Mathwise Academy | Grade 8 Test 1 - Whole Numbers

MATHWISE ACADEMY

Grade 8 Mathematics: Test 1 • Chapter 1: Whole Numbers

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
Calculators may NOT be used for Question 1 and Question 2. A non-programmable calculator may be used for Questions 3, 4, and 5 unless stated otherwise.
Round off final answers to two decimal places where necessary, unless specified otherwise.
Write neatly and legibly.

QUESTION 1: PROPERTIES AND OPERATIONS [10]

(Note: No calculators permitted for this question)

1.1 Calculate the value of the following using the correct order of operations:

$ 25 - 4 \times (7 - 4) + 18 \div 3 $

[3]

1.2 Use the distributive property to calculate the product without a calculator:

$ 6 \times 87 $

[2]

1.3 State the mathematical property demonstrated in each statement below:

1.3.1 $ 14 + (26 + 9) = (14 + 26) + 9 $ (1)
1.3.2 $ 45 \times 12 = 12 \times 45 $ (1)

1.4 State whether each statement is TRUE or FALSE:

1.4.1 $ 0 $ is the identity element for multiplication. (1)
1.4.2 $ 15 \div 0 $ is undefined. (1)
1.4.3 The set of natural numbers $ \mathbb{N} $ contains the number $ 0 $. (1)

QUESTION 2: PRIME FACTORS, HCF AND LCM [10]

(Note: No calculators permitted for this question)

2.1 Express the following numbers as products of their prime factors using exponential form:

2.1.1 $ 72 $ (2)
2.1.2 $ 120 $ (2)

2.2 Using your prime factorizations from Question 2.1:

2.2.1 Determine the Highest Common Factor (HCF) of $ 72 $ and $ 120 $. (2)
2.2.2 Determine the Lowest Common Multiple (LCM) of $ 72 $ and $ 120 $. (2)

2.3 Two large numbers are decomposed into prime factors:

$ A = 2^3 \times 3^2 \times 5^4 $
$ B = 2^2 \times 3^4 \times 7^1 $

Write down the HCF of numbers $ A $ and $ B $ as a product of prime factors.

[2]

QUESTION 3: RATIOS AND RATES [10]

3.1 Simplify the following ratios to their simplest form:

3.1.1 $ 45\text{ cm} : 1,5\text{ m} $ (2)
3.1.2 $ 0,4 : 0,16 $ (2)

3.2 Divide $ \text{R } 720 $ among Adam, Brian, and Caleb in the ratio $ 2 : 3 : 5 $.

[3]

3.3 A car travels a distance of $ 325\text{ km} $ in $ 2\text{ hours and } 30\text{ minutes} $ at a constant speed.

3.3.1 Calculate the speed of the car in $ \text{km/h} $. (1)
3.3.2 At this same speed, how long (in hours) will it take the car to travel $ 520\text{ km} $? (2)

QUESTION 4: CURRENCY AND PERCENTAGES [10]

4.1 The exchange rate between the South African Rand and the US Dollar is $ \$1 = \text{R } 14,80 $.

4.1.1 An online textbook costs $ \$35 $. Calculate its cost in South African Rands. (1)
4.1.2 Sipho has $ \text{R } 1\,110 $ to exchange for US Dollars before travelling. How many Dollars will he receive? (2)

4.2 A store purchases a school bag for $ \text{R } 240 $ and adds a markup profit of $ 35\% $. Calculate the selling price of the bag.

[2]

4.3 During a clearance sale, a jacket is sold for $ \text{R } 420 $ after a $ 25\% $ discount was deducted from the original price. Determine the original marked price of the jacket before discount.

[3]

4.4 In a mathematics test out of $ 60 $, Lerato scored $ 51 $ marks. Calculate her percentage score.

[2]

QUESTION 5: SIMPLE INTEREST AND HIRE PURCHASE [10]

5.1 Kagiso invests $ \text{R } 8\,000 $ in a savings account earning $ 7,5\% $ per annum simple interest for $ 4 $ years.

5.1.1 Calculate the total simple interest earned over the $ 4 $ years. (2)
5.1.2 Calculate the accumulated total value of the investment at the end of the term. (1)

5.2 Nomsa buys a computer system with a cash price of $ \text{R } 15\,000 $ on a hire purchase agreement. She pays a $ 20\% $ cash deposit upfront and finances the remaining balance over $ 3 $ years at a simple interest rate of $ 15\% $ per annum.

5.2.1 Calculate the deposit amount paid in Rands. (1)
5.2.2 Calculate the loan balance owed after paying the deposit. (1)
5.2.3 Calculate the total simple interest charged on the loan balance over the $ 3 $ years. (2)
5.2.4 Determine the total amount payable on the hire purchase agreement (including deposit). (1)
5.2.5 Calculate Nomsa's fixed monthly repayment installment. (2)

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 2 - Whole Numbers

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Chapter 1: Whole Numbers

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
Calculators may NOT be used for Question 1 and Question 2. A non-programmable calculator may be used for Questions 3, 4, and 5 unless stated otherwise.
Round off final answers to two decimal places where necessary, unless specified otherwise.
Write neatly and legibly.

