Year 8 Maths: Number, Percentages, Ratios & Rates
Rational and irrational numbers, recurring decimals, index laws, integers and fractions, percentage change, profit, loss and GST, ratios and best-buy rates. Worked examples, instant answer-checkers and exam-style practice.
Number, Percentages, Ratios and Rates
The number skills behind every other topic, from irrational numbers to working out the best buy.
Aligned to the Australian Curriculum v9 Year 8 Mathematics, Number (AC9M8N01–N05) and rates and ratios in Measurement (AC9M8M05, AC9M8M07), and the Victorian Curriculum Mathematics 2.0 Level 8.
00Start here
This guide covers the Year 8 number skills that everything else is built on: knowing what kind of number you have, working fluently with integers, fractions and powers, and using percentages, ratios and rates to solve real problems with money and measurement.
- Use real prices. At the supermarket, ask which pack is the better buy and why. That's the unit-price skill in section 08.
- Ask "how do you know?" For a 25% discount on $64, a good explanation is "one quarter is $16, so the sale price is $48".
- Let them use the checkers. The boxes marked Check tell your student instantly if an answer is right, so you don't need to mark.
Read the worked examples with a pencil in hand, cover the answer and try them yourself. Then use the answer-checkers in each section and the practice set in section 10.
01Types of number
| Type | Meaning | Examples |
|---|---|---|
| Natural numbers | Counting numbers | 1, 2, 3, … |
| Integers | Whole numbers, positive, negative or zero | −7, 0, 12 |
| Rational numbers | Can be written as a fraction of two integers | ¾, −2½, 0.6, 0.3, 5 |
| Irrational numbers | Can't be written as a fraction; decimal never ends or repeats | √2, √10, π |
| Real numbers | All rational and irrational numbers together: every point on the number line | All of the above |
The square root of a perfect square is rational: √49 = 7. The square root of any other whole number is irrational: √50 is between √49 = 7 and √64 = 8, closer to 7. Its decimal, 7.0710678…, never ends or repeats.
02Terminating and recurring decimals
Every fraction becomes either a terminating or a recurring decimal. Divide the numerator by the denominator to find out which.
| Fraction | Decimal | Type |
|---|---|---|
| 3/8 | 0.375 | terminating |
| 1/3 | 0.333… = 0.3 | recurring |
| 5/6 | 0.8333… = 0.83 | recurring |
| 2/11 | 0.181818… = 0.18 | recurring |
| 7/20 | 0.35 | terminating |
Simplify the fraction. If the denominator's only prime factors are 2 and 5, the decimal terminates. Any other prime factor (3, 7, 11, …) makes it recur. So 7/20 terminates (20 = 2 × 2 × 5), but 5/6 recurs (6 = 2 × 3).
The bar (or dots) sits over the digits that repeat. Write the repeating block once, then stop.
One repeating digit goes over 9; two repeating digits go over 99. So 0.4 = 4/9 and 0.27 = 27/99 = 3/11.
03Index laws
In 25, 2 is the base and 5 is the index. It means 2 × 2 × 2 × 2 × 2 = 32.
| Law | Rule | Example |
|---|---|---|
| Multiplying | am × an = am+n | 23 × 24 = 27 = 128 |
| Dividing | am ÷ an = am−n | 56 ÷ 54 = 52 = 25 |
| Power of a power | (am)n = am×n | (32)3 = 36 = 729 |
| Zero index | a0 = 1 (a ≠ 0) | 70 = 1, and 1,000,0000 = 1 |
| Product to a power | (ab)n = anbn | (2 × 5)3 = 23 × 53 = 1000 |
53 ÷ 53 = 50 by the division law. But any number divided by itself is 1. So 50 = 1.
- 23 × 24 is not 47: the base stays the same.
- 34 means 3 × 3 × 3 × 3 = 81, not 3 × 4 = 12.
- The laws only work when the bases are the same: 23 × 32 can't be combined into one power.
04Integers and fractions
Integers
- Subtracting a negative is adding: −6 − (−9) = −6 + 9 = 3.
- Multiplying or dividing: same signs give a positive, different signs give a negative. (−4) × (−5) = 20; −24 ÷ 3 = −8.
- Order of operations: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). Example: 10 − 3 × (−2)2 = 10 − 3 × 4 = −2.
Fractions
| Operation | Method | Example |
|---|---|---|
| Add / subtract | Make a common denominator first | ⅔ + ¾ = 8/12 + 9/12 = 17/12 = 1 5/12 |
| Multiply | Multiply tops, multiply bottoms (cancel first if you can) | ⅗ × 10/9 = 30/45 = ⅔ |
| Divide | Multiply by the reciprocal of the second fraction | 1½ ÷ ¾ = 3/2 × 4/3 = 2 |
| Mixed numbers | Change to improper fractions first | 2¼ × 1⅓ = 9/4 × 4/3 = 3 |
05Percentages
Per cent means "out of 100". Change between forms: 35% = 35/100 = 0.35.
| Task | Method | Example |
|---|---|---|
| Percentage of an amount | Multiply by the decimal | 18% of $250 = 0.18 × 250 = $45 |
| Express as a percentage | part ÷ whole × 100 | 27 out of 45 = 27 ÷ 45 × 100 = 60% |
| Increase | Multiply by (1 + rate) | $80 increased by 15% = 80 × 1.15 = $92 |
| Decrease | Multiply by (1 − rate) | $250 less 20% = 250 × 0.8 = $200 |
| Percentage change | change ÷ original × 100 | $40 to $50: 10 ÷ 40 × 100 = 25% increase |
| Find the original | Divide by the multiplier | After a 20% discount a jacket costs $96. Original = 96 ÷ 0.8 = $120 |
Percentage change is always calculated on the original amount. A price that rises by 10% and then falls by 10% does not return to where it started: $100 → $110 → $99.
