Study the diagrams below, then copy and complete the table:
Number of sides of polygon
Number of triangles formed
Sum of the interior angles
3
1
1 X 180º
4
2
2 X 180º
5
6
7
(b) Make a conclusion about the sum of the interior angles of a polygon with n sides.
Results:
The sum of the interior angles of a polygon with n sides is 180º(n - 2)
The sum of the exterior angles of a polygon is 360º
A regular polygon is a polygon is a polygon with equal sides and therefore equal angles too.
Every interior angle of a regular polygon with n sides =
(n  2).180
n
Every exterior angle of a regular polygon with n sides =
360
n
A polygon with 5 sides is called a pentagon, with 6 sides a hexagon, with 7 sides a pentagon,
with 8 sides a octagon, 9 sides a nonagon, with 10 sides a decagon.
EXERCISE 1
Find the sizes of the angles indicated by letters.
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EXERCISE 2
1.
Calculate, with reasons, the value of x (and/ or y) in each of the following polygons:
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2. Calculate:
(a) The sum of the of the interior angles of a hexagon.
(b) The size of one interior angle of a regular pentagon.
(c) The size of every interior angle of a regular nonagon ( 9 sides ).
(d) The size of every exterior angle of a regular octagon.
(e) The number of sides of a regular polygon if every exterior angle is 36º.
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(f) The size of every interior angle of the polygon in (e).
(g) The number of sides of a regular polygon if every interior angle is 135º.
(h) x, if the sizes of the interior angles of a polygon with 5 sides are:
x ; x + 10º ; 2x - 40º ; 2x – 60º and x respectively.
3. Calculate the sum of the interior angles of polygons with:
(a) 9
4.
(b) 12
(c) 18 sides
The exterior angles of regular polygons are equal to (a) 20º (b) 40º (c)15º.
For each polygon, calculate the number of sides and the sum of the
interior angles.
5.
How many sides does a regular polygon have if each interior angle is
equal to:
6.
(a) 150º
(b) 160º
(c) 170◄
Each interior angle of a regular polygon is four times the size of an exterior
angle of the polygon.
(a) Calculate the size of each interior angle
(b) How many sides does the polygon have?
Special quadrilaterals:
1. Parallelogram: We use the abbreviation parm. or ǁm
Definition:
A quadrilateral with both pairs of opposite sides parallel.
Properties:

Both pairs of opposite sides are equal

Both pairs of opposite angles are equal

The diagonals bisect each other

The diagonals bisect the area of the parallelogram

A parallelogram has no axis of symmetry
Rectangle:
Definition:
A rectangle is a parallelogram with one 90º-angle.
Properties:

Both pairs of opposite sides are equal
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
Both pairs of opposite sides parallel

Diagonals bisect each other

Diagonals bisect the area

All angles are 90º

The diagonals are equal

A rectangle has 2 axes of symmetry
Same as parallellgram
3. Rhombus: Definition: A rhombus is a parallelogram with 2 adjacent sides equal.
Properties:

Both pairs of opposite sides are parallel

Both pairs of opposite angles are equal

The diagonals bisect each other

Diagonals bisect the area

All sides are equal

Diagonals bisect each other perpendicularly ( i.e. at 90º )

The diagonals bisect the angles, i.e. P1  P2

A rhombus has two axes of symmetry
Same as parallelogram
etc.
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Square:
Definition:
A square is a rectangle with two adjacent sides equal.
Properties:

Both pairs of opposite sides are parallel

Diagonals bisect each other

All angles are 90º

Diagonals are equal

All sides are equal

Diagonals bisect each other perpendicularly ( i.e. at 90º

A rhombus has two axes of symmetry

Diagonals bisect the angles, forming 45º angles.
Kite:
Same as rectangle
Same as
the
Definition: A kite is a quadrilateral with two pairs of adjacent sides equal.
Properties:

Diagonals cut at 90º

One diagonal bisects the other, BO = OD

Diagonal AC bisects the angles

The kite has one axis of symmetry
rhombus
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Trapezium:
We use the abbreviation trap.
Definition: A trapezium is a quadrilateral with one pair of opposite sides parallel.

A trapezium has no axis of symmetry.
A
D
B
C
Exercise 3:
1. Which of the following quadrilaterals has the property given below;
Parallelogram, Rectangle, Rhombus, Square, Kite
(a)
Diagonals always equal
(b)
Diagonals always perpendicular
(c)
Diagonals bisect the area of the quadrilateral
(d)
Diagonals bisect the angles
2. Identify each of the following quadrilaterals:
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2.
(a)
(c)
Determine, with reasons, the values of a – g.
(b)
(d)
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4. Determine the values of all the variables in the following diagrams. First
identify the specific quadrilateral.
5. Determine the value of x ( with reasons ).
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6. ABCD is a parallelogram. BF = BE and AC║FE. Calculate x
7. ABCD is a parallelogram. Calculate A.
8. DEFG is a rhombus. Find x.
Exercise 4:
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1. Find the value of the unknowns in each case. (Each figure is either a rectangle or a square).
Measurements are in mm.
2. Find the values of the unknowns in each case. (Each figure is a parallelogram). Lengths are
in mm.
3. Find the values of the unknowns in each case. (Each figure is a rhombus). Lengths are in
mm.
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4. Find the values of the unknown sides and angles. All lengths are in mm.
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5. Find the values of the unknown sides and angles. All lengths are in mm.
6. Calculate the missing values in the following quadrilaterals:
7.
Identify each of the following quadrilaterals and then determine the value
of x.
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Exercise 5.
Calculate the value of x in each case. Round off your answers to two decimal places
where applicable.
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Exercise 6.
1. Use your knowledge of the properties of a kite to determine x, y and z in each of the
following kites. In each case state the properties you have used.
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2. Use your knowledge of the properties of parallelograms to determine x, y and z in each
of the following parallelograms. Explain your reasoning.
3. In the figure ABCD and ABFE are parallelograms. What deduction can you make about
DC and EF ? Explain.
4. Use you knowledge of the properties of a rhombus to calculate x, y and z
in each of the following cases. In each case indicate the property you
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have used.
5. Calculate the perimeter of each of the following quadrilaterals :
6. Use your knowledge of the properties of a trapezium to calculate the values of x and y.
Explain your reasoning.
7. Use your knowledge of the different kinds of quadrilaterals to calculate the values of x, y
and z in each of the following quadrilaterals. In each case state which properties you
are using.
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8. (a) Does a rhombus have all the properties that a parallelogram has ?
(b) Is a rhombus a special kind of parallelogram ?
(c) Is every rhombus a parallelogram.
(d) Is every parallelogram a rhombus ?
9. (a) Is every square a rhombus ?
(b) Is every rectangle a square ?
(c) Is every square a rectangle ?
(d) Is every rhombus a rectangle ?
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CHAPTER 12
AREAS
AREA FORMULAE :
1. Parallelogram : Area = b x h = base x perpendicular height.
(Base : a side of the parallelogram)
2. Triangle : Area = ½ b x  h = ½ base x perpendicular height.
3. Square :
Area = side x side = side2
4. Rectangle :
Area = l x b = length x breadth
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5. Rhombus :
Area = b x h = base x perpendicular height. (Like parallelogram) or Area =
½ d1 x d2 = ½ product of the diagonals
6. Kite : Area = ½ product of the diagonals
7. Trapezium :
8. Circle :
Area = ½.h . sum of the parallel sides =
Area =  r2
Circumference : 2  r
h
( a  b)
2
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Exercise 1:
1.
Calculate the areas of the following figures :
2. Write down the area of the folllowing diagrams in terms of a, b, and or c.
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3. Calculate the area of each of the following diagrams :
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4. ABCD is a rectangle. VD = 7 cm and the area of the rectangle is 63 cm 2. Calculate the
area of trapezium AECD.
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5. Use the diagram below and calculate :
(a) AE
(b) Area of ΔAED
(c) Area of AEBCD.
6. ABCD is a rhombus with AO = 4 cm. The area of the rhombus is 48 cm2. Calculate BD.
7. The area of the kite is 64cm2. Determine the value of x.
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8. The perimeter of the diagram is 48 cm. Calculate the area of the diagram.
Exercise 2 :
1. (a) Calculate the length of QR.
(c) Calculate the area of PQRS.
(b) Calculate the area of ΔSQR.
(d) Calculate the distance between PS and QR.
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2. Calculate the area of the following figures :
3. In parallelogram ABCD, AC = 12 cm, DP = BQ = 5 cm, and BC = 6 cm. Calculate : (a)
The area of ΔABC
(b) The area of ABCD
(c) The distance between AD and BC.
4. PQRS is a rectangle with vertices on a circle with centre O. The diagonal PR is a diameter
of the circle. If PQ = 72 mm and QR = 96 mm, calculate :
diameter PR.
(b) The area of the circle.
(c) The area of the shaded section.
(a) The length of the
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5. Calculate the areas of the following figures :
6. The perimeter of a square is 320 mm and its area is equal to that of a trapezium (fig. 1) in
which AD = 90 mm and DE = 40 mm. Calculate BC in millimetres.
7. In fig, 2 the area of the trapezium is 450 cm2. Calculate :
(a) The length of LP in millimetres (b) The perimeter of the trapezium in metres.
8. ABCD (fig. 3) is a kite with BD = 160 mm and the area of ΔABD = 48 cm2. If the
perimeter of the kite is 540 mm, calculate the area of the kite,
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9. The area of the kite in 9(b) is the same as that of the rectangle in 9(a). Which figure has
the smaller perimeter ? By how much do they differ ?
3. The circumference of the circle is 880 mm. If the area of the circle is the same as that of
rectangle ABCD and if BC = 280 mm, determine whether the perimeter of the
rectangle or the circumference of the circle is greater. What is the difference ?
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Exercise 3 :
1. Calculate the area of each of the following shapes.
2. Calculate the areas of ΔJKM and ΔMKL.
3. (a) If the area of ΔKMN is 36 mm2 and the base Mn = 8 mm, find the height.
(b) Copy the diagram and draw in the height.
M
N
K
4. Calculate the areas of the following shapes :
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5. Calculate the cost of gravelling a square playing- field of side 32 m at 60 c per square metre.
6. A room is as long as it is wide. Calculate its length if it has an area of 16 m2.
7. A flowerbed is in the shape of a rectangle of which the one side is 5 m and the other side is
3,5 m. A rectangular path around the bed has a width of 1,2 m. Calculate :
(a) The area of the ground to be cultivated. (b) The area of the path
(b) How many square tiles of side 200 mm will be needed to tile the path .
Exercise 4 :
1. Calculate the areas of the following quadrilaterals :
(a) A recangle having sides of 80 mm and 60 mm.
(b) A square with one side 170 mm.
(c) A rhombus with one side 100 mm and height 60 mm.
(d) A parallelogram with a base of 13 m and a height of 7 m.
(e) A kite with diagonals 78 mm and 36 mm.
(f) A trapezium having parallel sides 112 mm and 58 mm at a distance of 82 mm apart.
(g) A parallelogram with a height of 20 mm and the same base as a square with an area
of 900 mm2.
(h) A rhombus with a height of 12 mm and a perimeter of 96 mm.
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2. Copy and complete the following table :
Length of the two
Perp. Distance between
Area of trapezium
parallel sides
them
a)
8 cm and 15 cm
6 cm
………
b)
67 cm and 53 cm
100 cm
………
c)
15 cm and 9 cm
……….
96 cm2
d)
20 m and ……..
11 m
273 m2
e)
……………..
4 cm
40 cm2
3. Copy and complete the following table :
Diagonal
Diagonal
Area of kite
a)
72 mm
56 mm
………
b)
124 mm
85 mm
………
c)
………
96 mm
2 496 mm2
d)
18 m
……..
324 m2
e)
x
2x
…….
4. Copy and complete the following table :
Base
Perpendicular height
Area of parallelogram
a)
40 mm
35 mm
………..
b)
6,5 cm
33 mm
………..
c)
122 mm
……..
5 490 mm2
d)
………
63 mm
4 725 mm2
e)
12 mm
…….
135 mm2
f)
3x
X
………….
g)
…….
2x
4x
5. Calculate the area of the following figures (units in m) :
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6. Find the area of ΔRTV in figure (a) below (Lengths in cm).
(a)
(b)
7. Find the area of PQRS in figure (b) above. Measures in cm.
8. PQRS is a parallelogram. FR is perpendicular to SR. If the area of ΔFSR is 20 cm2, find
the area of ΔPSF.
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9. The perimeter of a parallelogram is 40 cm and the longer sides are 12 cm each. If its area is
60 cm2, find the distance between each pair of sides.
10. The two parallel sides of a trapezium are 8 cm and 17 cm long. If its height is 6 cm, find
its area.
11. The area of rectangle ABCD is 48 cm2. If the length of the diagonal AC is 12 cm, find the
breadth and the area of the rectangle.
12. A courtyard 9 m by 36 m is to be paved with cement tiles 450 mm by 450 mm. Calculate
the total cost at R113,65 per 100 tiles.
13. Calculate the areas of the following triangles. Do nr. (b) on two different ways. Lenghts are
in mm.
Exercise 5 :
1.
Calculate the area of the shaded parts in the diagrams below ;
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2.
Calculate the area and the perimeters of each figure. Round off your answer to two decimal
places where applicable.
3.
Calculate the area and the perimeters of each figure. Round off your answer to two
decimal places where applicable.
Page 158 of 268
4.
A square garden has an area of 121 m2. (a) Calculate the length of a
side. (b) Calculate what i twill cost to fence the garden, if fencing costs
R55,00 per metre.
5.
A farmer planted some grass on 2 ha of land. He sells the grass at R3,50 per m 2. Calculate
the total amount he can earn if he sells all the grass. ( 1 ha = 10 000 m2)
6.
A rectangular room is 3 m wide and 5,4 m long. Calculate the cost of carpeting the floor at
R45,50 per m2.
7. The floor area of a passage is 2,4 m2. The floor is tiled with square tiles, 200 mm by 200
mm. Calculate the number of tiles needed to tile the floor.
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8. A gravel path of width 600 mm runs around a rectangular lawn that is 18 m long and 15 m
wide. Calculate the perimeter of the outside edge of
the gravel path.
9.
A rectangle with an area of 98 cm2 is twice as long as it is wide. Find the measurements of
the rectangle.
Exercise 6:
1.
Calculate the areas of the following quadrilaterals : (Measurements in cm)
2.
The area of each figure is 64 cm2. Find the value of the unknown in each
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Sketch.
3.
Find the area of the following combined figures :
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4.
Find the area of ABCD in each of the following :
Page 162 of 268
CHAPTER 13
INTEREST AND FORMULAE
9.1 Substituting into formulae :
b 2  c 2 , b = 5 and c = 6, find a. (You may use a calculator)
Example : If a =
52  6 2
Solution: a =
25  36 =
=
61 = 7,81 (Correct to 2 decimal places)
EXERCISE 1. (Use your calculator where necessary)
1. If R =
V
with V = 8 and I = 4, find R.
I
2. If V = L x b x with L = 12, b = 14 and h = 2,5 find V.
3. a = πr 2 is the formula for the area of a circle. If r = 6, use your calculator to find the
area (to one decimal place)
4. If E =
1
2
GMm
and it is given that G = 10, M = 12,4 , m = 4,5 and r = 3, find F.
r2
5. If F =
6. If L =
mv 2 , find E given that m = 20 and v = 12.
Pr u
, find L if P = 510 , r = 8,5 and u = 2,5.
100
9.2 Changing the subject of a formula
Example: The area of a trapezium is given by the formula:
A =
1
2
(a + b)h
Find the value of a if area = 90 cm2, h = 7,5 cm and b = 14 cm
a
7,5
b = 14
Method 1: A =
1
2
(a + b)h
 2A = (a + b)h
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
2A
= a + b
h
a =
2A
- b
h
a =
2x90
- 14 = 10 cm
7,5
Method 2: A =
1
2
(a + b)h
Substituting the given values:
 2(90) = (a + 14)7,5

