index pages chapter 1 natural numbers 2 - 8 9 - 12 13 - … · 1.5.1 divisibility rule for 2: the...
TRANSCRIPT
INDEX
PAGES
CHAPTER 1 NATURAL NUMBERS …………….. 2 - 8
CHAPTER 2 VARIABLES ………………………… 9 - 12
CHAPTER 3 INTEGERS ………………………….. 13 - 20
CHAPTER 4 RATIONAL NUMBERS ……………. 21 - 31
CHAPTER 5 ALGEBRAIC EXPRESSIONS ……... 32 - 35
CHAPTER 6 SOLVING EQUATIONS ……………. 36 - 39
CHAPTER 7 SCIENTIFIC NOTATION …………... 40 - 43
CHAPTER 8 RATIO AND PROPORTION ……….. 44 - 50
CHAPTER 9 LINES AND ANGLES ……………….. 51 - 73
CHAPTER 10 TRIANGLES …………………………. 74 - 88
CHAPTER 11 AREA AND PERIMETER …………… 89 - 106
CHAPTER 12 VOLUME AND SURFACE AREA ….. 107 - 112
CHAPTER 13 FINANCIAL MATHEMATICS ……… 113 - 132
CHAPTER 14 DATA HANDLING …………………… 133 - 157
CHAPTER 15 PROBABILITY ……………………….. 158 - 164
CHAPTER 16 REVISION …………………………….. 165 - 190
CHAPTER 17 TESTS AND EXAM PAPERS ………... 191 - 234
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CHAPTER 1 NATURAL NUMBERS
1.1 NATURAL NUMBERS
N = {1 ; 2 ; 3 ; 4 ; . . .}
Properties of natural numbers:
1. There is a first number. 2. The set is infinite. 3. There is always a following number. 4. The numbers can be displayed on a number line. 5. Natural numbers can be divided into two distinct groups:
• Odd numbers: 1 ; 3 ; 5 ; 7 ; . . . and
• Even numbers: 2 ; 4 ; 6 ; 8 ; . . . 6. Natural numbers are ordered, i.e. they can be arranged from smallest to largest. 7. The following are always true: if a and b are natural numbers:
• a + b is a natural number.
• a × b is a natural number. 8. a – b and a ÷ b do not necessarily give natural numbers.
1.2 WHOLE NUMBERS OR COUNTING NUMBERS
N0 = { 0 ; 1 ; 2 ; 3 ; . . . }
1.3 THE NUMBER ONE
• Multiplication: 2 × 1 = 2 100 × 1 = 100 ½ × 1 = ½
• Division: 2 ÷ 1 = 2 2 ÷ 2 = 1 1 ÷ 2 = ½
EXERCISE 1:
Simplify:
(a) 1 + 1 + 1 (b) 1 × 1 × 1 (c) 19 ÷ 19 (d)
(e) 35 ÷ 1 (f) (g) 1 ÷ 7 (h)
(i) (j)
3
1.4 THE NUMBER ZERO
• Multiplication: 1 × 0 = 0 100 × 0 = 0 ½ × 0 = 0
• Division: Zero divided by a number = 0, e.g. 0 ÷ 1 = 0 0 ÷ 100 = 0
You cannot divide a number by zero. The answer is meaningless
(undefined).
e.g. 6 ÷ 3 = 2 2 × 3 = 6
6 ÷ 0 = ? ? × 0 = 6
EXERCISE 2:
Simplify:
(a) 0 ÷ 15 (b) (c) (d) 8 + (0 8)
(e) (f) (1 + 1) ÷ 0 (g) (h)
(i) (j) (k) (l)
1.5 DIVISIBILITY RULES: What is meant by divisibility? A natural number is divisible by another natural number when the quotient is a natural number. There may be no remainder.
1.5.1 Divisibility rule for 2: The last digit must be an even number, i.e. 0 ; 2 ; 4 ; 6 or 8.
1.5.2 Divisibility rule for 5: The last digit must be 0 or 5.
1.5.3 Divisibility rule for 10: The last digit must be 0.
EXERCISE 3:
Use the divisibility rules to determine whether the following numbers are divisible by 2; 5
and/or 10:
4
(a) 230 (b) 1 354 (c) 4 305 (d) 45 657
1.5.4 Divisibility rule for 4: The number formed by the last two digits must be divisible by 4. Ex. 1: 20 148 is divisible by 4 because 48 ÷ 4 = 12. Ex. 2: 33 406 is not divisible by 4 because 06 ÷ 4 = 1 remainder 2.
1.5.5 Divisibility rule for 8: The number formed by the last three digits must be divisible by 8. Ex. 1: 45 064 is divisible by 8 because 064 ÷ 8 = 8. Ex. 2: 95 916 is not divisible by 8 because 916 ÷ 8 = 114 remainder 4.
EXERCISE 4:
Without dividing, determine whether the following numbers are divisible by 4 and/or 8:
(a) 25 736 (b) 8 622 (c) 1 863 228 (d) 343 792
1.5.6 Divisibility rule for 3: The sum of the digits in the number must be divisible by 3. Ex. 1: 68 961 is divisible by 3 because 6 + 8 + 9 + 6 + 1 = 30 and 30 ÷ 3 = 10. Ex. 2: 3 541 is not divisible by 3 because 3 + 5 + 4 + 1 = 13 and 13 ÷ 3 = 4 remainder 1.
1.5.7 Divisibility rule for 6: The number must be divisible by 2 AND 3, i.e. the last digit must be even and the digit sum must be divisible by 3.
1.5.8 Divisibility rule for 9: The sum of the digits in the number must be divisible by 9.
EXERCISE 5
Use the divisibility rules to determine whether the following numbers are divisible by 2; 3; 4; 5;
6; 8 or 10
(a) 59 235 (b) 99 516 (c) 132 696 (d) 80 010
1.5.9 Divisibility rule for 11: The difference between the sum of the 1st, 3rd, 5th, etc. digits and the 2nd, 4th, 6th, etc. digits must be 0 or divisible by 11. Ex. 1: 73 810 is divisible by 11 because: 7 + 8 + 0 = 15 3 + 1 = 4 15 – 4 = 11 Ex. 2: 503 712 is divisible by 11 because: 5 + 3 + 1 = 9 0 + 7 + 2 = 9 9 – 9 = 0
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EXERCISE 6
1. Apply the divisibility test to determine which of the following numbers are divisible by 2; 3; 4; 5; 6; 8; 9; 10 or 11:
(a) 972 (b) 1 320 (c) 1 144 (d) 3 168
(e) 461 945 (f) 20 160 (g) 24 570 (h) 29 374 812
2. Complete the following sentences. (a) A number is divisible by 12 if the umber is divisible by . . . and . . . (b) A number is divisible by 15 if the number is divisible by . . . and . . . (c) All natural numbers are divisible by . . . (d) No numbers are divisible by . . .
3. True or false? (a) All odd numbers are divisible by 3. (b) All even numbers are divisible by 2. (c) Some even numbers are divisible by 5. (d) The sum of two odd numbers is odd. (e) The sum of two even numbers id even. (f) The product of a number that is divisible by 8 and a number that is divisible by 5 is divisible by 40.
1.6 FACTORS AND MULTIPLES The factors of a number are all the numbers that divide into this number without leaving a remainder. The multiples of a number are divisible by that number, i.e. can be divided by that number without leaving a remainder.
EXERCISE 7
1. Choose the numbers that are factors of 13 420: 2; 3; 4; 5; 6; 8; 9; 10; 11.
2. Write down all the factors of the following numbers.
(a) 60 (b) 84 (c) 72
(d) 105 (e) 63 (f) 99
3. Write down the first 5 multiples of the following numbers.
(a) 7 (b) 13 (c) 15
(d) 16 (e) 19 (f) 123
4. List all multiples of 7 between 25 and 68.
5. List all odd multiples of 3 between 20 and 40.
6
1.7 PRIME NUMBERS AND COMPOSITE NUMBERS A prime number is a natural number which has only two factors. Prime numbers = {2 ; 3 ; 5 ; 7 ; 11 ; . . .} A composite or compound number is a natural number which has more than two factors. Composite numbers = {4 ; 6 ; 8 ; 9 ; 10; . . .}
EXERCISE 8
1. How many prime numbers are there between 20 and 50?
2. How many of these are even?
3. List two prime numbers that are consecutive natural numbers.
4. Are there any other pairs of consecutive prime numbers? Explain.
5. Write down three consecutive composite numbers.
6. Write each of the following as the sum of two prime numbers.
(a) 10 (b) 18 (c) 24
(d) 46 (e) 68
7. Two prime numbers, like 3 and 5, where the difference between them is 2, are called
twin primes. Find the twin primes between the following pairs of numbers:
(a) 2 and 25 (b) 30 and 50 (c) 100 and 120
1.8 PRIME FACTORS The prime factors of a number are factors that are prime numbers.
EXERCISE 9
Factorise the following numbers into prime factors, then write each number as the product of its
prime factors:
(a) 77 (b) 76 (c) 184 (d) 315 (e) 504 (f) 5 040
(g) 2 700 (h) 3 969 (i) 20 449
(j) 5 005 (k) 168 (l) 14 400
1.9 HCF AND LCM The highest common factor of two or more numbers is the highest number that can divide into these numbers without leaving a remainder. The lowest common multiple of two or more numbers is the lowest number that is divisible by these numbers.
EXERCISE 10
7
1. Find the HCF of each of the following sets of numbers by listing the factors of each
number:
(a) 6; 44 (b) 42; 56 (c) 96; 108 (d) 15; 36 (e) 22; 55; 88 (f) 12; 18; 60
2. Find the LCM of the following sets of numbers by listing a few multiples of each number:
(a) 2; 7 (b) 9; 12 (c) 5; 8
(d) 4; 9 (e) 3; 5; 6 (f) 2; 3; 5
3. Use the method of prime factors to find the HCF and LCM of each group of numbers:
(a) 4; 8 (b) 15; 20 (c) 12; 9
(d) 6; 8; 12 (e) 12; 16 (f) 9; 12; 18
(g) 18; 24 (h) 27; 33 (i) 9; 27; 54
(j) 8; 24; 32
1.10 SQUARES AND CUBES
Square numbers = { ; ; ; ; ; . . .} = {1 ; 4 ; 9 ; 16 ; 25 ; . . .}
Cube numbers = { ; ; ; ; ; . . .} = {1 ; 8 ; 27 ; 64 ; 125 ; . . .}
EXERCISE 11
1. Calculate:
(a) (b) (c)
(d) (e) (f)
(g) (h) (i) 2
2. Calculate the following and try to spot a pattern:
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
3. Calculate the following. What do you notice about the answers:
(a) (b) (c)
(d) (e) (f)
4. Calculate:
(a) (b) (c)
(d) (e) (f)
5. Fill in the missing values:
(a) . . . × . . . = 25 (b) . . . × . . . × . . . = 8 (c)
8
(d) . . . = 36 (e) (f) 2
1.11 SQUARE ROOTS
The square root of a number is the number that when multiplies by itself, produces the
given product. e.g. = 4
EXERCISE 12
1. Find the square roots by inspection:
(a) (b) (c)
(d) (e) (f)
2. Simplify without a calculator:
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) ( (k) (
3. Calculate: (No calculator may be used)
(a) ( ) (b) (4 – 3( )
(c) (d)
(e) ) × 1 (f) –
1.12 FINDING SQUARE ROOTS AND CUBE ROOTS BY FACTORISATION
EXERCISE 13
Use factorisation to calculate:
1. (a) (b) (c)
(d) (e) (f)
(g) (h) (i)
2. (a) (b) (c)
(d) (e) (f)
(g) (h) (i)
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CHAPTER 2 ALGEBRA : VARIABLES
➢ An algebraic expression is the way we write the rule which defines the relationship between two sets of numbers.
➢ The replacement set is the set of numbers represented by the variable. ➢ Algebraic expressions contain terms which are made up of factors. ➢ Terms are separated by plus and/or minus signs. ➢ Factors are numbers separated by multiplication and/or division signs. ➢ The constant is the part of an expression that does not depend on the value of the
variable for its own value. ➢ Coefficients are the factors of products that appear in algebraic expressions. ➢ Powers enable us to write repeated multiplication in a shorter way.
EXERCISE 1:
Consider a cube, complete the following tables:
1.1
Number of cubes 1 2 3 4 5 10 n Number of faces
1.2
Number of cubes 1 2 3 4 5 10 n Number of edges
1.3
Number of cubes 1 2 3 4 5 10 n Number of vertices
2. Complete the following table:
x 1 2 3 4 5 6 n 3x 30 57
10
3. Complete the following:
x 1 2 3,6 7 8,1 10 23
4x + 1
3x – 1
37 – x
5x – 5
5(x – 2)
4. How many terms are contained in the following expressions:
4.1) a + b 4.2) a × b 4.3) 4x – 5y + 3z
4.4) 2x + 3y – 6p 4.5)
4.6) 3 × 12 4.7) 4.8) 8 × x × y × z
4.9) 5x + 7(2x – 4) 4.10) 5 – (3x + 5y – 8w + 3) + 16 + a
4.11) 6(4 + 5x) – 4x + 11(x + 2) 4.12) n + n + n + n + n + 3
5. Write down the coefficient of:
5.1) x in: 15ax 5.2) a in: 16a 5.3) in:
5.4) in: 5.5) in: 51 .16 5.6) x in: 24
6. Write down the coefficient of x in the following terms:
6.1) 6.2) 6.3)
6.4) 6.5) 6.6)
7. Complete the table below:
No. Expression Variable Variable Coefficient Constant
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
8. Write down an algebraic expression for each of the following:
8.1) a number 5 more than x
8.2) a number 5 times x
8.3) a number more than x
8.4) a number 4 more than 2 times x
8.5) the square of x
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8.6) x, increased by 3 and the answer multiplied by 5
8.7) 5 times the square of x
Example:
Multiply a number by 3 and add 6 to the answer. Multiply this answer by 5.
Expression:
Flow diagram: Any number → [
EXERCISE 2
1. Draw flow diagrams for the following:
1.1) A number is multiplied by 3, after which 12 is subtracted from the answer.
1.2) A number minus 7 is multiplied by 3 and then 6 is added.
1.3) Double a number and subtract 4 from it. Now divide your answer by 3.
2. Write these flow diagrams as expressions:
2.1) →
2.2)
2.3)
3. Natalie is now years old. Her father is 3 times as old as she.
3.1) How old is her father now?
3.2) How old will she be in 4 years time?
3.3) How old will her father be in 4 years time?
3.4) What will their combined age be in years time?
4. A plant is x cm high at the beginning of summer. It grows cm during the summer. The
gardener prunes off 7cm. How high is it now?
5. A pen costs R9 and a pencil R6. How much do I have to pay for x pens and pencils?
6. In a certain city, milk is delivered for R5,95 per litre plus R1,05 per delivery. What does it
cost to have x litres delivered?
7. A snail walks cm; crawls a further cm and runs the last x cm. How far did the snail
move?
8. Of the 200 people attending a concert, x are children. How many adults are there?
9. There are x 20c pieces and 50c pieces in a piggy bank. Determine the amount of
money in the piggy bank?
10. It costs R100 to transport 20kg of luggage, plus R19 for each kilogram over 20kg
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11. Write each of the following algebraic expressions as a flow diagram:
11.1) 11.2) 11.3) + 3 11.4)
EXERCISE 3
1. Determine the value of each of the following if:
1.1) 1.2) 1.3) 1.4)
1.5) 1.6) 1.7)
2. Determine the value of each of the following if:
2.1) 2.2) 2.3) 2.4)
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CHAPTER 3 INTEGERS
3.1 INTRODUCTION
Integers are positive and negative whole numbers
= { . . . ; -3 ; -2 ; -1 ; 0 ; +1 ; +2 ; +3 ; . . .}
or
= { . . . ; -3 ; -2 ; -1 ; 0 ; 1 ; 2 ; 3 ; . . .}
Number line:
EXERCISE 1
1. Use the number line to answer the following.
Which number is:
(a) 7 greater than 3 (b) 7 smaller than 3 (c) 7 greater than -5
(d) 7 smaller than -5 (e) 8 greater than 5 (f) 8 smaller than 5
(g) 5 greater than 0 (h) 5 smaller than 0 (i) 6 greater than -2
(j) 6 smaller than -2 (k) 4 greater than -1 (l) 4 smaller than -1
(m) 4 greater than 1 (n) 4 smaller than 1
2. Draw these numbers in your exercise book. Place each number in its correct position on
the number line.
3. Use > (greater than) or < (less than) to make the following sentences true:
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(a) -6 0 (b) 0 -3 (c) 1 -4
(d) -1 -5 (e) -1 1 (f) -2 -3
(g) 8 6 (h) -70 -54 (i) -118 -234
4. Write the following numbers in ascending order:
(a) 4 ; -2 ; -18 ; 2 ; 11 ; -15 ; 6 ; 40 (b) -18 ; 33 ; -34 ; 28 ; -29 ; 16
5. Give the next three numbers in each pattern:
(a) 3 ; 2 ; 1 (b) +3 ; + 2 ; +1 (c) -3 ; -2 ; -1
(d) -1 ; -3 ; -5 (e) 1 ; 3 ; 5 (f) -3 ; 0 ; 3
(g) +3 ; 0 ; -3 (h) -5 ; -1 ; 3 (i) 1 ; -2 ; -5
3.2 NUMBER LINES (GRAPHS OF INEQUALITIES)
Example:
(a) Write down all the possible values of x if x is an integer
(b) Draw a number line to represent the inequality
1. x > -2
2. x ≤ 1
3. -2 ≤ x < 4
1. (a) x = -1 ; 0 ; 1 ; 2 ; . . .
(b)
2. (a) x = . . . ; -2 ; -1 ; 0 ; 1
(b)
3. (a) x = -2 ; -1 ; 0 ; 1 ; 2 ; 3 (b)
EXERCISE 2
1. List (write down) all the possible values of x: 1.1 -5 < x < 6 1.2 4 < x < 6 1.3 7 ≥ x ≥ -1
15
1.4 8 ≤ x ≤ 12 1.5 x < -3 1.6 -7 ≤ x ≤ -3
1.7 x > 9 1.8 -10 ≥ x ≥ -16 1.9 12 < x
1.10 x < 3 1.11 -3 < x < 3 1.12 -2 ≤ x ≤ 2
2. Draw a number line to represent each of the above inequalities.
3. Write the solution for each of the following graphs:
4. What values of will make the following statements true?
Copy the number lines into your book. Use circles to indicate your answers.
16
3.3 ADDITION
When you add integers on the number line, start with the first number. Move to the left for
negative numbers and move to the right for positive numbers.
Example
1. (-8) + (+2)
(-8) + (+2) = -6
2. (+1) + (-3)
(+1) + (-3) = -2
EXERCISE 3
1. If necessary, use a number line to help you add the following expressions:
(a) (+1) + (-1) (b) (+3) + (-5)
17
(c) (-2) + (-3) (d) (-5) + (+2)
(e) (-3) + (0) + (+3) (f) (+1) + (0) + (-4) + (-1)
(g) (-4) + (+4) + (-1) (h) (-2) + (+1) + (+1) + (-3)
2. Calculate
(a) (+6) + (+7) (b) (+13) + (+9) (c) (+45) + (+56)
(d) (+34) + (+13) (e) (-6) + (-7) (f) (-13) + (-9)
(g) (-45) + (-56) (h) (34) + (-13) (i) (-6) + (+7)
(j) (+6) + (-7) (k) (+13) + (-9) (l) (-13) + (+9)
(m) (+45) + (-56) (n) (-45) + (+56) (o) (+34) + (-13)
(p) (-34) + (+13)
3.4 ADDITIVE INVERSES
When the sum of two numbers is 0, the two numbers are called additive inverses.
e.g. (+2) + (-2) = 0 +2 and -2 are additive inverses.
EXERCISE 4
1. Find the additive inverses of:
(a) 4 (b) -6 (c) +8
(d) 13 (e) -9 (f) -25
2. What number is just as far from 0 as:
(a) -51 (b) 77 (c) 83
(d) -10 (e) -18 (f) -10
3. Calculate:
(a) 11 + (71) (b) (-5) + (+5)
(c) 9 + (-9) + (-4) (d) (-7) +5 + (+7)
4. Use additive inverses and the properties of 0 to calculate the following:
Example: 16 + (-52) = 16 + (-16) + (-36)
= 0 + (-36)
= -36
(a) -7 + 89 (b) -34 + 17 (c) 22 + (-34) (d) 60 + (-85)
(e) 18 + 6 + (-12) (f) -34 + 22 + (-15) + 17
3.5 MULTIPLICATION
Complete the following tables:
(a) 4 × 3 = (b) 3 × 4 = (c) -4 × 3 = (d) 4 × 2 = (e) 2 × 4 = (f) -4 × 2 =
(g) 4 × 1 = (h) 1 × 4 = (i) -4 × 1 =
(j) 4 × 0 = (k) 0 × 4 = (l) -4 × 0 =
(m) 4 × -1 = (n) -1 × 4 = (o) -4 × -1 =
18
(p) 4 × -2 = (q) -2 × 4 = (r) -4 × -2 =
(s) 4 × -3 = (t) -3 × 4 = (u) -4 × -3 =
Summary
1. positive number × negative number = negative number
2. negative number × positive number = negative number
3. positive number × positive number = positive number
4. negative number × negative number = positive number
EXERCISE 5
Find the product:
(a) -2 × 0 (b) 5 × -4 (c) 0 × -9 (d) (-6) × (-9) (e) -3 × 11 (f) (+3) × (-7) (g) -12 × 12 (h) -6 × 8
(i) (-1) × (-81) (j) -8 × -3 (k) (l)
(m) (n) (o) (p)
(q) (r) (s) (t)
(u) (v) (w) (x)
(y)
3.6 DIVISION
Summary
1. positive number ÷ negative number = negative number
2. negative number ÷ positive number = negative number
3. positive number ÷ positive number = positive number
4. negative number ÷ negative number = positive number
EXERCISE 6
1. Find the quotient:
(a) -32 ÷ 4 (b) -35 ÷ (-7) (c) 38 ÷ 2
(d) 42 ÷ (-6) (e) -49 ÷ 7 (f) (-14) ÷ (-7)
(g) (h) (i)
(j) (k) (l)
2. Determine the value of x:
(a) = 8 (b) = -4 (c) = 3
(d) = -2 (e) = -1 (f) = -3
3. Calculate the value of the following expressions if x = -3:
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(a) (b) (c)
(d) (e) (f)
3.7 SUBTRACTION
Example
1. (+6) – (+2) = (+6) + (-2) = 6 – 2 = 4
2. (-6) – (+2) = (-6) + (-2) = -6 – 2 = -8
3. (+6) – (-2) = (+6) + (+2) = 6+2 = 8
4. (-6) – (-2) = (-6) + (+2) = -6 + 2 = -4
EXERCISE 7
1. Simplify:
(a) 4 – (+1) (b) 5 – (+2) (c) 3 – (+4)
(d) (+7) – (+12) (e) -6 – (+5) (f) -8 – (+7)
(g) 9 – (-3) (h) 2 – (-6) (i) -3 – (-2)
(j) -9 – (-5) (k) -3 – (-8) (l) -5 – (-5)
(m) 6 – (-6) (n) -8 – (+8) (o) 7 – (+7)
2. Calculate:
(a) 65 + (-56) (b) 46 – (-10) (c) 33 – (-61)
(d) -45 – (+21) (e) 53 – (-11) (f) -38 + (-41)
(g) 22 – (-19) (h) 22 – (+19) (i) -22 – (-19)
(j) -22 + (-19) (k) 37 + (-37) (l) -18 + (-18)
(m) -25 – (-25) (n) -52 + (+52) (o) 15 – (-15)
(p) -9 – (-14) (q) -9 – (+14) (r) 9 – (-14)
(s) 9 + (-14) (t) -28 – (+9) (u) -28 – (-9)
(v) 28 – (-9) (w) 28 + (-9) (x) -18 – (-23)
3. Simplify:
(a) -5 + (-6) – (-10) (b) 13 – (+28) – (-14)
(c) 3 + (-2) – (+9) + (+7) (d) -9 – (+25) + (-11) – (-3)
(e) 15 + (-8) + (-11) – (-5) + (+5) (f) -8 + (-3) + 11 – (+63) + 54
(g) -23 – (-2) + (-17) – (-19) (h) 98 – (-33) – (-28) – (-12)
(i) -64 + (-51) + (+21) + 41 – (+55) – (-31) + 15
4. Determine the value of n:
(a) n + (-12) = -16 (b) n + (-12) = 8 (c) -12 – n = 8
(d) n – (-12) = -16 (e) n – (-12) = 16 (f) -n – 12 = -24
3.8 MIXED OPERATIONS (ORDER OF OPERATIONS)
20
EXERCISE 8
1. Calculate:
(a) (b) (c) -8[5 + (-4)]
(d) -8[5 × (-4)] (e) -8(-5 × 4) (f) -8[-5 + (-4)]
(g) -8[-5 – (-4)] (h) -8(-5 – 4) (i) -8[5 – (-4)]
2. Determine the value of each expression if x = -4; -; -2; or -1:
(a) x (b) -x (c) 5x
(d) 3(x + 4) (e) 5 × -7x (f) -9x
(g) 3x + 12 (h) -x + (-3) (i) -2x + x
3. If x = 2; -2; 3; and -3, determine the values of these terms:
(a) x2 (b) 2x3 (c) -2x2
(d) (-x)2 (e) -x2 (f) -2x3
(g) 3x5 (h) x6
3.9 WORD PROBLEMS
EXERCISE 9
1. A man receives his monthly bank statement. The following amounts appear as withdrawals: -150; -24; -36; -110.
1.1 How much money, in rand, has he withdrawn altogether? 1.2 What is the balance on his bank statement if, before the withdrawals, he had R334 in his
account?
2. The temperature is -2°C and then drops a further 9°C. What is the new temperature reading? 3. A woman pays a cheque of R1 300 into her bank account, but then issues cheques to the value of R1 500. What is the amount of the overdraft that will be shown on her bank statement? 4. Sipho borrows R30 from his mother. He then borrows another R25. How much does he owe? 5. At midnight the temperature was -2°C. At midday it was 16°C. How much did the temperature rise between midnight and midday?
