Whole Numbers

Free

Grade 8

1. Properties of Whole Numbers and the Division Property of Zero

We revise the commutative, associative and distributive properties of whole numbers. A new rule in Grade 8 is the division property of zero — any number divided by 0 is undefined, because there is no number that, multiplied by 0, gives a non-zero result.

Colour key:propertiescalculationsundefined

The three properties

Commutative property — the order of numbers does not affect the result for addition and multiplication.

a + b = b + a a × b = b × a

Associative property — the grouping of numbers does not affect the result for addition and multiplication.

(a + b) + c = a + (b + c) (a × b) × c = a × (b × c)

Distributive property — multiplication distributes over addition and subtraction.

a(b + c) = ab + ac a(b − c) = ab − ac

Division property of zero

0 ÷ any number = 0

Zero divided by any non-zero number is always 0.

any number ÷ 0 = undefined

Division by zero is undefined — no number multiplied by 0 can give a non-zero result.

Why is division by zero undefined?

Ask yourself: what number × 0 = 12? There is no answer — anything multiplied by 0 is 0, never 12. Because we cannot find any answer, we say the result is undefined, not zero.

Worked Examples

1
Identify the distributive property: a(b + c) = ab + ac. Here a = 7, b = 15, c = 9.
2
Distribute: (7 × 15) + (7 × 9)
3
Calculate each part: 7 × 15 = 105 and 7 × 9 = 63
4
Add the parts: 105 + 63 = 168
5
Answer: 7(15 + 9) = 168 ✓

Answer7(15 + 9) = 168

1
Sipho claims 12 ÷ 0 = 0. To check, use the inverse operation: if 12 ÷ 0 = 0, then 0 × 0 must equal 12.
2
But 0 × 0 = 0, not 12. There is no number that when multiplied by 0 gives 12.
3
Therefore 12 ÷ 0 is undefined — Sipho is incorrect.
4
Answer: No — division by 0 is always undefined.

AnswerNo — 12 ÷ 0 is undefined

2. Prime Factors, LCM and HCF

We revise finding prime factors of 3-digit whole numbers, and use prime factorisation, inspection, or factorisation methods to find the LCM (Lowest Common Multiple) and HCF (Highest Common Factor) of numbers up to 3 digits.

Colour key:prime factorsHCFLCM

Key terms

Prime number

A number with exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13…

Prime factorisation

Writing a number as a product of its prime factors using a factor tree or repeated division.

HCF

Highest Common Factor — use the prime factors common to all numbers, each taken to its lowest power.

LCM

Lowest Common Multiple — use all prime factors that appear in any number, each taken to its highest power.

HCF vs LCM — quick rule

HCF → common factors, lowest exponents. | LCM → all factors, highest exponents.

Worked Examples

1
Factorise 120: 120 = 2 × 60 = 2 × 2 × 30 = 2 × 2 × 2 × 15 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
2
Factorise 300: 300 = 2 × 150 = 2 × 2 × 75 = 2 × 2 × 3 × 25 = 2 × 2 × 3 × 5 × 5 = 2² × 3 × 5²
3
Factorise 900: 900 = 2 × 450 = 2 × 2 × 225 = 2 × 2 × 9 × 25 = 2 × 2 × 3 × 3 × 5 × 5 = 2² × 3² × 5²
4
Find common factors with lowest powers: 2 appears in all three — lowest power is 2². 3 appears in all three — lowest power is 3¹. 5 appears in all three — lowest power is 5¹.
5
HCF = 2² × 3 × 5 = 4 × 3 × 5 = 60
6
Answer: HCF(120, 300, 900) = 60 ✓

AnswerHCF = 60

1
Factorise 18: 18 = 2 × 9 = 2 × 3²
2
Factorise 24: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2³ × 3
3
Find all factors with highest powers: 2 appears in both — highest power is 2³. 3 appears in both — highest power is 3².
4
LCM = 2³ × 3² = 8 × 9 = 72
5
Answer: LCM(18, 24) = 72 ✓

AnswerLCM = 72

Diagram

Factor tree for 120 branching down to its prime factors 2, 2, 2, 3 and 5, matching the worked example prime factorisation

3. Ratio, Rate and Proportion in Financial Contexts

We solve problems involving ratio (comparing quantities of the same kind), rate (comparing different kinds of quantities, often speed, distance or time), and increasing or decreasing a number in a given ratio. We also apply whole numbers, percentages and decimals to financial contexts including profit, loss, discount, VAT, budgets, loans, simple interest, hire purchase and exchange rates.

Colour key:profitlossVATratio

Key concepts

Ratio

Compares quantities of the same kind. Written as a : b or as a fraction. Always simplify by dividing by the HCF.

Rate

Compares quantities of different kinds. Common examples: km/h, R/kg, litres/hour.

Profit

Profit = selling price − cost price. Percentage profit = profit ÷ cost price × 100.

Loss

Loss = cost price − selling price. A loss occurs when the selling price is less than the cost price.