90 =
1
2
(a + 14)7,5
(multiplying both sides by 2)
180
= a + 14
7,5
 a
=
180
- 14 = 24 - 14
7,5
 a
=
10 cm
EXERCISE 2.
Change the subject of each of the following formulae. (Make the letter on
the right the subject of the formula)
1. A = L x b;
(b)
2. V = L.b.h
(b)
2. m = a - b
(a)
4. c = d - b
(b)
ab
2
(a)
5. v = u + at
6. m =
EXERCISE 3.
In question 1 to 7, evaluate the quantity in brackets using the values supplied in the curly
brackets in each case.
1. (t); v = u + at; (v = 120; u = 40; a = 5)
2. (v); E =
1
2
mv2; ( E = 21160; m = 20)
3. (r); A =  r2;
(A = 88) {Don’t forget the  key on your calculator; answer correct to one
decimal place}
4. (I); R =
V
; (V = 220; R = 11)
I
5. (r); S =
a
; (S = 20; a = 8)
1 r
6. (u); S =
u
(a + L); (S =98; a = 3; L = 25)
2
7. (h); A = 2  r(h + r);
(A = 110; r = 1,75) [correct to 2 decimals]
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8. Derive a formula for a from the formula: S =
n
(a + L) and hence evaluate a
2
if S = 38,5 n = 11 and L = 11,5
9.3 Simple interest:
The rule (formula) for calculating the simple interest on any principal
for any number of years (n) at r% per annum is:
S .I . 
P.r.t
100
Note: In these modern times with the high interest rates being demanded, money is seldom
borrowed or lent out at simple interest, but at compound interest with which we will deal in
the next section.
EXERCISE 4.
[All problems in this exercise are simple interest calculations with interest calculated per
annum(p.a.)]
1.
R450 is invested a 8% for 3 years. Find the interest earned.
2.
What sum of money should be invested at 8% for 3 years to earn R108
in interest?
3.
What rate of interest will earn R120 when R240 is invested for 4 years?
4.
Calculate the missing quantities (x) in the following table:
Principal
Interest
Rate
Time
a)
Rx
R450
15%
3 years
b)
R100
R140
X%
6 months
c)
R2000
R12500
12,5%
x years
d)
Rx
R50
7,5%
1 month
e)
R24000
Rx
19,5%
4,5 years
f)
Rx
R3836,25
16,5%
5,5 years
g)
R175,50
R182,52
16%
x years
5. Find the annual rate of interest if the simple interest on R1050 for 3 years and
6 months is R514,50.
6. An investor places R10 000 at 18% simple interest for 2 years. (a) What
would this amount to at the end of 2 years?
Now suppose he removed the amount after 1 year and reinvested that sum
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at 18% at another institution.
( b) What would he have in funds at the end of the second year?
c) What would have been the better investment?
9.4 Compound interest
In the simple interest calculations we have done, we calculated the amount by adding the
interest earned, over the agreed period, to the principal.
If this investment were over 5 years, we would calculate the interest for 5 years and add it to the
principal to give the amount.
If we calculated the simple interest at the end of 1 year and added it to the principal to give an
amount for the second year, to allow a second simple interest calculation, we would find that the
investment yield better results.
Interest calculated in this way is known as compound interest, (C.I.)
Compare the following two cases.
Case 1:
Determine the simple interest on R1 000 at 20% p.a. for 3 years.
S. I. =
1000x1x 20
= R600
100
What would the investment amount to after 3 years? Answer: R1 600.
Case 2:
Now consider an investment in which the interest is calculated at the end of each single year, and
re-invested.
After 1st year S.I. =
1000x1x 20
= R200; The amount at year end will be R1 200
100
Now calculate the S.I. on R1 200 for one year: S.I. =
1200x1x 20
= R240.
100
The amount for year end will be R1 200 + R240 = R1 440
The S.I for the 3rd year, will be =
1440x1x 20
= R288
100
Therefore final amount at end of 3rd year = R 1 728
Total C.I = R 1 728 –R 1000 = R728
This second case where interest is calculated by adding the interest to the principal(capital)
amount at the end of the year, so as to calculate interest for the next year, is an example of
compound interest. (C.I.)
Example 1: R400 is invested at 10% p.a. for 3 years.
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a) Calculate the amount if invested at simple interest.
b) Calculate the amount if invested at compound interest.
c) Which one will be the better investment?
Solutions:
a) S.I. =
P.n.r
100
=
400x3 x10
= R120;
100
 Amount = (400 + 120) = R520
b) Interest at the end of the first year =
Interest at the end of the third year =
400x1x10
= R40
100
484x1x10
100
After 3 years the amount = R484 + R48,40
= R48,40
= R532,40
c) C.I. will be a better investment as it will yield R12,40 more.
[R532,40 - 532,40 - R520]
Example 2: Calculate the C.I. on R300 for 3 years at 5% p.a.
(Check each step using the S.I. formula)
Principal for 1st year
:
R 300,00
Interest at 5%
:
R 15,00
Amount for 1st year or principal for 2nd year :
R315,00
Solution:
Interest at 5%
:
R 15,75
Principal for 3rd year
:
R 330,75
Interest at 5%
:
R 16,54
Amount after 3 years
:
R 347,29
Less original principal
:
R 300,00
 Interest
:
R 47,29 (To nearest cent)
EXERCISE 5
Compute the C.I. on the following:
1.
R200 at 4% p.a. for 2 years
2. R450 at 8% for 3 years
3. R64 at 5% p.a. for 3 years
4. R156 at 6% for 2 years
5. R1 024 at 7% p.a. for 3 years
6. R12 000 at 14% p.a. for 2 years
7. Find the difference between the simple interest and compound interest
on R424 at 6% p.a. for 3 years.
8. Which one is the better investment and by how much:
R 580 invested at 7% S.I. for 3 years or the same amount invested at
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6% compound interest for 3 years.
9. What is the difference in the simple and compound interest on R3 750
at 8% p.a. for 3 years?
10.
A town has a population of 2 450 and each year it increases by 5% of
what it was at the beginning of that year. What is the population at
the end of 3 years? (Give your answer correct to the nearest integer)
11. A farmer produces 1 200 bags of potatoes at the end of a certain
year. He plans to increase his production by 10% per annum from
then on. How many bags of potatoes will he be producing after 3
years? (Give your answer correct to the nearest bag)
12. Sipho wishes to purchase a new car costing R22 500. His father will lend
him all the money at S.I. at 12% p.a. for 5 years. The bank will lend him
the money at C.I. of 16% p.a. for 3 years. Which would be the better
deal?
13. Nikiwe moves into her new flat and buys R7 500 worth of furniture. The
company charges finance rates of 20% p.a. for 3 years at simple interest.
a) Calculate the interest over 3 years.
b) Determine the monthly instalments if she is to pay it all off in 36
months (3 years)
c) What would she pay back in total if the finance charges were 20%
p.a. C.I.?
14. Calculate the C.I. on R1 750 in the following cases: (Use a calculator)
a) At 12,5% for 3 years.
b) At 20,5% for 3 years
15. Calculate the amount at the end of 3 days if I invest R1000 000 at
19% p.a. S.I.
REVISION EXERCISE
1.
Given that y = 3x + 8, find y when: a) x = 7;
2.
If s = t2 - 2t, find s when:
3.
If w =
4.
A well-known formula for motion in physics is: s = ut +
a) t = 5
b) x = -7.
b) t = -5.
x 2  y 2 , find w when x = 5 and y = 12.
1
2
at2.
Find s if u = 4, t = 3 and a = 10.
5.
Solve for x in the following equations: (All letter symbols are non-zero)
Page 168 of 268
a) mx - a = 0
b) 14x - 5a = 3a + 4x
c) c - x = x - d
d) 5(x - a) = 2x - b
e) p(x - a) = 2ap
6. If A =
1
2
f) bx = abc - ac
b x h, make h the subject of the formula.
Now if v =
1
2
b x h x L, express v in terms of A and L.
7. If S = 2  rh, make r the subject of the formula.
Hence express V = 2  Rh - 2  rh in terms of R, S and h.
8. If S =
a
, make r the subject of the formula.
1 r
9. Find the simple interest (S.I.) for the following: Use your calculator.
(Give your answers correct to the nearest cent)
Principal
Rate (p.a.)
Time
a)
R4 000
12%
3 years
b)
R60 000
19%
2 12 years
c)
R1 750
15%
5 years 6 months
d)
R375
11,5%
9 months
e)
R1 500,50
12%
2 years and 3 months
f)
R65,20
8,5%
73 days
10. Calculate the compound interest for the following investments.
(Give the answers correct to the nearest cent)
Principal
Rate (p.a.)
Time
a)
R6 500
15%
3 years
b)
R25 000
21%
2 years
c)
R125 000
17%
3 years
11. Which would be the better investment over 3 years: R25 000 invested at 16%
p.a. , C.I. OR R25 000 invested at 17,5% p.a. S.I.?
12. When I opened my account at the Peoples’ Bank, the balance was exactly
R1 250. If the bank pays 15% p.a. on daily balance, how much will I have in
Page 169 of 268
my account by closing time, on the third day?
9.5 Depreciation:
Example:
A and B both receive R9 500. A invests his money at 16% simple interest. B buys
a motor car which depreciates annually by 16%. What is the difference between
their assets(R9 500) after 3 years?
Solution:
R9500x16 x3
= R4 560
100
The interest that A receives
=
Therefore A’s assets after 3 years
= R9 500 + R4 500 = R14 060
Depreciation for the first year
= R9 500 x
B’s principal for the second year
= R9 500 - R1 520 = R7 980,00
Depreciation for the second year
= R7 980 x
B’s principal for the third year
= R7 980 - R1 276,80 = R6 703,20
Depreciation for the third year
= R6 703,20 x
Assets at the end of the third year
= R6 703,20 - R1 072,51 = R5 630,69 : Person B
16
= R1 520,00
100
16
= R1 276,80
100
16
= R1 072,51
100
Difference in assets = R14 060 - R5 630,69 = R8 429,31
EXERCISE 6
1. The value of a motor car costing R12 400 diminishes by 12% ever y year. What will
its value be at the end of the third year?
2.
A and B both receive R10 800. A invests his money for 2 years at 10% compound
interest. B buys a car which depreciates at 10% yearly. What is the difference
between their assets after 2 years?
3. A and B both receive R12 500. A invests his money at 9% compound interest for 3
years and B which depreciates at 10% annually. What is the difference in the
value of their original R12 500 after 3 years?
4. A man buys a house for R85 000. He pays R40 000 cash. He borrows remaining
money for 3 years at 14% compound interest. What must he then pay to settle his
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debt?
5. If the value of a new car depreciates by 12% p.a. , what is the value of a R6 000
car after 3 years?
6. If a stroke of a pump removes 20% of the air in a cylinder, what percentage of the
air will be left after the third stroke?
7. Three brothers each inherited R250 000. A buys a flat that appreciates by 10%
per year. B buys a car that depreciates by 12% per year and C invests his
money at 9% C.I. per year. What will their assets be worth after 2 years?
CHAPTER 14
MORE FINANCIAL MATTERS
EXERCISE 1
Most people neat to take out a loan at some stage. One usually has to pay a deposit and then
the balance is paid by means of weekly, monthly or annual payments.
1. Calculate the deposit required for the following:
a) A deposit of 25% on a R2 399 TV.
b) A deposit of 20% on second hand car of R56 999.
c) A deposit of 35% on a fridge costing R3 999.
2
The following is an advert for a TV:
Total cost R2 499; Deposit 20%; Payments: R100 per month for 24 months or R185 per
month for 12 months.
a) Calculate the amount actually paid for the TV if you pay the deposit and the balance over
12 months;
b) Why would one want to consider paying the loan back over 24 months when you’ll be
paying more for the TV?
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3. A car is offered for sale in a garage with a price of R84 000 on the windscreen. John can
either pay cash or credit. If John buys the car on credit, he pays a deposit of 20% and 36
months instalment of R2 100.
a) Find the deposit he would have to pay.
b) Find the total price he has to pay if he takes the credit option.
If John pays cash he will pay the cash price, which will give him a
discount of 12,5% off the windscreen price.
c) What is the extra amount paid for credit compared with cash?
4. The same camera is available in two different shops at the same price of
R986,00 , but with credit terms. Shop A requires a deposit of 10% and
twelve monthly payments of R84,30.
Shop B requires a deposit of 15% and
ten monthly payments of R96,00. Which shop is offering the best deal?
5.
One can borrow money over a period of time. The table below shows the
monthly repayments. The amounts show how much must be paid back per
R1 000 that is borrowed:
Annual interest rate
a)
Loan period
10%
11%
12%
13%
12
87,92
88,38
88,85
89,32
24
46,14
46,61
47,07
47,54
36
32,27
32,74
33,21
33,69
48
25,36
25,85
26,33
26,83
Copy and complete the table below:
Amount borrowed
Loan Period
Interest Rate
R5 000
36 months
10%
R12 500
4 years
12%
R2 400
12 months
13%
R5 500
11%
Monthly