21
6. At 4 pm the temperature is 12°C. In the next 12 hours it falls 18°C. What is the temperature at 4 am? 7. A building has 6 floors below ground level and 45 floors above ground level. (a) How many floors does it have? (b) Mr Pinku parks on floor -2. His office is on floor 17. How many floors must he travel from his car to his office? Is there more than one correct answer?
CHAPTER 4 RATIONAL NUMBERS (Q)
4.1 WHAT IS A RATIONAL NUMBER?
DEFINITION:
A rational number is any number that can be written as a fraction, , where a and b are
integers and b≠ 0.
EXERCISE 1
1. Write as improper fractions
(a) 0 (b) 2 (c) -3
(d) 3 (e) -2 (f) 3
2. Write as mixed numbers or integers:
(a) (b) (c)
(d) (e) (f)
3. Write as fractions:
(a) 0,6 (b) 0,33 (c) -0,125
(d) -2,5 (e) 1,06
4. Write as decimal fractions:
(a) – (b) (c)
22
(d) – (e) (f)
(g) (h) (i) –
Summary
1. Rational numbers = Natural numbers, whole numbers, integers, fractions, mixed
numbers and some decimals.
2. The decimals must be terminating or recurring.
EXERCISE 2
1. Is each of the following statements true or false? If false, correct the statement:
(a) All natural numbers are also integers.
(b) All integers are also whole numbers.
(c) All whole numbers are also rational numbers.
(d) All rational numbers are also integers.
(e) is a rational number.
(f) 0 can be written as a rational number.
(g) -4 is bigger than -1.
(h) Every counting number is a rational number.
(i) All prime numbers are rational numbers.
(j) The sum of two rational numbers is also a rational number.
(k) The quotient of two rational numbers is also a rational number (division by 0
excluded).
(l) The product of a decimal number and a mixed fraction is a rational number.
(m) Between any two rational numbers there is at least one other rational number.
2. Choose from the list below those numbers that are:
(a) integers;
(b) rational numbers;
(c) counting numbers.
-1 ; ; ; 0; . ; ; 20
3. Write the rational number in four different ways.
4. Write down two rational numbers that meet these conditions.
(a) positive, greater than 3, less than 5, not an integer.
(b) negative, not a fraction, less than -100.
5. Write down two examples of each of the following:
(a) a positive rational integer.
(b) a negative rational number that is greater than -2 and not an integer.
6. What number lies half-way between each pair of numbers on the number line?
23
(a) 17; 16 (b) 25; 25,5 (c) -1 ; -2
(d) ; (e) -1 ; 3 (f) 1,05; 1,005
(g) 3; 5 (h) 15; 37 (i) ;
(j) 1 ; 2 (k) -3; -4 (l) 3,5; 3,55
7. Choose from the list all numbers that are not rational numbers.
; ; × ; 3 ; ; ; +
4.2 LIKE AND UNLIKE FRACTIONS
Like or similar fractions have the same denominator:
; - ; ;
Unlike fractions have different denominators:
; ; -
EXERCISE 3
1. Convert each of the following groups of unlike fractions to like fractions:
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
4.3 COMPARING FRACTIONS
We convert unlike fractions to like fractions when comparing fractions.
EXERCISE 4
1. Use the correct sign (<, > or =) instead of each box:
(a) (b) (c) - 1
(d) - (e) (f)
(g) (h) (i)
2. Write each of the following sets of fractions from largest to smallest:
(a) (b)
24
3. A javelin thrower is practising on the sports field. If her javelin travels and of the
field on each try, which of her throws covers the greatest distance?
4. Draw a line AB exactly 10 cm long.
(a) Measure the following lengths and mark the points on the line:
(i) AP is of AB (ii) AQ is of AB
(iii) AR is of AB (iv) AS is of AB
(v) AT is of AB (vi) AU is of AB
(b) Arrange these fractions from smallest to biggest:
4.4 FRACTIONS AND THE NUMBER LINE
EXERCISE 5
1. Place each of the following groups of fractions on a number line:
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
2. Draw number lines in your exercise book. Place the given numbers in their correct
positions on these number lines:
(a) (b) – ; (c) (d)
3. Draw a number line, indicating 0, 1 and 2. Show these numbers as points on the number
line:
4.5 EQUIVALENT FRACTIONS
EXERCISE 6
1. What number must you place in each box to make equivalent fractions?
(a) (b)
(c) (d)
2. Fill in the missing numbers to make equivalent fractions:
(a) (b) (c)
(d) (e) (f)
25
3. Use the following information to copy and complete the equivalent fractions below:
(a) (b) (c)
(d) (e) (f)
(g) (h)
4.6 ADDITION AND SUBTRACTION
4.6.1 FRACTIONS WITH THE SAME DENOMINATORS (LIKE FRACTIONS)
Examples:
1.
2.
= =
= =
OR =
= =
=
=
3.
=
= OR 4
= =
= =
= =
=
26
4.
= =
= =
= =
= OR =
=
=
=
EXERCISE 7
1. Calculate without using your calculator:
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
4.6.2 FRACTIONS WITH DIFFERENT DENOMINATORS (UNLIKE FRACTIONS)
Examples:
1.
= (LCM of 12 and 8 is 24)
=
=
2.
= =
= =
= =
= =
=
=
27
3.
= =
= =
= =
= =
= =
=
= =
4.
= =
= =
= =
= =
= =
= =
= =
=
EXERCISE 8
1. Work out the examples. Use your calculator only to test your answers:
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
(m) (n) (o)
28
(p) (q) (r)
4.7 MULTIPLICATION AND DIVISION
4.7.1 MULTIPLICATION
Example:
=
=
=
=
EXERCISE 9
Solve without using a calculator:
(a) (b) (c)
(d) (e) (f)
4.7.2 DIVISION
Example:
=
=
=
=
EXERCISE 10
Solve without using a calculator:
(a) 3 ÷ (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k) (l)
29
(m) (n)
4.8 SQUARE ROOTS AND CUBE ROOTS OF FRACTIONS AND DECIMALS
Examples
1.
2.
3.
EXERCICSE 11
(a) (b) (c)
(d) (e) (f)
(g) (h)
4.9 MIXED CALCULATIONS (ORDER OF OPERATIONS)
EXERCISE 12
Complete this exercise without using your calculator. Show all steps.
(a) (3 (b) (c) 1,5 ÷ (2 - )
(d) (e) (f)
(g) 8( (h)
30
(i) Calculate x if x = 10 – and b =
(j) Uncle Bob has 24 planks with which to build cupboards for his garage. He requires
planks to build one cupboard. How many cupboards can he make?
(k) The Mathematics teacher has corrected 3 question papers in the last hour. If she
has another 3 hours in which to work, how many more question papers will she be
able to mark? (l) Sam undertook to paint 5 rooms of a house. During the first 2 days he completed
2 rooms and earned R56.
(a) How many rooms does he still have to paint?
(b) How much money will he receive for the remainder of the work?
4.10 ADDING AND SUBTRACTING DECIMALS
EXERCISE 13
1. Estimate each answer to the nearest unit. Then add without using your calculator:
(a) 1,62 + 0,53 + 3,8 (b) 0,86 + 2,3 + 0,96
(c) 4,6 + 0,21 – 5,62 (d) 0,42 + 0,01 + 0,14
(e) 0,33 – 0,407 + 7 (f) -13 + 1,59 + 7,2
(g) -3,14 – 0,1 + 10 (h) -150 + 1,50 – 15,0
4.11 MULTIPLYING DECIMALS
• When multiplying decimals by powers of 10, we move the comma to the right. We
move the comma the same number of places as there are zeros in the power of 10. If
we don’t have enough digits, we add zeros.
Examples:
1. 5,643 × 100 = 564,3
2. 0,5 × 100 = 50
• When multiplying one decimal number by another decimal number, we find the number
of decimal places in the answer by adding together the number of decimal places in the
numbers you are multiplying.
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Example:
-1,25 × 4,3
-125
× 43
-375
-500 -1,25 × 4,3 = -5,375
-5,375
EXERCISE 14
1. Calculate without using a calculator:
(a) 3,1 × 0,1 (b) (-3,6)(-0,02) (c) (-22,6)(0,2)
(d) 0,04 × 0,4 (e) 0,03 × 0,02 (f) (-1,1)(0,2)(-0,2)
(g) 2,2 × 2 × 0,2 (h) 2,5a × 0,2 (i) (7,8x2)(0,03x3)
(j) (-17,9)(9,7) (k) (0,4)(0,01) (l) 4(0,01)
(m) 4(0,1) (n) (1,2)(-0,2)(0,2)(-0,1) (o) (-2,3x2)(-0,7)
2. Write down the answers without using a calculator:
(a) 3 × 10 (b) 3 × 100 (c) 3 × 1000
(d) 0,3 × 10 (e) 0,3 × 100 (f) 0,3 × 1 000
(g) 5,7 × 1 000 (h) 0,1 × 10 × 10 (i) 0,1 × 0,1
(j) 1,5 × 0,1 × 10 (k) 3,28 × 0,01 × 100 (l) 0,6 × 0,001 × 1 000
(m) 1,1 × 1 000 × 0,001 (n) 0,2 × 100 × 0,1 (o) 5 × 0,1 × 100
(p) 0,1 × 100 (q) 1 000 × 0,01 (r) 0,3 × 10 000
4.11 DIVIDING DECIMALS
• When dividing decimals by powers of 10, we move the comma to the left. We move the
comma the same number of places as there are zeros in the power of 10. If we don’t
have enough digits, we add zeros.
Examples:
1. 564 ÷ 100 = 5,64
2. 5 ÷ 100 = 0,05
• Dividing two decimal numbers:
0,5 ÷ 0,04
Method 1: Method 2:
0,5 ÷ 0,04 = 0,5 ÷ 0,04 =
= =
32
= =
= 12,5 = 12,5
Exercise 15
1. Calculate without using your calculator:
(a) 8,4 ÷ 2 (b) 6,9 ÷ 4 (c) 4,17 ÷ 10
(d) 0,03 ÷ 100 (e) 7,48 ÷ 20 (f) 94,5 ÷ 30
(g) 0,073 ÷ 0,5 (h) 0,875 ÷ 0,25 (i) 707 ÷ 0,07
2. Divide 7,4 by:
(a) 3 (b) 0,3 (c) 30
(d) 300 (e) 0,003 (f) 0,2
(g) 20 (h) 200 (i) 0,002
CHAPTER 5 ALGEBRAIC EXPRESSIONS
➢ Only like terms maybe added or subtracted from each other.
➢ Like terms are terms in which the variables are the same.
EXERCISE 1
1. Rewrite these expressions so that they are as short as possible:
1.1 1.2
1.3 1.4
1.5 1.6
1.7 1.8 . . . 26 terms
2. Simplify where possible: 2.1 2.2
2.3 2.4
2.5 2.6
2.7 2.8
2.9
➢ x5 = x. x. x. x. x
➢ x is to the fifth power
33
➢ x is called the base
➢ 5 is called the exponent
➢
➢ When two powers with the same base are multiplied, the answer can be written as a
power and it’s only necessary to add the exponents: xn × xm = xn+m
EXERCISE 2 1. Look at the example and complete the table:
Number Exponent Base Power
x
2. Write in short form: 2.1 2.2
2.3 ... ... 26 factors 2.4
2.5 2.6
2.7 2.8
3. Determine the value of: 3.1 3.2 3.3 3.4
3.5 3.6 3.7 3.8
4. Write as powers of 2: 4.1 8 4.2 64 4.3 4 4.4 16 4.5 128 5. Write as powers of 3: 5.1 9 5.2 27 5.3 243 5.4 81 5.5 3 6. Calculate the square root of each of the following: 6.1 4 6.2 25 6.3 36 6.4 9 6.5 49 6.6 1 6.7 81 6.8 64 6.9 121 EXERCISE 3 1. Complete the following: 1.1 1.2 1.3 1.4
1.5 1.6 1.7
2. Simplify: 2.1 2.2 2.3 2.4
34
2.5 2.6 2.7
3. Simplify: 3.1 3.2
3.3 3.4
3.5 3.6
3.7
4. Add the following expressions together: 4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
➢
➢
➢
➢
EXERCISE 4 1. Simplify by applying the above shown distributive property and adding where possible: 1.1 1.2
1.3 1.4
1.5 1.6
1.7 1.8
1.9 1.10
1.11 1.12
1.13 1.14
1.15
2. Place brackets in suitable places with the help of the distributive property: Example:
35
2.1 2.2 2.3
2.4 2.5 2.6
2.7 2.8 2.9
2.10 +
➢
➢
➢
➢ x =
EXERCISE 5 1. Simplify the following fractions:
1.1 1.2 1.3 1.4
1.5 1.6 1.7 1.8
1.9 1.10 1.11 1.12
2. Simplify
2.1 2.2 2.3
2.4 2.5 2.6
3. Simplify
3.1 3.2 3.3
3.4 3.5 3.6
3.7
4. A figure has a length of and a breadth of .
4.1 Find a formula for the perimeter of the figure. 4.2 Find a formula for the area of the figure. 4.3 If the area is square centimetres and the length is cm, give the breadth in
36
terms of .
4.4 Give the length in terms of and if the area is and the breadth is
5. Determine the value of for these values of :
5.1 = 2 5.2 = 3 5.3 = 1
6. Name the exponent of
6.1 6.2 6.3 6.4
CHAPTER 6 SOLVING EQUATIONS
➢ The value of the variable that makes the equation true is called the solution of the
equation.
➢ When we solve equations, we do addition and subtraction before multiplication and
division.
➢ Equations of the form , are called linear equations. The letter is the
coefficient of the variable , the letter is the constant of the equation.
➢
EXERCISE 1
1. Find the value of the variable (unknown) by dividing the coefficient of the variable in the
constant:
1.1 1.2 1.3
1.4 1.5 1.6
2. Use the given values of and substitute them in each of the expressions:
37
2.1 2.2 2.3
2.4 2.5 2.6
3. Solve for the unknown:
3.1 3.2 3.3
3.4 3.5 3.6
3.7 3.8 3.9
3.10 3.11 3.12
3.13 3.14 3.15
4. Solve for the unknown:
4.1 4.2 4.3
4.4 4.5 4.6
4.7 4.8 4.9
4.10 4.11 4.12
4.13 4.14 4.15
4.16 4.17 4.18
4.19 4.20 4.21
4.22 4.23 4.24
4.25 4.26 4.27
4.28 4.29 4.30
EXERCISE 2
1. Solve for , simplify it by removing the brackets:
1.1 1.2
1.3 1.4
1.5 1.6
1.7 1.8
1.9 1.10
2. Solve and test your answer:
2.1 2.2 2.3
2.4 2.5 2.6
2.7 2.8 2.9
2.10 2.11 2.12
2.13 2.14 2.15
3. Write an equation for each problem and solve it:
38
3.1 Think of a number, multiply it by 3 and add 14 to the product. The answer is 287.
What was the original number?
3.2 Find three consecutive numbers whose sum is equal to 24.
3.3 Find four consecutive numbers whose sum is 126.
3.4 There are four children in a family and there us a difference in age of 4 years
between each child. The sum of their ages is 36. How old is each child?
3.5 Six less than three times a number is 24 more than the number. Find the number.
3.6 A number is added to one quarter of a number. The answer is 15. What is the
number?
4. The Sithole family is going to watch soccer at a local stadium. Ticket prices for entry are
shown below:
Adults Students Children under 6 years
R17,00 R10,00 R3,00
Mrs. Sithole has to buy two adult tickets, three student tickets and one child’s ticket.
Write a number sentence to find the cost of the tickets. Then find the value of
5. A rectangle has a length cm and a breadth cm. The perimeter is 34cm.
5.1 Find the length and breadth
5.2 Calculate the area.
6. A rectangle has a length and a breadth The area is
Calculate the perimeter.
EXERCISE 3
1. Solve for :
1.1 1.2 1.3
1.4 1.5 1.6
2. Solve for
2.1 2.2
2.3 2.4
2.5 2.6
3. Solve for the unknown:
3.1 3.2
39
3.3 3.4
3.5
4. Complete the following table:
EXERCISE 4
1. It cost R17,00/hour to hire a bicycle at a coastal resort.
1.1 Draw up a table which will show the charge for the following hours:
Hours 1 2 3 4 5 6
Charge
1.2 Write a formula for calculating the cost in rand if the bicycle is hired for hours.
1.3 What will the cost to hire the bicycle for 13 hours?
1.4 How long did a man have the bicycle if he had to pay R153,00?
2. A man is 8 times older than his son. In 5 year’s time he will be 4 times older than his son.
How old is his son now?
3. Siphiwe scored goals more than Michael scored this season. They scored a total of 24
goals. How many goals did each boy score?
4. The length of a rectangle is longer than the breadth. If the perimeter of the
rectangle is , calculate the breadth.
5. Solve, using flow diagrams:
5.1 5.2
5.3 5.4
6. Solve for the unknown:
6.1 6.2
6.3 6.4
6.5 6.6
7. A number is multiplied by 3 and 13 is added to make the answer 43. What was the
original number?
40
8. There are 261 people at a fete. There are 16 more children than there are men and 7
more men than women. How many men are there?
9. A man walked a certain distance and rode 5 times as far. The total distance that he
travelled was How far did he walk?
10. In a class of 29 children, there are 7 more boys than there are girls. How many boys
and how many girls are there in the class?
11. The sum of three consecutive even numbers is 78. What are the numbers?
12. The sum of three consecutive odd numbers is 123. What are the numbers?
CHAPTER 7 MORE ABOUT NUMBERS
7.1 LARGE NUMBERS AND SCIENTIFIC NOTATION
NORMAL NOTATION SCIENTIFIC
NOTATION
BRITAIN/SA USA/FRANCE
1 000 000
1 000 000 000
1 000 000 000 000
1 000 000 000 000 000 000
1 ×
1 ×
1 ×
1 ×
1 million
1 milliard
1 billion
1 trillion
1 million
1 billion
1 trillion
SCIENTIFIC NOTATION (EXPONENTIAL NOTATION)
A universal way of writing very large or very small numbers.
The numbers are written in the form of a × where 1 ≤ a < 10 and n ϵ z
Examples
1. Write the following numbers in scientific notation:
(a) 315 000 000 = 3,15 ×
41
(b) 100 = 1 ×
(c) 550 000 = 5,5 ×
2. Write in normal notation (standard notation):
(a) 2 × = 20 000
(b) 1,32 × = 132
(c) 4,05 × = 405 000
3. Write the following numbers in words:
(a) 1050 000 000 000: one billion and fifty milliard
(b) 56 000 700 000: fifty-six milliard seven hundred thousand
(c) 1 071 000 000 000: one billion and seventy-one milliard
EXERCISE 1
1. Write the following numbers in scientific notation:
(a) 1 813 (b) 412 000 (c) 387 000 000
(d) 6 000 (e) 22 410 (f) 2 831,2
(g) 13 000 000 (h) 271,7 (i) 21 123
(j) 88 800 000 (k) 9 000 000 000 000 (l) 1 357 900
2. Write the following in numbers:
(a) one million (b) two milliard
(c) five billion (d) seven trillion
(e) 2½ million (f) 1,45 billion
3. Write in words:
(a) 2 000 000 (b) 130 000 000 (c) 580 000 000 000
(d) 2 500 000 000 (e) 5 000 000 000 000 000 (f) 1 000 000 000 000 000 000
4. Write in normal notation:
(a) 1,981 × (b) 9,01 × (c) 5,53 ×
(d) 1,075 × (e) 2,54 × (f) 9,8 ×
(g) 1,01 × (h) 3,14 ×
5. Write these numbers in scientific notation:
(a) 36 milliard
(b) Dinosaurs lived 120 000 000 years ago.
(c) In some plantations in the Eastern Transvaal there are 150 000 trees.
(d) 82 550 spectators cheered the Liverpool soccer team to victory.
(e) Swallows migrate about 11 000 kilometres every year.
(f) An oil tank contains 24 000 litres of oil.
(g) The Atlantic Ocean reaches a depth of 9 200 metres.
(h) There were 75 450 spectators at the Currie Cup rugby final.
(i) An oil well produces 400 000 barrels of oil per day.
42
7.2 RATIONAL AND IRRATIONAL NUMBERS
1. Decimal numbers may be divided into:
(a) Finite decimal numbers, eg = 0,125
(b) Recurring decimal numbers, eg
(c) Non-recurring infinite decimal numbers, eg 0,010000100001 . . .
2. Finite and recurring decimal numbers can be written as common fractions, they are
rational numbers.
3. Non-recurring infinite decimal numbers cannot be written as common fractions they
are irrational numbers
eg ; ; ; 3 + ; 5 × ;
EXERCISE 2
1. Rewrite these fractions as decimal numbers:
(a) (b) 1 (c)
(d) (e) (f)
2. Rewrite the following fractions as decimal fractions. What do you notice?
(a) ; ; ; (b) ; ; ; (c) ; ; ;
3. State in each case whether the number is a rational number:
(a) 2,3232323232. . . (b) 4,1 × 4 (c) –
(d) (e) 5 × (f) 2,01001000100001 . . .
4. Classify each number as a natural number, an integer, a rational number or an irrational
number:
EXAMPLE
-4: integer, rational number
: irrational number
(a) 3 (b) -9 (c) 0,7
(d) (e) – 16 (f) -
(g) 0,5 (h) (i) -5
(j) + 9
5. Calculate the following and round off to two decimal places where necessary:
43
(a) (b) (c) –
(d) 2 + (e) 3 × (f) ×
(g) × (h) (i) + 25
(j) ×
6. Solve for x and say whether the answer is a rational or an irrational number:
(a) 2x + = 0 (b) + = (c) -x = 1 + ( +
(d) (e) 3(x + ) = 18 (f) x = ( ( )
(g) 2x = (h) (i) 2x + =
7.3 ROUNDING OFF
Examples:
1. 0,0038439 rounded off to 3 significant figures is 0,00384
2. 80763 rounded off to 3 significant figures is 80800
3. 3165,507 rounded off to 4 significant figures is 3166
EXERCISE 3
1. Round off to the nearest metre:
(a) 3,6 m (b) 23,49 m (c) 123,501 m
(d) 200,901 m (e) 2,07639 km (f) 1,000168 km
2. Round off to the nearest hundredth of a second:
(a) 11,024 s (b) 0,08499 s (c) 2,00389 s (d) 19,995 s
3. Round off to the number of places after the comma indicated in brackets:
(a) 12,032 (2) (b) 59,085 (1) (c) 0,0048 (3) (d) 5,0034 (2)
4. Round off to the number of significant figures indicated in brackets:
(a) 9 021 (2) (b) 2,547 million (2) (c) 7,458 (3)
(d) 4,5498 × (3) (e) 923 981 (3) (f) 0,0032514 (2)
5. To how many significant figures are these numbers given?
(a) 23 000 mm (b) R13 020 (c) 0,00218
(d) 4 080 kg (e) 2,43 × km
44
CHAPTER 8 RATIO, RATE AND PROPORTION
8.1 COMPARING QUANTITIES
EXERCISE 1
1. Consider the following mixtures: (a)
Mixture 1 2 3 4 5 6 7 8
No. of spoons of sugar 6 4 4 3 6 5 1 3
No. of ml of water 200 100 200 100 300 200 20 50
No. of spoons of sugar for every 100 ml water
(b) Arrange the mixtures in order from most sweet to least sweet. 2. In a soccer tournament Kaizer Chiefs scored 14 goals in 7 matches, Orlando Pirates 15 goals in 5 matches, Ajax Cape Town scored 5 goals in 5 matches and Supersport United scored 8 goals in 2 matches.
45
(a) Which team scored the most goals?
(b) Which team scored the highest number of goals on average per match?
(c) By how many goals did Chiefs outscore Ajax Cape Town?
(d) How many times greater is the number of goals scored by Pirates than those
scored by Ajax Cape Town?
3. On a building site there are twice as many bags of cement as there are bags of sand. If there are 16 bags of cement, how many bags of sand are there?
8.2 RATIO
• A ratio is the comparison of two quantities.
• A ratio can be written as a fraction: a : b =
Example: Consider a grade 8 class that consists of 15 girls and 12 boys: Write down:
(a) the ratio of the number of boys to girls.
(b) the ratio of the number of girls to boys.
(c) the ratio of the number of boys to pupils.
(d) the fraction of pupils who are boys.
(e) the ratio of the number of girls to pupils.
(f) the fraction of pupils who are girls.
Answers:
(a) boys : girls = 12 : 15 = 4 : 5
(b) girls : boys = 15 : 12 = 5 : 4
(c) boys : pupils = 12 : 27 = 4 : 9
(d) fraction of boys = =
(e) girls : pupils = 15 : 27 = 5 : 9
(f) fraction of girls = =
EXERCISE 2
1.
46
2
Look at the pictures above. Write down the ratios below: (a) the ratio of buckets of water to bags of sand (b) the ratio of bags of sand to buckets of water (c) the ratio of buckets of water to the total number of items (d) the fraction of bags of sand of the total number of items (e) the fraction of buckets of water of the total 2. Study the two squares, then copy them and write the ratios below: 2 2 3 3 (a) the side of the small square to that of the larger square (b) the perimeter of the small square to that of the larger square (c) the area of the small square to that of the larger square 3. Write the following in their simplest form. Remember to make the units the same where necessary:
• Five litres of water is added to thirty litres of pure apple juice.
(a) What is the ratio of water to apple juice
(b) What fraction of the mixture is water?
(c) What fraction of the mixture is apple juice?
8.3 EQUAL RATIOS
• Before simplifying a ratio, the quantities being compared must have the same unit.
• The simplified ratio does not have a unit.
EXERCISE 3
1. Write each ratio in its simplest form:
(a) 8 : 12 (b) 24 : 12 (c) 33 : 55
(d) 16 : 4 (e) 5 : 10
2. Give each ratio in its simplest form:
(a) 80 minutes to 2 hours (b) 10 l to 25 000 ml
47
(c) 4 kg to 20 kg (d) 3 km to 120 m
(e) 50 c to R 1 (f) 254 sheep to 127 cattle
3. Determine the value of K in each of the following cases:
(a) = (b) = (c) =
(d) = (e) = (f) =
(g) = (h) = (i) =
(j) = (k) = (l) =
4. Look for groups of equal ratios in the list:
5. Complete the table in each example below:
(a) The apple juice and water are always mixed in the same ratio:
Water (litres) 10 15 22,5 65 81 105 490
Apple juice
(litres)
60
(b) Red and white paint have been mixed to give the same shade of pink:
Red paint (litres) 3 6 9 12 15
Apple juice
(litres)
4 40 200
(c) Flour and sugar are mixed in a recipe:
Flour per cup 3 15 21
Sugar per cup 1 2 3 10 15
(d) Oil and petrol are mixed in the ratio 20 : 1 for use in certain motorcycles:
Petrol (litres) 1 2 5 10 20
Oil (litres) 0,05
6. Mr Smith is taking the Standard 6 pupils on an excursion.
He plans to take along 2 litres of cooldrink for every 5 pupils. How many litres should he
buy for each class listed below?
