Mathwise Academy | Grade 9 Test 1 - Whole Numbers

MATHWISE ACADEMY

Grade 9 Mathematics: Test 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.
You may use an approved scientific calculator, unless stated otherwise.
To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: THE REAL NUMBER SYSTEM [10]

1.1 Consider the following set of numbers:

\( \left\{ -5 \quad ; \quad 0 \quad ; \quad \sqrt{16} \quad ; \quad \dfrac{22}{7} \quad ; \quad \sqrt{-9} \quad ; \quad 3,14159... \right\} \)

From the set of numbers above, write down:

1.1.1 One natural number. (1)

1.1.2 One integer that is NOT a whole number. (1)

1.1.3 One rational number that is NOT an integer. (1)

1.1.4 One irrational number. (1)

1.1.5 One non-real number. (1)

1.2 State whether the following statements are TRUE or FALSE. If false, provide the correct property:

1.2.1 \( 1 \) is the additive identity. (2)

1.2.2 The statement \( (5 + 3) + 2 = 5 + (3 + 2) \) illustrates the associative property of addition. (2)

1.2.3 Division of whole numbers is commutative. (1)

QUESTION 2: PRIME FACTORISATION, LCM & HCF [12]

2.1 Write each of the following numbers as a product of their prime factors. Write your final answers in exponential form:

2.1.1 \( 180 \) (2)

2.1.2 \( 252 \) (2)

2.2 Using your prime factorisations from Question 2.1, determine:

2.2.1 The Highest Common Factor (HCF) of \( 180 \) and \( 252 \). (2)

2.2.2 The Lowest Common Multiple (LCM) of \( 180 \) and \( 252 \). (2)

2.3 Application Problem:
Two different lighthouses flash their lights at regular intervals. Lighthouse A flashes every \( 12\text{ minutes} \), and Lighthouse B flashes every \( 15\text{ minutes} \). If they both flash together at 20:00, at what time will they next flash together again?

[4]

QUESTION 3: RATIO, RATE AND PROPORTION [14]

3.1 Simplify the following ratio into its simplest form:

\( 450\text{ ml} : 1,5\text{ litres} \)

[2]

3.2 Divide an amount of \( \text{R } 4\,800 \) between Anna and Ben in the ratio \( 3 : 5 \). Calculate how much money Ben receives.

[3]

3.3 A high-speed train travels a distance of \( 450\text{ km} \) in \( 2\text{ hours and } 15\text{ minutes} \). Calculate the average speed of the train in \( \text{km/h} \).

[3]

3.4 State whether each scenario represents Direct Proportion or Indirect Proportion, and solve the problem:

3.4.1 \( 5 \) identical textbooks cost \( \text{R } 850 \). Calculate the cost of \( 8 \) such textbooks. (3)

3.4.2 It takes \( 4 \) painters exactly \( 6 \) days to paint a large house. If all painters work at the same speed, how many days will it take \( 3 \) painters to paint the same house? (3)

QUESTION 4: PERCENTAGES & FINANCIAL MATHS [14]

4.1 Calculate the following:

4.1.1 Increase \( 250\text{ kg} \) by \( 12\% \). (2)

4.1.2 A pair of shoes marked at \( \text{R } 850 \) is sold at a discount of \( 20\% \). What is the final selling price? (3)

4.2 Matthew invests \( \text{R } 15\,000 \) into a savings account at a Simple Interest rate of \( 6,5\% \) per annum:

4.2.1 Calculate the total interest earned after \( 4\text{ years} \). (2)

4.2.2 What is the total accumulated value of the investment at the end of the \( 4\text{ years} \)? (1)

4.3 Sarah invests \( \text{R } 15\,000 \) into a different savings account offering a Compound Interest rate of \( 6,5\% \) per annum:

4.3.1 Calculate the total accumulated value of Sarah's investment at the end of \( 4\text{ years} \). (4)

4.3.2 Briefly explain why Sarah's final amount is higher than Matthew's, even though they invested the same amount at the same rate for the same time period. (2)

--- END OF PAPER ---

Mathwise Academy | Grade 9 Test 2 - Whole Numbers

MATHWISE ACADEMY

Grade 9 Mathematics: Test 2 • 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.
You may use an approved scientific calculator, unless stated otherwise.
To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: THE REAL NUMBER SYSTEM [10]