R256,36
Repayments
Page 172 of 268
R3 500
R6 200
b)
12%
R92,16
2 years
R288,98
Dylan buys a cell phone for R1 595. He pays a 15% deposit and takes out a loan for 12 months at
13% for the rest of the amount. How much does he have to pay back per month?
EXERCISE 2
VALUE ADDED TAX (VAT)
VAT in South Africa is currently at 14%. If you wish to work out the VAT to charge,
the simplest way is to multiply by 0,14. Why? If you wish to work out the total
amount including VAT, multiply with 1,14. Why?
1. Calculate the VAT on R1 500.
2. You do some work for someone and wish to pocket R500. How much money will you receive of
that R500?
3. You run a small business that makes paving slabs for a garden centre. Below is a list of the
dimensions of the slabs and their prices, excluding VAT.
Dimensions:
Price:
600 x 600
R20,00
300 x 600
R12,00
300 x 300
R8,00
A customer wants enough slabs to pave her patio using the design below:
Page 173 of 268
Find the cost of buying the slabs, including VAT.
EXERCISE 3
INCOME TAX
At some stage of your life you will probably have to fill in an income tax form declaring all your income.
You can calculate the tax you owe by using the table below.
Taxable Income
Rates of Tax
Where the taxable income does not exceed
R35 000
18% Percent of each R1 of the taxable income
- exceeds R35 001 but does not exceed
R6 300 plus 26% of the amount by which the
taxable income exceeds R35 000.
R45 000
- exceeds R45 001 but does not exceed R60 000
R8 900 plus 32% of the amount by the taxable
income exceeds R45 000
- exceeds R60 001 but does not exceed R70 000
R13 700 plus 37% of the amount by which the
taxable income exceeds R60 000
- exceeds R70 001 but does not exceed R200 000
R17 400 plus 40% of the amount by which the
taxable income exceeds R70 000
-exceeds R200 000
R69 400 plus 42% of the amount by which the
taxable amount exceeds R200 000
Worked example: If you earn R47 000 per annum, your tax will be R8 900 + (0,32 x 2000) = R9 540.
Calculate the tax payable if your income is:
1.
R27 000
2. R65 000
3. R130 000
4.
R400 000
Page 174 of 268
EXERCISE 4
BOND COSTS
At some stage in your life most of you will also buy your own house. Let us look at the financial
implications. A bond is a loan that you take out to cover the cost of your house. You pay interest to the
bank for the use of this money.
Here is a table to help you when you when calculating monthly repayments on a bond.
Interest %
Years
5
10
15
20
25
30
13,50
23,01
15,23
12,98
12,07
11,66
11,45
13,75
23,14
15,38
13,15
12,25
11,85
11,65
14,00
23,27
15,53
13,32
12,44
12,04
11,85
14,25
23,40
15,68
13,49
12,62
12,23
12,05
14,50
23,53
15,83
13,66
12,80
12,42
12,25
14,75
23,66
15,98
13,83
12,98
12,61
12,44
15,00
23,79
16,13
14,00
13,17
12,81
12,64
15,25
23,92
16,29
14,17
13,35
13,00
12,84
To calculate your monthly payments you do the following:
Monthly payment =
bondvalue
1000xfigureint able
Worked example: If you have a bond of R200 000, the interest rate is 14,50% and you wish to pay it off
over 20 years, find the monthly repayments.
Page 175 of 268
Monthly repayment =
R 200000
= R2 560
1000x12,80
1. If you have a bond of R400 000, the interest rate is 13,5% and you wish to pay it off over 20 years.
Find the monthly repayments.
2. If you have a bond of R450 000, the interest rate is 14% and you wish to pay it off over 25 years. Find
the monthly repayments.
3. If you have a bond of R450 000, the interest rate is 14% and you wish to pay it off over 25 years. Find
the monthly repayments.
4. If you have a bond of R1 000 000, the interest rate is 14,5% and you wish to pay it off over 20 years.
Find the monthly repayments.
a) Calculate your monthly repayments over 20 years.
b) How much have you actually paid for your R400 000 bond after those 20
years?
c) Say you fall on hard times, and decide to pay your bond back in 30 years.
What are your monthly repayments now?
d) What have you paid to the bank in those 30 years?
e) If you are highly paid and can afford to pay off your bond in 5 years, how
much do you end up paying the bank after 5 years?
f) What is your best option?
6. How much will your dream house cost? Assume you will get 100% bond, i.e. you
need to take a bond on the full purchase price. What are your monthly
repayments? What kind of job are you going to do to be able to afford your
dream house?
EXERCISE 5
FOREIGN EXCHANCE
The exchange rate at the time of writing are given below:
How many rand each unit of
currency is worth
One rand equals
0,198 dollars
Australia (Dollars $)
R5,04
0,116 Euros
Europe (Euros Є)
R8,61
0,081 Pounds
Great Britain (Pounds £)
R12,56
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0,149 Dollars
USA (Dollars)
R6,71
4,016 Rubles
Russia (Rubles)
R0,249
1. Use the exchange rates above and do the following calculations: (Answers correct
to 2 decimal places)
a) $20 = …..Pounds
c) 100 Euros = ….. Rands
2.
b) 20 Pounds = ……Rubles
d) 70 Aussie dollars = ….USA Dollars
A hat in Russia could be bought with US dollars or with Rubles. Which is the better
deal – 10 US dollars or 600 Rubles?
3.
The National Maths Olympiad costs R15 to enter. The Australian Matha
Competition costs 8 Dollars. Which is more expensive and by how much?
4.
The price on the back of a book is marked as 6,99 Pounds. It is sold in a local
bookshop for R135. It can be bought on the Internet for 6,99 Pounds plus a fee
of R50 for postage and packaging. Is it cheaper to buy the book on the Internet or
at the bookshop? (Show your working)
EXERCISE 6
Below is a table of exchange rates of the rand against some other currencies as
reported in 2005.
Foreign currency unit per rand
Rand per foreign currency
unit
Australian dollars
0,208
4,808
Botswana pula
0,714
1,401
British sterling
0,087
11,494
Canadian dollars
0,200
5,000
Euro
0,125
7,942
Hong Kong dollars
1,261
0,793
Indian rupees
6,804
0,124
Japanese yen
17,326
0,0576
Malawi kwacha
16,039
0,0632
New Zealand dollar
0,226
4,425
Page 177 of 268
Pakistan rupees
8,919
0,1122
Swiss francs
0,194
5,155
US dollar
0,155
6,435
Zambian kwacha
648,062
0,001534
Zimbabwean dollar
486,046
0,002057
Use the given table of exchange rates to answer the questions in this exercise.
1.
One barrel of brent crude oil costs $67 (US dollars)
(a) Calculate how much it would cost in South African rand.
(b) How much would it cost in Malawian kwacha.
(c) How much would it cost in Malawian kwacha.
(d) A South African petroleum company bought 1,5 million barrels of Brent
crude oil. Calculate the cost in rand terms.
2. (a) Find the current rand/dollar exchange rate and the price of Brent crude
oil from the newspaper, radio or TV, and calculate the cost in rand of
1.5
million barrels of Brent crude oil.
(b) Is oil more expensive in rand terms today, compared to 2005? The values
given in the table and question 1 are the costs in 2005.
(c) Calculate the percentage increase in Brent crude oil since 2005.
(d) Calculate the current petrol price, if petrol increased by the same percentage
as Brent crude oil since 2005. The petrol price in 2005 was R5,87 per litre.
Compare your answer to the actual current petrol price.
3. A Cape wine farmer exports 500 bottles of red wine to Britain. He is paid £6,50
per bottle. How much would he be paid in rand for the total consignment?
4. A United States business imports South African wood crafts which cost R850 000.
How much will they cost in US dollars?
5. A South African chain store imports hair dryers from Germany which cost Є12
each. They plan to sell them in their shops by marking up the cost price by 85%.
Calculate the selling price in rand.
6. CDs are imported from Europe at a cost of Є9,50 each. The mark-up on the
CDs in the South African Shops is 75%. Calculate the selling price in rand in the
Page 178 of 268
South African music stores.
7. A travel agent books a hotel in India for a customer in South Africa. The cost
is 1 900 rupees per person per night for dinner, bed and breakfast. Calculate
the cost in rand for two people for five nights.
8. Mrs. Khumalo works in Malawi as an accountant. She is paid 192 000 kwacha
per month. Her sister works in Canada and earns 4 500 Canadian dollars per
month. Determine who earns the most in rand terms.
9. Mr. Naidoo imports Swiss watches for the South African market at a cost of
R500 each.
(a) How much do the watches cost in Swiss francs?
(b) Calculate the selling price of each watch if he marks them up by 120 % ?
(c) Determine his profit in rand if he sells 50 watches.
10. Gold is trading at $435 per ounce. Determine how much revenue is received
in South African currency if 13200 ounces of gold is exported.
11. Photocopiers are imported from Europe at a purchase price of $7 500 each.
Import duty is calculated at 28 % of the purchase price and transport costs
are $550 for each of the photocopiers.
(a) Calculate the total cost in rand of importing each photocopier.
(b) Calculate the selling price in South African rand if the copiers are marked
up by 45 % on the cost of importing the photocopiers.
(c) Calculate the dealer,s profit if he sells 750 copiers.
Page 179 of 268
CHAPTER 15
STATISTICS
1.
Frequency Table
The frequency is the figure which represents the number of times a specific incident occurred.
Example:
The following are the marks (out of 10) which a group of 30 pupils obtained in a test:
5
6
4
8
5
1
2
9
5
5
2
3
3
4
8
7
9
2
3
1
5
4
6
10
9
6
7
6
5
6
Set up a frequency table and answer the following questions:
a) Which test mark appears most frequently?
b) Which test mark appears least frequently?
c) How many pupils scored 9 out of 10?
d) How many pupils scored 50% or more?
Solution:
e) Find the average of the class.
Frequency table:
Points gained
Tally
Frequency
1
11
2
2
111
3
3
111
3
4
111
3
5
1111 1
6
6
1111
5
7
11
2
8
11
2
9
111
3
10
1
1
Page 180 of 268
30
3
10
(Highest frequency) b) Least frequently:
: 1 pupil
10
10
a) Most frequently:
c)
9
: 3 pupils
10
d) 50% or more: 19 pupils e) average =
sumofscore
numberofscores
= 5,2
Mean, median and mode (Representative values)
The following were marks out of 20, obtained by 11 pupils in a geography test:
8;
9;
12;
12;
11;
16;
9;
7;
12;
12;
13
The Arithmetic mean or simply the mean is calculated as follows:
SUM OF SCORES
MEAN = ------------------------NUMBER OF SCORES
Mean =
121
= 11
11
To find the median of these scores, we first arrange the values in increasing numerical order,
repeating them where necessary:
7;
8;
9;
9; ,
11;
12;
12;
12;
12;
13;
16
The score in the middle is the median, hence here the median is 12.
[An uneven number of scores]
MEDIAN = MIDDLEMOST SCORE
[When we have an even number of scores, there is no middle figure. When this happens, we have
to take the mean of the middle two figures:
Example - 3; 4; 4; 6; 7; 8; 9; 10.
The Median =
67
= 6,5
2
The mode is the score which appears most often: Hence the mode is 12.
MODE = SCORE OCURRING MOST OFTEN
Page 181 of 268
EXERCISE 1
1.
Find the mean, median and mode of each of the following sets of data:
a) Test marks (out of 10): 3;
b) Ages (yrs): 13;
12;
5;
14;
8;
4;
15;
12;
8;
12;
5;
8;
12;
7;
6
14
c) Heights (m): 1,15; 1,10; 1,05; 1,12; 1,10; 1,11; 1,16; 1,11; 1,10; 1,08
2.
3.
Find the mean, median and mode of each of the following sets of data:
a) 107;
98;
100;
b)
23;
26;
25;
102;
25;
100;
21;
99;
23;
100;
102
25
John found the heights of the pupils in his class in centimetres, measured
correct to the nearest cm, to be:
145; 141; 140; 148; 152; 142; 150; 144; 143; 145; 144; 148; 148; 144; 142;
144; 147; 143; 150; 144; 146; 145; 146; 141; 151
a) What is the most common height?
tallest pupil?
c)
b) What is the height of the
What is the height of the shortest pupil?
d) If the pupils are arranged from tallest to shortest, with pupils of the
same height standing next to each other, what will the height of the
pupil standing in the middle, be?
e) What is the average height (to the nearest cm – use your calculator)
of pupils in the class be?
f) Give the correct names for the values found in questions a), d) and e).
The Range:
The mean, median and mode provide us with numbers which are representative of
the whole set of data, or which indicate where the centre of the set of data They
are referred to as measures of central tendancy.
It is, however, also important to have an idea of how “spread out” the data are.
The simplest way to do this is to subtract the lowest value in the se of data from the
highest. This difference is called the range.
RANGE = HIGHEST VALUE - LOWEST VALUE
Example: Calculate the range of the following set of data.
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Heights of pupils: 1,51m; 1,02m; 1,23; 1,57m; 1,04m; 1,32m; 1,15m; 1,24m; 1,31m; 1,42m.
