Whole Numbers
FreeGrade 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.
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
- 1Identify the distributive property: a(b + c) = ab + ac. Here a = 7, b = 15, c = 9.
- 2Distribute: (7 × 15) + (7 × 9)
- 3Calculate each part: 7 × 15 = 105 and 7 × 9 = 63
- 4Add the parts: 105 + 63 = 168
- 5Answer: 7(15 + 9) = 168 ✓
- 1Sipho claims 12 ÷ 0 = 0. To check, use the inverse operation: if 12 ÷ 0 = 0, then 0 × 0 must equal 12.
- 2But 0 × 0 = 0, not 12. There is no number that when multiplied by 0 gives 12.
- 3Therefore 12 ÷ 0 is undefined — Sipho is incorrect.
- 4Answer: No — division by 0 is always 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.
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
- 1Factorise 120: 120 = 2 × 60 = 2 × 2 × 30 = 2 × 2 × 2 × 15 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
- 2Factorise 300: 300 = 2 × 150 = 2 × 2 × 75 = 2 × 2 × 3 × 25 = 2 × 2 × 3 × 5 × 5 = 2² × 3 × 5²
- 3Factorise 900: 900 = 2 × 450 = 2 × 2 × 225 = 2 × 2 × 9 × 25 = 2 × 2 × 3 × 3 × 5 × 5 = 2² × 3² × 5²
- 4Find 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.
- 5HCF = 2² × 3 × 5 = 4 × 3 × 5 = 60
- 6Answer: HCF(120, 300, 900) = 60 ✓
- 1Factorise 18: 18 = 2 × 9 = 2 × 3²
- 2Factorise 24: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2³ × 3
- 3Find all factors with highest powers: 2 appears in both — highest power is 2³. 3 appears in both — highest power is 3².
- 4LCM = 2³ × 3² = 8 × 9 = 72
- 5Answer: LCM(18, 24) = 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: Comparing Quantities of the Same Kind
A ratio compares two or more quantities of the same kind — rand with rand, litres with litres, learners with learners. Because both sides are measured in the same unit, the unit cancels and a ratio has no unit of its own. It is written with a colon: 3 : 5.
Simplifying a ratio
Divide every part by their highest common factor, exactly as you would simplify a fraction. Both quantities must be in the same unit before you start:
24 : 36 → HCF is 12 → 2 : 3
150 : 90 → HCF is 30 → 5 : 3
1,5 m : 50 cm → convert first: 150 cm : 50 cm → 3 : 1
Sharing an amount in a given ratio
Add the parts to find how many parts there are altogether. For 3 : 4 : 5 that is 3 + 4 + 5 = 12 parts.
Find one part by dividing the whole amount by the total number of parts.
Multiply one part by each number in the ratio to get each share.
The check that always works
Add your shares back together. They must come to exactly the original amount — R210 + R280 + R350 = R840 ✓. If they do not, one of the multiplications is wrong, and you will find it in seconds.
Common mistake
Dividing by the number of people instead of the number of parts. Sharing R840 between three people in the ratio 3 : 4 : 5 is not R840 ÷ 3 — it is R840 ÷ 12, because the ratio says there are twelve equal parts, not three. Always add the ratio numbers first.
Worked Examples
- 1Total parts = 3 + 4 + 5 = 12
- 2One part = 840 ÷ 12 = R70
- 3Thabo = 3 × 70 = R210
- 4Lerato = 4 × 70 = R280
- 5Sipho = 5 × 70 = R350
- 6Check: 210 + 280 + 350 = R840 ✓
- 1Convert to the same unit first: 2,4 kg = 2 400 g
- 22 400 : 800
- 3Divide both by the HCF, 800
- 4Answer: 3 : 1 — a ratio has no unit ✓
- 1Total parts = 1 + 2 + 3 = 6
- 2One part = 240 ÷ 6 = 40 kg
- 3Stone is 3 parts: 3 × 40
- 4Answer: 120 kg ✓
Diagram
A bar of R840 divided into twelve equal parts, grouped 3, 4 and 5 to show each share
4. Rate: Comparing Quantities of Different Kinds
A rate compares two quantities of different kinds — kilometres with hours, rand with kilograms, litres with minutes. Because the two units are different, they do not cancel, so unlike a ratio a rate always keeps its units: 72 km/h, R14/kg, 5 ℓ/min. The word “per” is the giveaway.
