Overview
What you’ll learn
- Simplify ratios of two or three quantities, including those with different units.
- Divide a quantity in a given ratio, and find one part given another part.
- Use ratios to find missing quantities in equivalent proportions.
- Interpret and calculate rates, including unit rates.
- Use the relations distance = speed × time, speed = distance ÷ time and time = distance ÷ speed, converting between units of time.
In 30 seconds
- 1A ratio compares quantities of the same kind; simplify by dividing by the HCF: 24 : 36 = 2 : 3.
- 2Convert to the same units before forming a ratio: 40 min : 2 h = 40 : 120 = 1 : 3.
- 3Sharing in ratio m : n → find one part = total ÷ (m + n), then multiply.
- 4A rate compares different units, like $ per kg or L per minute — “per” means divide.
- 5The speed triangle: distance = speed × time, speed = distance ÷ time, time = distance ÷ speed.
- 6Map scale 1 : 50 000 → 1 cm on the map is 50 000 cm = 0.5 km on the ground.
Notes · 1 of 5
Working with ratios
A ratio compares parts to parts — simplify it like a fraction, but keep the colon.
A ratio says how many times one quantity is compared with another. The ratio 24 : 36 simplifies by dividing both parts by their HCF: 2 : 3. The order matters — 2 : 3 is not the same as 3 : 2 — and ratios compare like with like: boys to girls, not boys to total students (that second one is a fraction of the whole).
| Situation | Convert first | Simplified ratio |
|---|---|---|
| 40 minutes to 2 hours | 40 min : 120 min | 1 : 3 |
| 80 cents to $2 | 80 c : 200 c | 2 : 5 |
| 750 g to 1.5 kg | 750 g : 1 500 g | 1 : 2 |
Notes · 2 of 5
Finding unknown quantities from ratios
Two moves cover most questions: scale up equal ratios, or use the difference between parts.
- 1Spot the scaling
3 : 5 = 12 : ? — the first side grew ×4, so the second grows ×4: 20.
- 2Or use one known part
If 4 parts weigh 84 g, one part is 21 g — then build any quantity.
- 3Or use a difference
“Blue exceeds red by 12” with ratio 3 : 5 means 2 parts = 12, so 1 part = 6.
The difference between parts is the key to “there are 12 more…” questions: with ratio 3 : 5, the gap is 2 parts. Once one part is known, the total (8 parts) follows at once.
Notes · 3 of 5
Rates and unit rates
A rate compares different units; the unit rate turns every comparison into “per one”.
A rate compares quantities with different units: dollars per kilogram, litres per minute, words per minute. The word per signals division. The most useful form is the unit rate — the amount for exactly one unit — because it makes any comparison a one-step multiply.
| Information | Unit rate | Use it to find |
|---|---|---|
| 675 bottles in 3 min | 225 bottles/min | bottles in any number of minutes |
| $15 for 3 kg of rice | $5 per kg | cost of 8 kg → $40 |
| 330 copies at 24 copies/min | time = 330 ÷ 24 = 13.75 min | the minutes (then convert 0.75 min = 45 s) |
Notes · 4 of 5
Speed, distance and time
Average speed ties the three journey quantities together in one triangle.
Average speed = total distance ÷ total time. Rearranged: distance = speed × time, and time = distance ÷ speed. If a coach covers 260 km in 3 h 15 min, first write the time as a decimal: 3 h 15 min = 3.25 h (15 min = 0.25 h), giving 260 ÷ 3.25 = 80 km/h.
| Minutes | Hours |
|---|---|
| 15 min | 0.25 h |
| 30 min | 0.5 h |
| 45 min | 0.75 h |
| 1 h 30 min | 1.5 h |
Going the other way, a decimal answer like 2.5 h is 2 hours 30 minutes — the 0.5 multiplies by 60. Writing “2.5 h = 2 h 50 min” is a classic error: tenths of hours are not tens of minutes.
Notes · 5 of 5
Map scales
A scale is a ratio between drawing and reality — convert centimetres to kilometres carefully.
- 1Read the scale
1 : 40 000 means 1 cm on the map is 40 000 cm on the ground.
- 2Scale up
5 cm on the map → 5 × 40 000 = 200 000 cm on the ground.
- 3Convert to a friendly unit
200 000 cm = 2 000 m = 2 km.
Going backwards, convert the real distance into centimetres first: 1.2 km = 120 000 cm, so on a 1 : 40 000 map it is 120 000 ÷ 40 000 = 3 cm. The scale has no units because both lengths are measured in the same unit and cancel.
Revise
Key terms
The words examiners expect you to use precisely.
- Ratio
- A comparison of two or more quantities of the same kind, written with a colon, e.g. 3 : 5.