Solution: Highest value (height) = 1,57m
Lowest value
= 1,02m
Range = Highest value - lowest value = 1,57 - 1,02 = 0,55
The range is one measure of dispersion (or ‘spread-out-ness’)
EXERCISE 2
1.
Find the (i) mean; (ii) the range for each of the following sets of data:
a)
Marks out of 10: 2; 5; 7; 8; 6; 4; 0; 1; 4; 5; 4; 2; 0; 4; 3; 4; 5; 4; 3; 10
b)
Heights in cm: 151; 149; 136; 148; 153; 162; 145; 148; 150; 144; 139; 149;
148; 150; 153
c)
Number of matches in a box:
41; 40; 39; 45; 38; 43; 42; 41; 42; 40; 39; 42; 42; 44; 43; 43; 41; 41
2.
Calculate (i) the mean;
(ii) the range of each set of data.
Data set A: 32,4; 31,9; 32,5; 32,2; 32,4; 32,0; 32,4; 32,5; 31,8; 31,9
Data set B: 29,6; 32,0; 29,7; 32,1; 28,9; 31,9; 32,9; 43,3; 30,9; 31,4
Data set C: 32,2; 33,1; 34,3; 34,3; 32,5; 35,3; 20,9; 32,6; 34,1; 32,7
[This exercise points out that even though different sets of data may have the same mean, they can differ
greatly in nature. The additional statistic , range, does give an idea of how wide the spread of data is.
The value of the range does not, however, pick up important differences in the nature of the dispersion of
the data.]
3.
A Bag contains red, blue and green marbles. A boy is allowed to put his hand
into the bag, withdraw one marble and then put it back.
R B R B B B R G B B G R B B B R B B R B
(a) Make up a frequency table to show how often each colour is withdrawn.
(b) Which colour marble do you think there are (i) most of; (ii) least of in the bag?
(c) Draw a simple bar graph to illustrate the data.
4.
The number of children per family for each of the pupils in a certain Grade 9 class
is recorded in this set of data:
2 1 3 4 2 1 1 2 3 2 2 3 2 1 2 2 3 4 2 2 3 2 5 2 3
a)
Draw up a frequency table from these raw data.
b)
What is the most common number of children per family? (What is this
number known as?)
Page 183 of 268
c)
What kind of family could not be represented in the data and why not?
d)
What number of children in the class belong to families with four or more
children?
e)
5.
What percentage of pupils are the only child in a family?
There are five alternatives to a multiple choice question labelled: A, B, C, D
and E. The choices made by pupils answering the question were as follows:
A
B
A
E
B
A
C
A
E B
A
E
A
A
A
A
B
C
E
A
A
A
E
A
C
A
A
C
a)
Make up a frequency table from these data.
b)
Which alternative do you think is the correct one?
c)
What percentage of the class got the question right?
d)
Which alternative must have been easily recognised as wrong?
6. A grade 9 class was given a speed test in working out algebraic products. The
table shows how many pupils got a given number of products correct within
10 minutes.
Number correct:
0
1
2
3
4
5
6
7
8
9
10
Frequency:
0
0
0
1
3
6
10
12
6
2
0
(a) Draw a bar graph from the data.
(b) How many pupils are there in the class?
(c) What is the modal number of correct products for the class?
(d) What is the range of marks?
(e) What is the average mark? (Try to find a quick way of getting the total score
from the frequency table.)
(f) What percentage of pupils got above the median mark?
The bar graph below shows the frequency of the last digit of telephone
numbers, chosen from a telephone book. Study the graph and then answer
the following questions:
14
12
10
CY
7.
Page 184 of 268
8
6
4
2
0
1
2
3
4
5
6
7
8
9
LAST DIGIT
(a)
How many telephone numbers were used?
(b)
What is/are the code/s shown by the graph for the last digit?
(c)
Is there any overall trend shown by the bar graph? Could you suggest
some explanation for it?
(d)
8.
Do a similar investigation using your local telephone book.
The number of goals scored by various teams in the National Football League
during a certain season were as follows:
20
25
21
20
26
31
26
20
24
26
25
29
26
28
27
20
21
27
26
25
23
24
25
23
24
26
21
(a)
Draw a bar graph from the data.
(b)
What was the (i) modal number of goals scored? (ii) the median
20
number of goals scored? (iii) the range of the number of goals scored?
(c)
9.
How many teams scored more than 25 goals during this season?
Calculate the i) arithmetic mean, ii) the median, iii) mode and iv) the
range for each of the following groups of data. Discuss how useful each of
these statistics is in each case.
3.1
Three learners are trying out opening batsman in the school’s cricket team.
Their runs scored per match during the previous term are shown below:
a)
Tshepo:
45;
60;
b)
Brian:
60;
100;
c)
Chester: 24;
9.2
44;
30;
50;
50;
32;
48;
78;
0;
28;
9;
70
54;
50 ,
62
Nine learners receive the following amounts of pocket money each weak:
50c;
R1;
R1,50 75c;
R2,00
R5,00
75c;
R1,00
9.3 Ten workers in a factory receive the following monthly salaries:
R10,00
Page 185 of 268
R450; R1 500; R450; R2 000; R450; R550; R450; R3 000; R450; R500
INTERVALS AND BOUNDARIES:
In this unit you will be able to conduct an elementary survey and understand intervals and boundaries.
Example:
Class interval
Frequency
31 - 40
41 – 50
51 - 60
61 -70
71 - 80
81 – 90
91 - 100
2
8
12
19
11
3
1
To show exactly where one class interval ends and the next one begins, we need a class boundary.
Class interval
31 - 40
41 - 50
Class boundary
40,5
Halfway between 40 and 41 = Arithmetic mean of 40 and 41
=
Class interval
31 - 40
Class boundary
30,5
41 - 50
40,5
50,5
40  41
=
2
51 - 60
40,5
61 -70
60,5
71 - 80
70,5
80,5
81 -90
90,5
91 -100
100,5
Notice that the first class boundary is added at the start of the first class interval and the last class
boundary is added at the end of the last class interval.
EXERCISE 3
1. Histograms: Draw a histogram to display the data in this table.
Class interval
Class boundary
1,5 - 1,6
1,45
1,7 - 1,8
1,65
-0,2
1,85
1,9 - 2,0
2,05
2,1 - 2,2
2,25
+0,2
2. Class boundaries:
Find the class boundaries for each of the following class intervals:
a)
20 - 24 and 25 - 29
b) 10 - 13 and 14 - 17
c)
2,4 - 2,5 and 2,6 - 2,7
d) 75 - 79 and 70 - 74
e)
2,4 - 2,6 and 2,7 - 2,9
e) 1 - 9 and 10 - 18
f)
13 - 18 and 19 - 24
3.
Copy each of the following tables into your exercise book. Calculate the class
boundaries.
Page 186 of 268
a)
Class interval
18 - 22
23 - 27
28 - 32
33 - 37
1,1 - 1,5
1,6 - 2,0
2,1 - 2,5
2,5 - 3,0
Class boundary
b)
Class interval
Class boundary
4.
The heights (in m) of all learners in a grade 9 class are listed below:
1,47
1,51
1,28
1,31
1,48
1,53
1,25
1,65
1,71
1,54
1,67
1,38
1,45
1,46
1,37
1,63
1,57
1,40
1,39
1,48
1,59
1,61
1,70
1,75
1,48
1,35
1,29
1,69
1,73
1,42
1,51
1,43
1,62
1,38
1,52
1,54
1,48
1,56
1,62
1,55
1,47
1,57
(a)
Find the range.
(b)
Group the data into five class intervals.
(c)
Find the class boundaries.
(d)
Find the frequency for each class interval.
(e) Write down the modal class.
DRAWING HISTOGRAMS:
Some Grade 9 learners devised a general knowledge test just for fun. Here are
the results of the test:
Class interval
Class boundary
Frequency
31 - 40
30,5
41 - 50
40,5
1
50,5
4
51 – 60
60,5
8
61 - 70
70, 5
9
71 - 80
80,5
81 - 90
90,5
5
91 - 100
100,5
2
1
We can show this information on a histogram. A histogram looks very similar to a bar graph, but we plot
the frequency on the vertical axis and the class boundaries on the horizontal axis.
The modal class has
the tallest bar
Page 187 of 268
10
F
9
R
8
E
7
Q
6
U
5
E
4
N
3
C
2
Y
1
This bar, for example, tells us
that five learners scored
between 70,5 and 80,5 in the
test
0
Test marks:
30,5
40,5
50,5 60,5 70,5
80,5 90,5
100,5
Evaluation:
Write two sentences to describe the differences between a histogram and a bar graph. Check your
answer with a friend. Discuss your answers.
EXERCISE 4
1.
The histogram below shows the ages of all people working in a certain factory:
F
7
r
6
q
5
u
4
e
3
n
2
c
1
y
0
Age:
10,5
20,5
30,5 40,5 50,5
60,5
70,5
Page 188 of 268
a)
How many people have ages that lie in the modal class?
b)
How old are people in the modal class?
c)
How many people are between 30,5 and 40,5 years old?
d)
How many people are older than 50 years and 6 months?
2.
Here are the masses (in kg) of all the learners in a Grade 9 class:
Mass in kg
26-30
31-35
36-40
41-45
46-50
51-55
56-60
Boys
0
1
6
15
4
3
1
Fre-
Girls
2
5
13
7
2
1
0
quency
Total
2
6
19
22
6
4
1
Draw a histogram to show each of the following:
a) the boys’ masses;
b) the girls’ masses;
c) the masses of all the children in the class.
####################################################################################
REVISION EXERCISE
1. a) In a class of 48 learners, 28 play sport. What is the fraction of sport players?
b) What is the fraction of learners who don’t play sport?
c) What number do you get if you add these two fractions?
2.
For each of the following sets of data find:
i) the arithmetic mean
ii) the median
iii) the mode
iv) the range.
a)
1,2
2,2
3,2
b)
100
99
106
1,2
2,2
103
100
99
c)
2
3
4
5
5
5
9
10
d)
4
3
2
1
1
2
3
4
e)
1
2
3 ......... 1 000
110
5
3. The learners in Grade 9 wear shirts of the following sizes:
28
27
32
31
33
29
31
28
28
31
32
31
28
30
29
28
32
28
29
28
31
28
Calculate the i) arithmetic mean
ii) median
28
29
28
30
iii) mode and the iv) range for
this data. Discuss how useful each of these statistics is.
4. A market researcher for a car manufacturing, company finds that the annual
Page 189 of 268
income per family in a certain area is as follows:
108 000
120 000
72 000
125 000
95 000
110 000
80 000
87 000
100 000
125 000
90 000
93 000
120 000
100 000
85 000
120 000
a) Calculate the arithmetic mean, median, mode and range for this data.
b) Group the data into class intervals (use 8 class intervals).
c) Find the class boundaries.
d) Draw a frequency table.
e) Draw a histogram.
f)
Find the modal class.
************************************************************************************
PIE CHARTS:
A pie chart is a circular diagram. The whole circle, 360O, represents the subject being investigated, for
example the amount of fruit sold in greengrocers. The pie is then separated into sections (slices of the
pie) that make up the whole(3600). The area of a section represents its proportion to the whole.
So, if 25% of the fruit sold in the greengrocers were apples, then 25% of the pie would be labelled
apples’.
You can misrepresent data using pie charts in a number of ways:
1. If one section of the whole is left out, then it increases the percentage value of
the other sections that are displayed.
2. If the whole is not defined, then we do not know what the parts represent.
EXAMPLE: This sector represents: ( i) As a fraction:
24
1
=
of the total
360
15
1. Given:
(ii) As %:
24º
2. Given:
1
of a class has blond hair.
5
1
x 100% = 6,67%
15
Page 190 of 268
Therefore: (i)
1
x 360º = 72º
5
72%
Blond hair
(ii)
3.
1
x 100% = 20% is blond
5
In a class with 30 pupils there are 12 with blue eyes; 9 with brown eyes; 4 with
dark brown eyes and 5 with green eyes.
a) Draw a pie chart showing this information.
b) Calculate the fraction and percentage represented by each eye colour.
Solution:
Colour of eyes
Number of pupils
Angle at centre
Blue
12
12
x 360º = 144º
30
Brown
9
9
x 360º = 108º
30
Dark brown
4
4
x 360º = 48º
30
Green
5
5
x 360º = 60º
30
Totals:
30
360º
a)
Blue
144º
Page 191 of 268
b)
Colour
Fraction
%
Blue
12
2
=
5
30
12
x 100 = 40%
30
Brown
9
3
=
10
30
9
x 100 = 30%
30
Dark brown
4
2
=
15
30
4
1
x 100 = 13 %
3
30
Green
5
1
=
6
30
5
2
x 100 = 16
3
30
EXERCISE 5
1. Elize counted the number of hours of repeat programmes that were on
television in one week.
Channel
1
2
3
4
Hours of repeats
8
3
7
6
She calculated the angle of the first segment in the pie chart:
8
x 360º = 120º.
24
Page 192 of 268
Calculate the other angles and draw the pie chart.
2. In an election at a school, candidates for School President got these votes:
Palesa: 100;
Ingrid: 10;
Daniel: 260;
Hloni: 270;
Frances: 360
Show the above information in a pie chart.
**************************************************************************
CHAPTER 16
PROBABILITY
Sometimes we want to know the likelihood of an event occurring. If we rate an event which is
impossible as 0 and an event which is certain as 1, we can place the events on a scale.
Between 0 and 1 an event can be placed anywhere on the scale with the 50% chance in the
middle.
This is called a probability scale because we say that the likelihood, or probability, of an event is 1
if it is certain. The probability of an event with 50% chance is ⅟2.
Number of successful outcomes
Probability = -------------------------------------------------Total number of possible outcomes
Example 1.
A card is drawn from a pack of playing cards (52). What is the probability that the card is:
(a) A Queen
(b) A Club
(c) A Jack of Hearts
Solution:
(a) P(Queen) =
4
1
=
52 13
(b) P(Club) =
13 1

