Ratios & rates
This chapter has four main ingredients: ratios, rates, unit price, and the unitary method.
5.1 Big picture: why ratios and rates? ▶
Ratios compare same type quantities (students:teachers, blue:red tiles). Rates compare different units (kilometres per hour, dollars per hour, litres per minute).
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Design & maps use scale ratios like
1 : 100. -
Speed signs show rates such as
80 km/h. - Supermarket “best buys” rely on unit price (cost per 100 g, per L, per item).
5.2 Ratios: comparing same-kind quantities ▶
A ratio tells you how much of one quantity there is compared to another of the same kind, in the same units.
- Written with a colon:
3 : 5(“3 to 5”). - No units, and usually written in whole numbers in simplest form (divide both parts by the highest common factor).
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Equivalent ratios come from multiplying or dividing both sides by the same
number, e.g.
2 : 3and4 : 6.
5.3 Rates: different units in one package ▶
A rate compares different kinds of quantities, like distance and time.
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Common examples:
km/h,m/s,$ per hour,L/min. - A unit rate is “per 1” of something, e.g. $25 for 4 hours becomes $6.25 per hour.
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You can rearrange the rate relationship:
rate = first ÷ second,first = rate × second,second = first ÷ rate.
5.4 Unitary method & best buys ▶
The unitary method uses “value of 1 unit” as a stepping stone to everything else.
- Best buy: find cost per unit and pick the smallest.
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Reverse percentages: if
p%of a total is known, find 1%, then multiply up to 100%. - Works even when the new amount is more than 100% (e.g. after a pay rise).
5.5 Review & project ideas ▶
The review combines ratios, rates, unit prices and percentage changes in longer stories (concert crowds, theme parks, work shifts, successive discounts).
Treat it like a “boss level”: if you can connect ideas from earlier lessons, you’re in good shape.
Study game plan
- Pick one subtopic above and skim the notes.
- Switch to the quiz, tick that topic, and drill a few questions.
- Finish by trying an extended problem that mixes several ideas.
Visualising ratios, rates & unit price
Ratio diagrams
Here’s a simple picture of a ratio: two colours sharing one grid.
3 blue : 7 green ⇒ ratio of blue to green is 3 : 7.
Rate & unit price table
| Context | Total | Unit rate |
|---|---|---|
| Speed | 120 km in 2 h | 60 km per hour |
| Wages | $90 in 6 h | $15 per hour |
| Best buy | $4.80 for 300 g | $1.60 per 100 g |
All three examples use the same “per 1 unit” idea — just with different units.
How to use this app
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Practice ratios, rates and best buys in the quiz (inputs like
a:bfor ratios). - Use extended problems to practise explaining your reasoning, not just getting an answer.
a:b.
Numbers: just type the value (e.g. 24 or 24.5).
These questions are meant to feel a little “Year 7 boss fight”. You might combine several ideas at once: unit price plus percentage discount, or ratios plus total numbers.