(a) 25 pupils in Std 6A
(b) 30 pupils in Std 6B
(c) 20 pupils in Std 6C
7. Susan and Lulu pack apples during their school holidays. Susan packed 3 boxes in a
quarter-of-an-hour and Lulu packed 2 boxes in the same time. How many boxes did Lulu
48
pack in the time that Susan took to pack 16 boxes?
8.4 PROPORTIONAL DIVISION OR DIVISION OF QUANTITIES USING RATIOS
Example:
Two children are given pocket money in proportion to their ages, i.e. in the same ratio as
their ages. Sandy is 10 and her brother William is 8 years old.
How much does each child get if their parents put aside R72 for the children’s pocket
money?
Sandy : William = 10 : 8
= 5 : 4 (9 parts)
Sandy’s pocket money = of R72 = R40
Williams pocket money = of R72 = R32
EXERCISE 4
1. Divide:
(a) R150 in the ratio 2 : 3
(b) 39 kg in the ratio 1 : 2
(c) 36 mm in the ratio 2 : 7
(d) 35 litres in the ratio 2 : 5
(e) R45 in the ratio 2 : 3 : 5
2. 39 goldfish are separated into 2 tanks in the ratio 4 : 9. How many fish are put into each
tank?
3. 135 boys and girls are seated in an examination room in the ratio of 2 : 3.
(a) How many boys are writing the examination?
(b) How many girls are writing the examination?
4. Together Werner and Jerome pack 144 boxes of apples per day in the ratio of 5 : 7. How
many boxes does each boy pack in 5 days?
5. Divide a 14 m length of rope in the ratio of 3 : 4. How long will each piece be?
6. Frank and Pam spent R1 275 while they were on holiday. They spent it in the ratio of
14 : 9 : 28 on accommodation, food and transport respectively.
How much did they spend on each of the following?
(a) transport (b) accommodation (c) food
8.5 RATE
• A rate is a ratio of two quantities that have different units.
49
• Rate are usually expressed in a per unit form, where on number (usually the second
number), in the ratio is 1.
Examples: 1. Susan can type 120 words in three minutes. The rate = 120 : 3 = 40 : 1 This means that she can type 40 words in 1 minute, i.e 40 words per minute. 2. Thandiwe buys 5 cooldrinks for R24,95. The rate = 24,95 : 5 = 4,99 : 1 This means that she can buy one cooldrink for R4,99. EXERCISE 5 1. An aircraft travels at a speed of 550 km per hour. (a) How far will it travel in four hours? (b) How long does it take to cover 1 650 km? 2. A certain medicine for horses requires 5 ml for a 200 kg animal. How much is needed for:
(a) a 250 kg horse (b) a 125 kg foal?
3. Give the following rates in per unit form: (a) 295 km in 5 hours (b) 3 comic books for R41,91 (c) 4 tickets for R35 (d) 63 million CDs in 7 years (e) 10 oranges for R5 (f) A pipe pumps 15 (cubic metres) of water into a dam in 20 minutes
8.6 DRAWINGS: INCREASING AND DECREASING IN A GIVEN RATIO
Examples: 1. The sides of a square is 30 mm. What are the measurements of the new square, if a side of the original square is enlarged/increased in the ratio 2 : 1?
Scale = new square : original square = 2 : 1 = x : 30
50
This means that each side of the new square is twice as large as the original square. 2. Reduce/decrease the sides of the following triangle in the ratio 2 : 3. A 9 cm B 12 cm C Scale = new : original = 2 : 3 = AB : 9
AB = 6 cm BC = ? ; AC = ? 3. Increase in the ratio of .
60 mm Exercise 6 1. Here is a plan of a swimming pool: Measure the plan and use the scale to calculate the length and breadth of the pool?
Scale 1 : 600
Scale 1 : 600
2. Here is a plan of a house:
51
(a) Calculate the length and breadth of the kitchen.
(b) Find the area of the kitchen floor.
(c) What will it cost to tile the kitchen floor at R52,50 per
(d) Calculate the cost of carpeting the lounge at R80,75 per
3. Increase: (a) R25 in the ratio 8 : 5 (b) 120 in the ratio 5 : 3 (c) 1 hour 30 minutes in the ratio 7 : 5 4. Decrease: (a) 48 minutes in the ratio 5 : 6 (b) R12 in the ratio 3 : 4 (c) 45 litres in the ratio 4 : 9 (d) 300 marks in the ratio 2 : 3
CHAPTER 9 LINES AND ANGLES
Classification of Angles
1. A point is something that has position but no size.
•
2. A line is a straight, continuous arrangement of an infinite number of points. It has no
thickness but extends forever in two directions. We write it as PQ.
52
B
P Q
3. A ray starts at one endpoint and continues infinitely in one direction. We write it as AB.
•
A B
4. A line segment is a line with two endpoints. We write it as MN.
• •
M N
5. A plane is a flat surface that extends infinitely along both its length and breadth. It has
no thickness.
6. An angle is formed when two rays meet at a common end point, the vertex.
ray
- angle
vertex
ray
7. We name an angle by using the name of its vertex and one point on each of its sides. All
of the following are examples of naming an angle.
A
ABC
or
A C
C
A C D
1 and 2
or
1 2 and
or
B A C and C D
53
8. The measure of an angle or its size is an indication of the amount of turning that has
taken place. It is measured in degrees.
8.1 A revolution is completed when turning of 360° takes place.
8.2 A straight angle is formed when turning of 180° takes place.
•
8.3 A right angle is formed when turning of 90° takes place. Perpendicular lines are lines
that meet at 90°. The symbol means “is perpendicular to”. A small square in the
corner of any angle shows that it measures 90°.
P
PB AQ
A B Q
8.4 Angle between 0° and 90° are called acute angles.
8.5 Angles of between 90° and 180° are called obtuse angles.
8.6 Angles of between 180° and 360° are called reflex angles.
54
2
3
4 5
or
EXERCISE 1
1. Classify the following angles:
(a) 85° (b) 142° (c) 202° (d) 318° (e) 90°
(f) 45° (g) 360° (h) 135° (i) 180° (j) 337 ½°
(k) 75° (l) 15° (m) 165° (n) 179° (o) 100°
(p) 160°
2. Draw the following angles:
(a) Obtuse angle (b) Reflex angle (c) Revolution
(d) Right angle (e) Acute angle (f) Straight angle
3. Classify each of the following angles:
(a) (b) (c)
(d) (e) (f)
•
(g)
(h)
(i)
1
1
55
2 3 4
Adjacent angles
Adjacent angles have the same vertex and a common ray. They lie on either side of the common
ray.
Lets break that definition up into separate parts in order to understand it better:
1) Adjacent angles have a common ray.
correct incorrect
2) Adjacent angles lie on opposite sides of the common ray.
correct incorrect
3) Adjacent angles have a common vertex.
correct incorrect
4) Adjacent angles are not always the same size.
56
5) The sum of two adjacent angles is sometimes 90°.
If the sum of two (or more) adjacent angles add up to 90°, they are called
complementary angles – We call these angles each others complement.
e.g. 33° is the complement of 57° because 33° + 57° = 90°
+ = 90°
1
2
6) The sum of two diagonal adjacent angles is sometimes 180°.
If the sum of two (or more) adjacent angles add up to 180°, they are called
supplementary angles – We call these angles each others supplement.
e.g. 133° is the supplement of 47° because 133° + 47° = 180°
1 2 3
= 180°
EXERCISE 2
1. Write down the complement of each of these angles:
(a) 45° (b) 29° (c) 61°
(d) 90° (e) 30° (f) x°
(g) 0° (h) 90° - x° (i) 1°
2. Write down the supplement of each of these angles:
(a) 45° (b) 129° (c) 61°
(d) 90° (e) 180° (f) x°
(g) 0° (h) 90° - x° (i) 90° + x°
3. In this figure = 90°. Name all the numbered:
P
(a) acute angles
(b) obtuse angles 2 1 T
(c) straight angles 9
57
6
12
40°
118° 2
1
31°
3
58°
4
5 6
(d) complementary angles M 10
(e) supplementary angles 5 7
(f) adjacent acute angles 3 8
(g) adjacent obtuse angles 4 11
Q R
EXERCISE 3
Determine the size of all the angles in each of the following figures:
(1)
2)
3)
4)
58
7 8
9 38°
30°
1
70° 2
25° 3 30°
40°
270º
5)
6)
7)
8)
9)
5 4
50°
59
10)
11)
12)
13)
14)
50°
4
5
3
110°
40°
2
1 70°
80°
1
0
75° 9
8
7
20°
10°
60
15)
16)
17)
EXERCISE 4
Find the value of x and then the magnitude of the angles.
Combining geometry and algebra.
1)
x
60°
2x
150°
9 8
7
89° 70°
40°
6 120°
50°
61
2)
3)
4)
5)
6)
7)
x 100°
3x
x
x+20°
x 4x
𝑥
5x
𝑥
2x-40°
x+10°
5x
𝑥 25°
4x
𝑥
x
62
8)
9)
Intersecting lines and vertically opposite angles
Intersecting lines are lines that cross each other at one point.
When 2 straight lines intersect or cross, they form two pairs of vertically opposite angles.
Vertically opposite angles (vert. opp. ’s) are always equal in size.
That means:
For example:
x = 70° (vert. opp. )
70° x
1
2 4 3
2x+20°
x
x 3x
𝑥
63
EXERCISE 5
Determine the size of the numbered angles:
1)
2)
3)
4)
5)
6)
3 2
1 80° 20º
°
135°
45°
8 7
6
120º
4 5
145°
2
3
1 47°
64
110º
7)
8)
9)
10)
11)
40°
1
2 3
4
7
8 9
60°
120°
2
80°
4 3
145º
6
1
5
14°
8
29°
9
7
52°
45°
6
5
4
65
12)
13)
14)
Combining geometry and algebra
Example:
3x-10° 2x+40°
25°
7
6 5
80° 65°
4 3
2
1
55°
65°
5
6 7
66
2x + 40° = 3x - 10° (vert. opp. )
2x – 3x = 10° - 40°
-x = -50°
x = 50°
2x + 40° = 2(50°) + 40° = 100° + 40° = 140°
or
3(50°) - 10°
= 150° - 10°
= 140°
Exercise 6
Find x and then determine the size of the angles.
1)
2)
3)
70º
2
1 x x
1
2
x
x 3
4 5
6
30°
x
2x 1
67
x
4)
5)
6)
7)
Parallel lines
M N
K L
Parallel lines are lines with the same direction. We write MN║ KL
Parallel lines are lines that lie in the same plane and do not intersect.
3x-20° 140°
3x-50° x+20
°
x+20
° 70°
123°
1
2
3
x
x
68
x
Parallel lines are lines that are a uniform distance apart in a flat plane.
A line cutting across two or more other lines is called a transversal.
Parallel lines and a transversal forms three special types of angles:
(1) Corresponding angles
Corresponding angles are equal.
Pairs of corresponding angles are always on the same side of the transversal.
(2) Alternate angles
Alternate angles are equal.
Pairs of alternate angles are always on the inside of parallel lines and on alternate
(different) sides of the transversal.
(3) Co-interior angles
Co-interior angles are supplementary.
Pairs of co-interior angles are on the same side of the transversal inside the parallel
lines.
EXERCISE 7
Find the size of the unknown angles:
1) 2)
91°
b
a x
110°
69
r
3) 4)
5) 6)
7) 8)
9) 10)
11) 12)
80°
t
s
q
p
70° y x
f
e
d 35°
60
° c
25°
b
50° a
130°
p
q t
r s 130° 110°
20°
z
y
x
c b a 50°
110° r
q p
70
40º b
45º
13) 14)
15) 16)
17) 18)
19) 20)
21) 22)
1 = 2
4
70° 3
2
1 2
1
4 3
80°
50°
60°
d e
c b a
g
f h
40
°
b
c
a
30°
50°
30°
p r
q
110° x 70°
40°
y
50°
a
b x 112°
47° r
q p
100°
a
71
42º
23) 24)
25) 26)
27) 28)
29) 30)
31) 32)
33) 34)
29° 3
2
1 5
4
3 1
2
38°
70°
4 3
2
1
126°
3
2 1
1
136°
5
4 3
2
5
4 3
51°
2 1
50° 1
2
76°
1
2
54°
2
1
56°
4
3 2
1
72
5x
35) 36)
37) 38)
Combining Algebra and Geometry
EXERCISE 8
1) 2)
3) 4)
5) 6)
x - 30° 3x + 20° 3x -10°
60° + x
2x a b 3x +20°
x + 50°
x
2x b a x -10° 2x - 30°
p
135°
70°
4 3
2 1
130° 110° 5
4 3
2
1
57° 82°
3
2
1 78°
35° 4
3
2 1
130° 40°
5
4 3 2
1
31°
65° 5
4
3 2 1
73
K
7) 8)
9) 10)
11) 12)
13) 14)
15)
70° 50°
x
135°
H G F E
D B
x H
F E D
G C B
A
x + 22°
H
G
F
E
D C
B A
80°
x S
R Q
P
H F
G E
D C
B A x
20°
130°
G
D
E
F C
B
A
2x x
E
D
C B A z
y
x + 80° 3x
𝑥
2x + 10° x + 30° z
y
2x - 10° z y
x
x + 10° 3x
b a
74
Something extra EXERCISE 9
1) 2)
3) Are the following lines parallel or not (give reasons):
3.1) 3.2)
3.3)
CHAPTER 10 TRIANGLES
10.1 Types of triangles
What is a triangle?
❖ A triangle is a closed two-dimensional figure bordered by three straight lines. ❖ A triangle has three interior angles and three sides.
86°
85°
82°
80°
40°
122°
C
B R
S
Q 40°
55°
x
150°
130°
x
P
T
D
A
E
75
Equilateral triangle
All sides are the same length.
Isosceles triangle
Two sides are equal in
length.
Scalene triangle
All sides have different
lengths.
We can also name triangles by the size of their angles.
Acute-angled triangle
All the angles are acute.
Obtuse-angled triangle
One angle is obtuse.
Right-angled triangle
One angle is equal to 90°.
The side opposite the right
angle is called the
hypotenuse.
Example 10.1
Classify the following triangles according to angles and sides:
(a) (1) Acute-angled triangle (2) Scalene triangle
(b) (1) Right-angled triangle (2) Isosceles triangle
76
(c) (1) Acute-angled triangle (2) Equilateral triangle
(d) (1) Right-angled triangle (2) Scalene triangle
(e) (1) Acute-angled triangle (2) Isosceles triangle
(f) (1) Obtuse-angled triangle
(2) Scalene triangle
(f) (1) Obtuse- angled triangle (2) Isosceles triangle
EXERCISE 10.1
1. Classify the following triangles according to angles and sides:
1.1 1.2
1.3 1.4
1.5 1.6
77
A
B
C D
E
F
1.7 1.8
1.9
2. Classify the numbered triangles:
3. Draw rough sketches of these figures:
3.1 an acute-angled scalene triangle
3.2 a right-angled scalene triangle
3.3 an obtuse-angled scalene triangle
3.4 an acute-angled isosceles triangle
3.5 a right-angled isosceles triangle
3.6 an obtuse-angled isosceles triangle
3.7 an equilateral triangle
4. Examine the figure and identify all:
4.1 acute-angled triangles
4.2 obtuse-angled triangles
4.3 obtuse-angled triangles
4.4 isosceles triangles
4.5 equilateral triangles
4.6 scalene triangles
1
2
3 4
5
6
7
8
78
5. Study the figure below. List all the triangles that are:
(a) right-angled (b) isosceles (c) equilateral (d) scalene (e) obtuse-angled (f) acute-angled
6. Are the following statements true or false? Illustrate your answers with a sketch of each
triangle.
6.1 An equilateral triangle is always acute-angled.
6.2 An isosceles triangle is always acute-angled.
6.3 One of the interior angles of an isosceles triangle may be an obtuse angle.
6.4 One of the interior angles of an isosceles triangle may be a right angle.
6.5 An acute-angled triangle is always an equilateral triangle.
6.6 An acute-angled triangle is always an isosceles triangle.
6.7 A right-angled triangle is sometimes an equilateral triangle.
6.8 A right-angled triangle is sometimes an isosceles triangle.
10.2 General Properties of Triangles
a) The sum of the interior angles of a triangle. The three interior angles of any triangle always adds up to 180°
in ABC, ( )
Example 10.2
40°
60°
x
79
EXERCISE 10.2
1. Calculate the sizes of the angles marked with letters:
(a) (b)
(c) (d)
(e) (f)
48°
60°
50° 85
°
60° 71°
29° 58°
75°
45° 35°
110°
48
°
23°
80°
30°
80
v
(g) (h)
(i) (j)
Example 10.3
EXERCISE 10.3
1)
77°
110°
47°
63°
87°
81
2)
3)
b) The exterior angle of a triangle
The exterior angle of a triangle is formed outside the triangle by one of the sides of
the triangle and the produced adjacent side.
The exterior angle is equal to the sum of the opposite interior angles:
E E C C B B
A
A
82
Example 10.4
EXERCISE 10.4
Determine the sizes of the unknown angles in each of these sketches.
a) b)
c) d)
e) f)
140°
125°
32°
118°
30°
130° 100°
70°
51°
110°
45°
40°
30°
83
x
10.3 The theorem of Pythagoras
The theorem of Pythagoras:
In a right-angled triangle, the square on the hypotenuse equals the sum of the
squares on the other two sides.
hypotenuse2 = side2 + side2
The theorem of Pythagoras may be used in two ways in Mathematics:
❖ When the lengths of two sides of a right-angled triangle are known, the length of the third side can be calculated using the theorem and without construction or measurement.
❖ When the lengths of all three sides of a triangle are known, the theorem may be used to find whether the triangle is acute-angled, right-angled or obtuse-angled, without construction or measurement.
Example 10.5
x
NB!! Remember units.
8cm
15cm
side
side
hypotenuse
84
EXERCISE 10.5
1. Calculate the values of the variable:
(a) (b)
(c) (d)
(e) (f)
(g) (h)
6
8
15
17
12
16
3
5
6 10
10
8
12 5
85
x
(i) (j)
(k) (l)
(m) (n)
(o) (p)
(q) (r)
29
20
21 75
35
37
24
7
5 6
4
2
3
14
48
82
80
1
1,2
0,5
20
25
86
2x (s) (t)
Example 10.6
Will these triangles be acute-angled, right-angled or obtuse-angled triangles? (Measurements in
cm.)
SOLUTION Remember that the largest square is always formed on the longest side, so
the squares on the two shorter sides must be added together.
Figure 1 Figure 2 Figure 3
And
Square on AC is too small
because AC is too short so
∆ABC will be acute-angled.
And
Pythagoras’ theorem, applies
so
∆ABC will be right-angled.
and
Square on AC is too large
because AC is too long, so
∆ABC will be obtuse-angled.
EXERCISE 10.6
1. Decide whether each of these triangles is acute-angled, obtuse-angled or right-angled.
a) 150 mm; 200 mm; 250 mm
b) 150 mm; 360 mm; 390 mm
c) 200 mm; 210 mm; 280 mm
d) 110 mm; 350 mm; 370 mm
e) 70 cm; 60 cm; 50 cm
f) 720 mm; 750 mm; 209 mm
B
6
2,5 C
7
A
C
14 B
10,5
17,5
A
C 7 B
6 9
A
22
9
87
EXERCISE 10.7 – More uses of Pythagoras
1. In ∆ABC, Calculate the length of the third side in each of these cases.
Leave your answer in root form where necessary. a) AB = 120 mm and BC = 90 mm b) AB = 200 mm and BC = 150 mm c) AC = 510 mm and BC = 240 mm d) AB = 120 mm and AC = 350 mm e) AB = 100 mm and AC = 260 mm f) BC = 100 mm and AB = 260 mm g)
2. If a triangle has sides of 3 cm, 4 cm and 5 cm, is it a right-angled? 3. If a triangle has sides of 6 cm, 8 cm and 10cm, is it right-angled? 4. If a triangle has sides of 9 cm, 12 cm and 15 cm, is it right-angled? 5. What conclusion have you come to after obtaining these results?
6. In ∆ABC, Complete the table of Pythagoras triples (the measurements
of the sides of right-angled triangles expressed as natural numbers). Use, where necessary, the conclusion you came to in Question 3 above to make your task easier.
3 6 9 5 10 7 14 8 16 12 20
4 8 12 24 36 24 72 15 35 70
15 39 50 75 34 74 29
7. Calculate the exact lengths of the unknown sides.
a)
35mm
12m
m
5mm B
x
C
y
A
C
a
b A
B
c
88
b)
8. Calculate the perimeter of the figure.
9. Calculate the length (rounded-off) of the diagonal of:
a) a square with sides 40 mm long;
b) a rectangle with dimension of 56 mm by 65 mm;
c) a rectangular school hall, 25 m by 50 m.
10. Calculate the perimeter (rounded-off) of a square with a diagonal of 80 mm.
11. A rectangle, 16 cm long, has a diagonal 34 cm long. Calculate the exact perimeter of
the rectangle.
12. A ladder is 19,5 m long and is positioned in such a way that it reaches a window that is
18 m above the ground. Without moving the foot of the ladder, it is tilted over so that
the top now reaches a window 7,5 m above the ground on the other side of the road.
Calculate the width of the road.
13. Calculate the exact distance between the opposite vertices of a cube that has a side of
60 mm.
Revision exercise
1. Find in each of the following. Leave irrational answers in surd (√ ) form:
(a) (b)
9cm
3cm
6cm
8cm
R 22cm
5cm
10cm
Q S
T P
17cm
y
C D x B
15cm
A
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x x
(c) (d)
2. Find the sizes of the two unknown angles of this triangle:
3. The largest angle of a triangle is
A second angle is half this size, and the third angle is 15° smaller than the largest angle.
(a) Write each angle in terms of
(b) Write an equation that will help you to find the sizes of the angles. (c) Find the sizes of the three angles.
64°
10
10
4mm
6mm
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CHAPTER 11 AREA AND PERIMETER
Converting square units
Let’s find the area of the following parallelograms:
Ar
Area
We can convert this measurement of are to square centimetres in the following way:
So,
Similarly:
EXERCISE 11.1
1. A cutting machine has been programmed to cut out a triangular piece of wood. The coordinates of the triangle’s vertices are (1;1), (5;4) and (6;1).
(a) Draw the triangle on a Cartesian plane. (b) Calculate the area of wood cut out. (c) The actual measurement of the triangle is in square centimetres. Convert this
measurement to square millimetres. (d) If the triangle was cut from a rectangular piece of wood 5,2 cm long and 4,1 cm wide,
what is the area of the remaining piece of wood?
2. Convert the following measurements of area to square centimetres: (a) (b)
3. Convert the following measurements of area to square metres: (a) (b) (c)
We know that:
Therefore:
Which shows us that
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THE TRIANGLE:
h
b
P = sum of sides
=
A = ½ base × perpendicular
height
One of the sides
Height onto the
side which is used.
A = ½ b.h
Examples:
A =
= (Given: height 3, its base is 8)
=
=
Pythagoras
12 cm
13 cm C B
A
3 6
8
cm
12
b
h
h
b B C
A
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10
4
5 b
A = hb.2
1
=2
1.10.4
= 20 cm²
20 = 2
1(b.h)
(2
1of what is 20. Answer 40)
b = 8 cm
EXERCISE 11.2
1. Find the area and perimeter of each of the following triangles:
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2. Calculate the area of each triangle:
94
D
3. Calculate AD if the area = 12 cm² and BC = 4 cm: A
B C
4. Calculate: (a) Area of ∆ABC (b) BE
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5. Each of these triangles has an area of 120 cm². Calculate the lengths of the sides that are marked with small letters. All measurements are in centimeters:
6. Calculate the lengths of the line segments that are marked with small letters. All measurements are in millimeters.
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THE SQUARE AND RECTANGLE:
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Examples:
1. 4 cm 6 cm
P = 2l + 2b = 2 x 6 + 2 x 4 = 12 + 8 = 20 cm A = l x b = 6 x 4 = 24 cm²
2.
Area = 20 m² Length = 4 m What is the breadth? A = l x b 20 = 4 x b
4
20 = b
5 m = b
3.
Perimeter = 20 m Length = 4 m What is the breadth? P = 2(l + b) = 2l + 2b 20 = 2 x 4 + 2b 20 - 8 = 2b 12 = 2b 6 = b
EXERCISE 11.3
1. Calculate the area and perimeter of the following:
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2. Calculate the value of x:
THE CIRCLE
1 – A Circle is a set of points equidistant from a fixed point, which is the centre of the circle. 2 – A radius is any line segment stretching from the centre of a circle to any point on the circle. ( The plural is radii ) 3 – A chord is any line segment with endpoints on the circle. 4 – A diameter is any chord through the centre of a circle. 5 – An arc is any part of a circle.
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6 – A sector of a circle is enclosed by two radii and the included arc of a circle. 7 – A segment of a circle is enclosed by a chord and an arc of a circle. 8 – A semicircle is enclosed by a diameter and an arc of a circle.
Circle
P = 2πr or P = πd ( d : diameter)
A = πr²
Examples:
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EXERCISE 11.4 1. Calculate the circumferences of the circles which have the following radii. (a) 25 cm (b) 48 mm (c) π m 2. Calculate the circumferences of the circles which have the following diameters. (a) 50 mm (b) 250 mm (c) 20π cm 3. Calculate the diameters of the circles which have these circumferences. (a) 175,84 m (b) 1 570 cm (c) π m 4. The minute hand of a clock is 15 cm long and the hour hand is 10 cm long. How far does the tip of each hand move in one hour? 5. Examine the gears A, B and C. A has 40 teeth, B has 20 and C has 30.