52 4
(c) P(Jack of Hearts) =
1
52
Page 193 of 268
Example 2
There are 5 green balls, 2 red balls and 4 yellow balls. One ball is drawn from the bag. What
is the probability that the ball:
(a)Will be green (b) Will be red or yellow
Solution:
There are 11 balls in the bag. (a) P(green) =
5
11
(b) P(red or yellow) =
6
11
Tree Diagram: It is a useful diagram if two actions are carried out, one after the other.
Example 3
There are 5 yellow balls and 4 red balls in a bag. A ball is drawn from the bag, replaced and a
second ball is drawn. Determine the probability of drawing:
(a) 2 yellow balls
(b) 1 ball of each colour
Solution:
Outcome
Y
5/
Y
5/
Yellow; Yellow
9
4
/9
R
Yellow; Red
9
Y
5/
4
/9
Red; Yellow
9
R Red; Red
4
/
9
R
(a)
P(2 yellow) =
5 5 25
X 
9 9 81
(b) P (one of each colour) =
5 4 4 5 20 20 40
X  X =


9 9 9 9 81 81 81
Exercise 1
1. Estimate the probability in each of the following cases:
(a) The sun goes down at the end of the day.
(b) A newborn baby is a girl.
(c) You will throw a six on a die.
(d) You will take out a blue pen out of a pencil box containing a blue, a red
and a black pen.
Page 194 of 268
2. In a triangular one-day cricket series between Pakistan, India and South Africa,
the probability that either India or Pakistan will win is 0,6. What is the probability
that South Africa will win ?
3. A cheat tampered with a coin so that it was no longer fair. The probability of
getting heads is now twice that of getting tails. What is the probability of getting
heads ?
4. In a TV game show there are 240 people in the audience. If a person is picked at
random, the probability of a man being chosen is
5
. How many men are in the
8
audience ?
5. A card player has 14 cards in his hand including some kings. Another player
a card from the first one’s hand. If the probability of drawing a king is
1
, how
7
many kings are there in the first player’s hand ?
6. In a game at a school fête, a coin is dropped from above the centre of the board
shown. Each throw costs R 1,00. The person dropping the coin gets a small prize
and the R 1,00 back if the coin lands with its centre in the shaded region. Has the
board been well designed for the purposes of the fête? Give reasons for your
answer.
7. At another game at the fête, a metal disc is thrown from a distance onto a square
board with 1 m sides. A circle with a radius of 0,4 m is marked onto the board as
shown. The area between the circle and the edges of the board is painted. If the
disc misses the board, the throw is not counted.
(a) Calculate the probability of the disc falling with its centre in:
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(i) the circle
(ii) the painted area of the board.
(b) Players pay R 2 per throw. If they win they get a small price. On which
section of the board should a win be?
0,8 m 
1m
1m
8. A dice is thrown. Determine the probability of:
(a) Throwing a six
(b) Throwing a prime number
(c) Throwing an even number
(d) Throwing a number greater or equal to 4.
9. A ball is drawn from a box filled with balls as in the diagram below. Determine the
Probability of drawing (a) A white ball
(b) A green or a blue ball
(c) A
ball which is not red.
10. The number of matches in 10 boxes are: 48 ; 46 ; 49 ; 45 ; 44 ; 46 ; 47 ; 48 ; 45 ; 46
One box is drawn. What is the probability of drawing a box with:
(a) 49 matches
(b) 46 matches
(c) more than 47 matches
11. A bag contains 5 blue discs, 4 orange discs and 2 white discs. What is the
probability of drawing a disc that is:
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(a) Blue
(b) Either white or orange
(c) Either blue or white
(d) Not white
12. If a red and a blue dice are thrown simultaneously there are 36 possible
combinations. Complete the grid below to show the total of the two dice for
every possible throw. Now determine the probability, when throwing the 2 dice
simultaneously:
(a)
That the total is 12
(d) Of getting a double
(b) That the total is 9
(c) That the sum is at least 10
(e) That the total is more than 6
(f) Of getting a double and a total of more than 10.
13. There are 4 green balls and 3 white balls in a bag. A ball is drawn at random and
replaced. A second ball is drawn. Complete the following tree diagram.
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Now determine the probability that: (a) Both balls are green
white.
(b) Both balls are
(b) The first ball is green and the second ball is white.
14. A bag contains 4 white balls, 2 black balls and one pink ball. A ball is drawn at
random and replaced. What is the probability that:
(a) Both balls are pink
(b) Both balls are white
(b) The first ball is white, and the second ball is black.
(c) The tow balls are pink and white in any order.
15. A team consisting of two members has to be chosen. Only Ann, John, Nina, Sipho
and Thandi are available. The team is chosen by drawing 2 names simultaneously
from a hat.
(a) How many possible combinations are there? Determine the probability that:
(b) The team will consist of two girls
(c) 2 boys will be chosen
(d) john will be a member of the team
(e) The team will consist of one boy and one girl.
16. Look at question 15 again. This time the names are drawn one after the other: the
first draw for the A-member of the team and the second draw for the B-member.
Therefore(A;B). Now determine the probability of:
(a) The team consisting of two girls (b) The A-member being a boy (c) The Amember being a boy, and the B-member a girl (d) The team consisting of
one boy and one girl
(e) John being in the team
16. John is playing with the spinner shown in the diagram below. He spins it twice.
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(a) Copy and complete the tree diagram showing the probabilities for getting
white (w) and blue (b) on the respective branches.
(b) Now use the tree diagram to calculate the probability of getting: (i) ww
(ii)
bb (iii) bw.
w
w
b
w
b
b
17. David paints his dice so that it has 3 red (R) faces, 2 black (B) faces and one
white (W) face. He rolls his dice and flips a coin which randomly gives a head (H)
or a tail (T). (a) Draw a tree diagram showing the outcomes that are possible.
(b) Use this tree diagram to calculate the probability of getting: (i) RH (ii) WT
(iii) BH
18. Peter has 6 grey socks, six blue socks and four black socks in a drawer. The socks
are not made up into pairs. Just as he picks a black sock a power failure occurs.
What is the probability that Peter:
(a) will pull out a second black sock in the
dark? (b) Will end up with a pair of socks that are different in colour?
19. A set of cards is marked from 1 to 20. One of these cards is drawn at random.
What is the probability that this card is a multiple of either 3 or 5?
20. (a) What is the probability of getting two successive 6’s when a dice is thrown.
(b) What is the probability of getting a 3 and then a 5 when a dice is thrown.
Theoretical probability (Probability) versus Experimental probability (Relative frequency)
Probability =
Number of successful outcomes
-------------------------------------------------Total number of possible outcomes
Relative frequency
The number of times the outcomes did occur
= -----------------------------------------------------------The total number of trials
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Exercise:
1. A group of learners took turns to roll a dice. The outcomes of their experiment
were as follows: 3 ; 1 ; 1 ; 5 ; 6 ; 1 ; 4 ; 6 ; 4 ; 5 ; 1 ; 1 ; 3 ; 1 ; 5 ; 6 ; 5 ; 4 ; 1 ; 3
(a) Summarise the data in the following frequency table:
Outcomes of an experiment in which a dice is rolled ............... times
Number
Tally
f
Number
1
4
2
5
3
6
Tally
(b) If each member of the group had an equal number of turns, how large
could the group have been? List all possibilities.
(c) If the group had 5 members, how many turns did each member get?
(d) Calculate the theoretical probability of rolling:
(i) a three
(ii) an even number
(iii) a prime number
(e) Use your table and calculate the experimental probability of rolling:
(i) a three
(ii) an even number
(iii) a prime number
(f) Compare your answers to (d) and (e). Why do they differ?
NB. Every time you perform an experiment, you call this a trial. As the number of trials
increases, the experimental probability (relative frequency) moves closer to theoretical
probability.
f
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2. All 25 learners in a specific grade 9 class get one turn to roll the dice. The
results are summarized in the following table:
(a) What is the probability of rolling an even number.
(b) Use your table and calculate the relative frequency of an even number in
experiment above.
the
(c) What is the probability of rolling a prime number.
(d) Use your table and calculate the relative frequency of a prime number in the experiment
above.
CHAPTER 17
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CONGRUENCY AND SIMILIARITY
CONGRUENCY:
Two triangles are congruent if they are equal in every single respect (i.e. if two triangles are congruent the
three sides of the one are equal to the three sides of the other one and the three angles of the one are
equal to the three angles of the other). There are four cases where two triangles will always be congruent
to each other.
Case number one: If the three sides of one triangle are equal to the three sides of another triangle, the
two triangles will be congruent.
ΔABC Ξ ΔDEF [S ; S ; S]
Case number two: If two sides and the included angle of one triangle are equal to two sides and the
included angle of another triangle, the two triangles will be congruent.
ΔABC Ξ ΔDEF [S ;  ; S]
Case number three: If two angles and a side of one triangle are equal to two angles and the
corresponding side of the other triangle, the two triangles are congruent.
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ΔABC Ξ ΔDEF [  ;  ; S]
Case number four: If in two right-angled triangles, the hypotenuse and one other side of one triangle,
are equal to the hypotenuse and one other side of the other triangle, the two triangles will be congruent.
ΔABC Ξ ΔDEF [90º; H ; S]
EXERCISE 1
1.
State whether the following pairs of triangles are congruent or not. Give a reason for your
answer. If they are congruent, make sure you give the second triangle in proper order.
2.
In each of the following pairs of triangles two pairs of parts are marked equal. In each case write
down an additional pair of parts which must be equal to ensure congruency and state the case of
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congruency you are using. Give all possibilities.
EXERCISE 2
(i)
In each case state whether the pairs of triangles are congruent or not.
(ii)
If they are congruent, state the case of congruency and state the remaining pairs of equal
angles and pairs of equal sides.
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EXERCISE 3:
congruent.
State which condition for congruency makes the following pairs of triangles
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EXERCISE 4 :
Decide whether the following pairs of triangles are congruent or not Write the
triangles in the correct order and give the appropriate condition for congruency.
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EXERCISE 5 :
1.
State with reasons, whether or not the following pairs of triangles are
congruent.
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2.
In each case say whether the two triangles are congruent or not
necessarily congruent. Motivate your answers.
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4. In each case say whether the two triangles are definitely congruent, possibly congruent (but not
necessarily), or definitely not congruent. Give reasons for your answers.
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(a)
ΔABC and ΔPQR with A = 50º, AB = 5 m, AC = 3 m,
QR = 5 m.
R = 50º, PR = 3 m and
(b) ΔABC and ΔPQR with AB = QR, BC = PR and AC = PQ.
(c) ΔABC and ΔPQR with A  P and B  Q.
(d) ΔABC and ΔPQR with
and PR = 50 m.
B = 100º, Q = 100º, AB = 60 m, PQ = 60 m, AC = 50 m
(e) ΔABC and ΔPQR with AB = PQ, BC = QR and A  C  P  R.
(f) ΔABC and ΔPQR with B  Q , AB = PQ and AC = PR.
(g) ΔABC and ΔPQR with A  P , B  Q and AC = QR.
(h) ΔABC and ΔPQR with
A  R
and
AC = QR.
(i) ΔABC and ΔPQR with AB = PR, BC = PQ and C  Q  90 .
(j) ΔABC and ΔPQR with
4.
C  R  90 , BC = RP and AB = RQ.
State whether the two triangles are congruent and give reasons for your
statements. Name the triangles with their vertices in the correct order.
EXERCISE 6:
1.
Prove that AO = OD and BO = OC.
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2.
Prove that AB = CD and that AB║CD.
3.
Prove that PQ = PS and that QR = SR.
4.
Prove that AB = AC and B  C and BD = DC.
5.
O is the centre of the circle. Prove that AC = CB.
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6.
O is the centre of the circle. Prove that AB = CD.
7.
Prove that AD = BE.
8.
Prove that
9.
In the diagram below E1 = E 2 and
ΔFEG Ξ ΔHEG.
AC = EF
and BC = DF.
H  F. Prove that
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10.
Prove that: (a) ΔAOB Ξ ΔCOD
(b)
AB = DC
11.
Prove that AD = DB by using congruency. ( C1  C2 )
A
E
12.
(a) What type of triangle is ΔABC?
(b) Prove that ΔABE Ξ CBD.
(c) Prove that BED  BDE.
D
C
B
13.
In the diagram ΔDBE is an equilateral triangle. 1  2.
Prove that:
(a) ΔABE Ξ ΔCDB.
(b) AD = EC
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14.
Prove that (a) ΔABD Ξ ΔBAC
15.
In the diagram: AB = AC and DC = AE.
Prove that:
(a)
16.
If AB = DC
and
17.
if
ΔACD Ξ BAE
(b) ΔAOD Ξ ΔBOC
(b)
BAD  BEC
ABC  DCB , prove that AC = DB.
AB = XW, VXW  90  ABV , prove that ΔABV Ξ ΔWXV.
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18.
Use the diagram and prove that AB = AD.
EXERCISE 7:
1.
Prove that
AC = BD.
2.
Prove that
ACD  BDC .
3.
AB = AE. Prove that CE = BD.
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D
4.
Prove that BD = CE.
A
5.
AB = AE, AD = AC and
EXERCISE 8:
BAE  DAC. Prove that BC = DE.
(In this exercise you will need a construction)
1.
Prove that AD = BC.
2.
Prove that
AD = BC . ( Constr: AE  DC and
BF  DC)
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3.
Prove that AB = DC
and
4.
AC = BD. Prove that AB = DC.
5.
Prove that
B  C.
AB║DC.
(Note: There are three possible constructions which you might use
here. Do this example in three different ways.)
6.
Prove that PQ = PR. (Do this example in two different ways.
EXERCISE 9:
1.
Prove that EO = OF.
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2.
ABCD is a square. Prove:
(c)
3.
(a) ΔABQ Ξ ΔDCQ
(b)
ΔBPQ Ξ ΔQCR
ΔAPQ Ξ ΔDRQ
Prove that AD is the perpendicular bisector of BC.
A
4.
AB and CD bisect each other at E.
(a) Prove ΔACE Ξ ΔBDE.
(b) State the pairs of corresponding angles that are equal.
5.
PS bisects In ΔPQR, QPR and
PS  QR.
(a) Prove that ΔPQS Ξ ΔPRS.
(b) Calculate QR if PS = 12 cm and PQ = 13 cm.
P
1
2
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Q
6.
R
S
Given a circle with centre P and QR = SR.
(a) Prove that ΔPQR Ξ ΔPSR.
(b) State the pairs of corresponding angles that are equal.
7.
Using the Theorem of Pythagoras, find the lengths of KL and PR. Hence
Prove ΔKLM Ξ ΔPRQ using a condition of congruency other than 90º;H;S.
8.
ADB  ACB = 90º, and DA = CB.
Prove that:
(a)
(c)
ΔABC Ξ ΔBAD
(b)
DB = AC
ΔDAP Ξ ΔCBP
(d) DP.PB = CP.PA
9.
KLM  NLM = 90º and
L1  M 1 .
(a)
Prove that
ΔKML Ξ ΔNLM.
(b)
Calculate KM if LM = 30 mm and NM = 40 mm.
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10.
Prove that PQ = PR. (Hint: first prove ΔQPS Ξ ΔRPS)
P
Q
S
R
11.
(a) Is ΔABC Ξ ΔBCD?
(b) Is ΔABC │││ ΔBCD? Give reasons for
your answers
12.
AD = BC, D  C and DB  AC. Is ΔABD Ξ ΔEBC? Explain.
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13.
In the figure below, AB = AC, BD = EC and D1  E1 . Prove that
ΔABE Ξ ΔACD.
14.
O is the centre of both circles and
O1  O3 .
(a)
Why is OA = OD and OB = OC?
(b)
Prove that ΔAOB Ξ ΔDOC
(c)
Prove that AC = BD.
(d)
Prove that ΔAOC Ξ ΔDOB.
(b)
Prove ΔKON Ξ ΔMOP
EXERCISE 10:
1.
(a)
Prove ΔABC Ξ ΔDEF
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2.
Are the following pairs of triangles congruent or not. Explain your answer.
3.
(a) Prove ΔQSR Ξ ΔQTR
4.
5.
(a) Prove ΔPKR Ξ ΔTKR
(b)
Prove ΔABC Ξ ΔDCB
(b) Prove ΔEKF Ξ ΔGKH
Is this statement true? If two angles in a triangle are equal in size, the
sides opposite these angles will be equal in length. Prove your answer by
using congruency.
SIMILARITY
A. If two triangles are similar then:
(i) all pairs of corresponding angles are equal and
(ii) all pairs of corresponding sides are in the same proportion
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B. If two triangles will be similar if:
(i) The three angles of the one triangle are equal to the three angles of the
other triangle.
A
P
ΔABC ΙΙl PQR [ ; ;  ]
B
C
Q
R
(ii) The three sides of the triangles are in the same proportion.
A
2 cm
P
3 cm 4 cm
B
4 cm
6 cm
C
Q
R
8 cm
 PQ QR PR 2 
 AB  BC  AC  1 