QUESTION 1: PROPERTIES AND OPERATIONS [10]

(Note: No calculators permitted for this question)

1.1 Calculate the value of the following using the correct order of operations:

$ 40 - 5 \times (8 - 2) + 24 \div 6 $

[3]

1.2 Use the distributive property to calculate the product without a calculator:

$ 8 \times 64 $

[2]

1.3 Name the mathematical property demonstrated in each statement below:

1.3.1 $ (15 + 7) + 3 = 15 + (7 + 3) $ (1)
1.3.2 $ 29 \times 1 = 29 $ (1)

1.4 State whether each statement is TRUE or FALSE:

1.4.1 Division is a commutative operation. (1)
1.4.2 $ 12 \div 0 = 0 $ (1)
1.4.3 Subtraction is the inverse operation of addition. (1)

QUESTION 2: PRIME FACTORS, HCF AND LCM [10]

(Note: No calculators permitted for this question)

2.1 Express the following numbers as products of their prime factors using exponential form:

2.1.1 $ 90 $ (2)
2.1.2 $ 126 $ (2)

2.2 Using your prime factorizations from Question 2.1:

2.2.1 Determine the Highest Common Factor (HCF) of $ 90 $ and $ 126 $. (2)
2.2.2 Determine the Lowest Common Multiple (LCM) of $ 90 $ and $ 126 $. (2)

2.3 A certain natural number is a factor of $ 45 $, is less than $ 45 $, and is divisible by $ 3 $ but not by $ 9 $.

What is this number? (Show your reasoning).

[2]

QUESTION 3: RATIOS AND RATES [10]

3.1 Simplify the following ratio to its simplest form:

$ 600\text{ g} : 2\text{ kg} $

[2]

3.2 Determine the missing value in the following equivalent ratio:

$ 4 : 7 = 36 : ? $

[2]

3.3 Divide $ \text{R } 1\,200 $ among three partners in the ratio $ 3 : 4 : 5 $.

[3]

3.4 A water pump can fill a $ 4\,500\text{-litre} $ JoJo tank from empty in $ 15\text{ minutes} $.

3.4.1 Calculate the pumping rate in litres per minute ($ \text{l/min} $). (1)
3.4.2 At this same rate, how long (in minutes) will it take to fill a $ 1\,800\text{-litre} $ tank? (2)

QUESTION 4: CURRENCY AND PERCENTAGES [10]

4.1 The exchange rate between the South African Rand and the British Pound is $ £1 = \text{R } 21,50 $.

4.1.1 Khanya has $ \text{R } 3\,225 $ saved. How many British Pounds will she receive if she exchanges her money? (2)
4.1.2 You buy a book online for $ £18 $. Calculate the cost in South African Rands. (1)

4.2 A merchant buys a pair of sneakers for $ \text{R } 600 $ and sells them at a profit of $ 30\% $. Calculate the selling price of the sneakers.

[2]

4.3 During a "Back to School" sale, a bicycle is sold for $ \text{R } 1\,440 $ after a $ 20\% $ discount was applied. Determine the original marked price of the bicycle before the discount.

[3]

4.4 In a natural sciences project, David scored $ 45 $ out of $ 75 $. Calculate his percentage score.

[2]

QUESTION 5: SIMPLE INTEREST AND HIRE PURCHASE [10]

5.1 Thabo invests $ \text{R } 12\,000 $ at a simple interest rate of $ 8,5\% $ per annum for $ 5 $ years.

5.1.1 Calculate the total simple interest earned over the $ 5 $ years. (2)
5.1.2 Calculate the final accumulated value of the investment. (1)

5.2 An electronics store sells a smart TV for $ \text{R } 8\,500 $. A customer buys it on a hire purchase agreement. They pay a $ 10\% $ cash deposit upfront and pay off the balance over $ 2 $ years at $ 12\% $ per annum simple interest.

5.2.1 Calculate the deposit amount paid. (1)
5.2.2 Calculate the loan balance to be financed after the deposit is paid. (1)
5.2.3 Calculate the total simple interest charged on the loan balance over the $ 2 $ years. (2)
5.2.4 Determine the total amount payable on the loan (excluding the deposit). (1)
5.2.5 Calculate the fixed monthly repayment installment. (2)

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 1 - Integers, Exponents, Patterns, and Functions

MATHWISE ACADEMY

Grade 8 Mathematics: Test 1 • Integers, Exponents, Patterns, and Functions

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
Calculators may NOT be used for Question 1 and Question 2. A non-programmable calculator may be used for Questions 3 and 4.
Write neatly and legibly.

QUESTION 1: INTEGERS [12]

(Note: No calculators permitted for this question)

1.1 Calculate the value of the following without using a calculator:

1.1.1 $ -8 + (-5) - (-12) $ (2)
1.1.2 $ (-3) \times (+4) \div (-2) $ (2)

1.2 Evaluate the following:

$ (-4)^2 + (-2)^3 $

[3]

1.3 The temperature in Centurion drops from $ 5^\circ\text{C} $ by $ 12^\circ\text{C} $ overnight. Calculate the new temperature.