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Now complete the statements: (a) If A rotates in a clockwise direction then B rotates ……….. (b) If A rotates in a clockwise direction then C rotates ………. (c) If A completes 3 revolutions, then B completes ………. (d) If A completes 3 revolutions, then C completes ………. 6. The thickness of a ring is equal to the length of the radius of the inner circle. How many times is the outer circumference larger than the inner circumference? MIXED EXERCISE Examples:
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Exercise 11.5 1. Determine the perimeter and area of each of the following figures:
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104
105
2. In ∆ XYZ, XY = 8 cm; YZ = 5 cm and height XE = 4 cm. Calculate: 2.1 the area of ∆ XYZ 2.2 the length of DZ
3. A Ball has a diameter of 28 cm. It is rolled from point A to point B; AB = 43,98 cm. Calculate how many revolutions the ball makes to cover the distance. 4. Determine the area of the shaded section.
5. Determine the area of the shaded sector.
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6. A wheel covers a distance of 660 mm in 60 revolutions. Determine the radius of the wheel. 7. An athletics track consists of a rectangle and 2 semi-circles. The two lanes are 1 m apart.
7.1 Calculate the length of the outside lane. 7.2 Determine the distance between the starting positions of the two athletes
so that they run the same distance. 8. The diagram shows a rectangular wooden door with a rectangular glass window. 8.1 Determine the area of the glass. 8.2 Determine the area of the wood.
9. Diameter XY = 13 cm and XZ = 12 cm. Calculate the area of the shaded portion.
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10. Determine the shaded area.
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CHAPTER 12 VOLUME AND SURFACE AREA
VOLUME
The volume of a solid figure is the amount of space it occupies. The capacity of a hollow figure is the amount of substance it can hold. The volume of a prism and a cylinder is found by multiplying the area of the base by its height.
Prism/cylinder Base Area of base Volume
Cylinder Circle
Cube Square V =
Rectangular prism
Triangular prism
Fill in all the missing information in the table above. EXERCISE 12.1 Unless otherwise specified, all dimensions are given in centimetres. 1. Calculate the volume of the prisms or constructions.
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2. Cube-shaped cardboard boxes, 40 cm long, must be kept in a storeroom that has interior dimensions of 4m by 28m by 2,2m. 2.1 Calculate the volume of the empty store room. 2.2 Calculate the volume of a cardboard box. 2.3 Calculate the number of boxes that can be stored in the storeroom. 2.4 Calculate the maximum number of boxes that will fit on to the storeroom floor. 2.5 Calculate the maximum number of boxes that can be stacked on top of each other. 2.6 Calculate the actual number of boxes that can be stored in the storeroom. 2.7 Explain why the answers of 2.5 and 2.6 differ by 35 boxes. 3. These sketches depict prisms, all 500 mm high, that are used as flower pots. Other measurements are given in centimetres. Calculate which flower pot will be the cheapest to fill with compost.
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4. The sketch depicts a wooden chest that has walls that are 20mm thick. The exterior dimensions are given. Calculate: 4.1 the interior volume of the chest 4.2 the volume of wood needed to manufacture the chest (without a lid.)
5. Cube-shaped Turkish Delight sweets are packed by the sweet manufacturer into rectangular containers which measure 24cm by 18cm by 12cm. Calculate: 5.1 how many dozen sweets fit into each container if there are 4 layers in each container; 5.2 the price of a container at R1,70 per dozen sweets.
6. The sketch depicts a solid cylinder that fits exactly into a cubic container that has inside measurements of 42cm along one side. Calculate the amount of air in the container.
7. A cube with diagonals of 50mm fits exactly into a cylindrical container. Calculate the volume of the cylinder.
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8. A cylindrical-shaped piece of lead, 28cm high and with a radius of 11cm, is melted and remoulded into 8 lead cubes. Calculate the dimensions of the cubes.
Take π = .
SURFACE AREA
• The outside area of any shape is called the surface area.
• To determine the surface areas of an object, you must calculate the areas of each of its
faces.
• The easiest way to do this is to unfold the 3D solid figure. The flat 2D surface is called the net of
the 3D solid figure.
• The surface area is, therefore,
NOTE For any right prism, the surface area can be calculated as follows: Surface area = 2(area of base) + (perimeter of base × height)
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EXERCISE 12.2
1. (a) Draw nets of the prisms (i) to (iv) drawn below: (b) Write a formula to determine the surface area of each of the prisms. (c) Hence or otherwise, determine the surface area of each of the prisms.
2. The sketch depicts a rectangular prism.
2.1 Construct the prism. 2.2 Construct a net of the prism. 2.3 Calculate the volume of the prism. 2.4 Calculate the surface area of the prism. 2.5 Calculate the total length of the edges of the prism.
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3. Calculate the volume and surface are of each of the following solid figures:
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CHAPTER 13 PERCENTAGES AND FINANCIAL MATHS Percentages Percent means per hundred Examples:
1. 35% means 35 per 100
2. 100% means 100 per 100
3. 130% means 130 per 100
EXERCISE 1
1. Write the following as fractions and simplify, for example:
15% = 20
3
100
15=
(a) 30% (b) 75% (c) 200% (d) 850%
2. In a packet of 300 biscuits, 7% are broken. How many biscuits
are broken?
3. Express 8 as a percentage of 20.
4. If your bill at a restaurant is R 136,67 and you must tip the waiter 10% of
the amount, what will the tip amount to?
5. The following data were collected to show the local government how
many learners can afford to pay their school fees:
School A School B
Number of learners whose school fees are paid 2 400 2 000
Total number of learners in school 3 000 2500
(a) For each school, write the number of learners whose school fees
are paid as a fraction of the total number of learners in school.
(b) State the percentage of school fee payments for each school.
6. The salaries in a company range from R 4 000 to R 26 000 per month.
What would the salaries be after an increase of 5% to all?
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7. The National Development Agency (NDA) is an important funding
agency, working alongside government and civil society to reduce poverty. The
NDA planned to give out R 63 million during 2002/2003.
It planned to divide the money over a five year period as follows:
➢ 33% to early childhood development
➢ 24% to economic development
➢ 22% to rural development
➢ 14% to health
➢ 5% to human rights initiatives
Source: NDA Strategic Plan 2003/2006,www.nda.org.za
(a) Calculate the amount of money planned for each sector for 2002/2003.
(b) Do these amounts add up to the planned R 63 million set aside?
(c) Give an explanation for your answer in question (b).
8. The estimated population of Africa in 1995 was 580 million. Of these:
➢ 291 million people had average incomes below one US dollar a day.
➢ 43 million children were stunted as a result of malnutrition.
➢ 205 million were without access to health services.
➢ 249 million were without safe drinking water.
➢ 139 million youths and adults were illiterate in 1995.
Source: African Poverty at the Millenium, The World Bank, 2001
Write these numbers as a percentage of the total number of people in
Africa in 1995.
EXCHANGE RATES:
A country’s money is called its currency. South Africa’s currency is rands and cents.
Notes and coins are called cash. They come in different denominations. In South
Africa the smallest denomination was the 1c coin, but since it was discarded, the
smallest denomination is now the 5 c coin and the largest denomination is the R 200
note. Although we use cash for everyday transactions such as buying groceries, most
people keep their money in a bank account. They can withdraw it using an ATM card,
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or they may use money in other ways that do not involve cash, such as credit cards,
debit cards, cheques or even internet banking.
Different currencies have different values. The value of a currency is linked to many
different economic factors and it changes all the time. When you travel to another
country, you may need to buy money in the currency of that country. Rates which give
the value of one currency against other currencies are called exchange rates.
Exchange rates for each day are usually published in the newspaper or displayed at
banks. You can exchange money at foreign exchange bureaus, banks, airports and
hotels. There are different buying and selling rates for each currency. Most places
charge a fee for exchanging currencies, called a commission.
The following table of exchange rates put up at a local commercial bank, is used to do
the following calculations:
Currency We sell We buy
R/$ 1.00 6.20 1.00 6.11
R/₤ 1.00 11.60 1.00 11.40
R/₤ 1.00 4.75 1.00 4.82
3% commission on all transactions
Example:
1. I need $100. How much will it cost me in rands?
Solution: The bank is buying my rands, so I must pay R6,20 per dollar.
R6,20 X 100 = R 620,00
3% commission on this amount: 3% of R 620,00 = 60,181
620
100
3RX = .
R 620,00 + R 18,60 = R 638,60 will it cost me in Rand for $100.
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2. How many rands will I get for $500 ?
Solution: The bank is selling me rands, so I will receive R 6,11 per dollar.
R 6,11 X 500 = R 3 055,00
3% commission on this amount: 3% of R 3 055 = 65,911
3055
100
3RX =
R3 055,00 – R91,65 = R2 963,35 will I get for $500.
EXERCISE 2
Use the above table for exchange rates to solve the following problems:
1. Calculate how many rands it will cost (including commission) to buy:
(a) $1 000,00 (b) ₤1 000,00 (c) Є1 000,00
(d) $736,00 (e) ₤3 453,00 (f) Є2 458,00
2. Calculate how many rands you will get (after commission) for:
(a) ₤550,00 (b) $1 230,00 (c) Є5 638,00
3. Belinda is going on holiday to France and Italy, and she needs to buy
currency for these countries.
(a) What currency should she buy ?
(b) If she has R 15 460,00 to spend on her foreign currency, how much
will she get for her money after the commission charge ?
4. Ivan is from Spain. He is on holiday in Cape Town. He wnts to buy rands.
He has Є425,00. How much does he get after commission ?
5. Peter buys some pounds. He spends R 11 679,87 including a 3%
commission fee. How many pounds does he get?
6. Sabia went with her parents to the United Kingdom for a holiday. She
saved R 895 for the trip. She bought Pounds for that amount. On the trip
however, her parents paid for everything and she could return with her full
amount of money. How many rands will she get if she buys now rands for
her pounds ?
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EXERCISE 3
1. A foreign exchange desk at Heathrow airport in London had a South
African customer who wanted to buy ₤655. The bureau sold pounds at a
rate of R 11,31 per pound, and charged 2,25% commission on all
transactions. Calculate how much the South African customer had to
spend on his foreign currency.
2. Mr. Smith travels from Canada to Norway. The exchange rate is
approximately Є0,79 to $1,00, and he finds a bank that charges no
commission. If mr. Smith wants to buy Є950,00., how many dollars will
he need ?
3. Mr. Motsepe is visiting South Africa form Botswana. If the exchange rate
is R 1,32 per Botswana Pula, how many rands will he get for:
(a) Pula 4 500 (b) Pula 13 480 (c) Pula 8760
4. The picture shows prices outside a restaurant in London.
STARTERS WINTER SPECIAL
Vegetable soup ₤ 3.70 Vegetable lasagne with
soup starter
₤ 5.60 Prawn roll ₤ 5.35
Mixed Salad ₤ 4.25
MAIN DISHES DRINKS
Hamburger
(served with chips)
₤ 4,85 Coffee/tea ₤ 1.90
Soft drinks ₤ 1.85
Sausages and mash ₤ 3.90 Milkshake ₤ 2.15
(a) Using the prices in the picture, and what you have learned about different
currencies, estimate how much each of the following meals would cost in
rands:
(i) Prawn roll, sausages and mash and a cup of tea
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(ii) Hamburger and a milkshake
(iii) The winter special
(b) Now check your answer to (a) using a calculator. What was the
difference between your estimate and the actual answer?
5. (a) Amanda is traveling from South Africa to America. If the exchange
rate is $ 1 = R 7 35, and the bank is not charging commission, calculate
how many dollars Amanda can get for each of the following amounts:
(i) R 564,76 (ii) R 2 396 (iii) R 1 649,38
(b) When Amanda returns to South Africa, she has $ 346,80 left over.
How many rands can she get if the bank sells her rands at R 7,15 per Є1.
6. The Sibiya family from South Africa went to Spain on holiday. They
changed R 19 600 for euros at the exchange rate of R 7,45 for I euro.
The bank charged 1% of the money handled as a service fee. How many
euros did the Sibiya family get?
7. If one Botswana pula is equivalent to 0,73 rand and one rand is equivalent
to Zim $980, calculate:
(a) How many pula you would get for R10.
(b) How many Zim $ you would get for R10.
(c) Suppose a violin costs R3000. What would the price of the violin be in:
(i) pula (ii) Zim $
PROFIT AND LOSS
Cost: It is the money paid to buy or make goods, or to orovide services.
Income/revenue: The money received for selling goods or services
When the income is greater than the cost, the business makes a profit.
Income - cost = profit.
When cost is greater than the income, the business makes a loss.
Cost - income = loss.
Percentage profit = 100cos
Xtprice
profit.
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Percentage loss = 100cos
Xtprice
lost.
EXERCISE 4
1. Thandeka buys a machine to make candy floss, for R 536,00. To make the candy
floss she had to buy some sugar and essence, which cost her
R 780. She made 1 200 bags of candy floss and sells them at R 4 each.
(a) How much profit did she make?
(b) What was her percentage profit?
(c) What is the minimum amount for which she can sell her product not to
make a loss?
2. The cost price of a toy racing car is R 200. It was sold for for R 345,00.
Calculate: (a) the profit (b) the percentage profit
3. A chest of drawers that cost R 650 was sold at a loss of 12%. Calculate
the selling price of the chest of drawers.
4. A flat was sold for R 650 000 with a profit of 320%. Find the cost price of
the flat.
5. In 1980 a man bought a house for R 150 000. In 2005 he sold the house
for R 900 000. What was his percentage profit ?
6. A leather couch was bought for R 6 500 and sold at a profit of 21.8%.
Find: (a) the profit (b) the selling price of the couch
7. Sam bought a car for R 140 000. Nine years later, she sold it for R 74 000.
Calculate: (a) the loss made (b) the percentage loss
8. Copy and complete the following table:
Cost Price Selling Price Profit Loss
R 190 R 280
R 120 R 70
R 41 R 8
R 5 435 R 3 680
R 512 R 74
R 867 R 93
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BUDGETING:
Budgeting means estimating how much money you will need to spend at a particular
time or for a particular purpose. Budgeting helps us to make sure that we do not spend
more money than we have, or that we do not run out of money.
EXERCISE 5
1. Mandy earns R 5 300 per month. She wants to save R 400 per month.
She writes down her monthly expenditure.
Tax @ 18%: R 774
Cell phone bill: R 850
Rental : R 2 300
Electricity & water: R 300
Food: R 1 200
Entertainment: R 400
Petrol for her 2nd hand car: R 250
(a) How much money does Mandy have after she has paid tax?
(b) How much money does Mandy have after her expenses?
(c) What advice would you give Mandy about her budget?
2. The pie graphs below show the budgets for two different businesses.
(a) Identify each company’s greatest expense each month and calculate the
cost in rands.
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(b) Identify the similarities and differences between the two companies’
budgets. What do you think are the reasons for the similarities and
differences?
(c) Both the budgets above are calculated after wages have been paid.
Couriers for Africa has two staff members who each receive R 12 000
per month. Betty’s Home Bakes also has two staff members, but they
each receive R 8 000 per month. Work out the total monthly budget for
each company including wages.
(d) Redraw the pie charts including wages as one of the sectors.
BUYING ON CREDIT
When people want to buy expensive goods, they sometimes do not have enough
money to pay for the item immediately. In these cases, many people choose to buy on
credit. The shopper usually pays part of the price as a deposit and agrees to pay the
rest of the amount owed, or balance, in instalments over a fixed period of time. The
lender always charges interest on the unpaid amount.
There are different forms of credit:
❖ Banks offer credit accounts. They give each customer a credit limit, which is the
maximum amount that the customer may spend, and a credit card, which has the
customer’s account information stored on a magnetic strip. When the customer
uses the credit card to pay for a purchase, the salesperson swipes the card
through a card machine that reads the information on the card and contacts the
bank to confirm that the customer still has credit available. If there is credit
available, the bank will pay for the purchase and the customer must pay the bank
in monthly installments with interest.
❖ Many shops and businesses also offer in-store credit accounts. These work
similarly to credit cards, except they are only valid in a particular shop or group of
shops.
❖ Banks offer special loans, called mortgage loans, to people who are buying
property or homes. These are very large loans, often paid back over many
years.
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❖ Vehicle finance is the name given to loans for cars and other vehicles.
❖ A common form of buying on credit is called hire purchase. This is another way
of paying off the price of an item in instalments.
HIRE PURCHASE
When you buy an item on hire purchase (HP), you pay a deposit and sign an agreement
to pay fixed monthly amounts until the goods are paid for in full. You can take the
goods home but they technically belong to the shop until you have finished paying the
full amount. Another name for hire purchase is ‘ buying on terms’. To calculate the hire
purchase price, add the deposit to the total of the monthly payments. For example:
HP Price = deposit + instalments. R 899,99 + ( 6 x R 540) = R 4 139,99
Work out the interest using the formula: HP price – cash price = interest paid.
R 4 139,99 – R 3 599,00 = R 540,00, which is the interest the customer would pay.
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EXERCISE 6
1. For each of the following items, work out the hire purchase price:
(a)
(b)
(c)
2. For each item above, work out the total interest paid.
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3. Maggie buys a television on hire purchase. She pays a deposit of R 950
and 12 monthly instalments of R 420,00. Calculate the total HP price of
the television.
4. Helen buys a couch on HP. She pays a deposit of R 500 plus 18 monthly
instalments of R 350,00.
(a) Calculate the total HP price.
(b) If the cash price on the couch was R 2 999,99, calculate the total amount of
interest that Helen paid.
5. Jane bought this set of table and chairs on hire purchase. She paid a
deposit of R 1 350 and 12 monthly instalments. Calculate:
(a) how much interest she paid
(b) her monthly interest rate.
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6. Calculate: (a) the HP price of the stove
(b) the interest charged
(c) the interest as a percentage of the cash price
7. Calculate: (a) the deposit in rands
(b) the HP price
(c) the total interest
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8. The dishwasher below was sold on hire purchase at an interest rate of
60%. Calculate:
(a) the hire purchase price
(b) the total interest
(c) If R 1 000 deposit was paid, the monthly instalments if equal payments were
made over two years.
9. The cash price of a wardrobe is R 1 200. Agnes bought the wardrobe
on HP. She paid a deposit of R 250 and then paid 12 equal monthly
installments. The HP price was R 3 100. Determine:
(a) how much interest she paid
(b) her monthly installment amount.
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10. The local furniture store is selling a Sansonic 54 cm colourTV. The store offers
Two different payment options. You can either buy the TV at the cash price of
R 1 749 or you can buy it by taking out a hire purchase (HP) account at the
store. On HP, you have to pay R 180 deposit and then an additional 24
instalments of R 89 each. Calculate the difference between the HP price and
the cash price.
11. How much greater is the cost of the bicycle if you pay by instalments?
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12. A mini hi-fi is advertised as follows:
How much cheaper is the mini hi-fi if you pay cash ?
13. The furniture store Bargains For All has a fridge on sale at R 2 000.
The store owner decides to also sell the fridge on HP. The deposit is
20% of the cash price and there will be 24 monthly payments. The total
cost should be 25% more than the cash price. How much will each
monthly payment be?
14. Ingrid buys a used car for R29 000. She puts R10 000 down and
finances the rest through the dealer at 13% p.a. add-on interest. If she
agrees to make 36 monthly payments, find the amount of each payment.
How much extra has she paid?
DISCOUNT
Example 1: Calculate: (a) the discount (b) the selling price of a
washing machine marked R 3 150 less 10% for cash.
Solution: (a) Marked price (MP) = R 3 150
Discount = 3150100
10XR = R 315
(b) Selling price (SP) = R 3 150 - R 315 = R 2 835,00
Example 2. Calculate the selling price of the washing marked R 3150 less
10% for cash.
Solution: If the discount is 10%, the selling price is 90% of the marked price.
Selling price (SP) = 3150100
90XR = R 2 835,00
For sale: Metronic 3CD Hi-Fi
Cash Price: R 799
Or
Deposit: R 80 + instalments: 24 x R 39
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EXERCISE 7
1. Calculate (i) the discount (ii) the selling price of
(a) A radio marked R 400 less 7%.
(b) A stove marked R 5920 less 12,5%
(c) A lounge suite marked R 4 999,99 less 10%.
(d) A refrigerator marked R 5 495 less 15%.
(e) A camera marked R 592 less 2,5%.
2. Calculate the selling price ( Nett price)
(a) Marked Price = R 360 discount 10%.
(b) Marked Price = R 4 530 discount 3331 %
(c) Marked Price = R 1 852 discount 1221 %
(d) Marked Price = R 2 550 discount 20%
(e) Marked Price = R 1 200 discount 5%
3. An article marked for R 450 is sold for R 420. What was the percentage
discount?
4. An article was sold for R 756 after 10% was given. What was the marked price?
5. A bicycle is advertised for R 600 cash after a discount of 15% was
allowed. What was the marked price of the bicycle?
6. Study the advertisement below and calculate the following:
(a) The total area of the floor to be carpeted.
(b) The selling price of the carpeting per m².
(c) The original price ( i.e. the MP) of the carpeting per m².
(d) The original price ( i.e. the MP) to carpet the house.
(e) The price to the customer who decided to pay by 36 monthly instalments.
(e) The difference between the cash price and the price on terms to
carpet the house.
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3,
SIMPLE INTEREST
Banks make money by lending money to customers and charging interest on the loans.
The formula for simple interest is:
Total interest (i) = Principal(P) X rate(r) X time(t)
The principal is the amount of the loan. The rate is the percentage (fraction
over 100) at which the interest is charged, and the time is the number of years given to
repay the loan. The rate of interest is usually given as the interest rate per annum (per
year).
In stead of borrowing money from the bank on which you must pay interest, you can
also invest your money at the bank on which you will then earn interest.
Example 1: Michael borrows R 2 000 from the bank for two years at a rate of 6% per
annum. How much interest does Michael pay for the two years?
Solution: Total simple interest (SI) = principal X rate X time
= R 2 000 X .06 X 2
= R 240
UNBEATABLE OFFER
SAVE 331/3 % WITH THE HOME CARPET CO. CARPET YOUR ENTIRE HOME FOR ONLY R 537,50 CASH EXCL GST TERMS R 20,50 OVER 36 MONTHS NO DEPOSIT
Dining room passage
Lounge 3rd bedroom 2nd bedroom Main bedroom 4 m 2,5 m 2,5 m 3,5 m
3
,75 m
2,75
M
2,5
m
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NB: Percentage is a special rate in which the unit is 100.
6% means 100
6 = 0,06.
Example 2: Find the interest rate if R 18 interest is earned on an investment
of R 150 after a year.
Solution: Simple interest = P X r X t
18 = 150 X r X 1
r = 150
18X 100 ( to write the rate as a percentage)
= 12 %
EXERCISE 8
1. For each of the following loans, calculate:
(i) the simple interest due
(ii) the total amount paid back to the bank by the end of the loan period.
(a) R 2 4050 for three years at 10% per annum.
(b) R 5 800 over two years at 6% per annum.
(c) R13 000 over two years at 12,5 % per annum.
(d) R 34 600 over four years at 12% per annum.
(e) R 128 000 over five years at 8,25% per annum.
(f) R 60 000 for two and a half year at 8,5% per annum.
(g) R 6 000 for four months at 10% per annum.
(h) R 5 400 for 6 months at 8% per annum.
(i) R 4 000 for 10 months at 9% per annum.
(j) R 450 000 for twenty years at 7,5% per annum.
(k) R 15 500 for 2½ years at 5,5% per annum.
2. If you invest R 700 and receive R 35 interest after one year, calculate
the interest rate.
3. Investor A earns R 28 interest per annum on an investment of R 400.
Investor B earns R 48 interest per annum on an investment of R 600.
Which investment is the more profitable?
133
4. A person who invested R 4 000 withdraws the investment after a year and
receives a total of R 4 375. At what rate was the interest calculated?
5. Determine what percentage commission an agent receives if he earns
R 3 200 on a transaction of R 80 000.
6. Calculate the length of time for R 5 000 to earn R 1 000 interest at a 10%
simple interest p.a.
7. Calculate the length of time for R400 to earn R160 at 8% simple interest
p.a.
8. Peter inherited R20 000. He wants to buy a car for R35 000. How long
must he invest his money at 6,5% simple interest before he can afford the
car?
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CHAPTER 14
DATA HANDLING
The following concepts are important in data handling.
➢ Primary data are data that you have collected yourself.
➢ Secondary data are data that somebody else has collected.
➢ The population is the set of any large group of things that we are trying to
measure. The number of units in this set is called the population size.
➢ A sample is the set of units that you do get to study. Surveys rely on samples to
get their data.
➢ In the special case when data are collected on the entire population, we call a
census. Every five year the government counts all the people in the country.
This is called a census. A census uses a questionnaire to gather information.
Questions are asked about things such as employment, access to running water
and electricity, and refuse disposal. These data are then organized and
analyzed to help the authorities plan for the future.
Collecting data
Step 1
When you choose your topic to investigate, the next step is to design a question.
The question should either be answered using a questionnaire, observation or an
experiment.
(a) When you collect date by questionnaire,try to design the question as simple as
possible for eg. Just a yes or a no. Another way of doing this, is to give simple
choices from which people need to make only one choice, for example:
• How much pocket money do you get in one week?
❖ Less than R10
❖ Between R10 and R20
❖ More than R20
Avoid questions with an “or” statement in them, for example, ‘Do you like running or
swimming?’
Questions should be short and easily understood.
135
Questions should not favour one point or view over another. Such questions are
called leading questions.
(b) When collecting data by observation, we need to consider the time and place of
the observation and if this method is actually answering your question.
(c) when collecting data by experiment, we need to ask ourselves whether we are
generating enough results to show what is happening.
Step 2
Anything that distorts data so that it will not give a fairly representative picture of a
population is called bias. Choosing your sample by chance is the only method
guaranteed to be unbiased.
A traffic survey of how busy a particular road is, is biased if you only count the
number of cars that pass a point over three days between 1 pm and3 pm. Why do
you think this gives a biased sample?
ACTIVITY
(a) Think about what you know about bias. Then look at these example of questions in
a questionnaire. See if you agree with and can add to the following comments:
Question Comment
‘Cool’ kids like rock music. Do you like
rock music?
This is a leading question. The first sentence
should not be there because it implies you are
not ‘cool’ if you don’t like rock music.
It is important to learn science in school.
Tick one box.
Agree Disagree Don’t know
This question is good. It is clear and
simple to analyse.
What is your age?