1.1 Consider the following set of numbers:

\( \left\{ \sqrt{25} \quad ; \quad -3,4 \quad ; \quad \pi \quad ; \quad \dfrac{0}{5} \quad ; \quad \sqrt{-16} \right\} \)

From the set of numbers above, write down:

1.1.1 One integer that is NOT a natural number. (1)

1.1.2 One terminating decimal fraction. (1)

1.1.3 One irrational number. (1)

1.1.4 One non-real number. (1)

1.1.5 One prime number. (1)

1.2 Write down the name of the mathematical property illustrated by the following statement:

\( x(y + z) = xy + xz \)

[2]

1.3 Use the distributive property to calculate the following without using a calculator. Show all your working steps clearly:

\( 14 \times 12 + 14 \times 88 \)

[3]

QUESTION 2: PRIME FACTORISATION, LCM & HCF [12]

2.1 Write each of the following numbers as a product of their prime factors. Write your final answers in exponential form:

2.1.1 \( 264 \) (2)

2.1.2 \( 360 \) (2)

2.2 Using your prime factorisations from Question 2.1, determine:

2.2.1 The Highest Common Factor (HCF) of \( 264 \) and \( 360 \). (2)

2.2.2 The Lowest Common Multiple (LCM) of \( 264 \) and \( 360 \). (2)

2.3 Application Problem:
A farmer has \( 120 \) apples and \( 84 \) oranges. He wants to pack them into identical gift baskets so that each basket contains the exact same number of apples and the exact same number of oranges, with no fruit left over.

2.3.1 What is the maximum number of identical baskets he can make? (3)

2.3.2 How many apples and how many oranges will be in each basket? (1)

QUESTION 3: RATIO, RATE AND PROPORTION [14]

3.1 An amount of \( \text{R } 8\,400 \) is shared among three partners in the ratio \( 2 : 3 : 5 \). Calculate the amount of money the partner with the largest share receives.

[3]

3.2 A water tap can fill a \( 200\text{-litre} \) container in exactly \( 15\text{ minutes} \). Calculate the rate of water flow in litres per minute, rounded to two decimal places.

[2]

3.3 Solve the following proportion problems:

3.3.1 A butcher sells \( 6\text{ kg} \) of prime beef for \( \text{R } 750 \). Calculate the cost of \( 10\text{ kg} \) of the same beef. (3)

3.3.2 A team of \( 12 \) builders takes \( 8\text{ days} \) to build a brick wall. If they work at the exact same pace, how many days will it take a team of \( 16 \) builders to build the same wall? (3)

3.4 Increase the number \( 150 \) in the ratio \( 5 : 3 \).

[3]

QUESTION 4: PERCENTAGES & FINANCIAL MATHS [14]

4.1 The price of a concert ticket increases from \( \text{R } 400 \) to \( \text{R } 450 \). Calculate the percentage increase in the price of the ticket.

[3]

4.2 Financial Comparison:
Sipho has \( \text{R } 20\,000 \) to invest for a period of \( 3\text{ years} \). He has two options:

Bank A offers a Simple Interest rate of \( 7,5\% \) per annum.
Bank B offers a Compound Interest rate of \( 7\% \) per annum.

Show all calculations and advise Sipho on which bank will give him the best return on his investment at the end of the 3 years.

[7]

4.3 Hire Purchase:
Zinhle buys a new television set priced at \( \text{R } 8\,000 \) on a hire purchase agreement. She must pay a \( 15\% \) cash deposit upfront. The remaining balance is financed over \( 24\text{ months} \) at a simple interest rate of \( 10\% \) per annum.

Calculate Zinhle's fixed monthly instalment.

[4]

--- END OF PAPER ---

Mathwise Academy | Grade 9 Test 1 - Integers & Exponents

MATHWISE ACADEMY

Grade 9 Mathematics: Test 1 • Integers and Exponents

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 final answers with POSITIVE EXPONENTS unless stated otherwise.
Note: No calculators permitted for Question 1.
To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: INTEGERS [10]

Note: No calculators permitted for this question.