The three forms of one formula
Speed = Distance ÷ Time
Distance = Speed × Time
Time = Distance ÷ Speed
Getting the units to match
A speed in km/h must be paired with a time in hours and a distance in kilometres. Convert before you substitute:
30 minutes
= 0,5 hours
45 minutes
= 0,75 hours
2 h 15 min
= 2,25 hours
Unit price and the better buy
To compare two package sizes fairly, work out what one unit costs in each: price ÷ quantity. A 3 kg bag at R45 is R15 per kg; a 5 kg bag at R70 is R14 per kg, so the bigger bag is the better value. Comparing the two prices directly tells you nothing, because you are not buying the same amount — the unit price is what makes them comparable.
Average speed is not the average of the speeds
If a car travels 100 km at 100 km/h and then 100 km at 50 km/h, the average speed is not 75 km/h. Work it out properly: total distance 200 km, total time 1 + 2 = 3 hours, so the average speed is 200 ÷ 3 ≈ 66,7 km/h. Always divide the total distance by the total time.
Common mistake
Writing 2 hours 30 minutes as 2,30 hours. Minutes are out of 60, not out of 100, so 30 minutes is 0,5 of an hour and the time is 2,5 hours. Divide the minutes by 60, every time.
Worked Examples
- 1Speed = Distance ÷ Time
- 2= 180 ÷ 2,5
- 3Answer: 72 km/h — note the unit ✓
- 13 kg bag: 45 ÷ 3 = R15 per kg
- 25 kg bag: 70 ÷ 5 = R14 per kg
- 3Compare the unit prices: R14 < R15
- 4Answer: the 5 kg bag, by R1 per kg ✓
- 1Time taken: 08:20 to 11:05 is 2 h 45 min
- 2Convert to hours: 45 ÷ 60 = 0,75, so the time is 2,75 h
- 3Speed = 165 ÷ 2,75
- 4Answer: 60 km/h ✓
Diagram
The speed-distance-time triangle with D on top and S and T below, showing how to read off each of the three formulae
5. Increasing and Decreasing in a Given Ratio
“Increase R480 in the ratio 5 : 4” sounds strange the first time you meet it, but it is one short instruction: for every 4 you had, you now have 5. The ratio is a multiplier written as a fraction, and reading it in that order is the whole skill.
The rule
To change an amount in the ratio a : b → new = original × ab
The first number goes on top. That single fact decides whether the amount grows or shrinks.
Which way will it go?
a > b → it increases
Ratio 5 : 4 gives the fraction 54 = 1,25, which is more than 1, so the amount grows by a quarter.
a < b → it decreases
Ratio 2 : 3 gives 23 ≈ 0,67, which is less than 1, so the amount shrinks to two-thirds.
Increase R480 in the ratio 5 : 4.
Fraction = 54
New amount = 480 × 54 = 480 ÷ 4 × 5
= 120 × 5
= R600
Dividing first keeps the numbers small: 480 ÷ 4 = 120 is easier than 480 × 5 = 2 400 followed by a division. Both routes give the same answer.
Sanity-check before you write the answer
Ask yourself whether the answer should be bigger or smaller than what you started with, then look at what you got. R480 increased in the ratio 5 : 4 must be more than R480 — R600 passes. If your “increase” came out smaller, you turned the fraction upside down.
Common mistake
Reading “decrease R600 in the ratio 3 : 2” as a decrease. The first number is bigger than the second, so 600 × 32 = R900 is an increase — the wording and the ratio disagree, and the ratio always wins. To genuinely decrease R600 to two-thirds, the ratio must be written 2 : 3.