- Equivalent ratios
- Ratios that describe the same relationship, e.g. 3 : 5 and 12 : 20.
- Simplest form
- A ratio whose parts share no common factor greater than 1, e.g. 2 : 3.
- Proportion
- The statement that two ratios are equal, used to find missing quantities.
- Rate
- A comparison of quantities with different units, such as dollars per kilogram.
- Unit rate
- A rate expressed per single unit of the second quantity, e.g. $5 per kg.
Show 4 more key termsShow fewer key terms
- Average speed
- Total distance divided by total time for a journey.
- Scale (of a map)
- The ratio between a length on a map or drawing and the corresponding real length, e.g. 1 : 40 000.
- km/h
- Kilometres per hour, the standard unit of speed for vehicles.
- m/s
- Metres per second, another common speed unit.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Forming a ratio without matching units, e.g. writing 40 min : 2 h as 40 : 2.
The two numbers use different units, so the comparison is meaningless.
Instead: Convert first: 40 min : 120 min = 1 : 3.
Mistake: Sharing by dividing the total by the number of ratio terms, e.g. $126 in 4 : 5 split as $63 each.
The ratio describes parts, not people — there are 4 + 5 = 9 equal parts, not 2.
Instead: One part = 126 ÷ 9 = $14, giving $56 and $70.
Show 8 more mistakesShow fewer mistakes
Mistake: Reading 2.5 h as “2 hours 50 minutes”.
The decimal part is tenths of an hour: 0.5 h = 30 min, not 50 min.
Instead: Multiply the decimal by 60 to get minutes: 2.5 h = 2 h 30 min.
Mistake: Computing average speed as the average of two speeds, e.g. (60 + 40)/2 for two equal-distance legs.
The car spends more time on the slower leg, which drags the true average below 50 km/h.
Instead: Average speed = total distance ÷ total time; compute each leg’s time first.
Mistake: Forgetting to convert cm to km in map-scale answers, reporting 200 000 as the distance.
200 000 is the distance in centimetres, which answers a question nobody asked.
Instead: Finish the conversion: 200 000 cm = 2 km.
Mistake: Swapping the order in a rate division, e.g. dividing time by distance for speed.
Speed is distance per time — km per hour, not hours per km.
Instead: Match the units you want: km/h → km ÷ h; h → km ÷ km/h.
Mistake: Treating 5 minutes as 1/5 of an hour in speed working.
Hours split into 60 minutes, not 5, so 5 min = 5/60 = 1/12 h.
Instead: Divide minutes by 60: 15 min → 0.25 h, 5 min → 1/12 h.
Mistake: Reading the ratio 3 : 4 as “3/4 of the total”.
A ratio compares parts to parts; the fraction of the whole has the total number of parts underneath.
Instead: 3 : 4 means the first share is 3 out of 3 + 4 = 7 parts: 3/7 of the total.
Mistake: Reading 0.2 h as “20 minutes”.
The decimal is tenths of an hour; minutes come from multiplying by 60.
Instead: 0.2 h = 0.2 × 60 = 12 min.
Mistake: Cancelling a ratio by subtracting the same number from both parts, e.g. “simplifying” 15 : 9 to 6 : 3.
Ratios simplify by dividing, not subtracting — 15 : 9 = 5 : 3, and 6 : 3 is a different relationship.
Instead: Divide both parts by their HCF: 15 : 9 = 5 : 3.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Simplify ratios
Simplify the ratio 24 : 36.
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 6Foundation
- 5Standard
- 4Challenging
15 questions worth 54 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 7 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 7.
Mark scheme
PROChapter 7 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 2Coming soonWeighted Assessment 2 (WA2)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 1 Math Chapter 7 about?
Chapter 7 covers ratios of two or three quantities (simplifying, equivalent ratios and sharing in a ratio), rates and unit rates in daily life, the speed-distance-time triangle, and map scales.
How do I share an amount in a given ratio?
Add the ratio parts to get the total number of parts, divide the amount by that total to find one part, then multiply each ratio term by one part. Check by adding the shares back to the original amount.
How do I calculate average speed?
Divide the total distance by the total time, including any stops. Do not average the speeds of different legs — for example, 60 km/h and 40 km/h over equal distances gives 48 km/h, not 50 km/h.
How does a map scale work?
A scale of 1 : 40 000 means 1 cm on the map equals 40 000 cm (0.4 km) on the ground. Multiply map centimetres by the scale for real distances, and convert real distances to centimetres before dividing by the scale.
When is this chapter tested?
Ratio, rate and speed questions usually appear in Weighted Assessment 2 and the end-of-year exam, often as word problems. Every school sets its own scope, so check your school’s assessment outline.
Up next
Chapter 8: Angles, Triangles and Polygons