Overview
What you’ll learn
- Write ratios involving whole numbers, fractions and decimals in simplest form.
- Divide a quantity in a given ratio, including three-part ratios, and find quantities before a ratio changed.
- Use map scales to convert between map distances/areas and real distances/areas.
- Distinguish direct from inverse proportion and solve problems with the unitary method.
- Calculate average rate and average speed as total over total.
- Convert speeds between km/h and m/s.
In 30 seconds
- 1A ratio compares quantities of the same kind; simplest form divides every part by the HCF, and ratios of fractions or decimals are cleared first.
- 2To share an amount in the ratio a : b, find one part as amount ÷ (a + b), then multiply by each number of parts.
- 3A map scale 1 : n means 1 cm on the map is n cm in reality; areas scale by n², so 1 cm² on the map is n² cm² on the ground.
- 4Direct proportion: both quantities rise and fall together, y = kx. Inverse proportion: one rises as the other falls, xy = k.
- 5The unitary method finds the value of one unit first; it solves every proportion question in this chapter.
- 6Average speed is total distance ÷ total time — never the average of the two speeds.
- 7Convert speeds with × 5/18 from km/h to m/s and × 18/5 from m/s to km/h.
- 8A rate compares different units (e.g. $ per kg); keep the units through the working and quote them in the answer.
Notes · 1 of 5
Ratios in simplest form
Clear fractions and decimals, then divide by the HCF — the order matters.
A ratio a : b says how many times one quantity is compared with another. Ratios have no units — 45 min : 1 h must become 45 : 60 before simplifying, because both parts need the same unit. Simplify by dividing both parts by the HCF: 45 : 60 = 3 : 4.
- 1Match the units
Convert everything to the same unit: 0.8 kg : 400 g → 800 : 400.
- 2Clear decimals and fractions
Multiply both parts to remove them: 1.2 : 0.45 → 120 : 45 (multiply by 100); 2/3 : 5/6 → 4 : 5 (multiply by 6).
- 3Divide by the HCF
120 : 45 → 8 : 3. Stop when the only common factor is 1.
A ratio also behaves like fractions of the total: in the ratio 3 : 5 the first share is 3/8 of the total and the second is 5/8. That viewpoint powers every sharing question.
Notes · 3 of 5
Map scales: distance and area
A scale of 1 : n shrinks lengths by n and areas by n².
A map scale written 1 : 50 000 means every 1 cm on the map represents 50 000 cm in reality — 500 m. To use it, convert the map measurement to real measurements with the factor n, then change units: centimetres to kilometres divide by 100 000.
Areas are two-dimensional, so each direction scales by n and the area scales by n². A 1 : 50 000 map has area scale 1 : 2 500 000 000 — a 6 cm² park on the map covers 6 × 250² m² at 1 : 250. This is the single most-forgotten idea in the topic: squaring the linear scale is a marking point on its own.
Notes · 4 of 5
Direct and inverse proportion
One quantity scaling up with another is direct; one scaling down as the other grows is inverse. The unitary method solves both.
| Clue in the question | Type | Structure |
|---|---|---|
| More of A needs more of B (5 kg costs more than 2 kg) | Direct | y = kx; double one, double the other |
| More of A needs less of B (more workers, fewer days) | Inverse | xy = k; double one, halve the other |
| Fixed total to share at a constant rate | Inverse | work = rate × time stays constant |
- 1Find one unit
5 kg of rice costs $42, so 1 kg costs $42 ÷ 5 = $8.40.
- 2Scale to the asked quantity
8 kg costs $8.40 × 8 = $67.20.
- 3Sanity check the direction
More kilograms should cost more — direct proportion behaves this way.
Rate questions are proportion with mixed units: 450 words in 12 minutes is a typing rate of 37.5 words per minute, or 2 250 words per hour. Quote the compound unit in the answer — a bare number does not say what it measures.
Notes · 5 of 5
Speed: average speed and unit conversions
Average speed is total distance over total time, and 72 km/h is exactly 20 m/s.
Speed is the rate of covering distance: speed = distance ÷ time. When a journey has several legs, the average speed is the overall rate: average speed = total distance ÷ total time, computed with every pause included in the time.
Unit conversion rides on 1 km/h = 1000 m ÷ 3600 s = 5/18 m/s. So to convert km/h to m/s multiply by 5/18, and to go back multiply by 18/5. Two conversions worth memorising as anchors: 72 km/h = 20 m/s and 15 m/s = 54 km/h.
| km/h | m/s |
|---|---|
| 18 | 5 |
| 36 | 10 |
| 72 | 20 |
| 90 | 25 |
| 108 | 30 |
Revise
Key terms
The words examiners expect you to use precisely.
- Ratio
- A comparison of two or more quantities of the same kind, written a : b with no units.
- Simplest form
- The equivalent ratio whose parts share no common factor above 1.
- Map scale
- The ratio 1 : n between a map length and the real length it represents.
- Area scale factor
- The square of the linear scale factor; areas on a 1 : n map scale by 1 : n².
- Direct proportion
- A relationship where both quantities increase or decrease together in the same ratio, y = kx.