06Profit, loss, discount and GST
- Profit = selling price − cost price. Loss = cost price − selling price.
- Profit as a percentage of cost = profit ÷ cost price × 100. A shirt bought for $40 and sold for $50 makes $10 profit, which is 25% of the cost.
- Mark-up is the amount added to the cost price to get the selling price; a 60% mark-up on a $25 item gives 25 × 1.6 = $40.
- GST is 10% on taxable sales in Australia (not every purchase attracts GST). For a taxable price before GST, multiply by 1.1. To find GST in a wholly GST-inclusive taxable price, divide by 11. A taxable $66 bill includes $6 GST, so the price before GST was $60.
A shop buys sneakers for $90 each and marks them up 50%. In a sale it takes 20% off the marked price. What is the profit on each pair?
Marked price = 90 × 1.5 = $135. Sale price = 135 × 0.8 = $108. Profit = 108 − 90 = $18, which is 18 ÷ 90 × 100 = 20% of the cost.
07Ratios
- A ratio compares quantities of the same kind in a fixed order: 3 cups flour to 2 cups milk is 3 : 2.
- Simplify by dividing every part by the highest common factor: 12 : 18 = 2 : 3; 4 : 6 : 10 = 2 : 3 : 5.
- Make the units the same first: 50 cm : 2 m = 50 : 200 = 1 : 4.
- Dividing in a ratio: share $360 in the ratio 2 : 3 : 4. Total parts = 9; one part = $40; shares are $80, $120 and $160. Check: they add to $360.
- Using a ratio: a cordial is mixed 1 : 7 with water. For 400 mL of cordial you need 400 × 7 = 2,800 mL of water.
08Rates
A rate compares two different quantities, so it always has units: 80 km/h, $24.50 per hour, $8.40/kg, 3 L/min.
| Rate | Formula | Example |
|---|---|---|
| Speed | speed = distance ÷ time | 280 km in 3.5 h = 80 km/h |
| Distance | distance = speed × time | 2 h 15 min at 60 km/h = 2.25 × 60 = 135 km |
| Pay | pay = rate × hours | 12 h at $24.50/h = $294 |
| Unit price | price ÷ quantity | 500 g for $4.20 = $8.40/kg |
Coffee: 500 g for $4.20 or 1.2 kg for $9.60? Unit prices: 4.20 ÷ 0.5 = $8.40/kg and 9.60 ÷ 1.2 = $8.00/kg. The 1.2 kg pack is the better buy.
72 km/h = 72,000 m per 3,600 s = 20 m/s. Shortcut: km/h ÷ 3.6 = m/s.
09Flashcards
10Exam-style practice
34 marks. Show working.
1. Classify each as rational or irrational: √81, √20, 0.7, π, −3⅕. (3)
2. Without dividing, decide whether 7/40 and 4/15 terminate or recur. Explain. (2)
3. Simplify, leaving your answer as a power: (a) 3⁴ × 3⁵ (b) 10⁸ ÷ 10³ (c) (7²)⁴ (d) 6⁵ × 6 ÷ 6⁰ (4)
4. Evaluate −8 + 3 × (−4) − (−6)² ÷ 9. (2)
5. Calculate 3⅓ ÷ 1¼, giving a mixed number. (2)
6. A phone costing $1,150 is discounted by 12%. Find the sale price. (2)
7. After a 15% pay rise, Sam earns $920 a week. What did Sam earn before? (2)
8. A trader buys 40 hats for $300 and sells them all at $12 each. Find the profit and the profit as a percentage of the cost. (3)
9. A plumber's invoice is $385 including GST. How much GST is included, and what was the price before GST? (2)
10. Concrete is made from cement, sand and gravel in the ratio 1 : 2 : 3. How much sand is needed for 4.8 tonnes of concrete? (2)
11. A car travels 210 km in 2 hours 20 minutes. Find its average speed in km/h. (2)
12. Which is better value: 750 mL of shampoo for $6.90 or 1.25 L for $10.75? Show unit prices. (3)
13. A shop raises a $50 price by 20%, then puts it on sale at 20% off. Is the final price $50? Explain. (3)
14. Explain why 0.9 is equal to 1. (2) Challenge
11Checklist: can you…
- classify numbers as integers, rational or irrational and place them on a number line?
- estimate a square root between two whole numbers?
- convert fractions to decimals and tell whether they terminate or recur?
- use and explain the index laws, including the zero index?
- calculate with negative integers using the order of operations?
- add, subtract, multiply and divide fractions and mixed numbers?
- increase or decrease by a percentage using a multiplier?
- calculate percentage change and find an original amount?
- solve profit, loss, mark-up, discount and GST problems?
- simplify ratios and divide a quantity in a ratio?
- calculate and convert rates, and find the best buy?