[2]

1.4 Replace the $ * $ with $ > $, $ < $, or $ = $ to make the statement true:

$ (-5) \times 2 \quad * \quad -10 $

[1]

1.5 Arrange the following list of integers in descending order (from largest to smallest):

$ 4 \quad ; \quad -2 \quad ; \quad 0 \quad ; \quad -8 \quad ; \quad 5 $

[2]

QUESTION 2: EXPONENTS [13]

(Note: No calculators permitted for this question)

2.1 Simplify the following using the laws of exponents. Leave your answers in exponential form where applicable:

2.1.1 $ x^3 \times x^4 \times x $ (2)
2.1.2 $ \dfrac{2^8 \times 3^5}{2^5 \times 3^4} $ (3)

2.2 Simplify the following expression:

$ (3x^2y^3)^2 $

[3]

2.3 Simplify the following, showing all your steps:

$ 5x^0 + (5x)^0 $

[2]

2.4 Calculate the final value of the following expression:

$ (-1)^{15} + (-1)^{10} $

[3]

QUESTION 3: NUMERIC AND GEOMETRIC PATTERNS [13]

3.1 Consider the following number sequence:

$ 2 \quad ; \quad 7 \quad ; \quad 12 \quad ; \quad 17 \quad ; \quad \dots $

3.1.1 Write down the next two terms in the sequence. (2)
3.1.2 State the rule for this sequence in words. (2)
3.1.3 Determine the general rule for the $ n\text{-th} $ term ($ T_n $) of the sequence. (2)
3.1.4 Use your rule to calculate the value of the $ 40^{\text{th}} $ term. (2)
3.1.5 Which term in the sequence is equal to $ 87 $? (2)

3.2 Consider the number pattern:

$ 3 \quad ; \quad 6 \quad ; \quad 12 \quad ; \quad 24 \quad ; \quad \dots $

3.2.1 What is the constant ratio for this sequence? (1)
3.2.2 Write down the next two terms. (2)

QUESTION 4: FUNCTIONS AND RELATIONSHIPS [12]

4.1 The rule for finding $ y $ is given by the equation $ y = -3x + 4 $.

Determine the output values ($ y $) for the input values $ x \in \{-2; 0; 5\} $.

[3]

4.2 Determine the input value ($ x $) if the output value ($ y $) is $ -17 $ for the rule $ y = 4x - 1 $.

[3]

4.3 The input ($ x $) and output ($ y $) values of a function are given in the table below:

$ x $	$ 1 $	$ 2 $	$ 3 $	$ 4 $	$ 10 $
$ y $	$ 1 $	$ 4 $	$ 7 $	$ 10 $	$ ? $

4.3.1 Determine the algebraic rule for this function in the form $ y = \dots $ (2)
4.3.2 Calculate the missing $ y $-value when $ x = 10 $. (2)

4.4 A flow diagram takes an input $ n $, applies the rule $ (n \div 2) + 5 $, and produces an output $ p $.

Calculate the value of $ p $ if $ n = 12 $.

[2]

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 2 - Integers, Exponents, Patterns, and Functions

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Integers, Exponents, Patterns, and Functions

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
Calculators may NOT be used for Question 1 and Question 2. A non-programmable calculator may be used for Questions 3 and 4.
Write neatly and legibly.

QUESTION 1: INTEGERS [12]

(Note: No calculators permitted for this question)

1.1 Calculate the value of the following using the correct order of operations:

$ 15 - (-3) \times (-4) + (-10) \div 2 $

[3]

1.2 Use the commutative and associative properties of multiplication to calculate the following:

$ -5 \times 13 \times (-2) $

[2]

1.3 Substitute the given values and evaluate the expression:

If $ x = -2 $ and $ y = -3 $, calculate the value of $ x^2 - y^2 $.

[3]

1.4 A student's bank account is overdrawn and has a balance of $ -\text{R } 150 $.

The student makes a deposit of $ \text{R } 400 $, followed by a withdrawal of $ \text{R } 320 $. Calculate the final balance of the bank account. Show your calculation.

[4]

QUESTION 2: EXPONENTS [13]

(Note: No calculators permitted for this question)

2.1 Simplify the following algebraic expressions. Leave your answers with positive exponents:

2.1.1 $ (2x^3y^2)(-3x^2y^4) $ (3)
2.1.2 $ \dfrac{15a^6b^3}{-3a^2b^3} $ (3)

2.2 Evaluate the following expression:

$ 4x^0 + (4x)^0 $

[2]

2.3 Calculate the final value of the following:

$ (-2)^4 - (-2)^3 $

[3]

2.4 Solve for $ x $ by inspection:

$ 3^x = 81 $

[2]

QUESTION 3: NUMERIC AND GEOMETRIC PATTERNS [13]

3.1 Consider the following number sequence:

$ 5 \quad ; \quad 11 \quad ; \quad 17 \quad ; \quad 23 \quad ; \quad \dots $

3.1.1 Write down the next two terms in the sequence. (2)
3.1.2 Determine the general rule for the $ n\text{-th} $ term ($ T_n $) of the sequence. (2)
3.1.3 Calculate the value of the $ 50^{\text{th}} $ term. (2)

3.2 Consider the following geometric number pattern:

$ 3 \quad ; \quad -6 \quad ; \quad 12 \quad ; \quad -24 \quad ; \quad \dots $

3.2.1 State the constant ratio for this sequence. (1)
3.2.2 Write down the next term in the sequence. (1)

3.3 A pattern of small squares is constructed using matches:

Pattern 1 (4 matches)

Pattern 2 (7 matches)

Pattern 3 (10 matches)

3.3.1 How many matches will be required to build Pattern 4? (1)
3.3.2 Determine the rule ($ T_n $) for the number of matches used in Pattern $ n $. (2)
3.3.3 Which pattern number will be constructed using exactly $ 61 $ matches? (2)

QUESTION 4: FUNCTIONS AND RELATIONSHIPS [12]

4.1 A flow diagram takes an input value, multiplies it by $ -3 $, and then adds $ 5 $ to produce the output $ y $.

Determine the output values ($ y $) for the input values $ x \in \{-2; 0; 4\} $.