VAT

VAT = rate × price excl. VAT. In South Africa VAT is 15%. Total price = price + VAT.

Increase/decrease in a ratio

To increase in ratio a : b: new = original ÷ b × a. To decrease: new = original ÷ a × b.

Speed formula

Speed = Distance ÷ Time. Rearrange to find distance (Speed × Time) or time (Distance ÷ Speed).

Price per unit (better value)

Divide price by quantity to find the unit price. Compare unit prices to see which option is the better buy.

Sharing an amount in a ratio

Add the ratio parts to find the total parts. Divide the amount by the total parts to find one part, then multiply by each ratio number.

Increase vs decrease in a ratio

When a number is increased in ratio a : b (a > b), divide by the smaller part and multiply by the larger part. When decreased (a < b), divide by the larger and multiply by the smaller.

Worked Examples

1
Profit = selling price − cost price = R260 − R200 = R60
2
Percentage profit = profit ÷ cost price × 100 = 60 ÷ 200 × 100
3
= 0.30 × 100 = 30%
4
Answer: Profit = R60 and percentage profit = 30% ✓

AnswerProfit = R60; Percentage profit = 30%

1
VAT amount = VAT rate × price excl. VAT = 0.15 × R150 = R22.50
2
Total price = price excl. VAT + VAT = R150 + R22.50 = R172.50
3
Answer: R172.50 ✓

AnswerTotal price = R172.50

1
The ratio 5 : 4 means we increase from 4 parts to 5 parts (the new value is larger).
2
New amount = original ÷ smaller part × larger part = R480 ÷ 4 × 5
3
= R120 × 5 = R600
4
Answer: R600 ✓

AnswerNew amount = R600

1
Write the formula: Speed = Distance ÷ Time.
2
Substitute the values: Speed = 180 km ÷ 2.5 h.
3
Calculate: 180 ÷ 2.5 = 72.
4
Answer: The average speed is 72 km/h ✓

AnswerAverage speed = 72 km/h

1
Find the unit price of the 3 kg bag: R45 ÷ 3 = R15 per kg.
2
Find the unit price of the 5 kg bag: R70 ÷ 5 = R14 per kg.
3Compare: R14 per kg is less than R15 per kg, so the 5 kg bag is the better buy.
4
Answer: The 5 kg bag gives the cheaper price per kilogram ✓

AnswerThe 5 kg bag is cheaper (R14/kg vs R15/kg)

1
Find the total parts: 3 + 4 + 5 = 12.
2
Find the value of one part: R840 ÷ 12 = R70.
3
Thabo's share = 3 × R70 = R210. Lerato's share = 4 × R70 = R280. Sipho's share = 5 × R70 = R350.
4
Check: R210 + R280 + R350 = R840 ✓

AnswerThabo = R210, Lerato = R280, Sipho = R350

Next: Numbers and the Real Number System

Back to Grade 8

Whole Numbers

Free

Grade 8

1. Properties of Whole Numbers and the Division Property of Zero

Colour key:propertiescalculationsundefined

The three properties

Commutative property — the order of numbers does not affect the result for addition and multiplication.

a + b = b + a a × b = b × a

Associative property — the grouping of numbers does not affect the result for addition and multiplication.

(a + b) + c = a + (b + c) (a × b) × c = a × (b × c)

Distributive property — multiplication distributes over addition and subtraction.

a(b + c) = ab + ac a(b − c) = ab − ac

Division property of zero

0 ÷ any number = 0

Zero divided by any non-zero number is always 0.

any number ÷ 0 = undefined

Division by zero is undefined — no number multiplied by 0 can give a non-zero result.

Why is division by zero undefined?

Ask yourself: what number × 0 = 12? There is no answer — anything multiplied by 0 is 0, never 12. Because we cannot find any answer, we say the result is undefined, not zero.

Worked Examples

1
Identify the distributive property: a(b + c) = ab + ac. Here a = 7, b = 15, c = 9.
2
Distribute: (7 × 15) + (7 × 9)
3
Calculate each part: 7 × 15 = 105 and 7 × 9 = 63
4
Add the parts: 105 + 63 = 168
5
Answer: 7(15 + 9) = 168 ✓

Answer7(15 + 9) = 168

1
Sipho claims 12 ÷ 0 = 0. To check, use the inverse operation: if 12 ÷ 0 = 0, then 0 × 0 must equal 12.
2
But 0 × 0 = 0, not 12. There is no number that when multiplied by 0 gives 12.
3
Therefore 12 ÷ 0 is undefined — Sipho is incorrect.
4
Answer: No — division by 0 is always undefined.

AnswerNo — 12 ÷ 0 is undefined

2. Prime Factors, LCM and HCF

Colour key:prime factorsHCFLCM

Key terms

Prime number

A number with exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13…

Prime factorisation

Writing a number as a product of its prime factors using a factor tree or repeated division.

HCF

Highest Common Factor — use the prime factors common to all numbers, each taken to its lowest power.