ΔABCΙΙΙ ΔPQR
(iii) If two sides of the two triangles are in the same proportion and the included
P
angle equal.
A
9 cm
3 cm
B
50º
ΔABC ΙΙΙΔPQR
4 cm
C
Q
50º
⁰
12 cm
R
PQ QR 3

 and the included B = included Q
AB BC 1
NB: The labelling must indicate which angles correspond. We can also read off the
corresponding sides i.e. from ΔABC │││ ΔQRP we can deduce that
AB BC AC


.
If you find that two pairs of corresponding angles of two triangles are
QR RP QP
equal, then the third pair will also be equal because the sum of the angles of every triangle is
180º, and so the triangles are similar.
Similar triangles will vary in size, but they will have the same shape.
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Example:
Find the lengths of the unknown sides if ΔABC ΙΙΙ ΔPQR
Solution:
ΔABC ΙΙΙ ΔPQR

PQ PR QR


AB AC BC

60 x 80


30 25 y

And
2 80

1
y
2
x

1 25
[Given]
[ Sides in proportion ]
 x = 50 units
2y = 80
(By cross multiplication)
 y = 40 units
EXERCISE 11:
1.
Find the lengths of the unknown sides in the following pairs of similar
triangles.
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2.
(i) Which of the following pairs of triangles are similar?
(ii) If similar, write the similarity in the correct order.
3.
(i) Which of the following pairs of triangles are similar?
(ii) If similar, write the similarity in the correct order.
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4.
In ΔABC,
F = 84º.
5.
B = 59º. In ΔDEF, D = 59º and
Prove that ΔABC ΙΙΙ ΔEDF.
In ΔABC, A = 110º and B = 30º.
F = 140º.
6.
A = 37º and
In ΔDEF, E = 30º and
Prove that ΔABC is not similar to ΔDEF.
In ΔABC, AB = 4 cm; BC = 4 cm and AC = 6 cm. In ΔDEF,
DE = cm; EF = 6 cm and DF = 9 cm. Prove that ΔABC ΙΙΙ ΔEDF.
7.
(a) Prove that ΔABC ΙΙΙ ΔAED.
(b) Calculate EB and CD.
B
D
2
1
6 cm
9 cm
A
12 cm
1
14 cm
E
8.
(a) Prove that ΔBCD │││ ΔACE
C
(b) Calculate a, b and c.
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9.
(a) Prove that ΔPMN ΙΙΙ ΔLMK.
(b) Calculate d and e.
10.
(a) Prove that ΔPQR ΙΙ ΔRSQ.
(b) Calculate f.
EXERCISE 12. Find the lengths of the sides marked from c to j. All lengths are given in cm.
(a)
x
y
c
9
12
8
y
x
15
d
(b)
3
4
g
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2
h
10
(c)
6
i
j
i-1
6
4
2.
Find the lengths of the marked sides of these pairs of similar triangles:
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3.
Is ΔABC similar to ΔFDE? Explain your answer.
D
B
55º
95º
A
4.
30º
C
30º
E
F
Are ΔPQR and ΔXYZ similar? Explain your answer.
X
P
12
6
Q
R
Y
4
5.
5
3
Z
8
In each case say whether the statement is true or false. Give reasons:
(a) If in triangles ABC and PQR, A  P, B  Q and C  R,
the triangles are similar.
(b) If in triangles ABC and PQR, A  P, B  Q and C  R, the
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triangles are congruent.
(c) Two congruent triangles are also similar.
(d) Two similar triangles are also congruent.
6.
In the figure, triangles ADE and ABC are similar.  A = 30º and
A
ADE  B =45º.
(a) What is the size of  C ? Explain.
(b) What is the size of AED ? Explain.
(c) If AD = 30 mm, DB = 50 mm and
D
x
E
DE = 25 mm, calculate the length of BC.
B
7.
x
The two triangles ABC and DEF are similar.
(a) Calculate the length of DF.
(b) Find the ratio between the perimeters of the two triangles. What do
you notice?
8.
Which of the following pairs of triangles are similar. Give reasons.
9.
Each of the following pairs of triangles is similar. Calculate the value of x.
C
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CHAPTER 18
VOLUME AND SURFACE AREA
1.
Volume of prism = Area of base X height
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2.
External surface area of prism:
(a) Only the sides = Perimeter of base X height
(b) Prism without lid = Perimeter of base X height + area of base X 1
(c) Prism with lid = Perimeter of base X height + Area of base X
Volume
1
Vol = Area of base X height
H
Vol =  r2H
Total external area
Perimeter of base XH
+ Area of base X 2
= 2  rH + 2  rH
Cylinder
2
Vol = area of base X height
Perimeter of base X H +
H
L
b
= LXbXH
Rectangular prism
3
= 2(L + b) X H + 2 X L X b
Vol = Area of base X Height
= side X side X side
4
Area of base X 2
Perimeter of base X height + area of base X
height
= 4 X side X side (H) + 2Side2
Cube
= side 3
c
Vol = Area of base X Height
Perimeter of base X H + area of base X 2
= ⅟2bh X H
= (b + c + d) X H + = ⅟2bh X 2
d
b
H
Triangular prism
= 4side2 + 2side2 = 6side2
= (b + c + d) X H + bh
Exercise 1:
1.
Determine the (a) Volume and (b) total surface area of each of the
following prisms in terms of a, b, c, x, y, H etc:
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2.
Calculate (a) the volume and (b) the total external area of each of the
following diagrams:
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4 A wooden cylinder has a square hole bored out of it as shown in the diagram below.
Calculate the volume of the remaining wood.
4. (a) Calculate x and y.
(b) Which of the containers A or B has the greater volume?
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5 Determine the total area of canvass used to make the the tent below.
6 Find the volume in the trough in the diagram below.
7.
The figure below shows a coffee-can with a plastic lid. Calculate:
(a) The volume of the can.
make the can.
(b) The area of the metal needed to
(c) The area of the plastic lid.
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7 Find the total external area of the following figure:
8 Determine the area of the carpet on the stairs.
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9 Twelve condensed milk cans are packed into a box. Each can has a diameter of 4 cm and it
also has a height of 4 cm. Find the volume of the remaining space in the box.
EXERCISE 2
Calculate (a) the surface area, and (b) the volume of the following shapes. Round off your
answer to two decimal places where applicable:
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EXERCISE 3;
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1.
2.
Look at the plan for the roof of a house below:
(a)
Work out the height (h) of the supporting beam for the roof .
(b)
Calculate the area of the whole roof including the triangular gables.
(c)
Calculate the volume of the triangular prism formed by the roof.
(To 2 decimals)
A solid metal cylinder has a radius of 10 mm and a height of 300 mm.
(a) Calculate the volume metal in the cylinder.
(b) Calculate the surface area of the cylinder.
(c) If the density of the metal is 20 g/mm3, calculate the mass of the
cylinder.
3.
A cube has a volume of 1728 cm3. Calculate:
(a) The length of one side of the cube. (b) The surface area of the cube.
4. The dimensions of the rectangular base of a water tank are 53 mm and 48
mm. The height of the water in the tank is 120 mm. All this water is poured
into an empty cylinder and the water rises to a height of 120 mm in this
cylinder. Calculate the radius of the cylinder.
5.
A cool drink manufacturer decides to market his cool drink in a cylindrical
container which will hold 500 ml of cool drink and which is 120 mm high.
(a)
What will the radius of the container be.
(b)
Calculate how much material will be needed to make one such
container.
6. The figure below shows the cross-section of a gutter. The cross-section has
the shape of an isosceles trapezium and the dimensions are shown.
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(a) Let one meter of this gutter be filled with water. How many litres of water
will it hold?
(b) If the outer surface of one meter is painted, calculate the area that is
painted.
7.
39,408 litres of fuel is poured into a cylindrical container, 70 cm high and
with a radius of 14 cm. How far from the top of the cylinder does the fuel
lie?
EXERCISE 4:
1.
All dimensions not specified are in cm.
Calculate the volumes of the following prisms, or combinations of prisms:
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2.
The sketch shows a solid wooden box, with sides 20 mm thick and the
external dimensions given. Calculate:
(a) The volume of the inside of the box
(b) The volume of wood needed to make the box without a lid.
3.
The sketch shows a cylinder which fits exactly into a cubic container with
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Sides 42 cm (Inside dimensions). Calculate the volume of air left in the
container.
4.
A cylindrical piece of lead, 28 cm high and with a radius of 11 cm, is
melted down and shaped into 8 cubes Calculate the dimensions of
the cubes. Take  
EXERCISE 5:
1.
22
.
7
All measurements not specified are in centimetres.
Calculate the surface area of the solids:
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2.
A rectangular hole, 50 cm by 40 cm, is cut right through a wooden
cylinder, which is 60 cm high and has a diameter of 140 cm. Calculate