[3]

4.2 The relationship between $ x $ and $ y $ is given by the formula: $ y = \dfrac{x}{2} - 4 $.

4.2.1 Calculate the output if the input is $ x = -6 $. (2)
4.2.2 Calculate the input $ x $ if the output is $ y = 1 $. (2)

4.3 The input ($ x $) and output ($ y $) values of a function are given in the table below:

$ x $	$ 1 $	$ 2 $	$ 3 $	$ 4 $
$ y $	$ -1 $	$ 1 $	$ 3 $	$ 5 $

Determine the algebraic rule for this function in the form $ y = \dots $

[3]

4.4 The formula for a specific function is given as $ y = x^2 - 3x $.

Calculate the value of $ y $ when $ x = -4 $.

[2]

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 1 - Algebraic Expressions and Equations

MATHWISE ACADEMY

Grade 8 Mathematics: Test 1 • Algebraic Expressions and Equations

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
Write neatly and legibly.

QUESTION 1: ALGEBRAIC EXPRESSIONS - BASICS & SUBSTITUTION [10]

1.1 Consider the algebraic expression: $ 4x^2 - 5x + 8 $

1.1.1 How many terms are in this expression? (1)
1.1.2 Write down the constant term. (1)
1.1.3 State the coefficient of $ x $. (1)

1.2 Evaluate the following expression if $ a = -2 $ and $ b = 3 $:

$ a^2 - 2ab + 3b^2 $

[4]

1.3 Write down an algebraic expression for each of the following statements:

1.3.1 Four more than three times a certain number $ x $. (1)
1.3.2 The square of the product of $ x $ and $ y $. (2)

QUESTION 2: SIMPLIFYING ALGEBRAIC EXPRESSIONS [13]

2.1 Simplify the following algebraic expressions:

2.1.1 $ 3a^2b - 5ab^2 - 7a^2b + 2ab^2 $ (3)
2.1.2 $ -3x(2x - 5) - (2x^2 - 3x) $ (4)
2.1.3 $ \dfrac{15x^3 - 10x^2}{5x} $ (3)

2.2 Determine the simplified answer for the following:

$ \sqrt{9x^{16} + 16x^{16}} $

[3]

QUESTION 3: ALGEBRAIC EQUATIONS [15]

3.1 Solve the following equations for $ x $. Show all your steps clearly:

3.1.1 $ 3x - 5 = 16 $ (2)
3.1.2 $ 5x - 2 = 6 + 4x $ (3)
3.1.3 $ 7(x + 2) = 6(x + 1) $ (4)
3.1.4 $ \dfrac{2x}{3} = 4 $ (3)

3.2 Solve the following exponential equation for $ x $:

$ 3^{x-1} = 9 $

[3]

QUESTION 4: PROBLEM SOLVING WITH ALGEBRA [12]

4.1 The sum of three consecutive odd numbers is $ 39 $.

Set up an equation and determine the three numbers.

[5]

4.2 The length of a rectangle is $ 5\text{ cm} $ more than its breadth. The perimeter of the rectangle is $ 38\text{ cm} $.

Calculate the length and the breadth of the rectangle.

[4]

4.3 Jason states that $ x = -1 $ is the solution to the following equation:

$ \sqrt{5x + 6} = x $

Show whether Jason is correct or incorrect by evaluating both sides of the equation.

[3]

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 2 - Algebraic Expressions and Equations

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Algebraic Expressions and Equations

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
Write neatly and legibly.

QUESTION 1: ALGEBRAIC EXPRESSIONS - BASICS & SUBSTITUTION [10]

1.1 Consider the algebraic expression: $ 3x^3 - 5x^2y + 4xy - 7 $

1.1.1 How many terms are in this algebraic expression? (1)
1.1.2 Write down the constant term. (1)
1.1.3 State the coefficient of the $ x^2y $ term. (1)

1.2 Evaluate the following expression if $ a = -1 $ and $ b = 2 $:

$ 2a^2 - 3ab + b^2 $

[4]

1.3 Write down an algebraic expression for each of the following statements:

1.3.1 The square root of the sum of $ x $ and $ y $. (1)
1.3.2 Subtract the product of $ 4 $ and $ x $ from $ 10 $. (2)

QUESTION 2: SIMPLIFYING ALGEBRAIC EXPRESSIONS [12]

2.1 Simplify the following algebraic expressions:

2.1.1 $ 5a^2b - 3ab^2 - 2a^2b + 7ab^2 $ (2)
2.1.2 $ (-3x^2y)(-4xy^3) $ (2)
2.1.3 $ -2x(3x^2 - 4x + 1) - x(x^2 - 3x) $ (4)
2.1.4 $ \dfrac{12x^4y^3 - 8x^3y^2}{-4x^2y} $ (4)

