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# Year 8 Maths: Number, Percentages, Ratios & Rates
- URL: https://same-page.ghost.io/year-8-maths-number-percentages-ratios/
- Published: 2026-11-02T06:34:11.000Z
- Updated: 2026-11-07T06:23:43.000Z
- Description: 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.
- Author: Same Page
- Tags: Year 8, Maths, Free sample

Same Page · Year 8 Maths

# 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.

**Less than a coffee a month.** Founding families pay **A$3.71 a month** (**A$44.50 billed yearly** for the first year, then A$89 a year). One subscription covers the whole household and every Year 8–10 guide. This guide is free to read in full. [Claim the founding offer](https://same-page.ghost.io/founding-families/) [Or pay monthly, A$9.99](https://same-page.ghost.io/#/portal/signup)

[Start here](#start)[Types of number](#types)[Recurring decimals](#decimals)[Index laws](#index)[Integers & fractions](#ops)[Percentages](#pct)[Profit, loss & GST](#money)[Ratios](#ratio)[Rates](#rates)[Flashcards](#cards)[Practice](#practice)[Checklist](#check)

## 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.

For parents 
1. **Use real prices.** At the supermarket, ask which pack is the better buy and why. That's the unit-price skill in section 08.
2. **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".
3. **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.

How to use this guide

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    |

−3−2−101234−2½−½⅓ = 0.333…√2 ≈ 1.414π ≈ 3.142● rational● irrational

Rational and irrational numbers both have exact places on the number line. Original Same Page diagram.

Square roots

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.

Between which two whole numbers does √30 lie? Type the smaller one.

## 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 |

The quick test

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.

Extension: recurring decimal to fraction

One repeating digit goes over 9; two repeating digits go over 99\. So 0.4 \= 4/9 and 0.27 \= 27/99 = 3/11.

Write 5/8 as a decimal.

## 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  |

Why a⁰ = 1

53 ÷ 53 \= 50 by the division law. But any number divided by itself is 1\. So 50 \= 1.

Common mistakes
- 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.

Simplify 4³ × 4² ÷ 4⁴ and evaluate.

Evaluate (2³)² + 9⁰.

## 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: **B**rackets, **I**ndices, **D**ivision and **M**ultiplication (left to right), **A**ddition and **S**ubtraction (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              |

Evaluate −12 ÷ (−3) + (−2) × 5.

Evaluate 2½ − 1⅔ as a decimal to 2 decimal places.

## 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 |

Exam tip

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.

Start $100Add 10%: $10$110Take 10% off:$9910% of $110 is $11, not $10.$100 → $110 → $99

An equal percentage rise and fall use different starting amounts, so $100 finishes at $99\. Original Same Page diagram.

A $64 game is reduced by 25%. What is the sale price in dollars?

A town's population grows from 12,500 to 13,750\. What is the percentage increase?

## 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.

Worked example

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.

A meal costs $49.50 including GST. How much of that is GST, in dollars?

## 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.

$96 ÷ 8 equal parts = $12 per partAna: 5 parts5 × $12 = $60Ben: 3 parts3 × $12 = $36Check: $60 + $36 = $96

Eight equal parts divide $96 in the ratio 5 : 3, giving $60 and $36\. Original Same Page diagram.

Share $96 between Ana and Ben in the ratio 5 : 3\. How much does Ana get, in dollars?

## 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                 |

Best buy

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.

Converting rates

72 km/h = 72,000 m per 3,600 s = **20 m/s**. Shortcut: km/h ÷ 3.6 = m/s.

A tap fills a 45 L bucket in 2.5 minutes. What is the rate in L/min?

Convert 90 km/h to m/s.

## 09Flashcards

## 10Exam-style practice

34 marks. Show working. Show / hide all answers

1\. Classify each as rational or irrational: √81, √20, 0.7, π, −3⅕. (3)

√81 = 9 rational; √20 irrational; 0.7 \= 7/9 rational; π irrational; −3⅕ = −16/5 rational.

2\. Without dividing, decide whether 7/40 and 4/15 terminate or recur. Explain. (2)

40 = 2³ × 5, only 2s and 5s, so 7/40 terminates (0.175). 15 = 3 × 5 includes a 3, so 4/15 recurs (0.26).

3\. Simplify, leaving your answer as a power: (a) 3⁴ × 3⁵ (b) 10⁸ ÷ 10³ (c) (7²)⁴ (d) 6⁵ × 6 ÷ 6⁰ (4)

(a) 3⁹ (b) 10⁵ (c) 7⁸ (d) 6⁶ (6 = 6¹ and 6⁰ = 1).

4\. Evaluate −8 + 3 × (−4) − (−6)² ÷ 9\. (2)

−8 + (−12) − 36 ÷ 9 = −8 − 12 − 4 = **−24**.

5\. Calculate 3⅓ ÷ 1¼, giving a mixed number. (2)

10/3 ÷ 5/4 = 10/3 × 4/5 = 40/15 = 8/3 = **2⅔**.

6\. A phone costing $1,150 is discounted by 12%. Find the sale price. (2)

1150 × 0.88 = **$1,012**.

7\. After a 15% pay rise, Sam earns $920 a week. What did Sam earn before? (2)

920 ÷ 1.15 = **$800**.

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)

Revenue 40 × 12 = $480\. Profit = 480 − 300 = **$180**. 180 ÷ 300 × 100 = **60%**.

9\. A plumber's invoice is $385 including GST. How much GST is included, and what was the price before GST? (2)

GST = 385 ÷ 11 = **$35**; before GST = **$350**.

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)

6 parts; one part = 0.8 t; sand = 2 × 0.8 = **1.6 t**.

11\. A car travels 210 km in 2 hours 20 minutes. Find its average speed in km/h. (2)

2 h 20 min = 2⅓ h. 210 ÷ 7/3 = 210 × 3/7 = **90 km/h**.

12\. Which is better value: 750 mL of shampoo for $6.90 or 1.25 L for $10.75? Show unit prices. (3)

6.90 ÷ 0.75 = $9.20/L; 10.75 ÷ 1.25 = $8.60/L. The **1.25 L** bottle is better value.

13\. A shop raises a $50 price by 20%, then puts it on sale at 20% off. Is the final price $50? Explain. (3)

50 × 1.2 = $60; 60 × 0.8 = **$48**. No: the 20% discount is taken from the larger price, $60, so it removes $12, more than the $10 that was added.

14\. Explain why 0.9 is equal to 1\. (2) Challenge

One repeating digit goes over 9, so 0.9 \= 9/9 = 1\. Or: ⅓ = 0.3, and 3 × ⅓ = 1 while 3 × 0.3 \= 0.9.

## 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?

**Less than a coffee a month.** Founding families pay **A$3.71 a month** (**A$44.50 billed yearly** for the first year, then A$89 a year). One subscription covers the whole household and every Year 8–10 guide. Liked this guide? [Claim the founding offer](https://same-page.ghost.io/founding-families/) [Or pay monthly, A$9.99](https://same-page.ghost.io/#/portal/signup)

**Sources and credits.** Curriculum alignment: content descriptions from the Australian Curriculum Version 9.0 © Australian Curriculum, Assessment and Reporting Authority (ACARA), licensed under CC BY 4.0 (creativecommons.org/licenses/by/4.0). Topic codes from the Victorian Curriculum F–10 © VCAA are cited for alignment only. ACARA and VCAA do not endorse this guide. All explanations, diagrams and questions are original Same Page material. Same Page is a study aid: check your school's course outline. Spotted an error? Email atlasknoxbuilds@gmail.com and we'll fix it.