Worked Examples
- 1The ratio 5 : 4 means multiply by 54
- 2480 ÷ 4 = 120 (one part)
- 3120 × 5 = 600
- 4Answer: R600 — bigger than R480, as an increase should be ✓
- 1The ratio 3 : 5 means multiply by 35
- 2350 ÷ 5 = 70
- 370 × 3 = 210
- 4Answer: 210 g — smaller, because 3 < 5 ✓
- 1Ratio 4 : 6 → multiply by 46 = 23
- 2900 ÷ 6 = 150 (flour per person)
- 3150 × 4 = 600 g
- 4Answer: 600 g, enough for 4 people ✓
Diagram
Two bars showing R480 becoming R600 in the ratio 5 : 4, with the four parts of R120 marked and a fifth part added
6. Profit, Loss and Discount
A shop buys goods at the cost price and sells them at the selling price. The difference between the two is a profit if the shop sold for more, and a loss if it sold for less. Everything in this section comes from that one comparison.
The formulae
Profit = selling price − cost price
Loss = cost price − selling price
% profit = profit ÷ cost price × 100
% loss = loss ÷ cost price × 100
Percentage of what?
Percentage profit and percentage loss are always worked out on the cost price — what the shop paid — not on the selling price. A jacket bought for R200 and sold for R260 makes R60 profit, which is 60200 = 30%, not 60260 ≈ 23%. This is the single most common error in the whole section.
Discount
A discount is a reduction off the marked price, and it is a percentage of the marked price. There are two equally good routes:
Two steps
20% of R850 = R170, then R850 − R170 = R680.
One step
A 20% discount leaves 80%, so R850 × 0,80 = R680 directly.
Working backwards from a marked-up price
If an item is sold at a 25% profit for R500, then R500 represents 125% of the cost price. So the cost price is 500 ÷ 1,25 = R400 — divide, do not take 25% of R500. Whenever a price already includes a percentage, you divide by the growth factor to get back.
Worked Examples
- 1Profit = selling price − cost price = 260 − 200 = R60
- 2% profit = profit ÷ cost price × 100
- 3= 60 ÷ 200 × 100
- 4Answer: profit R60, which is 30% ✓
- 1A 15% discount leaves 85% of the marked price
- 22 400 × 0,85
- 3Answer: R2 040 — the discount itself was R360 ✓
- 1Loss = cost price − selling price = 3 200 − 2 720 = R480
- 2% loss = 480 ÷ 3 200 × 100
- 3= 0,15 × 100
- 4Answer: 15% — measured on what he paid, not what he got ✓
Diagram
A cost-price bar of R200 with a R60 profit block added to reach the R260 selling price, labelled 30% of the cost price
7. VAT and Budgets
VAT — Value Added Tax — is a tax added to most goods and services in South Africa. The rate is 15%. Almost every price you see on a shelf already includes it, which is why knowing how to work both forwards and backwards matters.
The two directions
Adding VAT: price including VAT = price excluding VAT × 1,15
Removing VAT: price excluding VAT = price including VAT ÷ 1,15
Why you divide, not subtract
A shelf price of R230 already contains the VAT, so it represents 115% of the original price — not 100%. Taking 15% off R230 gives R195,50, which is wrong. Dividing gives 230 ÷ 1,15 = R200, and checking forwards confirms it: 200 × 1,15 = R230 ✓. Whenever a price already includes a percentage, divide by the factor to get back to where it started.
R150 excluding VAT → 150 × 1,15 = R172,50 including VAT
R460 including VAT → 460 ÷ 1,15 = R400 excluding VAT
The VAT itself in that second example = 460 − 400 = R60
Budgets
A budget is a plan for money: what comes in, what goes out, and what is left. The arithmetic is easy — the skill is in laying it out clearly and reading it.
| Item | Amount |
|---|---|
| Income | R8 400 |
| Rent | R3 200 |
| Food | R2 600 |
| Transport | R1 100 |
| Balance | R1 500 |
A positive balance is a surplus — money left to save. A negative one is a deficit — the household is spending more than it earns, which cannot continue. Questions often ask what fraction or percentage of the income each item takes: rent here is 3 200 ÷ 8 400 ≈ 38%.