QUESTION 3: ALGEBRAIC EQUATIONS [15]

3.1 Solve the following equations for $ x $. Show all your steps clearly:

3.1.1 $ 4x - 7 = 13 $ (2)
3.1.2 $ 2(3x - 1) = 4x + 6 $ (4)
3.1.3 $ \dfrac{3x}{4} - 2 = 7 $ (3)
3.1.4 $ 4x - (x - 2) = 5 - 2x $ (3)

3.2 Solve the following exponential equation for $ x $:

$ 2^{x+1} = 16 $

[3]

QUESTION 4: PROBLEM SOLVING WITH ALGEBRA [13]

4.1 A mother is $ 30 $ years older than her daughter. In $ 5 $ years' time, the mother will be $ 3 $ times as old as her daughter.

Let the daughter's current age be $ x $. Set up an equation and calculate their current ages.

[5]

4.2 The length of a rectangle is twice its breadth. The perimeter of the rectangle is $ 48\text{ cm} $.

Calculate the length and the breadth of the rectangle.

[4]

4.3 A man bought $ 5 $ pens and $ 3 $ notebooks. A notebook costs $ \text{R } 10 $ more than a pen.

If he paid $ \text{R } 150 $ in total, what is the cost of one pen?

[4]

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 1 - Common Fractions, Decimal Fractions, and Graphs

MATHWISE ACADEMY

Grade 8 Mathematics: Test 1 • Common Fractions, Decimal Fractions, and Graphs

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
You may NOT use a calculator for Question 1 and Question 2.
Write neatly and legibly.

QUESTION 1: COMMON FRACTIONS [17]

(Note: No calculators permitted for this question)

1.1 Calculate the following and leave your answer in its simplest form. Show all steps:

1.1.1 $ \dfrac{3}{4} - \dfrac{1}{6} + \dfrac{2}{3} $ (3)
1.1.2 $ 2\dfrac{1}{2} \times 1\dfrac{1}{3} $ (3)
1.1.3 $ 3\dfrac{1}{3} \div 4\dfrac{2}{5} $ (3)

1.2 Calculate the following, applying the correct order of operations:

$ \sqrt{\dfrac{16}{25}} - \left(\dfrac{1}{2}\right)^3 $

[4]

1.3 Rewrite the improper fraction $ \dfrac{23}{5} $ as a mixed number.

[2]

1.4 Calculate $ \dfrac{5}{6} $ of $ \text{R } 360 $.

[2]

QUESTION 2: DECIMAL FRACTIONS [17]

(Note: No calculators permitted for this question)

2.1 Write $ 0,075 $ as a common fraction in its simplest form.

[2]

2.2 Round off $ 14,896 $ to two decimal places.

[1]

2.3 Calculate the following without a calculator. Show your working clearly:

2.3.1 $ 12,45 + 3,8 - 5,09 $ (3)
2.3.2 $ 0,4 \times 0,12 $ (2)
2.3.3 $ 24,5 \div 0,7 $ (3)

2.4 Calculate the following directly:

$ \sqrt{0,0064} + (0,3)^2 $

[3]

2.5 A car travels $ 12,5\text{ km} $ on a single litre of petrol.

How far will the car travel on $ 4,2\text{ litres} $ of petrol?

[3]

QUESTION 3: GRAPHS [16]

3.1 Consider the Cartesian plane:

3.1.1 Write down the coordinates of a point $ P $ if it starts at the origin, translates $ 4 $ units left, and then $ 3 $ units up. (2)
3.1.2 In which quadrant does the point $ R(5 ; -2) $ lie? (1)

3.2 A plumber charges a fixed call-out fee of $ \text{R } 150 $ and an additional $ \text{R } 60 $ per hour of work. The cost $ C $ (in Rands) can be represented by the equation $ C = 150 + 60h $, where $ h $ is the number of hours worked.

3.2.1 If a graph were drawn to represent this equation, would the data be considered discrete or continuous? Explain your answer briefly. (2)
3.2.2 Calculate the total cost for $ 4 $ hours of work. (2)
3.2.3 If the plumber's total bill is $ \text{R } 450 $, calculate how many hours the plumber worked. (3)

3.3 The temperature $ T $ ($\text{in } ^\circ\text{C}$) of a room is recorded over a day. It starts at $ 10^\circ\text{C} $ at 06:00, rises steadily to a maximum of $ 25^\circ\text{C} $ at 14:00, stays constant until 15:00, and then drops steadily to $ 18^\circ\text{C} $ at 18:00.

3.3.1 What is the maximum temperature recorded during this period? (1)
3.3.2 During which specific time interval is the temperature increasing? (2)
3.3.3 During which specific time interval is the temperature constant? (1)
3.3.4 If you were to draw this graph, would the trend line from 15:00 to 18:00 be linear or non-linear? Give a reason for your answer based on the description. (2)

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 2 - Common Fractions, Decimal Fractions, and Graphs

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Common Fractions, Decimal Fractions, and Graphs

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
You may NOT use a calculator for Question 1 and Question 2.
Write neatly and legibly.