1.1 Calculate the value of the following without using a calculator. Show all your steps:

[4]

\[ \frac{-12 - (-4)(-3)}{-2} - (-3)^2 \]

1.2 Evaluate the following algebraic expression if \( x = -2 \), \( y = -1 \), and \( z = 3 \):

[3]

\[ x^2 - yz \]

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

[3]

\[ \sqrt[3]{-64} + \sqrt{100 - 36} \]

QUESTION 2: LAWS OF EXPONENTS [18]

2.1 Simplify the following expressions using the laws of exponents. Leave your answers with positive exponents:

2.1.1 \( x^3 \times x^4 \times x^{-2} \) (2)

2.1.2 \( \dfrac{12a^5b^2}{-3a^2b^6} \) (3)

2.1.3 \( (2x^2y^3)^3 \) (3)

2.1.4 \( 3x^0 + (3x)^0 \) (2)

2.1.5 \( \dfrac{(-2xy^2)^2 \times 3x^4y^{-1}}{6x^5y^3} \) (5)

2.2 Write the bases as prime numbers and then simplify:

[3]

\[ \frac{2^5 \times 9^2}{6^3} \]

QUESTION 3: SCIENTIFIC NOTATION [10]

3.1 Write the following numbers in scientific notation:

3.1.1 45 000 000 (2)

3.1.2 0,00045 (2)

3.2 Write \( 3,2 \times 10^5 \) in standard decimal form (as an ordinary number).

[2]

3.3 Calculate the following and leave your final answers in scientific notation:

3.3.1 \( (2 \times 10^4) \times (3 \times 10^5) \) (2)

3.3.2 \( \dfrac{4,5 \times 10^{-3}}{1,5 \times 10^2} \) (2)

QUESTION 4: EXPONENTIAL EQUATIONS [12]

4.1 Solve for \( x \) in the following exponential equations:

4.1.1 \( 3^x = 81 \) (2)

4.1.2 \( 2^{x-1} = 16 \) (3)

4.1.3 \( 5^x = \dfrac{1}{125} \) (3)

4.1.4 \( 2 \cdot 3^x = 54 \) (4)

--- END OF PAPER ---

Mathwise Academy | Grade 9 Test 2 - Integers & Exponents

MATHWISE ACADEMY

Grade 9 Mathematics: Test 2 • Integers and Exponents

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 final answers with POSITIVE EXPONENTS unless stated otherwise.
Note: No calculators permitted for Question 1.
To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: INTEGERS [10]

Note: No calculators permitted for this question.

1.1 Calculate the value of the following without using a calculator. Show all your steps:

[4]

\[ \sqrt{144} - (-2)^3 + \frac{-15}{-3} \]

1.2 Evaluate the following algebraic expression if \( a = -3 \), \( b = 2 \), and \( c = -1 \):

[3]

\[ a^2 - bc \]

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

[3]

\[ -5 - (-4)(-2) + (-3)^2 \]

QUESTION 2: LAWS OF EXPONENTS [18]

2.1 Simplify the following expressions using the laws of exponents. Leave your answers with positive exponents:

2.1.1 \( 2x^4 \cdot 3x^{-2} \cdot x^0 \) (2)

2.1.2 \( (3a^2b^3)^2 \) (3)

2.1.3 \( \dfrac{15p^3q^{-2}}{5p^5q} \) (4)

2.1.4 \( \dfrac{(2x^2y)^3 \times x^4}{4x^{10}y^3} \) (4)

2.2 Write the bases as prime numbers and then simplify:

[5]

\[ \frac{5^4 \times 8^2}{10^3} \]

QUESTION 3: SCIENTIFIC NOTATION [10]

3.1 Write the following numbers in scientific notation:

3.1.1 0,000078 (2)

3.1.2 145 000 000 (2)

3.2 Write \( 6,1 \times 10^{-4} \) in standard decimal form (as an ordinary number).

[2]

3.3 Calculate the following without a calculator and leave your final answers in scientific notation:

3.3.1 \( (4 \times 10^3) \times (2 \times 10^5) \) (2)

3.3.2 \( \dfrac{9 \times 10^6}{3 \times 10^2} \) (2)

QUESTION 4: EXPONENTIAL EQUATIONS [12]

4.1 Solve for \( x \) in the following exponential equations:

4.1.1 \( 2^x = 64 \) (2)

4.1.2 \( 3^{x+2} = 1 \) (3)

4.1.3 \( 7^{2x-1} = 49 \) (3)

4.1.4 \( 3 \cdot 5^x = 375 \) (4)

--- END OF PAPER ---