13-15 15-17 17-19
This question is confusing. The groups
overlap. Where, for instance, do you make
your tick if you are 17.
(b) make corrections to the biased and unclear questions.
136
10
8
6
4
2
ORGANISING DATA
Step 3
Do the survey and record it
Ask each person in your sample the same question. Record the answers. You do not
need to write down who said what.
For example, if your question is: ‘Do you buckle up when traveling in a vehicle?’ and
the people in your sample had a choice between answering ‘Always’ or ‘Sometimes’ or
‘Never’, you can use a tally chart to record their answers.
Do you buckle up when traveling in a vehicle? Always Sometimes Never
Number of learners اااا
ااا
اااا
اااا
اااا
Step 4
Data can be displayed using a table.
Do you buckle up when traveling in a vehicle? Always Sometimes Never
Number of learners 8 10 4
Data can also be displayed on a graph:
Bar graph to show how many learners buckle up
No o
f le
arner
s
Alw
ays
Som
etim
es
Nev
er
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Step 5
From your survey you can conclude that only 22
8 or 36,36% of the learners in your
sample wear a seat belt at all times when traveling in a vehicle. From this you can
decide, if your sampling was unbiased, that only about 36% of the learners in your
school wear seat belts at all times when traveling in a vehicle.
GRAPHING DATA
A graph is a very powerful visual tool and is often used when displaying sets of data.
Different types of graphs are: Bar graphs, histograms, line graphs, pie charts, stem
and leaf plots and scatter plots
A: BAR GRAPHS AND DOUBLE BAR GRAPHS
Bar graphs are useful for plotting categorical data ( data that fall into categories ).
There is one bar for each category and the height of the bar shows how many times
each piece of data occurs. We use bar graphs to represent data that takes single values
such as different makes of cars or the number of people in different jobs.
The number of times each piece of data occurs is called the frequency.
Examples:
1. The seriousness of road traffic injuries involving children is reduced in cases where
seat belts are worn.
The table below provides data from a survey on Youth Risk Behaviour that was
conducted in 2002 by the Medical Research Council.
Percentage of high-school learners who
Always wear seat belts when driven by someone else
National EC FS GT KZN LP MP NC NW WC
14,3 13 14,3 10,2 11,4 19,7 16,8 12,7 18,5 14,9
These data can be represented by a bar graph.
138
We use double bar graphs when we want to compare two categories
2. In another survey on high-school learners wearing seat belts the following date were
obtained:
Percentage of high-school learners who always
wear seat belts when driven by someone else
Grade 8 Grade 9 Grade 10 Grade 11
Boys 18,4 13,8 16,2 14,2
Girls 16,5 11,3 12,8 9,9
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EXERCISE 1
1. One hundred learners were asked about the types of books they were reading. The
table gives information about their answers.
No. of learners
reading non-fiction
No. of learners
reading fiction
No. of learners
reading no books
Total
Girls 31 23 6 60
Boys 25 5 10 40
Draw a double bar graph of this information Show the number of boys and girls on the
vertical axis and each category on the horizontal axis.
2. The history teacher collected the following marks from 20 learners in a test.
10 ; 1 ; 10 ; 8 ; 6 ; 3 ; 5 ; 2 ; 2 ; 5 ; 1 ; 6 ; 7 ; 7 ; 5 ; 7 ; 1 ; 0 ; 8 ; 5
(a) Copy and complete this table:
Marks 0 1 2 3 4 5 6 7 8 9 10
No. of learners 1 3
(b) Draw a bar graph of the information. Show the marks on the horizontal axis and the
number of learners in each category (frequency) on the vertical axis.
Boys
Girls
140
HISTOGRAMS
We use histograms to represent data that is measured on a scale, for example, mass or
length. Such data is called continuous data. We divide the scale in class intervals and
each measurement can only fall in one class interval.
Example: The table shows the mass of 200 avocados in different class intervals.
Represent the data in a histogram.
Class interval Frequency
51 – 100- g 5
100 – 150- g 15
150 – 200- g 70
200 – 250- g 60
250 – 300- g 30
300 – 350- g 20
Use the histogram to answer the following questions:
141
(a) How many avocados had a mass between 150 and 200 g ? Ans: 70
(b) How many avocados had a mass less than 150g ? Ans: 20
(c) What fraction of the avocados had a mass more than 250 g ? Ans: 4
1
200
50=
THE BROKEN LINE GRAPH
A line graph is usually used to show the change and the direction of change over time.
A line graph shows what happens to data as time changes. Usually, the measurements
are taken at regular time intervals. The time intervals are always shown on the
horizontal scale. The data values can be read from the vertical scale.
Sometimes we call a line graph a line series graph. The graph of a line series can show
whether there are:
• Trends ( a tendency to increase, decrease or remain the same over a period
of time.
• Random variations (small unpredictable movements)
• Seasonal variations ( a pattern that repeats regularly and predictably)
Example:
Traffic police gathered the following information about accidents caused by driving while
under the influence of alcohol. The information, as shown in the table below, was
collected over a four week period. Draw a line graph to represent the data.
Day Number of accidents
Monday 58
Tuesday 64
Wednesday 87
Thursday 54
Friday 182
Saturday 205
Sunday 148
142
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EXERCISE 2
1. Look at the following average climate statistics for Bethlehem in the Free State
for the period 1980 to 1990. On the same set of axis, draw two line graphs, one
showing the average daily maximum temperatures and the other showing the
average daily minimum temperatures. Show the months on the horizontal axis
and the temperatures on the vertical axis.
Month Average daily max (ºC) Average daily min (ºC)
January 27 13
February 26 13
March 24 11
April 22 7
May 19 2
June 16 -2
July 16 -2
August 19 1
September 22 5
October 23 8
November 25 10
December 26 12
2. The graph below illustrates the Standard Bank rand/US dollar rates of exchange
over the 10 consecutive business days in July and August 1999 as shown in the table
below.
Day 1 2 3 4 5 6 7 8 9 10
Rand rate per US$ 6,20 6,19 6,23 6,23 6,14 6,16 6,24 6,28 6,28 6,24
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The graph shows clearly the fluctuation (or changes) in these rates of exchange. These
fluctuations are very important to importers and exporters , especially if a deal is
pending.
(a) On which day in this period was the rand::
(i) strongest (ii) weakest against the US dollar?
(b) From day 5 to 8 was the rand strengthening or weakening against the US dollar?
(c) A South African importer is buying equipment from the USA to the value of
US$125 756. If she transfers the money at :
(i) the most favourable
(ii) the least favourable rate quoted in the table, calculate how much the equipment will
cost in rands.
(d) How much can the importer in (c) save if she transfers the money at the most
favourable rate?
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STEM-AND-LEAF DIAGRAMS
Stem plots or stem-and-leaf diagrams are used to represent numerical data.
Example: Draw a stem-and-leaf diagram of the following numbers:
46; 33; 67; 53; 61; 52; 69; 41; 70; 32; 45; 38; 50; 62; 61; 52; 66; 39; 51; 72;
59.
Answer: To display the data in a stem-and-leaf diagram, follow the following steps:
Step 1 Find the least and greatest
number. Identify the tens digits
in each
The least number, 32, has 3 tens.
The greatest number, 72, has 7 tens.
Step 2 Draw a vertical line and write the
tens digits from least to greatest
to the left of the line. These
form the stems.
3
4
5
6
7
Step 3 Write the units digits for each
value to the right of the line.
These form the leaves.
Stem Leaf
3 3 2 8 9
4 6 1 5
5 3 2 0 2 1 9
6 7 1 9 2 1 6
7 0 2
Step 4 Order the leaves in each row
from the least to the greatest.
Stem Leaf
3 2 3 8 9
4 1 5 6
5 0 1 2 2 3 9
6 1 1 2 6 7 9
7 0 2
NB. If you want to rank a set of three digit numbers, use the first two digits as
the stem.
146
Exercise 3
1. This set of data represents the ages of 40 women currently serving in parliament:
37 46 34 60 32 46 47 41 51 53 58 38
52 63 48 45 55 52 36 42 60 46 52 32
55 30 58 54 55 29 52 53 37 47 50 38
59 61 55 40
Show these data on a stem-and-leaf plot.
2. Maggie recorded the number of people taking the bus to town each morning for
three weeks:
27; 58; 24; 20; 43; 35; 78; 58; 53; 34; 36; 38;
45; 54; 47
Make a stem-and-leaf plot of the data.
PIE CHART
Pie charts are used to show comparisons. ‘The slices of the pie’ are called the sectors.
They show how the whole is divided up into different parts.
Example:
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 to show the information.
(b) Calculate the fraction and percentage represented by each eye colour.
Solution:
Colour of eyes Number of pupils Angle at centre
Blue
12 º360
30
12X = 144º
Brown 9 360
30
9X = 108º
Dark brown 4 º360
30
4X = 48º
Green 5 º360
30
5X = 60º
Total: 30 360º
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(a)
(b)
Colour Fraction %
Blue
5
2
30
12= 100
30
12X = 40%
Brown
10
3
30
9= 100
30
9X = 30%
Dark brown
15
2
30
4= 100
30
4X = 13
31 %
Green
6
1
30
5= 100
30
5X = 16
32 %
EXERCISE 4
1. The following table shows the area of the continents of the world and their
populations. Antarctica is excluded because there are no permanent inhabitants.
Continent Area in millions of km² Populations in millions (1991)
Africa 30,259 670
Asia 44,045 3 234
Europe 10,517 697
North America 24,237 432
South America 17,840 302
Australia 7,682 17
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(a) Draw two pie charts to represent this data, one the compare the areas and one
to compare the populations.
(b) Compare the two pie charts and comment on the population density in each
continent.
2. There are 48 pupils in a certain Grade 8 class. In a survey on their favourite drink
The following data was obtained:
Milo: 4 Cacao: 8 Tea: 12 Coffee: 24
Represent the data in a pie chart and label your pie chart properly.
3. There are 36 pupils in a class. The numbers belonging to the respective sports-
houses of the school are given in the following table:
House White Red Green Blue
No. of pupils 7 9 12 8
Construct a pie chart to represent these data.
4. Carefully study the pie chart below and answer the questions based on the chart;
Use a protractor to measure the angles.
Favourite subject: Grade 8
(a) List the subjects in decreasing order of popularity.
(b) If there are 24 pupils in this Grade 8 class, work out, for each subject, how
many pupils consider it their favourite.
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5. The income of a farm over a year broke down as follows: 30% from dairy
products; 20% from vegetables; 10% from fruit; 40% from maize.
Draw a pie chart to illustrate these facts.
SCATTER PLOTS
We use a scatter plot to compare two sets of data. A scatter plot can be used to show if
two sets of data are related.
Correlation is a measurement of how strong the relationship is between two sets of
data. When one variable decreases while the other increases, there is a negative
correlation. If the points are also in almost a straight line, we say there is a high
negative correlation. The dots in the scatter plot below show a high negative
correlation between fuel left and distance traveled.
When there is no correlation between the variables, we say there is no correlation, as in
the scatter plot below.
When two variables both increase, there is a positive correlation between them. If the
points are nearly a straight line, we say there is a positive correlation.
150
The might be outliers in the set of data. These are striking exceptions to the overall
pattern. A reason for the outlier on the above positive correlation graph in the second
month, could be that the baby had feeding problems or was ill.
We can draw in a line that best fits the data ( as in the graph below). A line of best fit
can be used to make predictions. For example, we can estimate how much the baby
will weigh the next month.
Example:
The teacher asked the learners in his Grade 8 class how many books they read in the
past month. The answers that he obtained are represented in the following table:
7 2 0 3 2 4 0 5 1 3
1 4 1 2 0 1 2 3 0 2
1 1 4 1 1 0 0 0 2 3
(a) Organize the data so that you can see how many learners read 0 books, how
many read 1 book, how many 2 books, etc.
151
(b) Represent this information on a scatter plot.
(c) Draw a straight line (line of best fit), through the scatter plot.
(d) Is there an increasing or declining tendency?
Solution:
(a) Frequency table:
No. of books read No. of learners Total no. of learners
0 ││││ ││ 7
1 ││││ │││ 8
2 ││││ │ 6
3 ││││ 4
4 │││ 3
5 │ 1
6
7 │ 1
Total: 30
(b)
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(c) Drawing a straight line through this plot will have the following effect:
(e) There is a declining tendency- fewer learners read many books.
EXERCISE 4 1. The following table shows the “provincial equitable share” of the 1998/1999 budget and the population of each province.
Province Population in millions Allocation in billions(R)
Eastern Cape 6,5 15
Gauteng 7 14,1
Kwazulu-Natal 8,7 17,6
Limpopo 5,4 11,1
Western Cape 3,7 9,5
North West 3,4 7,2
Free State 2,8 6
Mpumalanga 3,0 5,5
Northern Cape 0,7 2,1
(a) Show this information on a scatter diagram, labeling each point with the initials of
the province so that you can identify them.
(b) Draw the line of best fit and comment on the type of correlation.
(c) Does the scatter diagram indicate that the population size of each province is an
important factor in allocating the provinces’ shares.
153
Sum of all data items
Total number of data items
2. These are the English and Mathematics marks for a class of 14 learners:
Eng 60 55 40 90 83 76 34 59 64 28 96 62 70 32
Maths 64 51 45 80 74 86 21 63 50 40 68 51 49 26
(a) Draw a scatter plot of this information. (b) Draw a line of best fit and determine whether there is a correlation between a
learner's Maths mark and her English mark.
MEASURES OF CENTRAL TENDENCY
In many situations it is helpful to describe data using a single number that is most
representative of the entire set of data. We call such a number a measure of central
tendency.
Average and spread are the two most useful measures of central tendency.
The most common measures of average are mean, median and mode.
The most common measure of spread is range.
MEAN
The mean is also known as the arithmetic mean. It can be defined as the sum of
values divided by the number of values.
Mean = -----------------------------------
Example 1: A netball team scored the following goals in their first seven matches;
10; 15; 6; 7; 5; 40; 12
Mean = 7
95
7
12405761510=
++++++ = 13,6 goals.
When there are extreme values (such as the goal of 40 in this example), the extreme
value (outlier) pulls the mean to one side.
154
MEDIAN
The median (middle most number) of a set of values is found by placing the values in
order of size and then choosing the value in the middle.
Example 2: Consider the goals of the netball team in example 1 above.
First write them in order: 5; 6; 7; 10; 12; 15; 40
The median is 10 goals
If there are two numbers in the middle, the median is halfway between then. If the team
had played eight matches and scored 12 goals in the eight match, we would determine
the median as follows:
5; 6; 7; 10; 12; 12; 15; 40
Median = 2
1210+ = 11 goals
MODE
The mode of a set of data is defined as the item that occurs most frequently.
Example 3. A netball team scored the following goals in their fist 8 matches:
10; 15; 6; 7; 5; 40; 12; 12
Find the mode.
Solution: In order the number of goals are: 5; 6; 7; 10; 12; 12; 15; 40
The mode is 12 goals
RANGE
The range of a set of data values is the difference between the highest and lowest
values.
Example 4; Consider the number of goals of the netball team in example 3.
Solution: Range = 40 – 5 = 35
Example 5: The English teacher collected the following marks from all his learners in an
English grammar test: 1; 7; 0; 10; 8; 7; 3; 3; 5; 7; 6; 8; 7; 10; 7; 7; 6; 7; 5; 6
(a) Draw up a frequency table
(b) Draw a bar graph of the information
155
(c) Find the mean.
(d) Find the median
(e) Find the mode
(f) Find the range
Solution:
(a)
Marks obtained 0 1 2 3 4 5 6 7 8 9 10
Number of learners 1 1 0 2 0 2 3 7 2 0 2
(b)
Bar graph of English Grammar test marks
(c) To find the mean: The total number of learners is 20. The total number of marks
scored by these 20 learners is:
0X1 + 1X1 + 2X0 + 3X2 + 4X0 + 5X2 + 6X3 + 7X7 + 8X2 + 9X0 + 10X2 = 120
Mean = 20
120 = 6
(d) To find the median: Arrange the set of data from smallest to largest.
0 ; 1; 3 ; 3 ; 5 ; 5 ; 6 ; 6 ; 6 ; 7 ; 7 ; 7 ; 7 ; 7 ; 7 ; 7 ; 8 ; 8 ; 10 ; 10
The value between the 10th and 11th entry is 7.
Median is 7
(e) From the graph you can see that the value that appears most is 7.
the mode is 7
(f) The difference between the highest and the lowest values is 10 – 0.
the range is 10
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EXERCISE 5
1. The Grade 8 Mathematics teacher collected the following percentages from a recent
test: 26 ; 63 ; 54 ; 36; 44 ; 32 ; 60 ; 58 ; 38 ; 52 ; 55 ; 53 ; 47 ; 55 ; 55
(a) Write the data in order from smallest to greatest.
(b) Find the mean.
(c) Show the data on a stem-and-leaf diagram.
(d) use your stem-and-leaf diagram to find the following:
(i) median (ii) mode (iii) range
2. Arrange each of the following sets of data in order from least to greatest number and
then find: (i) The mean (ii) the median (iii) the mode (iv) the range
(a) 0 ; 0 ; 1 ; 2 ; 3 ; 3 ; 4 ; 5
(b) 4 ; 4 ; 4 ; 6 ; 6 ; 7 ; 7 ; 9 ; 10
(c) 4 ; 5 ; 3 ; 6 ; 1 ; 8 ; 4 ; 2 ; 7 ; 2 ; 1 ; 6 ; 9 ; 6 ; 5 ; 4 ; 3
3. Put the following sets of data on a stem-and-leaf diagram and determine for each set
of data: (i) the mode (ii) median (iii) range
(a) 16 ; 25 ; 34 ; 38 ; 27 ; 22 ; 69 ; 88 ; 65 ; 45 ; 42 ; 63 ; 69 ; 77
(b) 19 ; 28 ; 25 ; 26 ; 33 ; 51 ; 29 ; 43 ;56 ;29 ; 42 ;15 ;16 ;55 ;14 ; 30 ; 66 ;27 ;28 ; 29
3. The following sales at the school’s tuck shop are recorded over the 10-minute break:
Money spent
5 6 8
6 0 2 2 5 7 9 9
7 3 4 6 8 8 8
8 0 2 5 6 7
9 1
n = 21
5 6 represents R 5,60
(a) List the sales in order of size, from least to greatest.
(b) Determine the mean.
(c) How many learners bought goods worth more than the mean?
(d) How many customers did the shop have in those 10 minutes?
157
4. These are the shoe sizes of female learners in a Grade 8 class:
4 ; 5 ; 5 ; 5 ; 521 ; 5
21 ; 6 ; 6 ;6 ; 6 ; 6 ; 6 ; 6
21
Calculate: (a) the mean (b) median (c) mode (d) range
5. These are the test results for a Grade 8 General Knowledge competition:
Group A 51 53 45 57 48 49 53 53 56 62
Group B 50 45 57 48 49 53 53 56 58 59
(a) Calculate for each set of data:
(i) the mean (ii) median (iii) mode (iv) range
(b) Compare the two groups’ performances in the competition.
6. John noted the time he spent studying each day for one and a half weeks before an
examination:
Day Sun Mon Tues Wed Thur Fri Sat Sun Mon Tues Wed
Time (minutes) 30 140 100 90 150 100 65 40 130 100 95
(a) Calculate the difference between his longest and shortest time of study. (The range)
(b) What was the most common time interval he spent studying per day during the one
and a half weeks? (The mode)
(c) Calculate the total time he spent studying and then the mean time studying per day.
7. Using the following set of scores: 9; 9; 8; 8; 7; 6; 6; 5; 4; 4; 4; 4; 3; 3; 1 Find:
(a) the median (b) The mode (c) The mean
(d) The mean if each score is multiplied by 5.
(e) The mean if 10 is added to each score.
8. The symbols below show the ratings of a random sample of 13 learners for a Maths
project from A = excellent to E = weak:. B; C; D; C; A; E; D; B; C; E; E; D; C
Find: (a) The mode (b) The median.
8. Peter did a study on the distance (in kilometers) between the homes od the learners
in his class and the school. He recorded his data in the following table:
45 15 18 19 36 18 19 20 6 20 12 8
6 40 8 5 23 4 5 6 2 5 9 9
12 5 15 16 9 8 43 15 6 9 6
(a) Organize the data in a stem-and-leaf diagram.
(b) Determine the median
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(c) Determine the mode
(d) Calculate the mean
(e) What is the distance of the learner furthest from school?
(f) What is the distance of the learner closest to school?
(g) What is the range of the data?
159
The number of equally likely outcomes
The number of equally likely outcomes
CHAPTER 15 PROBABILITY
In this chapter you must know the following concepts:
Fair: Every possibility has the same chance of occurring.
Outcome: A number between 0 and 1, inclusive (or between 0% and 100%)
That measures how likely it is for a chance event to happen. On the two
extremes, events that can’t happen have a probability of 0
and events that are certain to happen have a probability of 1.
Random sample: A sample chosen from a population in such a way that
each member of the population has an equal chance of being chosen.
Relative frequency: The number of times an event happens in a statistical
Experiment divided by the number of trials conducted.
Sample space: The set of all the possible outcomes of an event, usually
written down as an organised listing of possible outcomes.
Trial: Experiment.
Probability or chance, is a way of telling the likelihood of an outcome
that might take place. The type of event is called an event.
The probability of an event = ---------------------------------------------------------
We write the probability of an event as P(event).
Probabilities can be shown on a probability scale. At one end, events that can’t happen
have a probability of 0. At the other end, events that are certain to happen have a
probability of 1. Probabilities can be written as fractions, decimals of percentages.
impossible unlikely equal chance likely certain 0 ½ 1 Examples:
1. What is the chance of getting a 3 when rolling a dice?
Solution: There are 6 possible outcomes, namely: 1; 2; 3; 4; 5; 6
P(3) = ---------------------------------------------------------
The number of favourable outcomes
The number of ways to throw a 3
160
X
The number of possible equally likely outcomes
I I I I I
Therefore on the probability scale this has an unlikely chance.
60 6
1
62
63 6
4
65 6
6
2. A set of cards is numbered 1; 2; 3; 4; 5; 6; 7; 8; 8; 9; 10; 11; 12
If you must draw a card without looking, what is the probability of choosing a
number less than 5 ?
Solution: There are four numbers lees than 5, therefore there are four
favourable outcomes. There are 12 cards in total, which means there are 12
possible equally likely outcomes.
P(less than 5) = ---------------------------------------------------------------
= 12
4 =
3
1
EXERCISE 1
1. This fair spinner is spun. Is the spinner more likely to land on a black or an
unshaded block? Give a reason for your answer.
2. Write the letters of the word probability on individual cards and put them
in a bag. A letter is selected at random and then returned to the bag.
(a) What is the probability that the letter selected is a b?
(b) What is the probability that the letter selected is a vowel?
(c) What is the probability that the letter selected is a p ?
(d) What is the probability that the letter selected is not an I?
The number of favourable outcomes
161
3. There is only a 15% chance of success in an experiment. What is the
probability of failure as a percentage and as a decimal?
4. (a) Write down all the possible equally likely outcomes in rolling a die?
(b) What is the probability of getting a:
(i) 4 (ii) prime number (iii) 9
(iv) number less than 9 (v) 4 or a 5 (vi) factor of 6
(c) What is the probability of not getting a 4?
5. If you roll a die 100 times, what will the probability be to get:
(a) a 2? (b) A number bigger than 6 (c) an odd number
6. The names of 5 learners, Ansie, Betty, Andrea, Sannie and Naomi are
placed in a hat and one name is drawn at random. What is the
probability that it:
(a) begins with a (b) is Betty or Naomi (c) has at least 5 letters
7. A bag contains 5 red, 3 blue and 2 green balls. If one ball is drawn at
random, what is the probability that it is:
(a) red (b) green (c) blue
(d) red or green (e) not green
8. There are 24 balls in a bag, some are red, some are green and some blue.
If one ball is drawn at random,, the probability of drawing a red ball is 3
1
and the probability of drawing a red or a green ball is 4
3. What is the
probability of drawing:
(a) a green (b) a blue (c) a red or a blue ball?
9. What is the probability that someone’s birthday is on:
(a) 1 May if he is born in May
(b) 5 February if she is born in February
(c) 31 April if he was born in April
(d) A Monday
(e) 29 February if it is 2008
(f) 29 February if it is 2010
162
Total number of trials conceded
Number of observed occurrences of the
event Total number of trials
RELATIVE FREQUENCY
Relative frequency is a measure of a chance happening in
actual experiments
Relative frequency = --------------------------------------------------- The probability of an event A occurring, is written as P(A). We know now that the
probability of an event can be calculated by dividing the number of favourable outcomes
by the number of possible outcomes. This is also called the theoretical probability. It
differs from the relative frequency. The relative frequency is determined by performing
the experiment a lot of times.
Relative frequency = ------------------------------------------------------
The relative frequency is not a good predictor of the chance of the event’s happening
unless the number of trials is very large. The more trials, the closer the relative
probability will come to the actual probability.
Example:
Kiara wants to test the relative frequency of getting a 3 when throwing a dice by doing
an experiment. She takes a dice and rolls it 50 times. She writes down her
results and draws up a table.
2 3 6 1 1 5 1 3 6 5
2 5 3 6 3 1 4 3 1 1
4 2 4 6 2 4 2 2 3 2
1 1 5 4 2 1 2 2 6 6
1 2 4 2 5 1 3 5 1 1
Dice number Frequency
1 13
2 12
3 7
4 6
5 6
6 6
The number of times an event happen
163
Number on dice 1 2 3 4 5 6
Relative frequency
13/50
12/50
7/50
6/50
6/50
6/50
The relative frequency of throwing a 3 is 50
7 = 0,14. But the probability of
throwing a 3 is 6
1 = 0,17. The larger the sample, the more likely the relative
frequency will be closer to the probability of the event.
EXERCISE 2:
1. Look at the following spinner and answer the questions that follow:
(a) Give the sample space of the experiment.
(b) Is it equally likely for the spinner to spin black, grey and white? Why?
(c) Determine the probability of white coming up on the spinner.
164
(d) Determine the probability of each of the other two colours coming up on the
spinner.
Vanessa spins the spinner 30 times and recorded her outcomes as follows:
Colour Outcomes Total
Black │││││ │││││ │││││ ││ 10
Grey │││││ │││ 8
white │││││ │││││ │││││ │││││ │ 12
(e) Determine the relative frequency of each of the three colours coming up on the
spinner. How does the relative frequency compare with the probability of each
colour coming up?