QUESTION 1: COMMON FRACTIONS [16]

(Note: No calculators permitted for this question)

1.1 Calculate the following and leave your answers in their simplest form. Show all steps:

1.1.1 $ 1\dfrac{1}{4} + \dfrac{2}{3} \times \dfrac{9}{8} $ (4)
1.1.2 $ 3\dfrac{1}{2} \div 1\dfrac{1}{6} $ (3)

1.2 Calculate the following, showing all your working:

$ \sqrt{\dfrac{16}{25}} - \left(\dfrac{1}{2}\right)^3 $

[4]

1.3 A shopkeeper decides to increase the price of a bicycle, which currently costs $ \text{R } 450 $, by $ 20\% $.

Calculate the new price of the bicycle.

[3]

1.4 Express $ 12\dfrac{1}{2}\% $ as a common fraction in its simplest form.

[2]

QUESTION 2: DECIMAL FRACTIONS [17]

(Note: No calculators permitted for this question)

2.1 Write $ \dfrac{7}{8} $ as a decimal fraction.

[2]

2.2 Round off $ 34,865 $ to one decimal place.

[1]

2.3 Calculate the following without a calculator. Show your working clearly:

2.3.1 $ 0,4 \times 0,12 \times 100 $ (3)
2.3.2 $ 12,5 \div 0,05 $ (3)

2.4 Calculate the following directly:

$ \sqrt{0,0144} + (0,2)^3 $

[4]

2.5 A delivery truck uses $ 6,5\text{ litres} $ of fuel to travel $ 100\text{ km} $.

Calculate how many litres of fuel the truck will use to travel $ 350\text{ km} $.

[4]

QUESTION 3: GRAPHS [17]

3.1 Consider the equation $ y = 3x - 4 $:

3.1.1 Calculate the output value ($ y $) if the input value ($ x $) is $ 2 $. (2)
3.1.2 Calculate the input value ($ x $) if the output value ($ y $) is $ 11 $. (2)

3.2 The volume of water $ V $ (in litres) in a $ 500\text{-litre} $ tank is recorded over a period of $ 50\text{ minutes} $.

• The tank starts completely empty.

• Water is pumped in at a steady, constant rate until the tank contains $ 300\text{ litres} $ exactly $ 20\text{ minutes} $ later.

• The water level then remains constant for $ 10\text{ minutes} $.

• Finally, a valve is opened, and the water is drained at a constant rate until the tank is completely empty $ 20\text{ minutes} $ later.

Based on the scenario above, answer the following questions:

3.2.1 How much water is in the tank at the very start ($ 0\text{ minutes} $)? (1)
3.2.2 During which specific time interval is the volume of water in the tank constant? (2)
3.2.3 If you were to draw this graph, would the data be considered discrete or continuous? Explain your answer briefly. (2)
3.2.4 Calculate the rate at which the water is pumped into the tank during the first $ 20\text{ minutes} $ (in litres per minute). (2)

3.3 Consider the Cartesian plane:

3.3.1 Write down the coordinates of point $ A $ if it starts at the origin, translates $ 3 $ units to the right, and then $ 5 $ units down. (2)
3.3.2 In which quadrant does the point $ B(-2 ; 6) $ lie? (1)
3.3.3 State whether the point $ C(0 ; 4) $ lies on the $ x $-axis or the $ y $-axis. (1)
3.3.4 If point $ D $ is currently at $ (4 ; -1) $ and it translates $ 2 $ units to the left, what are its new coordinates? (2)

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 1 - Data Handling: Probability

MATHWISE ACADEMY

Grade 8 Mathematics: Test 1 • Data Handling: Probability

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
Leave all probability answers as fractions in their simplest form unless specified otherwise.
Write neatly and legibly.

QUESTION 1: PROBABILITY CONCEPTS AND THE SCALE [10]

1.1 Define the term "Sample Space" in the context of probability.

[1]

1.2 State whether the following statements are TRUE or FALSE:

1.2.1 The probability of any event always lies between $ 0 $ and $ 1 $ (inclusive). (1)
1.2.2 If an event has an even chance (50-50) of happening, its probability is $ 0,05 $. (1)

1.3 Assign a probability decimal (from $ 0 $ to $ 1 $) to each of the following events:

1.3.1 The probability of a human being growing wings and flying tomorrow. (1)
1.3.2 The probability of tossing a fair coin and it landing on Heads. (1)
1.3.3 The probability of rolling a $ 7 $ on a standard 6-sided die. (1)
1.3.4 The probability of tomorrow being Tuesday if today is Monday. (1)

1.4 A jar contains $ 3 $ red marbles and $ 3 $ blue marbles.

1.4.1 What is the probability of picking a red marble? Write your answer as a fraction, a decimal, and a percentage. (3)

QUESTION 2: THEORETICAL PROBABILITY [18]

2.1 A standard 6-sided die is rolled. Determine the probability of rolling:

2.1.1 the number $ 4 $. (1)
2.1.2 an odd number. (2)
2.1.3 a prime number. (2)
2.1.4 a natural number less than $ 7 $. (1)

2.2 A card is drawn at random from a standard pack of $ 52 $ playing cards (excluding jokers). Calculate the probability of drawing:

2.2.1 a club. (2)
2.2.2 the King of Hearts. (2)
2.2.3 any red card. (2)

2.3 A letter is chosen at random from the word $ \text{STATISTICS} $. Determine the probability of choosing:

2.3.1 the letter $ \text{S} $. (2)
2.3.2 a vowel. (2)
2.3.3 the letter $ \text{T} $ or the letter $ \text{I} $. (2)

QUESTION 3: BAGS AND PREDICTIONS [12]

3.1 A bag contains $ 5 $ red balls, $ 3 $ blue balls, and $ 4 $ yellow balls. One ball is drawn out of the bag at random.