LCM

Lowest Common Multiple — use all prime factors that appear in any number, each taken to its highest power.

HCF vs LCM — quick rule

HCF → common factors, lowest exponents. | LCM → all factors, highest exponents.

Worked Examples

1
Factorise 120: 120 = 2 × 60 = 2 × 2 × 30 = 2 × 2 × 2 × 15 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
2
Factorise 300: 300 = 2 × 150 = 2 × 2 × 75 = 2 × 2 × 3 × 25 = 2 × 2 × 3 × 5 × 5 = 2² × 3 × 5²
3
Factorise 900: 900 = 2 × 450 = 2 × 2 × 225 = 2 × 2 × 9 × 25 = 2 × 2 × 3 × 3 × 5 × 5 = 2² × 3² × 5²
4
Find common factors with lowest powers: 2 appears in all three — lowest power is 2². 3 appears in all three — lowest power is 3¹. 5 appears in all three — lowest power is 5¹.
5
HCF = 2² × 3 × 5 = 4 × 3 × 5 = 60
6
Answer: HCF(120, 300, 900) = 60 ✓

AnswerHCF = 60

1
Factorise 18: 18 = 2 × 9 = 2 × 3²
2
Factorise 24: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2³ × 3
3
Find all factors with highest powers: 2 appears in both — highest power is 2³. 3 appears in both — highest power is 3².
4
LCM = 2³ × 3² = 8 × 9 = 72
5
Answer: LCM(18, 24) = 72 ✓

AnswerLCM = 72

Diagram

Factor tree for 120 branching down to its prime factors 2, 2, 2, 3 and 5, matching the worked example prime factorisation

3. Ratio, Rate and Proportion in Financial Contexts

Colour key:profitlossVATratio

Key concepts

Ratio

Compares quantities of the same kind. Written as a : b or as a fraction. Always simplify by dividing by the HCF.

Rate

Compares quantities of different kinds. Common examples: km/h, R/kg, litres/hour.

Profit

Profit = selling price − cost price. Percentage profit = profit ÷ cost price × 100.

Loss

Loss = cost price − selling price. A loss occurs when the selling price is less than the cost price.

VAT

VAT = rate × price excl. VAT. In South Africa VAT is 15%. Total price = price + VAT.

Increase/decrease in a ratio

To increase in ratio a : b: new = original ÷ b × a. To decrease: new = original ÷ a × b.

Speed formula

Speed = Distance ÷ Time. Rearrange to find distance (Speed × Time) or time (Distance ÷ Speed).

Price per unit (better value)

Divide price by quantity to find the unit price. Compare unit prices to see which option is the better buy.

Sharing an amount in a ratio

Add the ratio parts to find the total parts. Divide the amount by the total parts to find one part, then multiply by each ratio number.

Increase vs decrease in a ratio

When a number is increased in ratio a : b (a > b), divide by the smaller part and multiply by the larger part. When decreased (a < b), divide by the larger and multiply by the smaller.

Worked Examples

1
Profit = selling price − cost price = R260 − R200 = R60
2
Percentage profit = profit ÷ cost price × 100 = 60 ÷ 200 × 100
3
= 0.30 × 100 = 30%
4
Answer: Profit = R60 and percentage profit = 30% ✓

AnswerProfit = R60; Percentage profit = 30%

1
VAT amount = VAT rate × price excl. VAT = 0.15 × R150 = R22.50
2
Total price = price excl. VAT + VAT = R150 + R22.50 = R172.50
3
Answer: R172.50 ✓

AnswerTotal price = R172.50

1
The ratio 5 : 4 means we increase from 4 parts to 5 parts (the new value is larger).
2
New amount = original ÷ smaller part × larger part = R480 ÷ 4 × 5
3
= R120 × 5 = R600
4
Answer: R600 ✓

AnswerNew amount = R600

1
Write the formula: Speed = Distance ÷ Time.
2
Substitute the values: Speed = 180 km ÷ 2.5 h.
3
Calculate: 180 ÷ 2.5 = 72.
4
Answer: The average speed is 72 km/h ✓

AnswerAverage speed = 72 km/h

1
Find the unit price of the 3 kg bag: R45 ÷ 3 = R15 per kg.
2
Find the unit price of the 5 kg bag: R70 ÷ 5 = R14 per kg.
3Compare: R14 per kg is less than R15 per kg, so the 5 kg bag is the better buy.
4
Answer: The 5 kg bag gives the cheaper price per kilogram ✓

AnswerThe 5 kg bag is cheaper (R14/kg vs R15/kg)

1
Find the total parts: 3 + 4 + 5 = 12.
2
Find the value of one part: R840 ÷ 12 = R70.
3
Thabo's share = 3 × R70 = R210. Lerato's share = 4 × R70 = R280. Sipho's share = 5 × R70 = R350.
4
Check: R210 + R280 + R350 = R840 ✓

AnswerThabo = R210, Lerato = R280, Sipho = R350

Next: Numbers and the Real Number System

Back to Grade 8