(f) What would you expect the relative frequency for each of the colours be if
Vanessa spins the spinner a 1000 times? Why?
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. 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 8
5. How
many men are in the audience ?
4. A card player has 14 cards in his hand including some kings. Another
player draws a card from the first one’s hand. If the probability of
drawing a king is 7
1, how many kings are there in the first player’s hand ?
5. A certain class decides to have a raffle. They sell 100 tickets.
(a) How many possible winning tickets are there?
(b) The class teacher buys 1 ticket. What is the probability of the teacher
winning?
(c) The RCL Representative buys 2 tickets. What is the probability of
her winning ?
(d) Pinky buys 10 tickets. What is the probability of Pinky winning ?
165
(e) How many tickets should you buy to have a 20% chance of winning
the raffle?
6.(a) How many letters are there in the words:
(i) hippopotamus (ii) hypotenuse
(b) List the possible outcomes if you choose a letter at random from the
Words: (i) hippopotamus (ii) hypotenuse
(c) How many p’s are there in each of the words?
(d) What is the probability of choosing a vowel at random from each of the
two words?
(e) How many vowels are there in each of the two words?
(f) What is the probability of choosing a ‘p’ at random from each of the two
words?
(g) How many consonants are there in each of the two words?
(h) What is the probability of choosing a consonant at random from each
of the words?
7. A set of 20 cards has one card each of the numbers from 1 to 12, two
cards numbered 15, three cards numbered 20 and three cards numbered
25. Find the probability of each event:
(a) P(7) (b) P(multiple of 5) (c) P(prime)
8. Two surveys include questions about access to running water in houses
in an urban village.
Survey A Survey B
No. with access to piped water 23 160
No. of homes in survey 33 400
(a) Calculate the relative frequency of homes with access to piped
water in each survey.
(b) Which survey is likely to better represent the situation in the whole
village?
166
CHAPTER 16
REVISION
EXERCISE 1
1. Determine whether the following sets of numbers are closed under the
indicated operations:
a) The set of odd numbers under addition
b) [0; 1] under subtraction
c) The set of multiples of 7 under multiplication
d) The set of perfect squares under multiplication
e) The set of composite numbers under addition
2. Determine the following by means of prime factors:
a) the H.C.F.(highest common factor) of 112 and 210;
b) the L.C.M.(lowest common multiple) of 18 and 24;
c) 784
d) 3 1728
3. If # indicates the operation: add the square of the first number to the
square root of the second number, determine :
a) 9 # 4; b) 4 # 9; c) Is # commutative?
4. Write down the algebraic expressions for the following:
a) The cost of x apples if 1 apple costs y cents
b) The number of minutes in z hours
c) The number of pupils present in a class of 35, if x pupils are absent
d) The next even natural number after n, if n is odd.
e) The sum of 3 consecutive natural numbers, x + 1 being the smallest
f) An even number
g) A multiple of 3
5. If x = 3, y = 2 and z = 1, find the values of:
a) x2y2 b) x2 + y2 + z2 c) (x + z)y
d) 3xy + 4yz e) (x + y)(y + z) f) x3 - y3
167
6. Simplify fully:
a) 4ab + 5bc + 6ab + bc b) 3x3y2z X 4xyz X 2x2y
c) bca
cba2
423
5
50 d) 3a2b(4a + 5b + 2c)
e) ab
abba
5
1520 22 +
7. Simplify:
a) 3x2 X 2x3 b) 2
25
8
16
xy
yx c) (3a2b)3
d) 32
223
)2(
)(2
qp
qp e) 2m2(m + 2) + m(2m2 + 1)
f) 2
23243
3
393
ab
abbaba ++
8. Determine x and y in each figure, giving reasons:
a) b) c) x d) 37o x
4x
3x 2y 45o y
x 70O
9. Draw a sketch of:
a) (i) an acute angle; (ii) a reflex angle
b) a pair of adjacent complementary angles
c) a pair of adjacent supplementary angles
d) a pair or supplementary angles which are not adjacent
10. Use a ruler and compasses only, to construct the following angles:
a) 60o b) 30o c) 90o d) 150o
168
11. Draw any obtuse angled triangle (use a ruler, make the triangle fairly large)
Drop perpendiculars (use a ruler and compasses) from the vertices of the triangle
to the opposite sides (or sides produced)
Do the perpendiculars you have drawn (extended if necessary) pass through a
common point?
2
12. Study the figure and then name: 3 A1
a) 6 pairs of corresponding angles 4 2
b) 7 pairs of vertically opposite angles 3 B1
c) 4 pairs of co-interior angles 3 2 4
d) 6 pairs of alternate angles 4 C1
5 6
13. Say which pairs of lines are
parallel in the following figures, giving reasons: R 50o F
a) A E b) E
60o
F H 70o 110o S 110o G H T M L 130o H A L
c) E F M R N
169
EXERCISE 2 1. Write down algebraic expressions for:
a) The number of months in m years n months
b) The change (in cents) if Rx is paid for an article costing y cents
c) The number of seconds in x minutes
d) The difference between the product of p and q and their sum
2. Simplify:
a) 7a - (-2a) b) -3x + 4x -(-5x)
c) (-4a2b) X (-3ab2) d) (-1)5 X (-1)3 X (-a)2
e) 222
1054
4
40
cba
cba− f) -3(2a - 4b - 5c)
g) (-4x3y2)2 h) 7a - 2a(5b - 1)
i) (7a - 2a)(5b - 1) j) 6x(2x - 1) - 2
k) 6x(2x - 1)(-2) l) -(3a - 4b -1)
m) xy
yxyx
2
810 432
−
−− n) 8a - 12a X 2
o) -7xx + 4x + 2x2 - 3x p)
−
x
x 442
3. a) Add 2x - y + 3; -3x + 4y - 5; -x - 4
b) From 2a - 3b + c subtract 6a - n2b - c
c) Divide abc
abcbcacba
4
846 23322
−
+−−
d) Determine the product of -2abc and 3a - 2b -c
e) Simplify: 3x(x + 2y) -2x(2x -2y)
4. Solve the following equations and check your solutions.
a) 2x - 15 = 3 - 4x b) 3(m + 2) = 2(m + 3)
c) 3
52 −p = -7 d) 15 -2(1 - y) = 3
5. a) The third side of an isosceles triangle is 15 mm longer than the other
two equal sides. What is the length of each side if the perimeter is
150 mm?
170
b) There are toffees, chocolates and boiled sweets in a bag. There are 3
more toffees than chocolates and twice as many boiled sweets as tof-
fees. If there are 69 sweets altogether, how many of these are choco-
lates?
6. Calculate the values of x and y in each of the following:
a) b) c)
x 51o 72o 81o
x y
½x 43o x y 59o
d) e) R
x 56o 80O
X
x
35o 114o y S T
7. Determine which of the following pairs of triangles are congruent. If they are con-
gruent, state the condition which applies.
a) b) c)
E P S V W
G
F Q R T U
H
d) E e) f) V
L M
T U
H
F G R P W
171
g) L T
78O
M 37O 78O N R 37O S
8. State whether each number is real and rational OR real and irrational:
a) -3 b) 5
2 c) -1
4
1 d) 18
e) 169 + f) 2536− g) 0,6 h) 4
12
i) 9,0
9. Between which two integers do the following irrational number lie?
a) 3 b) 50 c) 160 d) 8000
10. Determine the length of the hypotenuse in a right angled triangle in which
the other two sides measure 2m and 3m.
11. Simplify:
a) 4 - 3 X 3
1 b)
½¼
1
+ c)
2
2
1
3
2−
d)
6
53x
xx+
e) 2m - 3
2m f) ½ X 4 -
21
2
g) 1,6 X 0,3 h) 0,36 0,012 i) 0,36 120
j) (0,11)2 k) (-0,2)3 l) 0,5 X 2,52
172
12. If x = -4 and y = ½, evaluate:
a) x - 2y b) -2xy c) 2x2y
d) 2xy2 e) (2xy)2
13. Convert to a decimal fraction:
a) 8
7 b) -
25
2 c)
6
25 d)
7
3
EXERCISE 3
1. a) Factorise 36 completely b) List the factors of 36
c) List the prime factors of 36 d) List the first 2 multiples of 36
e) Determine the H.C.F (highest common factor) of 36 and 40
2. Simplify fully:
a) 4a2 + 5a - 6 + 3a2 - 6a - 8 b) 7x2 - (-3x2)
c) -4x2y X 13xy2 X -2xy d) 52
432
8
16
cab
cba
e) -3a(2a - 4b + 5c) f) 24
6524
6
126
ba
baba
−
+−
g) z
yx
3
4 23
X yx
z4
2
12
9 h) (-1)7 X (-1)6 X (-2)2
i) 0,6ab X 0,03a2b2 j) ab
ab
1
k) 0,0064 + 0,25 l) 64
4
m) 64
4 n)
4
5x +
2
3x
3. Simplify fully:
a) 5a(2a - a2) - (6a + 2a2)3a b) 7x - 5x(2 - 4x)
4. From the sum of 3y2 - 2y - 3; -4y - 2y2 + 5 and -1 + y - 6y2
subtract y
yyy
4
1248 23 −−
173
5. Show that (a - b)2 = a2 - 2ab + b2 if a = 2 and b = 3
6. Simplify:
a) 5
3
1½ +
b) 6
43
c) 0,75 (0,05)2
d) (0,3)2 + (0,4)2 - 2 X 0,3 X 0,4 X 0,5
7. Solve and check the following equations:
a) 5 - 2x = x + 2 b) 3 -2(x - 1) = 3x - 15
c) 4
82 −y = 2y + 4 d) 1 -
3
2 m− = m + 5
8. What ratio compares:
a) R270 and 50c b) 500 m2 and 1 ha
9. Divide: a) R270 in the ratio 5 : 6 : 7
b) 1,5 liter in the ratio 2 : 4 : 9 (answer should be written as ml)
10. a) A recipe uses 250g margarine and 500 g flour. If I wish to increase the
recipe in the ratio 5 : 2, how much margarine and how much flour will I need?
b) The cost of a certain brand of pocket calculators has decreased by 8% this
year. If they cost R15 at the beginning of the year, what do they cost now?
11. Which is the faster rate: 100m in 12 seconds OR 42 km in 1½ hours?
12. a) If I buy litre bottles of cold drink for 75 c per litre and sell 200 ml cups for
20 c each, what percentage profit will I make on the cost price?
b) If I take into consideration that the cups cost me 2 c each and I get 25 c
deposit back on returning empty bottles, what is my actual percentage profit
on the money I paid out – assuming there was nothing wasted? Give your
answer correct to the nearest percent.
13. a) Find the (i) mean, (ii) median, (iii) mode of the following set of data:
(IQ’s of a grade 8 class)
102 110 147 135 98 95 105 120 110 90
115 120 125 105 95 128 115 100 95 100
b) If 100 is taken as ‘average’ intelligence would you describe the class as
average on the whole?
174
14. Rockridge High takes pupils from three Primary Schools, A, B and C, in its
vicinity . There are 180 pupils in grade 8. An analysis of which Primary
School each comes from, resulted in the pie chart shown. Study the pie chart
and then answer the questions:
a) Which Primary School was the main feeder of pupils?
b) How many pupils came from each of the respective feeder schools?
c) What % of grade 8 pupils came from:
(i) Primary School A? (ii) Primary School B?
15. Draw a bar graph to show how Jane spent her pocket money during the month
of October. (Use graph paper)
Spent on Amount
Drinks and Sweets R5,50
Comics R1,50
Doll’s Clothes R8,00
Presents R9,00
Savings R2,00
175
16. Two boys investigating the number of peas in the pods of 2 kg of peas their
mother bought at the market, got the following set of data:
(number of peas per pod)
2 5 2 4 3 6 4 5 5 2 6 3 6 5 4
5 3 2 5 4 5 3 2 5 6 5 5 4 4 5
3 5 4 1 3 2 4 3 4 5
a) Make a bar graph to show how many pods there were which contained
respectively: 2, 3, 4, 5 and 6 peas.
b) Calculate the: (i) mean number of peas per pod
(ii) modal number of peas per pod
(iii) median number of peas per pod
c) Which of the statistics in (b) is the most suited to this experiment? Why?
17. ABCD is a quadrilateral in which angle A = angle B = angle C and
D =150o. Find the measure of angle B.
18. Calculate the sizes of the angles indicated by letters.
≠ 19. PQ and RS are two line segments intersecting at O. State what kind of
quadrilateral PORS forms under each of the following conditions:
a) OP = 0Q, RO ≠ OS, PQ ┴ RS;
b) OP ≠ OQ, OR ≠ OS, POS > 90O;
c) OP = OQ, OR = OS, POR < 90O;
d) OP = OQ, OR = OS, PQ ≠ RS and PQ ┴ RS;
e) OP = OR = OQ = OS, PQ ┴ RS;
f) OP = OR = OQ ≠ OS, PQ ┴ RS.
176
20. Calculate the lengths of the sides marked x and y in the kite shown
below.
21. In the figure, triangles ABC and CDE are congruent, and
angle B = angle D = 90o. BAC = 72o. If B, C and D lie in the
same line (BC ≠ CD) show that ACE is right-angled and isosceles.
22. Given that 1, 2, 3 and 4 are straight lines and 3 // 4
( 1 is not // 2)
Using only the angles given below, name the following:
a) one pair of adjacent angles which are not supplementary;
b) one pair of vertically opposite angles;
c) three angles of which the sum is 180o;
d) one pair of equal corresponding angles;
e) one pair of equal alternate angles;
f) one pair of non adjacent angles which are supplementary;
177
g) one pair of corresponding angles which are not equal;
h) a pair of complementary angles.
23. Determine the values of x, y and z in each figure (giving reasons)
178
24. State whether the following pairs of lines are parallel or not. If they are
parallel, give the necessary reason(s) why you say so.
25. In the school grounds, there is a rectangular lawn 50m by 40m. Pupils
are supposed to walk on the path around the sides of the grounds. What
distance would Sam save by walking diagonally across the grounds?
26. Complete the following table: [Redraw it in your script]
Formula for perimeter Formula for area
Rectangle
Square
Parallelogram
Rhombus
Kite
Trapezium
Circle
179
27. Calculate (All measurements are in mm):
a) the areas of the figures given below;
b) the value of x in each case.
28. Calculate the area of the figure (measurements in mm)
29. Calculate a) the area and b) the perimeter of the following figure:
[ 7
22 ]
30. Triangles ABC and DEF have the same area, and bases of 9 units and
12 units respectively. If their heights differ by 2 units, find their heights.
31. Rectangle B has an area of 60 mm2 less than rectangle A, and a breadth
of 8 mm more than A’s. If A’s length is 10 mm and B’s length is 9 mm,
find their breadths.
180
EXERCISE 4
MULTIPLE CHOICE REVISION
Five possible answers, a) b) c) d) e), are given for each question. Find
the correct answer and only write down the corresponding letter that indicate your
answer, next to the question number. Every question has only one correct answer.
1. 4
2 is equivalent to - a) ½ b) 2 c)
3
6 d) 1 d)
8
5
2. If x + 4 = 7, x will be - a) 11 b) -3 c) 3 d) 7 e) 4
3. 10
7 = ? : a) 10,7 b) 0,7 c) 1,07 d) 0,07 e) 0,007
4. 3 + 0,243 = ? a) 0,3243 b) 32,43 c) 3,423 d) 3,243 e) 3,432
181
5. Which one of the following is false?
a) 0,6 > 0,5 b) 10,39 > 11 c) 12,0 = 12 d) 0,89 < 0,9
6. In figure ABCD in a parallelogram. Find x.
a) 52o b) 25o c) 41o A x 41o B
d) 63o e) 650 72o 42o
B C
7. In the figure ABCD is a rhombus. The magnitude of C1 is -
a) 35o b) 40o c) 45o A B
d) 50o e) 55o
50O 1
D 2 C
8. a + a + a = ? a) a3 b) 3a c) 3a3 d) 2a3 e) None of these
9. 2a + 3b = ? a) 5ab b) 6ab c) 2ab d) 2a + 3b e) none of these
10. 2a X -3a = ? a) 6a2 b) 5a2 c) 6a d) -6a2 e) none of these
11. (ab)2 = ? a) a2b b) ab2 c) a2b2 d)2ab e) 2a2b2
12. –(-2a)(-3b) = ? a) -6ab b) 6ab c) 5ab d) -5ab e) 6a2b2
13. 2ab2 + ab2 + a2b = ? a) 4a2b2 b) 3ab2 + a2b c) 3a2b + ab2
d) 4a2b e) 4ab2
14. 2(3x + 2y) = ? a) 10xy b) 6x + 2y c) 3x + 4y d) 6x + 4y
15. -6x - (-4x) = ? a) 24x b) -10x c) -2x d) 10x e) 0
16. (4a2 + 16a) 4a = ? a) a + 4 b) 16a c) 4a2 + 9
d) 16a + a e) a2 + 4a
182
17. (4 X 9 + 4) - (6 + 2 X 4) = ? a) 8 b) -8 c) 0 d) -2 e) 26
18. In the figure, ABC is a straight line.
x = ? a) 10o b) 20o c) 30o X
d) 40o e) 80o 100O 3X
A B C
19. Arrange the following in order from least to greatest:
(i) 12
5 (ii)
4
1 (iii)
24
14
a) (ii) (i) (iii) b) (iii) (ii) (i) c) (i) (ii) (iii)
d) (ii) (iii) (i) e) (i) (iii) (ii)
20. p2q2 + 3pq2 + 4pq2 - p2q2 is equal to:
a) 7p2q4 b) -12p4q8 c) 7pq2 d) 9p2q8 e) none of these
21. 2(p3q)2 is equal to:
a) 4q6p4 b) 4p2q c) 2p5q2 d) 2p6q2 e) 4pq
Study the pie chart carefully and then answer questions 23 and 24.
23. Which of the following combinations accounts for
the greatest proportion of the whole represented? 44
a) A & C b) A & B c) B & C d) A & D
24. Which percentage of the total belongs to C?
a) 45% b) 25% c) 12½% d) 10% e) 5%
25. 24 3
2 of 6 is equal to :
a) 4 b) 864 c) 196 d) 216 e) 6
26. 2pq - (3pr - 4ps) is equal to -
a) 3p b) 2pq - 3pr - 4ps c) -5prs d) 2pq - 3pr + 4ps
D
A
C
B 45o
183
27. In pq(b - c), the factors are -
a) p, q and (b – c) b) p, q, b and c c) (b – c)
d) p and q e) cannot be resolved into factors
29. For questions 29 - 33: Select from the list below, the label that describes the
angles in each of the following diagrams -
a) complimentary b) adjacent supplementary c) corresponding angles
d) alternate angles e) co-interior angles
34. 0,001 km is equal to -
a) 1 m b) 100 mm c) 0 dm d) 10 m e) none of these
35. 3,21 3 = ?: a) 9,63 b) 1,07 c) 3.07 d) 1,7 e) 10,7
36. Which of the following is not a factor of 840?
a) 4 b) 7 c) 15 d) 17 e) 24
37. Which of the following in NOT a prime number?
a) 61 b) 57 c) 47 d) 31 e) 23
38. The HCF (highest common factor) of 8x2y2z4 and 3xyz2 is -
a) 3xyz2 b) 8xyz2 c) 24xyz d) 24x2y3z2 e) xyz2
39. The LCM (lowest common multiple) of 8x2y2z4 and 3xyz2 is -
a) 8x2y3z4 b) 24x2y4z4 c) 3xyz2 d) 24x4y4z2 e) 3xyz
40. Twice a certain number increased by 5, is 20. The number is -
a) 10 b) 12 c) 15 d) 25 e) none of these
41. 8
3 expressed as a decimal, is -
a) 3,8 b) 0,375 c) 0,38 d) 0,0375 e) 0,3
184
42. The recurring decimal 0,1111….., when expressed as a fraction, is -
a) 11
1 b)
3
1 c)
9
1 d)
9,0
1 e)
1,1
1
43. The perimeter and area of a rhombus of side 50 mm with one diagonal is
equal to 60 mm, are -
a) 200 mm : 1200 mm2 b) 100 mm : 2 400 mm2
c) 200 mm : 2 400 mm2 d) 200 mm; 3 000 mm2
e) 200 mm; 4 800 mm2
44. The perimeter and area of a circle of radius 7 cm are respectively : [ = 7
22]
a) 44 cm, 154 cm2 b) 22 cm, 154 cm2 c) 44 cm, 616 cm2
d) 154 cm; 44 cm2 e) none of these
45. Solve for x: 2x = -8 : a) -4 b) 4 c) -10 d) -6 e) -4
1
46. Solve for x : x - 9 = 2: a) 9
2 b) 11 c) -7 d) 7 e)
9
2−
47. The mean, median and mode (in this order) of the following set of data
2 3 4 3 7 3 6 , are :
a) 3; 4; 3 b) 3; 3; 4 c) 4; 3; 3 d) 4; 4; 3 e) 4; 3; 4
48. What is the value of the expression: (a –b) –c(2a + b)
if a = 2, b = -3 and c = -4
a) 9 b) 3 c) 1 d) 12 e) 5
49. What is the value of the expressing
2
2
r
pq if p = -4; q = 3 and r = -2
a) 6 b) -6 c) 9 d) -9 e) none of these
50. 9
2
3
2 = ? : a)
27
4 b)
9
4 c)
3
1 d)
81
8 e)
2
1
EXERCISE 5
1. Express each fraction as a percentage:
(a) 5
1 (b)
4
9
185
2. Express each percentage as a decimal:
(a) 52% (b) 149% (c) 13% (d) 0,6%
3. Express each decimal as a percentage:
(a) 0,8 (b) 0,63 (c) 1,33 (d) 0,05
4. How much is 22% of R 298?
5. The average selling price of houses in rural areas in 2004 rose to R 532 000
from R 440 000 in the previous year. How much was the percentage increase?
6. First Investors Bank offers 7% interest per year.
(a) How much interest would you get in 1 year on R 350?
(b) How much money would you have altogether?
7. Which earns more in a year: R 1 500 at 11% per year or R 1 800 at 9% per year?
8. Calculate the interest on R 17 500 for a four-month period at 1221 % per year.
9. Palesa is a keen golfer. She has been saving for a new set of clubs
The cash price is R 1 340 for the set. But there are also three HP offers available:
They are: HP offer 1: No deposit, 18 monthly instalments at R 112 per month.
HP offer 2: No deposit, 24 monthly instalments at R 91 per month.
HP offer 3: R 220 deposit, 36 monthly instalments at R 56 per month
How much more than the cash price would each of the HP schemes cost?
10. Tshepo has family everywhere. For his birthday, he got 20 dollars from his
grandfather.in the USA and 20 pula from his aunt in Botswana. Tshepo
checked the exchange rates for that day at the bank:
❖ 6,17 rand to the dollar
❖ 0,81 pula to the rand
How much did he get altogether?
EXERCISE 6
1. A bureau de change is selling rands at R 11,28 to ₤1,00. The bureau charges 2%
commission on exchanges. Calculate how many rands you would get, after the
commission fee, for the following amounts:
(a) ₤50,00 (b) ₤485,00 (c) ₤1 225,00
186
2. The same bureau de change is buying rands at Є0,21 per R1. Calculate how
many Euros you would get, after the commission fee, for the following amounts:
(a) R 5 600 (b) R 568,98 (c) R 1 345,65
3. Five visitors arrive in South Africa from various parts of the world. The table below
shows the amounts that they want to convert to rands, and the exchange rate.
(a) Find the rand equivalent of each amount.
(b) The agent charges 1,5% commission on foreign exchange. Deduct this
amount from each total.
Country of origin Currency Exchange rate Amount
Jamaica Jamaican dollar (J$) J$1 = R0,09 J$16 000
U.S.A. US dollar (US$) US$1 = R 7,42 US$430
Canada Canadian dollar (CAN$) CAN$1 = R 5, 48 CAN$720
Germany Euro (Є) Є1 = R 8, 32 Є450
Japan Yen (¥) ¥1 = R 0,0653 ¥34 560
4. A furniture store sold a fridge for R 3 650 and made a profit of 15%. Calculate the
cost price of the fridge.
5. A shop owner bought and sold two articles, A and B. Article A was sold for
R 2 100 and made a profit of 50%. Article B was sold for R 2 860 and made a
profit of 46%.
(a) Calculate the cost price of each article.
(b) Which one was the more expensive article?
(c) On which article did the shop owner make bigger profit?
6. The cash price of an washing machine is R 4 200. If bought on hire purchase,
the buyer must pay a deposit of R 30% plus 12 monthly instalments of R 325.
(a) The hire purchase price.
(b) The total interest paid.
7. Sydney bought a second hand motor bike for R 8 599 and after two years sold it
for R 6 350..
(a) What loss did Sydney make?
(b) what percentage loss did he make?
187
8. (a) Calculate the simple interest on a loan of:
(i) R 5 500 over 3 years at 10,25% per year.
(ii) R 29 000 over 6 years at 13,5% per annum.
(b) For each loan above, find the total amount paid to the bank by the end of the
specified period.
EXERCISE 7
1. If $ 1 buys R 5,85 and ₤1 buys R 12,28 work out the values of the following
amounts in rands:
(a) $ 475,95 (b) ₤ 12,80 (c) $ 15,00 (d) ₤ 1865
2. Work out whether the seller made a profit or a loss on each item. Then work
out the percentage profit or loss ( rounded off to the nearest 0,5%) and the actual
profit or loss.
(a) I bought a motor car for R 325 000 and I sold it for R298 000.
(b) I bought an apartment for R215 000 and I sold it for R489 000.
3. Julian bought a television on hire purchase. The cash price of the television
was R3899.99 but Julian paid a deposit of 25%, followed by 12 instalments of
R 341,25.
(a) What was Julian’s deposit amount?
(b) What is the total H.P. price?
(c) How much interest did Julian pay?
4. Calculate how much water this cattle trough can hold:
188
4. 36 teams travel to a soccer tournament. They use the following as their main
means of transport:
Plane: 12 Bus: 14 Train: 9 Car: 1
Construct a pie chart to illustrate the data.
5. The pie chart below shows the relative masses of different types of food
eaten by a cat in one month.
(a) Calculate the fraction of the cat’s total diet each month that is made
up of dry food.