3.1.1 What is the probability of drawing a blue ball? (2)
3.1.2 What is the probability of drawing a red OR a yellow ball? (2)
3.1.3 What is the probability of drawing a green ball? (1)
3.1.4 What is the probability of drawing a ball that is NOT red? (2)

3.2 A spinner is divided into $ 8 $ equal-sized sectors, numbered $ 1 $ to $ 8 $.

3.2.1 Determine the probability of the spinner landing on a multiple of $ 3 $. (2)
3.2.2 If the spinner is spun $ 400 $ times, predict how many times you would expect it to land on a multiple of $ 3 $. (3)

QUESTION 4: EMPIRICAL PROBABILITY (RELATIVE FREQUENCY) [10]

4.1 A student throws a standard 6-sided die $ 60 $ times to test if it is fair. The results are recorded in the table below:

Number on die	$ 1 $	$ 2 $	$ 3 $	$ 4 $	$ 5 $	$ 6 $
Frequency (Throws)	$ 8 $	$ 12 $	$ 10 $	$ 9 $	$ 15 $	$ 6 $

4.1.1 Calculate the relative frequency (empirical probability) of rolling a $ 5 $. (2)
4.1.2 What is the theoretical probability of rolling a $ 5 $? (2)
4.1.3 Calculate the relative frequency of rolling an even number during this experiment. (3)
4.1.4 Explain briefly why the relative frequency calculated in 4.1.1 is not exactly equal to the theoretical probability in 4.1.2. (2)
4.1.5 What will likely happen to the relative frequency if the student rolls the die $ 6\,000 $ times instead of $ 60 $ times? (1)

--- END OF PAPER ---

Mathwise Academy | Grade 8 Test 2 - Data Handling: Probability

MATHWISE ACADEMY

Grade 8 Mathematics: Test 2 • Data Handling: Probability

Time: 1 Hour

Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
Leave all probability answers as fractions in their simplest form unless specified otherwise.
Write neatly and legibly.

QUESTION 1: PROBABILITY CONCEPTS AND THE SCALE [10]

1.1 Define the term "Event" in the context of probability.

[1]

1.2 Assign a probability decimal (from $ 0 $ to $ 1 $) to each of the following events:

1.2.1 The probability of rolling an $ 8 $ on a standard 6-sided die. (1)
1.2.2 The probability of tossing a fair coin and it landing on Tails. (1)
1.2.3 The probability of drawing a red ball from a bag containing exactly $ 10 $ red balls. (1)
1.2.4 The probability of the sun rising in the West tomorrow morning. (1)

1.3 State whether the following statements are TRUE or FALSE. If false, briefly correct the statement:

1.3.1 The probability of an event can be calculated to be $ 1,5 $. (2)
1.3.2 If you toss a single fair coin once, the sample space is $ \{\text{Heads}, \text{Tails}\} $. (2)

1.4 What is the total sum of the probabilities of all possible equally likely outcomes in a sample space?

[1]

QUESTION 2: THEORETICAL PROBABILITY [16]

2.1 A cash bag contains $ 20 $ five-rand coins, $ 30 $ two-rand coins, and $ 10 $ one-rand coins. Find the probability that one coin selected at random from the bag is:

2.1.1 a one-rand coin. (2)
2.1.2 a two-rand coin. (2)
2.1.3 a one-rand OR a five-rand coin. (2)

2.2 A letter is chosen at random from the word $ \text{REPRESENTATIVE} $. Determine the probability of choosing:

2.2.1 the letter $ \text{E} $. (2)
2.2.2 a vowel. (2)

2.3 A card is drawn at random from a standard pack of $ 52 $ playing cards (excluding jokers). Calculate the probability of drawing:

2.3.1 any black card. (2)
2.3.2 the Queen of Spades. (2)
2.3.3 a "face card" (i.e. a Jack, Queen, or King of any suit). (2)

QUESTION 3: NUMBERED BALLS AND PREDICTIONS [12]

3.1 Ten numbered balls (numbered $ 1 $ to $ 10 $) are placed in a bag. One ball is taken out of the bag at random. Determine the probability of selecting a ball:

3.1.1 numbered $ 7 $. (1)
3.1.2 with an even number. (2)
3.1.3 with a prime number. (2)
3.1.4 with a number greater than $ 10 $. (1)

3.2 Probabilities can be used to make predictions:

3.2.1 A standard 6-sided die is rolled $ 1\,200 $ times. Predict how many times you would expect to roll a multiple of $ 3 $. (3)
3.2.2 The same die is rolled $ 600 $ times. Predict how many times you would expect to roll the number $ 4 $. (3)

QUESTION 4: EMPIRICAL PROBABILITY AND HIGH-ORDER CHALLENGE [12]

4.1 A survey of $ 80 $ people was conducted to find out their favourite flavour of frozen yoghurt. The results are as follows: Chocolate ($ 25 $ people), Vanilla ($ 15 $ people), Strawberry ($ 30 $ people), and Mint ($ 10 $ people).