- Inverse proportion
- A relationship where one quantity increases as the other decreases, keeping their product constant, xy = k.
Show 4 more key termsShow fewer key terms
- Unitary method
- Solving a proportion by first finding the value of one unit.
- Rate
- A comparison of quantities with different units, such as dollars per kilogram.
- Average speed
- Total distance travelled divided by total time taken.
- Constant
- The unchanging value in a proportional relationship (k in y = kx or xy = k).
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Simplifying a ratio whose parts have different units, e.g. 45 min : 1 h → 45 : 1.
Ratios compare like with like; the units must match before any dividing.
Instead: Convert first: 45 min : 60 min = 3 : 4.
Mistake: Dividing only one part of a ratio by the HCF.
A ratio changes value unless every part is divided by the same number.
Instead: 45 : 60 → 9 : 12 → 3 : 4 — divide all parts each time.
Show 10 more mistakesShow fewer mistakes
Mistake: Sharing by multiplying the total by each ratio number, e.g. $480 × 2 and $480 × 3.
The shares must add up to the total; $960 + $1440 does not.
Instead: Divide the total by the number of parts first: $480 ÷ 5, then multiply.
Mistake: Using the area scale 1 : n for lengths, or the linear scale 1 : n for areas.
Lengths scale by n but areas by n² — one direction each.
Instead: On a 1 : 250 plan, 1 cm becomes 2.5 m but 1 cm² becomes 6.25 m².
Mistake: Reading a 1 : 50 000 scale as "1 cm = 50 km".
50 000 cm is 0.5 km, not 50 km.
Instead: Convert carefully: 50 000 cm = 500 m.
Mistake: Treating "more workers, fewer days" as direct proportion.
The product of workers and days is fixed; adding workers shortens the job.
Instead: Inverse proportion: 6 × 18 = 9 × 12 = 108 worker-days.
Mistake: Averaging two speeds to get an average speed, e.g. (50 + 40)/2 = 45 km/h.
The legs take different times, so they do not contribute equally.
Instead: Average speed = total distance ÷ total time; the van example gives 44 km/h.
Mistake: Forgetting to include rest stops in the total time of a journey.
Average speed covers the whole door-to-door duration, moving or not.
Instead: Total time includes every pause; total distance excludes retraced none.
Mistake: Converting km/h to m/s by multiplying by 18/5.
That converts the wrong way: 72 km/h would become 259.2 m/s.
Instead: km/h → m/s multiplies by 5/18; m/s → km/h multiplies by 18/5.
Mistake: Adding ratios term by term when combining, e.g. a:b = 1:2 and b:c = 3:4 means a:c = 3:8 by direct cross-read.
The shared quantity b must be written with the same number of parts in both ratios before combining.
Instead: a:b = 3:6 and b:c = 6:8, so a:b:c = 3:6:8 and a:c = 3:8 — after re-scaling b.
Mistake: Answering a rate question with a bare number and no unit.
A rate is a compound quantity; the number alone does not say what is per what.
Instead: Quote the unit: 37.5 words/min, $8.40/kg, 44 km/h.
Mistake: In changing-ratio problems, applying the change to the old ratio numbers directly, e.g. 3 : 5 plus 12 red becomes 15 : 5.
The 12 is a number of marbles, not a number of parts.
Instead: Model one part with k: (3k + 12) : 5k, then use the new ratio to solve for k.
Check yourself
Quick quiz
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Simplify ratios
45 : 60 in simplest form is
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Practise
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20 questions worth 90 marks, each with a model answer, mark scheme and examiner tips.
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Chapter 2 Practice Paper
30 marks · 40 min · 4 pages
5 multiple-choice questions and structured questions with answer lines. Chapter 2 (N2 Ratio and proportion, N4 Rate and speed).
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Model answers with a mark for each creditworthy point — for marking at home or in class.
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Questions
Frequently asked questions
What does Sec 3 E-Math Chapter 2 cover?
It covers syllabus sub-topics N2 and N4 of O-Level 4052 / SEC G3 K310 in full: ratios including fractions and decimals, sharing in a ratio, map scales for distance and area, direct and inverse proportion, average rate and average speed, and km/h to m/s conversion.
How do I tell direct and inverse proportion apart quickly?
Ask whether making the first quantity bigger should make the second bigger or smaller. Bigger-with-bigger is direct (more rice, more cost). Bigger-with-smaller is inverse (more workers, fewer days). The unitary method then finishes either type.
Why is the average speed not the average of the speeds?
Because the legs take different amounts of time, so they count unequally. Average speed is defined as total distance over total time; averaging the two speeds only works in the special case where the two legs take exactly the same time.
When do I square a map scale?
Only for areas. A 1 : 50 000 scale means lengths shrink 50 000 times, so areas shrink 50 000² = 2 500 000 000 times. Every map-area question is testing whether you remember to square.
When is this chapter usually tested in school?
Ratio, rate and speed usually sits in Term 1, so it anchors Weighted Assessment 1 and returns in the end-of-year examination. Check your school’s assessment outline for exact scope.
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Chapter 3: Percentage and Personal Finance