(b) Altogether, the cat eats 12 kg of food each month. How much of each
type of food does it eat?
6. A certain high school has 90 learners in Grade 8 classes. Each learner was
asked to find out on which day of the week they were born. Here are the results:
Day of the week Number of learners Angle at centre
Sunday 9 36
Monday 12
Tuesday 18
Wednesday 15
Thursday 15
Friday 11
Saturday 10
(a) How many sectors will be needed to show this data on a pie chart?
189
(b) Copy the table and complete it.
(c) Draw the pie chart to display the data.
7. Jane scored 34 ; 40 ; 35 ; 33 ; and 39 out of 50 for five tests.
(a) Calculate her average (mean) mark out of 50.
(b) What is her mean percentage.
(c) What is her median percentage?
(d) Write down the range of the marks scored by Jane?
8. The mean percentage in a test written by 20 learners was 64%. A new girl
joined the class and wrote the same test, scoring 85%. Calculate the mean
percentage for the whole class of 21 learners.
9. Nine learners scored the following percentages in a test.
80% ; 82% ; 30% ; 35% ; 80% ; 10% ; 20% ; 88% ; 25%
(a) Calculate the mean of the data.
(b) A learner must score 40% to pass. How many learners failed the test?
(c) Do you think the mean is a good representative value of the data?
Give a reason for your answer.
(d) Determine the mode of the data.
(e) Is the mode a good representative value to use for the data?
(f) Determine the median of the set of data.
(g) Do you think the median is a good representative value of the data?
EXERCISE 8
1. Determine whether the following numbers are divisible by 5 and by 6. Give
a reason for your answer.
(a) 3 215 (b) 42 420 (c) 342 006
2. Paul needs 2 m of rope. He has 1 m 75 cm of rope. What fraction of the rope
does he still need?
3. Write 70c as a fraction of R2,80.
190
4. James’s father contributed 7
2 of the money he needed for his movie ticket and
his mom gave him 5
1 of the price. What fraction of his movie ticket does he
have to pay?
5. John spends 3
1 of the day at school and
8
3 of the day sleeping. What fraction
of the day remains?
6. Arrange the values in descending order:
(a) 5
42 ; 290% ; 2,95 ; 200% ;
100
209
(b) 1,07 ; 136% ; 10
17 ;
3
1
7. Calculate:
(a) 7
2 of one day (b) 2)
5
12( (c) 2)
8
31(
8. Eric and Martin share 21 sweets between them in the ratio 4:3.
(a) How many sweets does each one get?
(b) What fraction of the sweets does martin get?
EXERCISE 8
1. Simplify:
(a) a² x a3 (b) 2a² x 3a3 (c) (x6)2 (d) (2a3)²
(e) at . ap (f) 2(5x²y)²y² (g) 4(5x²y²)y²
(h) k
k
3
6 3
(i) am
a
24
6 3
(j) 22
10
)2(
16
a
a
(k) xn+2.x2.xn+1 (l) 3a² + a² (m) 4p + 3m -2p
(n) 3a² + 2a + 2a² -a (o) (2a²)²(3m3) (p) 4(2a + 3b)
(q) 4
23
4
)4(3
p
p (r)
43
3 ²)²)²(²(
ma
amma (s)
3
23
3
)6(2
y
yxy
191
2. Simplify:
(a) ²
)2(3 55
x
xx + (b)
)²2(
²4
a
a (c)
)²4(
32 66
ab
ba
(d) 61016 ba (e) 2(3a5)2 (f) (5a²)(3a3) + (2a3)(2a)
(g) 3 968 ba . (3a²)² (h) 3a²b + 4ab² + 2²b (i) 3 3
3
27
a
a
3. Write each answer in the form of an algebraic expression:
(a) What is the square of three times a number?
(b) What is three times the square of a number?
(c) The number 9 is added to a number and the sum is divided by 3.
(d) What is the sum of two consecutive numbers; x is the smaller number?
(e) Write down a number, multiplied by two, and the product subtracted from 11.
(f) A boy is now x years old; his father is three times as old as he is. How old
will his father be in five years’ time?
(g) An athlete runs 1 km on Mondays. On Tuesday, he runs twice as far plus
three km.
❖ How far does he run on Tuesdays?
❖ What is the total distance that he runs on a Monday and a
Tuesday?
4. How many terms in each expression?
(a) 12x + 2(3x + 5) X 2x (b) xx
x
352
)1(3
−
+
5. What is the sum of: 2(2y-3x) and 6xy + 4y ?
6. Subtract -6x + 3 from -12x² + 4x + 6
7. Substitute a = 2; b = -1 ; c = -3 into each expression below and calculate the
answer:
(a) (a + b)² (b) abc (c) ab²c (d) a²b – c²
(d) 3a - 2b + 5c (e) 3(a² - b²) – (a² + b²) (f) (b + c)a
8. Solve these equations:
(a) 5n - 3 = 3n + 9 (b) b – 2 = 5b – 18
(c) 4(k + 1) - k = 8k - 20 (d) 4(y – 2) = 3(y + 1)
192
CHAPTER 17
TESTS AND EXAM PAPERS
EXAM PAPER NOVEMBER Total: 125
PAPER A(1)
Question 1
Are the following statements true or false?
1.1 All prime numbers are odd
1.2 The product of an even number and an odd number is always even.
1.3 The square of 16 is 4
1.4 The sum of two negative numbers is positive
1.5 If x = -5, the x2 = -25 [5]
Question 2.
Simplify each of the following without using a calculator:
2.1 3 - 4 + 7 - 6 + 9 2.2 15a - 28a - 14a
2.3 -4(-9 - 6) 2.4 (-1)10
2.5 -6(2b - 4b) 2.6 7m.4m
2.7 (3x + 7) + (4x - 8) 2.8 xy
yx
2
8 2
− [14]
Question 3.
3.1 Complete the following diagrams:
a) 4 ?
-2 ? (3)
? 12
x 3 + 3
193
b) 6,5 ?
-2 ? (3)
? -16
c) 12 ?
(4)
? -16
3.2 Write down an algebraic expression for each of the flow diagrams above. (8)
[18]
Question 4.
The mass of an empty truck, which is used to transport bags filled with sand, is 2 720
kg. Each bag of sand has a mass of 80 kg.
4.1 What is the total mass of the truck and its load if it carries 52 bags of sand? (2)
4.2 If the total mass of the truck and its load is 5 280 kg, how many bags of sand(2)
are on the truck?
4.3 Complete the table:
Bags of sand 5 10 15 20 25 30 35 40
Total mass in kg
(4)
4.4 If the total mass of the truck and its load should not exceed 6 500 kg, what is
the maximum number of bags of sand that it may carry? (3)
[11]
Question 5.
5.1 Replace each of the following expressions by an equivalent expression in its
simplest form. Show all your work.
a) 4a _ 2 + 9a - 0,5 - 10a + 4 b) (10x2 - 3x) + 4x(2x - 5)
+ 7 x 2
- 2 x 3 + 2 x 4
194
c) 3(4x - 5) - 3(4x + 8) d) (30x2y + 18xy2) - 6xy2
(11)
5.2 Simplify (without a calculation):
a) -3(-4a)3 b)
+
4
1
3
7 -
8
13
c) (-a2b)(-2ab2) d) -43
2 -
5
7
e) 2
a -
5
a +
4
3a f)
ab
aabba
6
31824 23 +−
g) 3
2x -
−
8
5
4
31
xx h) 168100 yx
i) 3 3927 ba− j) 1444
k) 3 027,0 (32) [43]
Question 6.
6.1 Write down all the prime numbers between 24 and 48. (2)
6.2 Write the expression x2 - 5x3 + 2x + 4 + 3x in descending powers of x.
a) Write down the co-efficient of x3
b) What is the constant term?
c) How many terms are in the expression? (4)
6.2 If x = -4 and y = ½, determine the value of each of the following:
a) x - 2y b) (2xy)3 (4) [10]
Question 7.
Solve each of the following equations:
7.1 3x - 5 = 7 7.2 2(2x + 5) = -2(1 + x)
7.3 7 - 2y = -5y + 4 7.4 -2
x + 5 = 3 [10]
Question 8.
8.1 Write the following ratios in their simplest form:
a) 35 : 49 b) 250g : 1,5 kg
c) 2,5 m : 75 cm (6)
195
8.2 Divide R51 between Ronny, Elize and Jack in the ratio of 2 : 7 : 8. (4)
8.3 A certain biscuit recipe, which produces 10 portions, requires 500 g butter and
1 kg flour. If I want to adjust the recipe so that it only produces 4 portions, how
much butter and flour will then be required? (4)[14]
######################################################################
PAPER B (1) NOVEMBER Total: 125
Question 1.
Complete the following statements:
1.1 The identity element of multiplications is _________________________
1.2 The expression (3xy + 3x2) + 2(x - 1) has _______ terms.
1.3 The additive inverse of -3x is __________________________________
1.4 3,14 is a ______________________ number.
1.5 Any number raised to the power 0, is ____________________________ [5]
Question 2.
2.1 Replace each of the following expressions with an equivalent expression in its
simplest form.
a) 5x + 4 + 3x - ½ + 7x - 4x (2)
b) (2m + 8) - (5m - 2) (3)
c) 2(3x + 8) - 3(12x + 15) (4)
2.2 Draw flow diagrams for each of the following algebraic expressions:
a) 3x - 6 b) 3(4x + 2) (5)
2.3 The cost of a long distance train ticket is calculated as follows: R20,60 for
any journey, plus 30 cents per kilometer.
a) Write down a formula to calculate the cost. (2)
b) What is the constant term? (1)
c) Complete the following table:
Number of km 320 712 3480
Cost (Rands) 116,60 506,60 1578,20
(4)
196
d) Draw a flow diagram to represent the situation. (2) [23]
Question 3.
3.1 What is meant by “the solution of an equation”? (2)
3.2 Without solving the equation -12 + 3x = 2(3x - 7), show that x = 3
2 (5)
3.3 Solve the following equations:
a) 8 = 40 - 2x b) 2
x = 3
c) 3(x + 5) - 4 = 12 + x d) 3 - 2(x - 1) = 3
459 −x (12)
3.4 Compile an equation and determine the value of the unknown:
a) 3 times a certain number increased by 4 equals 37 (4)
b) A boy travels the first part of his journey to school by bicycle. The dis-
tance he then travels by bus, is 5 times as long as the first part. The total
distance to the school is 18 km. How far does he travel by bicycle? (4)
[27]
Question 4.
4.1 Express 11 880 as a product of its prime factors. (4)
4.2 What is the smallest number divisible by 1; 2; 3; 4 5 and 6? (2)
4.3 How many prime numbers are there between 10 and 48? (2)
4.4 If you start with 4 and count in threes, you get the following sequence:
4; 7; 10; 13; ……
a) Use your calculator to determine if 76 is a number in this sequence.
b) What is the value of the 63rd number in the sequence?
c) What is the position of 376 in the sequence? (6)
4.4 Consider the expression 2x3 - 3x2 + 6 - 5x.
a) What is the co-efficient of x?
b) What is the exponent of x in the second term?
c) What is the constant term?
d) Write the expression in ascending powers of x. (4)
[18]
197
Question 5.
5.1 Simplify without using a calculator:
a) 3a + 2b _ a + 4b b) 3a2b(4a - 5b + 2c)
c) bca
cba2
23
9
45 d) 2x(x + 4) -5x(2x - 3y)
e) (2ab3)3 f) (-4a2b3) x (-3a4b2)
g) 2
23253
3
393
ab
abbaba +− h)
4
3x +
5
3x -
10
3x
i) 5
2a
−
7
15
14
5 aa j)
12
664,0
y
x
k) 312
36
434
648
y
bxb (32)
5.2 If a = 4
3, b = -3 and c = 4, determine the value of
+
82
ca 2 + b2c (5)
[37]
Question 6.
6.1 Determine the rate in each of the following cases:
a) A pipe that fills a dam delivers 30 m3 in 40 minutes (m3/minute) (2)
b) Toothpaste is sold at R3,50 for 70 ml. (ml/R) (2)
6.2 Niven and Ainsley inherit R990 from a wealthy family member. For every R5
Niven receives, Ainsley receives R6. How much does each receive? (3)
6.3 The number of pupils in a school increases in the ratio 5 : 6. If there are 780
pupils in the school after increase, how many were there originally? (3)
6.4 A bricklayer make a concrete mix as follows: 5 parts sand; 4 parts cement and
3 parts of stone. How many parts of cement and stone must he use respectively
if he uses 8 parts of sand? (5)
[15]
********************************************************************************************
198
PAPER A(2) NOVEMBER 75 MARKS
Question 1
Are the following statements true or false?
Supplementary angles add up to 90º.
If two straight lines intersect, then the sum of the vertically opposite angles is 180º.
The size of each angle in an equilateral triangle is 60º
A median of a triangle is perpendicular to a side of the triangle.
The formula for the volume of a cylinder is r²xheight. (5)
Question 2
2.1 The diameter of a 50 cent coin is 2,2 cm. calculate to one decimal place:
(a) The circumference of the coin. (3)
(b) The area of the coin. (3)
2.2 Determine the area of the circle which is not covered by the triangle.
(6)
Question 3
3.1 Show by means of sketches:
(a) A pair of vertically opposite angles.
(b) A pair of complementary angles.
(c) A pair of corresponding angles. (3)
3.2 Calculate the value of x in each of the following figures. Show all your
steps and give reasons.
199
Question 4
4.1 Calculate the values of x and y. Give reasons for your answers.
200
Calculate the area of PQRS:
S
(8)
Question 5
5.1 The minute hand of a church clock is 400 mm long and the hour hand is
90 mm. Calculate the distance between the endpoints of the two hands at
3 o’clock.
(3)
5.2 Calculate the total area of canvas required to manufacture a triangular
tent with dimensions as shown in the sketch. Give answer correct to
two decimal places.
two decimal place
(5)
201
5.3 The figure shows a cylindrical can of cooldrink with the dimensions of the
net of the curved section.
(a) Calculate the radius of the can.
(b) Calculate the volume of the can.
(4)
[75]
PAPER A(3) NOVEMBER 120 MARKS
1. Simplify:
(a) a2.b3.ab2 (b) 2a + 3b + 3a - b (c) 3x + 2x + x
(d) (3x)(2x)(x) (e) 3x + 4x 2x (f) (3x + 5x) 2x
(g) x.x + x.x.x + x.x.y (h) (2a + 6a) 2 + (7a – a) 3
(i) 2a + 6a 2 + 4a 2a (j) a
aaa
3
235 −+
(k) 12x3y2z4 4xyz3 (l) 2x3y2z2 X 3xy2z2
(m) 2x(3x2)2 (n) abc + 2cab - cba
2. Consider the expression: x3 + 4x2 - 3x + 7
(a) What kind of order is it in?
(b) What is the coefficient of the third term?
(c) What is the degree of the expression?
(d) What is the index of the third term?
(e) What kind of expression is it?
202
3. (a) Find a formula for each of the following and then give it in its simplest form:
(All measurements are in millimeters.
The length of carpet (L) needed to run from A to B on this flight of stairs.
(b) The total length of wire (L) needed to make the skeleton of this cube.
(c) The length (L) of the side of a square if its perimeter is 12a.
4. If x = 2, y = -2 and z = -4 find the value of each of the following expressions:
(a) x2 + y2 + z2 (b) x2y (c) xy2
(d) (x + y)2 (e) (2xy)3 (f) (z – x)(z + x)
5. Solve for x:
(a) 2x + 5 = 13 (b) 124
=−x
(c) 4x - x + 2 = 8
6. Complete the following statements:
(a) An angle greater than 90º and smaller than 180º is called a(n) ………angle.
(b) A triangle with two sides of equal length is called a(n) ………..triangle.
(c) A quadrilateral with only one pair of opposite sides parallel is called a(n) ……..
(d) A plane figure with ten sides is called a(n) ………
(e) A prism with eight surfaces is a(n) ……..prism.
(f) A pyramid with eight surfaces is a(n) ……..pyramid.
(g) A regular polygon is ………..
(h) In 40 minutes, the minute hand of a clock turns through ………degrees.
(i) In 5 hours, the hour hand of a clock turns through ………degrees.
203
7. Calculate the sizes of the angles marked a to f. Do not measure.
9. (a) Express 11 880 in powers of its prime factors.
(b) What is the smallest natural number by which 11 880 must be multiplied,
to make a number divisible by 21?
(c) Find the value of x which make the four-digit number 4x58 divisible by:
(i) 3 (ii) 11 (iii) 99
10. The row of numbers 1X2X3, 2X3X4, 3X4X5, 4X5X6, ………is obtained in a
systematic way.
(a) What will the next number be?
(b) What will the 17th number be?
(c) What will the nth number be?
11. (a) John is x years old and Peter is two years older. Andrew is twice as old as
John’s and Peter’s combined ages. Express Andrew’s age in terms of x.
(b) If the ages of all three add up to 24 years, make an equation and solve
it to find John’s age.
PAPER A(4) 120 MARKS
1. (a) Simplify the expression 4x2 + 8x3 - 5 + 3x + 4x - 3x3 + x2 , writing
your answer in descending powers of x.
(b) Look at the simplified expression:
(i) What is its degree?
(ii) What is the coefficient of the x term?
(iii) How many terms has it got?
204
2. Simplify:
(a) 3a + b + 2a + 3b (b) (3a2)(5a)(a) (c) a2.b.a5.b4
(d) 2x(4x2)2 (e) (3x + 5x) 2 (f) 3x + 4x 2
(g) ab
ba
4
8 3
(h) xy + x.x.y + yx (i) 5a + 3a.b
(j) (x –x)y + 2(x + x) (k) 5(x2 + 3x2) (l) 2xy + 3y2 + 5xy – xy + 4x2 + y2
3. (a) y = 2x2 + 1. Find the values of y if: (i) x = 0 (ii) x = 1 (iii) x = 2
(b) If a = 0, b = 1 and c = -2, find the value of each of the following:
(i) ab + bc (ii) bc2 (iii) (b + c)2
(iv) a2 + b2 + 2c2 (v) (c – b)(c + b)
4. (a) Write down the next two terms in each of the following sequences:
(i) 2 ; 5 ; 8 ; 11 ; …….. (ii) 3 ; 9 ; 27 ; 81 ; ……..
(iii) 1 ; 4 ; 9 ; 16 ; ……… (iv) 0 ; 3 ; 8 ; 18 ;……….
(b) Write a formula for the nth term in each sequence. Each formula must start
with Tn = ………..
5. (a) The sketch shows the skeleton of a rectangular prism. Write a formula for the
length of wire (L) needed to make this skeleton. Give answer in simplest form.
(b) Find the length of wire used if x = 9, y = 10 and z = 15.
6. (a) Write down the two-digit number with tens-digit x and units-digit y.
(b) Write down the number with its digits reversed.
(c) If y = 2x, express the numbers in (a) and (b) in terms of x.
(d) If it is further given that the two numbers add up to 132, find the
larger number. (Make and equation and solve it)
7. Find:
(a) All the prime numbers smaller than 40.
205
(b) All the ways in which 36 can be expressed as the sum of two primes.
(c) The smallest natural number which is divisible by 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8
and 9.
8. Complete the following statements:
(a) A reflex angle is ……………….
(b) Between 3o’clock and 7o’clock the hour hand of a clock moves through
……………….. degrees.
(c) A wheel has 24 spokes. The size of the angle between each adjacent pair
of spokes is …………….. degrees.
(d) 108º is a(n) …………… angle.
(e) A triangle with all its sides equal is called a(n) ……………… triangle.
(f) A hexagonal pyramid has ……………… vertices.
(g) A triangle with one angle 100º and all its sides unequal is called a(n)
……………………… triangle.
9. Draw a rough sketch to show what is meant by each of the following, putting
in all necessary markers:
(a) A right-angled isosceles triangle (b) A parallelogram
(c) A chord of a circle (d) A regular pentagon.
(e) A major segment of a circle.
10. In the figure ABC = CBD. Write down the name of one angle which
will be equal to:
(a) ABE + ABC (b) ABE + CBD
(c) CBD + ABC + ABE (d) 2ABC + reflexEBD
(e) reflexEBA - CBD
206
11. Write down the sizes of the angles marked a to f. Do not measure.
PAPER A(5) 120 MARKS
1. Simplify:
(a) a + a + a (b) 6a – a (c) 3x2.2x (d) a + a.a
(e) (3x2y3)(2xy2) (f) 2
6
2
8
x
x (g) (3a – a).a (h) (4x)2
(i) 3a.5a – a.2.a (k) 3x2y + 4x2 – x2y – 4 (l) xxx
xx2.)2(
2
17 2
3
66
−−
2. (a) Arrange the expression 4x6 + 5x2 – x + 7x3 + 2 in ascending powers of x.
(b) What is the coefficient of the x term?
(c) What is the degree of the expression?
(d) How many terms has it got?
3. y = x2 - 5x = 6. Find the values of y if: (a) x = 0 (b) x = 1 (c) x = -2
4. In the row of numbers 1 + 2 + 3 + ……., the sum of the first n numbers is
2
2 nn +. What is the sum of the first 80 numbers?
5. In a singles tennis tournament, every player plays against every other player.
(a) How many matches are there if there are six players?
(b) How many matches are there if there are n players?
207
6. (a) Write out this pattern and complete it:
12 + 1 + 2 = 4 = 22
22 + 2 + 3 = 9 = 32
32 + 3 + 4 = …. = …..
42 + …..+…..= ….=……
52 + ….+……= ….=……
62 + ….+……=……=…..
72 + 7 + 8 = ….=…..
n2 +….+…..= (….)2
(b) Use the pattern to calculate the value of 992 + 99 + 100. Show your method.
7. Using markers to show equal line segments and equal angles, draw the net of:
(a) a prism with a regular hexagonal base (b) a pyramid with a square base.
8. If xy = 63 and y + 2 = 9, find the value of:
(a) x (b) (x – 1)(x + 1) (c) x2 + y2 (d) (x + y)2
9. Share R 120 between A, B, C and D so that A gets twice as much as B,
C gets R16 more than A and D gets three times as much as B. (Make as equation
and solve it)
10. (a) Express 16, 24 and 40 as powers of their prime factors.
(b) Find the smallest number which will leave a remainder of 7 when it is
divided by 16 or 24 or 40.
11. (a) Find the values of y if the five-digit number 12x7y is divisible by 5.
(b) Find the values of x if 12x7y is divisible by 55.
12. In the figure:
(a) What kind of angle is OED ?
(b) What kind of triangle is ∆AOF ?
(c) What kind of triangle is ∆BCD ?
(d) What kind of figure is BAED ?
(e) What kind of figure is ABDEF ?
208
13. Write down the sizes of the angles marked a to f. Do not measure.
14. Write down a formula P , the perimeter of each of these figures, then simplify it.
PAPER A(6) 120 MARKS
1. (a) Write the expression x2 + 5x3 + 2x + 4 + 3x4 in descending powers of x.
(b) Write down the term with the largest coefficient.
(c) Write down the index of the x term.
(d) What is the constant term ?
(e) find the value of the expression when x = 2.
2. Simplify where possible:
(a) 4x + 2x + x (b) (4x)(2x)(x) (c) 6x4 2x3
(d) a + a.b + 2a (e) x
xxx
3
43 −+ (f) 3x +
x
xx
3
4 −
209
(g) (a + a).b + 2a (h) a2.a3.a (i) 3x2 + 3x
(j) x.x.x + x.x (k) 2x(3x2)2 (l) 3a2.5b2
(m) 5x2 + 4x3 – 2x2 – x3 + 5 + x (n) 2a + a a + 2a + 6a 2a
(o) (2a + a) a + (2a + 6a) 2a
3. (a) In a row of numbers, the nth term is (3n – 1). Write down the first 3 terms.
(b) In the row numbers 3 + 6 + 9 + 12 + …..
(i) What is the 100th term ? (ii) What is the nth term ?
4. Complete the following statements:
(a) If the wind turns from north to southeast it turns through …..º.
(b) A prism with 12 edges is a(n) ………. Prism.
(c) A pyramid with 12 edges is a(n) ……..pyramid. It has …….vertices
and ………. faces.
(d) If you place two equilateral triangles side by side the figure you get is a(n) …….
(e) If you place three equilateral triangles side by side the figure you get is a(n) …….
(f) If you place 4 equilateral triangles side by side you get a(n) ……… or a(n)
………. This could also be used as the net of a(n) …….
5. Find the size of the angles marked a to e without measuring them.
210
6. Use this sketch to help you to write (a + b)2 in another way in algebra.
7. (a) A prism has an n-sided base. Write down:
(i) the number of vertices (ii) the number of edges (iii) the number of faces.
(b) If the number of vertices plus edges is 60 , what kind of prism is it ?
(c) Is it possible for the number of vertices plus edges of a prism to be 72 ?
Explain.
PAPER B(1) 150 MARKS
1. (a) Add: 2x3 – 8x2 + x - 3 (b) Subtract: -2a - 5b + 3c
-x3 + 7x – 4 -a + 4b - 2d
3x3 + 5x2 - 10x -1 --------------------------
----------------------------
-------------------------
-----------------------------
(c) Multiply (2x2 – 3x – 5) by -2x2y
(d) Divide (24a2b2 - 12ab2 + 36a2b) by -12ab
2. Simplify:
(a) (2ab)(-3a2b)(-ab) (b) 2(-2a3)2 (c) 4a2 - 4a + a2 + 3a
(d) 63
2
2
aaX
a (e)
33
22
6
18
ba
ba
− (f) 3 – (a – 2)
(g) (x + y)y - (2y)(y) (h) a
aba
5
510 − (i) 2(a – 2) – (a – 2)
3. If x = 3, y = -2 and a = 0, find the values of:
211
(a) 3xy2 (b) 2x – 3y (c) x-y (d) x2 - y2
(e) a
x (f) (x – y)2 (g)
y
x (h)
x
a
4. Solve for x:
(a) 3x – 5 = 2x + 4 (b) 4x + 3 + 3x - 7 = 5x - 9
(c) 4(x – 2) - (x – 3) = 5x (d) 3x - 1,4 = 2,8
5. Complete the following statements:
(a) A triangle with two sides of equal length and one angle of 90º is called a(n) ……
(b) A pyramid with nine vertices has a(n) …..base.
(c) The complement of (90º - x) is ……..
(d) The formula for finding the volume of a cylinder is ……….