4.1.1 Calculate the relative frequency (empirical probability) of a person choosing Strawberry. (2)
4.1.2 Calculate the relative frequency of a person choosing Mint OR Vanilla. (2)
4.1.3 If $ 400 $ people are surveyed at a similar event, predict how many people will choose Chocolate as their favourite flavour. (3)

4.2 Algebraic Probability Challenge:
One ball is selected at random from a bag containing $ 12 $ balls. Suppose that there are $ x $ red balls in the bag.
You add a further $ 6 $ red balls to the bag. The probability of now selecting a red ball is doubled.
Set up an equation and determine the original number of red balls ($ x $).

[5]

--- END OF PAPER ---

INSTRUCTIONS AND INFORMATION

QUESTION 1: PROPERTIES AND OPERATIONS [10]

QUESTION 2: PRIME FACTORS, HCF AND LCM [10]

QUESTION 3: RATIOS AND RATES [10]

QUESTION 4: CURRENCY AND PERCENTAGES [10]

QUESTION 5: SIMPLE INTEREST AND HIRE PURCHASE [10]

Solution: 1.1 Order of Operations

1.1 Evaluating \( 25 - 4 \times (7 - 4) + 18 \div 3 \)

Solution: 1.2 Distributive Property

1.2 Calculating \( 6 \times 87 \)

Solutions: 1.3 Mathematical Properties

1.3.1 \( 14 + (26 + 9) = (14 + 26) + 9 \)

1.3.2 \( 45 \times 12 = 12 \times 45 \)

Solutions: 1.4 True or False

1.4.1 Identity Element for Multiplication

1.4.2 \( 15 \div 0 \)

1.4.3 Natural Numbers and Zero

Solutions: 2.1 Prime Factorisation

2.1.1 Prime Factors of \( 72 \)

2.1.2 Prime Factors of \( 120 \)

Solutions: 2.2 HCF and LCM

2.2.1 Highest Common Factor (HCF)

2.2.2 Lowest Common Multiple (LCM)

Solution: 2.3 HCF from Decomposed Factors

2.3 HCF of \( A \) and \( B \)

Solutions: 3.1 Simplifying Ratios

3.1.1 \( 45\text{ cm} : 1,5\text{ m} \)

3.1.2 \( 0,4 : 0,16 \)

Solution: 3.2 Dividing in Ratio

3.2 Dividing \( \text{R } 720 \) in ratio \( 2 : 3 : 5 \)

Solutions: 3.3 Speed and Travel Time

3.3.1 Speed of the Car

3.3.2 Time to Travel \( 520\text{ km} \)

Solutions: 4.1 Exchange Rates

4.1.1 Cost of \( \$35 \) Textbook in Rands

4.1.2 Dollars for \( \text{R } 1\,110 \)

Solution: 4.2 Markup on School Bag

4.2 Selling Price with 35% Markup

Solution: 4.3 Original Marked Price

4.3 Reversing a 25% Discount

Solution: 4.4 Test Percentage Score

4.4 Calculating Percentage

Solutions: 5.1 Simple Interest

5.1.1 Total Simple Interest

5.1.2 Accumulated Total Value

Solutions: 5.2 Hire Purchase

5.2.1 Cash Deposit

5.2.2 Loan Balance Owed

5.2.3 Simple Interest on Balance

5.2.4 Total Amount Payable

5.2.5 Monthly Instalment

INSTRUCTIONS AND INFORMATION

QUESTION 1: PROPERTIES AND OPERATIONS [10]

QUESTION 2: PRIME FACTORS, HCF AND LCM [10]

QUESTION 3: RATIOS AND RATES [10]

QUESTION 4: CURRENCY AND PERCENTAGES [10]

QUESTION 5: SIMPLE INTEREST AND HIRE PURCHASE [10]

Solution: 1.1 Order of Operations

1.1 Evaluating \( 40 - 5 \times (8 - 2) + 24 \div 6 \)

Solution: 1.2 Distributive Property

1.2 Calculating \( 8 \times 64 \)

Solutions: 1.3 Mathematical Properties

1.3.1 \( (15 + 7) + 3 = 15 + (7 + 3) \)

1.3.2 \( 29 \times 1 = 29 \)

Solutions: 1.4 True or False

1.4.1 Commutativity of Division

1.4.2 \( 12 \div 0 = 0 \)

1.4.3 Subtraction and Addition

Solutions: 2.1 Prime Factorisation

2.1.1 Prime Factors of \( 90 \)

2.1.2 Prime Factors of \( 126 \)

Solutions: 2.2 HCF and LCM

2.2.1 Highest Common Factor (HCF)

2.2.2 Lowest Common Multiple (LCM)

Solution: 2.3 Factor Puzzle

2.3 Determining the Unknown Factor

Solution: 3.1 Simplifying Ratio

3.1 Simplifying \( 600\text{ g} : 2\text{ kg} \)

Solution: 3.2 Equivalent Ratio

3.2 Solving \( 4 : 7 = 36 : ? \)