the surface area of the wooden block.
3. The heights of two right prisms are 40 cm and 16 cm. The bases are squares
with sides of 15 cm and 6 cm respectively. Determine these ratios:
(a) The heights
(b) The area of the bases
(c) The surface areas
(d) The volumes
(e) Are the prisms similar? If so, determine the scale
factor and explain what it means.
4.
Calculate the surface area and volume of these solids:
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5.
The sketch below shows two similar cylinders, P and Q. (a) Calculate the
scale factor. (b) Calculate the height of the larger cylinder. (c) Find
the ratio of the volumes. (d) Find the ratio of the surface areas.
6. You have two cubes A and B. The side length of cube A is 8 cm. The
volume of cube B is 217 cm3 more than that of cube A. Calculate the
scale factor of the two cubes.
7.
Calculate the volumes of each of the following prisms:
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8.
Calculate (a) the volume (b) the total surface area of the figures below:
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CHAPTER 19
TRANSFORMATIONS
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In grade 9 we do two types of transformations namely translations and reflections.
A. Translations
If A is the point (2 ; -1), the image of A under the translation rule (x + 2 ; y – 3) is
A’(4 ; -4).
Exercise:
1. Find the image of A(3 ; -2) and B(-2 ; -3) under the rule (x -1 ; y – 2).
2.
In the sketch below find the translation rule for ΔABC to ΔA’B’C’
3. Determine the co-ordinate rule of translation for each figure.
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4. (a) Plot the following points on graph paper: A( 1 ; 2 ); B( 2 ; -1 ) and C(-1 ; 3 )
(b) Obtain their image points under the following translations:
(i) ( x – 2 ; y + 1)
(ii) ( x + 2 ; y – 3)
5. Determine the images of A( - 3 ; 2), B( 0 ; - 3) and C( 3 ; -4) under the
following translations:
(i) (x + 2 ; y + 3)
(ii) (x – 3 ; y + 5)
(iii) (x – 4 ; y – 7)
(iv) (x + 5; y – 2)
6. Determine the co-ordinate rule of translation for each figure:
7. Given that: A(2 ; 0), B(4 ; 1) and C(3 ; 3). Determine:
(a) The image of ΔABC under the rule (x -5 ; y + 2) and mark this A’B’C’.
(b) The image of A’B’C’
(c)
under the rule (x + 2 ; y – 4) and mark this A’’B’’C’’.
Compare ΔABC with A’’B’’C’’ and determine the co-ordinate rule of
translation from the first one to the last one.
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B: REFLECTIONS
(a)
A’(- 4 ; 3) is the image of A(4 ; 3) when it is reflected over the y-axis.
A’’( 4 ; - 3) is the image of A(4 ; 3) when it is reflected over the x-axis.
(b) A’(4 ; 2) is the image of A(2 ; 4) if reflected over the line y = x and B’(-2 ; -3)
is the image of B( -3 ; -2) if reflected over the line y = x.
The general rule for reflection over the line y = x is: (x ; y)
(y ; x)
The general rule for reflection over the line y = -x is: (x ; y)
(-y ; -x)
EXERCISE 2:
1.
Determine the image points of A(2 ; 3), B( - 2 ; 3), C( - 3 ; -1) and D(4 ; - 2)
when they are reflected over:
(a) The x-axis
(b) The y-axis
(c) The line y = x
(d) The line y = -x
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(e) The line y= 3 (f) The line y = -4 (g) The line x = 3 (h) The line x = -2
2. Describe the following transformations:
3. Plot the following points on graph paper. A(1 ; 1); B(6 ; 7) and C(2 ; 6)
Then reflect the given triangle:
(a) In the x-axis
4.
(b) In the y-axis (c) In the line y = x
(d) In the line y = -x
Write the following transformations in algebraic notation:
(a) Δ2 to Δ3 (x ; y) → ( ; )
(b) Δ3 to Δ4
(x ; y) → ( ; )
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(c) Δ1 to Δ2 (x ; y) → ( ; )
5.
Draw the image according to the given rule. Identify each type of transformation
in words. (a)
(x ; y) → (x ; y-2)
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(b) (x ; y) → ( -x ; y)
(c) (x ; y) → (y ; x)
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(d) (x ; y) → (I – x ; y – 3)
(e) (x ; y) → ( -y ; -x)
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6. In the following complete the coordinates that transform Δ1 to Δ2
(a) (x ; y) →( ; )
(b) (x ; y) →( ; )
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REVISION EXERCISES
EXERCISE 1.
1. Write the following common fractions as recurring decimals:
a)
3
7
b)
2
11
2. Write the following recurring decimals as common fractions:
a) 0, 8383……….
b) 0,12323……..
3. Arrange the following real numbers in ascending order:
a) 1,3;
4.
4
;
3
2
b) -1,3; -
4
; - 2
3
Graph each of the following sets on the number line:
a) {x: 2x + 1 ≤ 0}
b) {y: -5 < y ≤ 2}
c) {z: z ≤ -1 or z ≥ 1}
d) {x: -8 < x ≤ -1; x  Z}
5.
a) x -3 X x2 X x X x-1
b) 23 x 2-1 x 2-4
c) a2b x ab2 x 3a3b
d) 2(3x4)2
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e) (2a3b)2
6.
[m, n ≠ 0]
Write the following numbers in scientific notation:
a)
0,000 000 000 042
b)
9 200 000 000 000
4,7 x104  3,8 x103
5,1x104  3,8 x103
7.
Calculate the value of:
8.
Multiply by inspection:
9.
(4m 2 n)(2mn 3 )
(2mn) 4
f)
a) (2a + 5)2
b) (2a + 5)(2a - 5)
c) (2a + b)(3a - 2b)
d) (5a - b)2
e) (2a + 5)(a - 3)
f) (4x - 3y)(4x + 3y)
Simplify:
a) (x - 2)2 + (x + 1)(x - 3)
b) 2(x - 3)(x - 1) - (x - 2)2
c) (2x - y)(x - 2y) - (x + 2y)2 - (x - 2y)(x + 2y)
10.
Calculate the value of 8x3 - 4x2y + 2xy2 + y2 when :
x =
1
and y = -1
2
11. Show that (x - y)3 = x3 - 3x2y + 3xy2 - y3, when:
a) x = 2 and y = 3
b) x = 0 and y = -
1
2
c) x = -1 and y = -2
12. Simplify: (a, x, y ≠ 0)
a)
13.
 12a 3  18a 2  36a
3a 2
b)
15x 2 y 2  5 xy
 5 xy
Factorise:
a) 25 - a2
b) x2 - 11x + 18
c) x2 + 11x + 18
d) x2 - 3x - 18
e) x2 + 7x - 18
f) 2x2 - 8y2
g) 3x - 12xy2
h) 2x3 + 6x2 - 8x
i) a2b - 3ab - 4b
j) x(x - 7) + 6
k) (x - 1)2 - (x - 1) - 12
l)
14. Simplify:
a)
9a 2 b 6 {a, b > 0}
12x 4 y 3
c)
4x3 y 4
b) (3x2y3)2
d)
( 2 x 2 y ) 2 .(3 x 3 y ) 2
25x 4 y 6
3 x  2  3 x 1
3 x  3 x2
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15. I wish to make a container in the shape of a rectangular prism. The container must hold
60 cubic units.
a) Write down all the possible dimensions of such a prism if all the dimensions are
natural numbers. (There are 10 different possibilities)
b) Which of the dimensions in a) will produce a box with the smallest surface
area?
16. a) Find the volume and the surface area of the triangular prism :
6
3
4
5
b) If the dimensions of the triangular base are doubled, what will the volume and
the surface area be?
c) If the dimensions of the triangular base are each multiplied by k, what will the
volume be?
17.
I wish to make a cylinder with a volume four times the volume
r
h
of a cylinder alongside, but with the same height, h.
What must the radius be in terms of r?
EXERCISE 2
1
a)
Copy and complete the table below. The first example has been done for
you.
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Number
N
Z
Q
Q’
R
X
4
2
-4
0
3
- 25
7
1
8
22
7
3
Π
. .
b)
(i) Convert 0, 3 5 to common fraction form.
(ii) Convert 2
1
to decimal fraction form.
7
(iii) Is it possible to convert 0,101 001 000 1 ……. to common fraction form?
Give reasons for your answer.
2. Re-write the following with positive exponents: { x ≠ 0}
(a) 3x-2
(b) (3x)2
(c)
(
2 2
)
3
(d) (3x-1)-1
3. Simplify: {a , b ≠ 0}
(a)
4a3 x 3a4
(d)
3(a 4 ) 3
3a 4 .a 3
(b)
a2b-3 x 2a-3b2
(e)
(a 2 b) 3 .( 2a 3 b 4 ) 2
a 2 b 3 .( 4a  2 b 5 ) 3
(c)
(f)
4. Write the following using interval notation:
(a)
{ x: x ε R,
0 < x < 5}
(b)
{ x: x ε R, -2 ≤ x ≤ 2}
(3a 4 ) 2
3(a 4 ) 2
2 3 .3 2 .2 .3 3
9 2 .6 3
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(c)
5.
6.
{ x: x ε R, 1 ≤ x ≤ 10}
(d)
{ x: x ε R, -4 ≤ x ≤ 0 }
Write the following using set builder notation:
(a) [ -2 ; 2 ]
(b) the set of natural numbers between 1 and 10.
(c) ( -3 ;  )
(d) the set of even numbers between 20 and 30
(e)
(f) the set of multiples of 3 greater than 12
[ -4 ; 5 )
State whether or not each each of the following expressions is a
polynomial. State the degree of each polynomial:
(a)
7.
2x2 - 3x + 1
(b) (x2 + 2)2
(a) Multiply and then simplify:
(c) x +
1
x
(d) -5
( 3x - 2 )( x2 - 2x + 6 )
(b) Subtract 6xy - 2x2 + 3y2 from the product of (x + 2y) and (5x - y).
8.
Simplify:
(a) (a + 3)(a - 2) - (a - 5)2 - 2(a - 1)(a + 1)
(b) (i)
9.
9( x  2) 2 y 4 , ( x > 2 )
8a 3 b 2
, ( a > 0, b ≠ 0 )
2ab 6
Factorise fully:
(a) 121 - m2
(b) x2 + 3x + 2
(e) 3a2b + 3b (f) 3a2b - 3b
10.
(ii)
(c) x2 - 2x - 3 (d) x4 - 81
(g) 2x2y2 - 4x2y - 8xy2 (h) x3 - 5x2 + 6x
(a) Which one of the cylinders below has the greater capacity?
(b) Which cylinder has the greater surface area? Show all your work.
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11.
In the rectangular prism alongside, BC = 9 units,
CD = 12 units, CF = 12 units and BH = 12 2 units.
Calculate the lengths of BD and DH.
12.
Simplify:
(a)
13.
2a 3b