Common mistake
Subtracting 15% to remove VAT. It gives an answer that is close enough to look right and is always slightly too small. Check by adding the VAT back: if your excl-VAT answer × 1,15 does not return the original price, you subtracted when you should have divided.
Worked Examples
- 1VAT = 15% of 150 = 0,15 × 150 = R22,50
- 2Total = 150 + 22,50
- 3Or in one step: 150 × 1,15
- 4Answer: R172,50 ✓
- 1R6 900 is 115% of the price excluding VAT
- 2Excluding VAT = 6 900 ÷ 1,15 = R6 000
- 3VAT = 6 900 − 6 000 = R900
- 4Check: 6 000 × 1,15 = R6 900 ✓
- 1Total expenses = 3 200 + 2 600 + 1 100 = R6 900
- 2Balance = 8 400 − 6 900 = R1 500 surplus
- 3Food as a percentage = 2 600 ÷ 8 400 × 100
- 4Answer: R1 500 left over; food takes about 31% ✓
Diagram
A bar of R200 excluding VAT with a 15% block added to reach R230, and an arrow back showing the division by 1,15
8. Simple Interest, Hire Purchase and Exchange Rates
Interest is the price of borrowing or the reward for saving. In Grade 8 all interest is simple interest: it is worked out on the original amount every year, so the same amount is added each year and the growth is a straight line. Compound interest, where the interest itself earns interest, comes in Grade 9.
Simple interest
I = P × i × n
A = P + I
P is the amount you start with, i is the rate as a decimal, n is the time in years.
Interest or amount?
I is the interest earned; A is the total in the account. If a question asks how much interest was earned, stop at I. If it asks for the value of the investment, add the principal back. Read the last line of the question again before you write your answer.
Hire purchase
Hire purchase is how many households buy a fridge or a stove: pay a deposit now, then pay off the rest in equal monthly instalments. Interest is charged on the loan only — the part you still owe after the deposit.
Loan = cash price − deposit. Only this reduced amount earns interest.
Interest = loan × rate × years, then total repayable = loan + interest.
Instalment = total repayable ÷ number of months — that is years × 12, not years.
Exchange rates
An exchange rate is a rate like any other: it tells you how many rand one unit of another currency is worth. If R1 = $0,055, then $1 costs about R18,18. Going from rand to dollars you multiply by the rate; going back you divide. The safest check is a rough estimate — R2 000 must come to a few hundred dollars, so an answer of $36 000 is obviously the wrong way round.
Turning months into years
The rate is per annum, so n must be in years before you substitute: 18 months is 1,5 years, 9 months is 0,75, and 30 months is 2,5. Substituting 18 instead of 1,5 makes the interest twelve times too big.
Worked Examples
- 1P = 4 000, i = 8% = 0,08, n = 3
- 2I = P × i × n = 4 000 × 0,08 × 3 = R960
- 3A = P + I = 4 000 + 960
- 4Answer: interest R960, total R4 960 ✓
- 1Deposit = 10% × 8 000 = R800
- 2Loan = 8 000 − 800 = R7 200
- 3Interest = 7 200 × 0,12 × 2 = R1 728
- 4Total repayable = 7 200 + 1 728 = R8 928
- 5Months = 2 × 12 = 24
- 6Answer: 8 928 ÷ 24 = R372 per month ✓
- 1Each dollar costs R18,50
- 2Number of dollars = 3 700 ÷ 18,50
- 3Answer: $200 — a rough check: R3 700 ÷ R18 ≈ 200 ✓
Diagram
A cash price bar split into a deposit and a loan, with the loan extended by its interest and then divided into monthly instalments