(e) The supplement of the complement of 40º is …….º
(f) The smallest angle between the hands of a clock at 5 0’clock is …….º
(g) A parallelogram with adjacent sides equal is called a(n) ……….
6. Find the sizes of the angles marked a to e, in that order. Write all steps and give
reasons for all your steps.
7. Draw a rough sketch of this figure and fill in the sizes of all the other angles.
212
8. (a) Using 7
22= , find the area of the shaded region in each of the following figures:
(b) Find the perimeter of the shaded region in question 8 (a) (iii)
9. (a) The sketch shows the skeleton of a cube made with a piece of wire 108 cm
long. What is the length of each edge of the cube?
(b) Find (i) the volume (ii) the total surface area of a cube with these dimensions.
(c) If 216 cm of wire was used instead of 108 cm, what would the new volume be?
10 (a) Write down the nth term in each of these sequences:
(i) 1 ; 2 ; 3 ; 4 ; 5 ………… (ii) 1 ; 4 ; 9 ; 16 ; 25 ; ……….
(iii) 2 ; 4 ; 6 ; 8 ; 10 ; ……… (iv) 1 ; 3 ; 5 ; 7 ; 9 ; ……..
(v) 2x ; 3x2 ; 4x3 ; 5x4 ; 6x5 ; ………..
(b) Change (i) m minutes to seconds (ii) x cents to rands
(iii) k kilometers to metres.
11. (a) A river flows at the speed of 4 km/h. If a boat travels at a speed of x km/h
in still water: (i) what is the speed when traveling upstream?
(ii) What is its speed when traveling downstream?
(b) If its speed when traveling downstream is twice as fast as its speed when
traveling upstream, find its speed in still water. (Make an equation and
solve it)
213
12. (a) This table shows a relationship between x and y.
x 1 2 3 4 5 6 50
y 2 6 12 20 30 a b
(i) Study the pattern and find the values of a and b.
(ii) Write down the formula which shows the relationship between x and y.
(b) Simplify: ( Show all steps )
(i) 6
1
2
1+ (ii)
12
1
6
1
2
1++ (iii)
20
1
12
1
6
1
2
1+++
(c) Write down and complete the following pattern:
2
1
2
1=
.......6
1
2
1=+
........12
1
6
1
2
1=++
.......20
1
12
1
6
1
2
1=+++
......30
1
20
1
12
1
6
1
2
1=++++
(d) Predict the value of ........30
1
20
1
12
1
6
1
2
1+++++ to n terms. What is the nth
term?
PAPER B(2) 150 MARKS
1. (a) Add: 2x2 - 3x + 1 (b) Subtract: 2a – 3b + 4c
3x2 + 5x - 4 3a – 5b - d
-x2 - x - 3 -----------------------
----------------- -----------------------
-----------------
(c) Multiply: (3a – 4b + 2c) by -2ab
(d) Divide: (12x2y2 - 15x2y) by -3x2y
214
2. Simplify:
(a) 2x + 3y – x – y (b) (-2a2)3 (c) xy
xyyx
4
84 22 −
(d) (2a)(3a)(-a)(-4) (e) 2a(a – 3b) (f) (a – 3b) – 2a
(g) 2
22
8
8
4
4
y
xy
xy
yx− (h)
3
2
4
aX
a (i)
3
2
24
aaa+
(j) 6
5
24
3 aaa−+ (k) 5a2 – 4a + 3a2 – 7a3 – a3 + a
3. If m = , n = -3 and x = -2, find the values of:
(a) m² - n² (b) (c) (d) 4m²nx
4. Solve for x:
(i) 3x + 8 = 5x + 13 (ii) 2(x – 3) – (x + 1) = 5x – 4
(iii) 8x – 2 – 4x = 3x – 5 + x + 1
5. Translate the following sentences into algebraic language:
(a) The sum of x and y is greater than the product of p and q.
(b) The square of y is equal to three times y.
(c) The square of the sum of a and b is less than the sum of the squares of c and d.
(d) The cost of p apples at q cents each is R5.
(e) The sum of three consecutive odd numbers, if the first number is (2x – 1), is y.
6. Complete the following statements:
(a) The supplement of 70º is ……..º.
(b) An obtuse angle is defined as ……………………..
(c) An angle of 207º is a(n) ……….angle
(d) Each angle of an equilateral triangle is ………..º
(e) The three interior angles of a triangle add up to ……….º
(f) If two complementary angles are in the ratio 2:3, the largest angle is ……º
(g) If two supplementary angles are in the ratio 2:3, the smallest angle is …..º
215
7. Draw a rough sketch of this figure and fill in the sizes of all the other angles.
8. Make an equation and solve it to find the value of x in each of the following
figures. All steps and reasons must be given.
9. Using π = 3,14 calculate the area of each of the following figures:
10. (a) Calculate the internal volume of:
(i) A cylindrical container with internal radius 4,5 cm and internal
height 45 cm, using π = 3,14.
(ii) A cubical box with internal length 27 cm.
(b) How many balls of 9 cm diameter can be fitted into each container
in 10(a) ?
216
11. The length of a rectangle is twice its breadth. If its length is decreased by
five units and its breadth increased by 3 units it becomes a square. Find
the area of the rectangle. Hint: Let the breadth be x units.
12. Write down the three-digit number with its hundreds-digit x, its tens-
digit y and its units-digit z.
PAPER B (3)
1. Simplify:
(a) (2x2)(-3x3)(4x4) (b) ab
abba
3
3²6 3 −
(c) -3a(2a2 – 3ab + 4b2) (d) (-3x3)2
(e) -7a + 2a2 – 5 = 3a + a2 + a (f) x8
0
(g) a
aa
5
32 + - (2a – 3a) (h) 2(a – 3) – 3(a + 2)
(i) 44
3
3
2 aaa
− (j)
2
33
2
4.2
x
xx
2. (a) Arrange these expressions in descending powers of x and add the
expressions together:
1 - x - 3x2 ; 2x - 5 - x2 ; 2x2 - x + 2 ; 1 - x2
(b) Subtract: 2a - 3b - c from 2b - a - c
(c) Multiply: 1 - 2x + x2 by -x
(d) Divide: a + b by -1
3. If x + y + 3 = 0 and y = -2, find the value of:
(a) 2xy (b) y
x (c) x - y
4. Solve for y:
(a) 2y - 7 = 5y - 4 (b) 2(y - 4) = 2y - 8 (c) 2
1
3−=
y
217
5. (a) A man is x years now. How old will he be in x years time?
(b) The wheel of a bicycle makes x revolutions to cover a distance
of y m. What is the circumference of the wheel?
(c) The perimeter of a square is 8x cm. What is its area?
(d) How many hours will it take to walk x km at y km/h ?
(e) If the area of a circle is πx2, what is its circumference ?
6. I think of a number, add two, multiply the result by 5 and then subtract 35.
The result is four times my original number. Find the original number.
7. Find the size of x in each of the following figures, giving all steps and
all reasons:
8. Draw a rough sketch of this figure and fill in the sizes of all the other angles.
218
9. (a) Find (i) the volume (ii) the total surface area of a rectangular
prism 10 cm x 8 cm x 6 cm.
(b) The base of a triangular prism has dimensions as shown in the sketch.
Its volume is the same as the volume of the rectangular prism in (a).
Find its height.
5 cm
8 cm
PAPER B(4)
1. State the number of terms in each of the following expressions:
(a) a + b ÷ c (b) (4a - 3b)c
(c) ab2c - 2ab - 5 (d) 3
54 cba −+
2. Simplify:
(a) a + b - a - 2b (b) 2ab(a - 2b)
(c) bba
22
24−
−
− (d)
2
224
−
−− bba
(e) a + 2b - 2(a - 3b) (f) 5
2
52
aaa+
(g) (-2a)3 + 3(-a)3 (h) (a - 2b + 3c)2a
(i) 5 + 4a - 3a3 + 7a - 6a2 - a2 - 5a3 - 2
219
3. Solve for x:
(a) 2
1
5−=
x (b) 9x = 5x
(c) 2x - 5 = 8x - 13 (d) 2(x – 1) – (3x – 2) = 4(x + 2)
4. If X = 2a2 - 5a + 1, Y = 4a - 3 - a2, Z = 3 - a - a2,
find X + Y - Z.
5. If a = 1, b = -3, c = 0 and d = 2
1, evaluate:
(a) b2 - d2 (b) c
b (c)
d
ac
(d) (b - a)2 (e) a3 - bd + c
6. Say whether the following statements are always true, sometimes true
or never true.
(a) (90 - 2y)º and 2yº are complementary angles.
(b) A square is a regular quadrilateral.
(c) The sum of two acute angles is less than 180º.
(d) In any triangle, the greatest angle lies opposite the longest side.
(e) If a transversal cuts two straight lines then the co-interior angles
are supplementary.
(f) The supplement of 3
2 of a right angle is 30º.
(g) The exterior angle of a triangle is equal to the interior opposite angle.
(h) A triangle with sides 4 cm, 5 cm and 9 cm cannot be constructed.
220
7. Calculate the value of x in each of the following figures, by making an
equation and solving it.
8. Copy the sketch below and fill in the sizes of all the other angles.
9. Using π = 3,14, find the area of the shaded part of each of the following
figures:
221
10. A rectangular swimming pool is 25 m long, 20 m wide and 1,5 m deep.
Find: (a) its volume (b) how many kilolitres of water it will hold
(c) how long it will take to fill the pool if it is being filled by a pipe which
delivers 250 litres per minute.
PAPER C (1)
1. Simplify:
(a) 3a - 5b - 4a +b (b) -2(-2a)3
(c) x
xxx
4
1684 23
−
+− (d) a - b - (a + b)
(e) (2x2y)(3xy2)(xy) (f) 3abc - 2bca - 5bac
(g) 3x(x - 4) (h) (4a + 5a) ÷ 3 + 2a + 6a ÷ 2
(i) 164100 yx (j) 4
3
52
aaa+−
2. (a) arrange these expressions in descending powers of x and then add
the three expressions together:
5x - 2x3 + x2 - 4 ; 3 - 7x2 - 4x3 ; 8x2 - 3x - 5 + 6x3
(b) Subtract: 2a - 3c + d from -3a - 4b - 3c + 2d
3. If x = -2, y = -3 and z = 2
1, find the values of:
(a) x2y (b) x2 - y2 - z2 (c) (x - y)2
(d) yx
yx
+
− (e) 2x - 3y2 - xz
4. Solve for x:
(a) 4(x - 2) = -8 (b) 4(x - 2) = x - 2 (c) 2
34=
x
(d) 2(x - 5) - (3x + 1) = 4(2x - 3) + 8
5. Mary has three times as many stamps in her collection as her friend Jenny.
If she gives 18 stamps to Jenny she will have twice as many stamps
as Jenny. How many stamps has each girl got ?
222
6. Write down a formula for finding (i) the perimeter (ii) the area of each of
the following figures, using the letters on the sketches. Then simplify
if possible:
7. Write down the formulae for finding (a) the perimeter (b) the area of the
shaded part of this figure in terms of x and π. Then simplify if possible.
8. Complete the following statements:
(a) A triangle with an angle of 108º and all its sides of different lengths is
called a(n) ………………………
(b) A polygon with 8 sides is called a(n) ………………
(c) The supplement of 55º is ……….º
(d) In 35 minutes, the minute hand of a clock turns through …..º
(e) A triangle with all sides equal is called a(n) ………and each of its
angles is ………º.
(f) A prism with 10 vertices is a(n) ………….. prism.
(g) If the angles of a triangle are in the ratio 1 : 1 : 2, it is a(n) ……..,
…………….. triangle.
223
E
9. Find the sizes of the angles marked a to e in that order.
10. (a) Express 324 and 512 as powers of their prime factors.
(b) Hence find (i) 324 and (ii) 3
512
(c) In ∆ABC , B = 90º , a = 24 mm and b = 30 mm. Calculate:
(i) c (ii) the area of ∆ABC.
(d) Find (i) the side (ii) the total surface area of a cube with volume
512 cm3.
11. (a) In ∆ABC, a = 10 cm, b = 24 cm and C = 90º. Calculate:
(i) c (ii) the ratio a : b : c
(b) In ∆ABC, A : B : C = 5 : 12 : 13. Calculate C.
(c) Compare your results in (a) with your results in (b). What can you
deduce ?
12. Explain why it is not possible to construct the triangles with the
measurements below:
13. (a) The length and breadth of a rectangle are in the ratio 3:2. If the
length were decreased by 4 units and the breadth increased by 4
units it would be a square. Find its length, breadth and area.
(b) If the length of a rectangle in (a) is decreased in ratio 5 : 8 and its
breadth increased in ratio 5 ; 4, find the new length, breadth and area.
224
(c) What is the ratio of the new area to the old area ?
14. An athlete runs the 100 m sprint in 10,8 s. What is his speed in km/h ?
PAPER C(2)
Simplify:
1. (a) -(2x3y)2 (b) 3a - 2a - 5a + 7a
(c) (2x2y)(3x3)(4x4y) (d) 129a
(e) 42
3 abab (f)
−
−
2332
aaaa
(g) x
yx
6
)3)(2(
−
− (h) x(3x - 2)
(i) 3
027,0 (j) 2
22
4
8
4
4
b
ab
a
a
−−
(k) 3
2
5
3
2
aaa−+ (l) 3(x - 2) - 2(x - 1)
2. (a) Add: -2x + 3y - 4z (b) Subtract: 3x2 - 4x + 2
3x - 2z -2x2 + 5x - 1
-5x - 7y + z ______________
______________ ______________
______________
(c) Multiply: 2x2 - 3x + 4 by -3xy
(d) Divide: 18x3 - 12x2 + 15x by -3x
3. (a) If x = 2y and 32=
z, find
(i) the value of x + y + z (ii) the ratio of x : y : z
(b) Show whether the following statements are true or false when
x has the given value:
(i) 4x - 1 = 3 when x = 4
3 (ii) 2x2 – 3 = 5 when x = -2
(iii) 3
2+x <
3
2 when x =
4
1
225
4. (a) Write each of the following as an algebraic sentence:
(i) The product of a and b is equal to five times p.
(ii) The square of x is smaller than the cube root of y.
(iii) The sum of the squares of a and b is greater than the square
root of c.
(b) Express: (i) m minutes in hours (ii) b kilograms in grams
(iii) the cost in rand of 2x books at 5y cents each
(iv) the lengh of a side of a cube whose volume is 8x3 cm3,
in terms of x.
5. Calculate the sizes of the angles marked a to e, in that order.
6. In the sketch, .xECAFBCEBF ===
(a) Find the value of x.
(b) Now make a copy of the sketch and fill in the sizes of all the angles.
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7. AC is the diagonal of square ABCD.
(a) Express the area of ABCD in terms of x.
(b) Express AC2 in terms of x.
(c) If AC = 4 cm, find the area of ABCD.
(d) AC is the diagonal of the square. Write a formula for finding the area of a
square in terms of its diagonal (d).
(e) If d = 18 cm, find the area of ABCD and the side of the square.
8. ∆ABC is the base of a triangular prism with height 85 mm. Calculate:
(a) AC (b) the volume of the prism
(c) the total surface area of the prism.
A
24 mm
B 45 mm C
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9. (a) Two towns A and B are 480 km apart but on the map they are 32 mm
apart. Towns P and Q are 1080 km apart. How far apart will they be on
the map ?
(b) What is the scale of the map ?
10. (a) The following tables show a relationship between x and y. Write down
a formula which expresses the relationship in each case, in the form
y = ……. Then use it to find the missing values a and b
in each table.
(i)
x 2 7 9 12 b
y 5 10 12 a 17
(ii)
x 5 20 22,5 25 b
y 3 12 13,5 a 1,2
(iii)
x 4 25 49 144 b
y 2 5 7 a 13
(iv)
x 2 6 8 10 b
y 12 4 3 a 36
(b) Which of the tables in (a) if any, represent a direct proportion?
(c) Which of the tables in (a) if any, represent an inverse proportion?
PAPER C(3)
1. (a) Arrange the following expressions in descending powers of x and then add:
3x - 7 - 2x2 ; 2 + 5x2 - 3x ; -x2 + 6 - 2x
(b) Subtract: (5a + 4b - 3c) from (8a - 3b + 4c)
(c) Multiply: (2a - 3b - 4c) by -2a2b
228
(d) Divide: 2x4 - 3x3 + x2 by - x2
2. Simplify:
(a) 4x - 3y + 2x + y (b) x
xx
4
84 32
−
−
(c) -(-3b2)2 (d) 3681x
(e) 3
2
2
xx− (f) (-3a2b)(-4ab3)(abc)
(g) 2(3a - 2) - (3a - 2) (h) 2
33
4
8.4
x
xx
(i) 24
3
3
2 aaX
a (j)
312627 ba−
3. Find, by inspection, the values of x which make the following equations true.
Some of the equations have more than one solution.
(a) x(x - 5) = 2(x - 5) (b) 02
3=
−
−
x
x
(c) (x - 2)2 = 9 (d) x3 = -8 (e) 2(x - 3) = 2x - 6
4. Complete each of the following statements. Write the full statement not just the
missing part:
(a) 2a - 4b = 2( ) (b) x - y - z = x - ( )
(c) 2a - 3b = 2−
(d) a + 2ab - b2 = a + b( )
5. (a) A cricketer’s batting average is found by the formula A = tn
r
−,
where r is the number of runs, n is the number of innings and t is
the number of times not out. Find his average if he scores 522 runs
in 18 innings, three times not out.
(b) F = 5
1609 +C is the formula for converting temperature in degrees
Celsius to degrees Fahrenheit. Convert the following to Fahrenheit:
(i) 100º C (b) -10º C (iii) 24º C
(c) If n < 41, (n2 + n + 41) gives a number which is always prime.
Check to see if this is true when: (i) n = 1 (ii) n = 2 (iii) n = 7
229
(d) Show that if n = 41, (n2 + n + 41) gives a number which is
not prime.
(e) (i) The formula for finding the area of a sphere is 4 r2. Assume
that the earth is a perfect sphere with radius 6400 km. Calculate
its surface area correct to the nearest million km2 and express your
answer in scientific notation using = 3,142.
(ii) If 10% of the earth’s surface is habitable and the world’s popu-
lation is approximately 5,3 milliard, calculate the area of land available
per person in m2.
6. (a) I have 12 coins, x of them are ten-cent pieces and the rest are fifty-
cent pieces.
(i) How many fifty-cent coins have I got ?
(ii) What is the total value of my money in cents ?
(b) Find:
(i) the volume of a rectangular prism which is 24x cm by 9x cm by 8x cm.
(ii) the length of the side of a cube with the same volume as this prism.
(iii) the total area of all the faces of this cube.
(c) After the nth day of December, what fraction of December remains ?
7. A furniture dealer stocks four-legged and three-legged tables which are stacked
together in his storeroom. He knows he has 48 tables in stock, but he doesn’t
know how many of each kind. Instead of unpacking them he finds it easier to
count the legs. He counts 172 legs. How many of each kind did he have in
stock? Hint: Let the number of four-legged tables be x.
8. Complete these statements:
(a) In a right angled triangle the other two angles are ……….
(b) If a wheel makes 10 revolutions per minute, in one second a spoke will
turn through ………..º
(c) If the angles of a triangle are in the ratio 2 : 3 ; 4 the angles are
………..º , …………º and ………º.
(d) In ∆ABC, ABC = 108º, a = c = 5 cm therefore ∆ABC is a(n)
……………triangle.
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(e) The formula for finding the area of a circle is …………..
9. Find the values of x and y in each of these figures:
10. Make a neat copy of this sketch and fill in the sizes of all the other angles.
11. (a) If it takes 4 h to travel 340 km, how long will it take to travel 544 km ?
(b) If a car travels at an average speed of 90 km/h, it takes 4½ h to complete
a journey. If the average speed was increased to 100 km/h how long
would the same journey take ?
(c) (i) Find the cash price of an umbrella marked R14,80 less 15 % discount.
(ii) What scale factor did you use ?
(d) Tom, Dick and Harry share their great-aunt Mathilda’s estate of R36 800
in the ratio 2:2
11:
3
1, respectively. What is Dick’s share ?
PAPER C(4)
1. Simplify:
(a) (-2a2)3 (b) (-2a3)2 (c) 54
35
3
12
ba
ba−
(d) 4
46
4
128
a
aa − (e)
4
46
4
)12(8
a
aa − (f) 1616a
231
(g) 3
2.
2
aa (h)
3
2
2
aa+ (i)
3
2
2
aa
(j) 2(a - 3b) - 3(a + b)
2. (a) What must be added to 2a2 - 3a + 4 to get 3a2 - 4a + 7 ?
(b) What must a - 3b + 4c be multiplied by to get -2a2 + 6ab – 8ac ?
(c) What must 4a2bc – 12ab2c be divided by to get -a + 3b ?
(d) Find the sum of 3x - 4y + z and -2x - y - z.
3. If a = 2, b = -3 and c = 2
1 find the values of:
(a) -2a2 (b) (-2a)2 (c) 22 ba +
(d) (a + b)2 (e) ac - c
a (f) 24 ba +
4. (a) Show whether the following statements are true or false when x has
the value given:
(i) 2(x - 5) - 3(x + 1) = 4 if x = 17
(ii) 2
3−x = 2x + 1 if x =
3
5−
(b) Solve for x: 3(x - 2) + 5 = 3x - 1
5. The following statements are incorrect. Rewrite each statement correctly
by changing just one thing:
(a) The complement of 35º is 145º.
(b) A pentagon has six sides.
(c) A reflex angle is an angle bigger than 270º but smaller than 360º.
(d) A triangle with three equal sides is called isosceles.
(e) A rhombus is a regular quadrilateral.
6. Make an equation to express the relationship between a, b and c in each
of the following figures:
232
7. Make a neat copy of this sketch and fill in the sizes of all other angles
8. Find the sizes of the angles marked a to d in the figures below:
9. (a) a farmer has 880 m of fencing material. What area can be enclosed if
he makes (i) a square enclosure (ii) a circular enclosure using
7
22= ? Which has the greatest area ?
(b) A rectangular prism is 27 m long, 18 m wide and its volume is
5 832 m3. Find its height.
(c) Find the length of the side of a cube with a volume of 5 832 m3.
10. (a) A car travels 386 km in 4 h.
(i) What is the average speed in km/h ?
(ii) At the same rate, how far should it go in 6 h?
(iii) At the same rate, how long should it take to cover 442 km ?
233
(b) A car completes a journey in x hours if it averages y km/h. How long will
the same journey take if it averages z km/h ?
(c) An examination paper is marked out of 240 but it should be out of 300.
(i) What scale factor must be used to convert each mark ?
(ii) Convert a mark of 186 out of 240 to a mark out of 300.
(iii) Convert a mark of x out of 240 to a mark out of 300.
11. In a multiple choice mathematics competition some questions are given 3 marks,
some 4 marks and some 5 marks. Julie got twice as many 3 mark
questions correct as 5 mark question and two more 3 mark questions than 4
mark questions. Her total score was 68. How many 3 mark questions did she
get correct ?
REVISION EXERCISE
1. Tebogo did a survey of how many marbles his friends had. The results are:
15 ; 10 ; 9 ; 12 ; 12 ; 8 ; 20 ; 14 ; 9 ; 11
(a) What is the mean number of marbles ?
(b) What is the range of the data ?
(c) What is the mode of the data ?
2. The two pie charts below show the nutritional content of cheese slices and
cheese spread:
234
(a) Write a few sentences about what you see when you look at the two pie
charts. (for example, what do you notice about water content and protein
content ?)
(b) Measure the segments in both pie charts and give the information
(in degrees) in a table.
(c) Draw a pie chart to illustrate the information in the table below:
Nutritional content of cheese and tomato pizza
Water Protein Fat Carbohydrate Other
52% 9% 12% 25% 2%
3. The tennis club has 120 tennis balls: 30 balls are still brand new, 40
are old, and the rest are good (they have not been used for more than
one hour)
(a) Display the data on a graph.
(b) Draw a pie chart to display the information,
4. Below is the harvest for one year of apples (to the nearest hundred) from
twelve apple trees:
800 2400 1800 4000 600 1900
2300 100 2200 2600 1400 1700
(a) What is the range of the data ?
(b) Calculate the mean of the data.
5. A few classes took part in a memory experiment. They looked at ten objects
and then had to write down as many of them as they could remember after
the objects had been removed. The results for two classes are in the table
below:
Class Number of objects remembered
8a 4 8 7 7 5 7 8 7 7 10 7 7 8 6 7 6 7 7 8 9
8b 8 7 7 9 9 6 8 7 9 7 8 9 8 7 4 7 8 8 9 7 10 7 8 9 9
(a) What is the range for 8a, and for 8b ?
(b) Find the mean, median and mode for each class.
235
(c) Which class do you think did better at remembering objects ? Give a reason
for your answer.
6. If a = -2, b = -3 and c = 1, calculate the following without the use of a
calculator. Show all your steps.
(a) a - b (b) a + 2(b - c) (c) (a - b)2
(d) a2 - b2 (e) 2a - 3b2 - ac (d) ac
ba 32 +
7. The price of an article increases from 50c to 55c and then again from 55c
to 60c.
(a) Find the percentage increase in each case.
(b) What is the total percentage increase on both price increases, i.e. from
50c to 60c?
8. A bag contains one green, five red and three yellow balls. A ball is drawn at
random. Find the probability that it is: (a) red (b) yellow (c) green
One more red ball and one more green ball are added to the bag. What is the
probability of selecting a red ball now?
9. Susan has eight cards numbered 2, 2, 2, 3, 3, 4, 4, 6. A card is selected
at random and then put back. The trial is repeated 200 times. How many
times would you expect her to pick: (a) A card with 2 on it (b) An even
numbered card. (c) A card with a value of 3 or higher.
10. Calculate the value of:
(a) 5
32 +a if a = 7 (b)
x
yx
2
43 + if x = 2 and y = 5
(c) a - a
ab −2 if a = 4 and b = 8 (d)
5
yx − if x = 13 and y = 19
(e) b
ba
−
− 2 if a = 9 and b = -2 (f)
t
t
2
)4(2 + if t = -5
11. Simplify without a calculator:
(a) (2
12
8
15 + ) - ( )
3
12
2
12 + (b) 2 - ( )
2
11
3
1+ (c) -3 - (2 - ½)
THE END
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