b 2a
2a 3b
x
b 2a
(c)
2a 3b

b 2a
Simplify:
a 1 a  2

(a)
2a
4a
14.
(b)
4x 2  x 4x  1
.
(b)
4x  1 4x 2  x
5a ab 2

(c)
ab 2 5b 2
Solve:
(a) 4x - 2 - 3x + 8 = -2x - 4 + 5x
(b)
x x
 2
2 3
(c) 3 - 2(2x - 1) = 5(3 - x)
(d)
3
5x  2
( 2 x  7) 
3
5
3
(e) (x - 3)2 - (x - 2)2 = (x - 4)2 - (x - 1)2
15.
Solve the following equations. Give the answers correct to two decimal places if necessary:
(a)
2( x  3) 3( x  2)
1

1
5
10
5
(b)
(b) 3(x + 1)2 - (x - 4)2 = 2(x + 3)2 + 4(x - 3)
3 3 2
1
 

x 4 3x 1
x
2
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16.
(a) Tom, Dick and Harry have a total of R144. Tom has R12 less than Harry.
Harry has R48 more than Dick. How much do they each have?
(c)
Plane A takes off at 10:00 and plane B takes off at 10:20. When will plane B have
flown the same distance as plane A, if their speeds are 225 km/h and 285 km/h
respectively.
17.
(a) Write the ratio 2
1 3
: 3 in its simplest form.
4 5
(b) Mr Jones works for 8 hours a day, 5 days a week. What is the ratio of
the time that he works every week to the time that he does not work.
(d) If 144 : 48 = 294 : x, find x.
18.
(a)
A
120
60
x
40
z
B
96
48
8
Y
64
If the values of A and B in the table above are in proportion, find
the values of
x , y and z.
(b) (i) If two dozen eggs cost R 4,80, how much will half a dozen eggs cost?
(ii) How many eggs will you be able to buy for 80 cent?
19.
Determine which of the following pairs of triangles are congruent. If they
are congruent , state the condition which applies.
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20.
(a) Calculate the size of x in degrees.
(b) Under which condition(s) will ΔABD be similar to ΔCAD?
(c) Under which condition(s) will ΔABD be congruent to ΔCAD ?
21.
In the figure below, you are given ΔEFG and all six of its dimensions.
Which of the other triangles are congruent to ΔEFG? Give your
answer(s) in the form: ΔEFG Ξ ΔABC [ ;; S ].
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22.
Calculate the lengths of the sides of ΔXYZ marked x and y.
23.
In ΔABC, AB = AC, BX  AB. Prove, giving reasons, that:
(a) ΔYBC Ξ ΔXCB
(b) ΔOBC is an isosceles triangle.
THE END
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Exercise 3
Thick red Classroom Maths P 398
1. Give an example of:
(a) A natural
REVISION EXERCISE
1.
Oxford successful Mathematics p. 150 nr 1 and pgs ---155
2.
Kagiso Maths P 124 nr 3
3. Kagiso Maths P 124 nr 4
4. maths in our world p 74
5. maths in our world p 79
6. maths in our world p 85 +88 + 91 + 92
7. Classroom Mathematics p 326 n.1
8. . Classroom Mathematics p 330 n.2
8. . Classroom Mathematics p 345 n.1
8.
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