Grade 6 · Grade 6: Number, Ratio and Percent · Ratio, Rate and Percent
Ratios, Rates and Unit Rates
Compare quantities with ratio language, build equivalent-ratio tables and find the unit rate that makes shopping decisions easy.
Learning objectives
- Describe a relationship between two quantities using ratio language.
- Build tables and double number lines of equivalent ratios.
- Calculate a unit rate and use it to solve rate problems.
- Decide which of two offers is better value.
AERO Mathematics alignment
AERO.M6.NS.1
Understand the concept of a ratio and use ratio language to describe a relationship between two quantities.
AERO.M6.NS.2
Understand the concept of a unit rate a/b associated with a ratio a:b (b ≠ 0) and use rate language in context.
AERO.M6.NS.3
Use ratio and rate reasoning to solve real-world and mathematical problems using tables, tape diagrams, double number lines and equations.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
A shop sells 3 notebooks for 18 riyals. Another sells 5 of the same notebooks for 28 riyals.
Think about it
Which shop is the better value, and how can you prove it with numbers rather than a feeling?
Hint: Find the price of exactly one notebook in each shop.
EXPLAIN
Ratio, rate and unit rate
A ratio compares two quantities by division. We write the ratio of 3 notebooks to 18 riyals as 3:18, or as the fraction 3/18. A rate is a ratio between quantities measured in different units, such as riyals per notebook. A unit rate tells you how much of one quantity matches exactly one of the other.
Equivalent ratios
Multiplying or dividing both parts of a ratio by the same non-zero number gives an equivalent ratio. Adding the same amount to both parts does not: 3:18 and 4:19 are not equivalent.
| Notebooks | Cost (QAR) | Cost per notebook |
|---|---|---|
| 3 | 18 | 6 |
| 6 | 36 | 6 |
| 5 | 28 | 5.60 |
| 10 | 56 | 5.60 |
The second shop charges 5.60 per notebook, so it is the better value. The unit rate turned a messy comparison into a single number.
Common misconception
A ratio is not a subtraction comparison. 3:18 does not mean a difference of 15; it means each notebook accounts for one part in six of the total cost.
Three tools for ratio problems
- Ratio table: scale both columns by the same factor.
- Double number line: line up the two quantities and step along together.
- Equation: write cost = 5.60 × notebooks and substitute.
INVESTIGATION
Try it yourself
Collect three real prices from a shop or a delivery app for the same product in different pack sizes. Build a ratio table, calculate each unit rate and decide which pack is the best value. Then explain when the cheapest unit rate is still not the best choice.
Watch
Ratios and Rates
Math Antics · 8:49
A clear picture of ratio language before you build your own equivalent-ratio tables.
Before you watch: Is a ratio comparing by subtraction or by division?
- Give two ratios equivalent to 3:5.
- What is the unit rate for 150 km in 3 hours?
Interactive simulation · PhET
Proportion Playground
Use the paint and necklace screens to build ratios that look and taste the same. Record each equivalent ratio you find in a table.
While you explore
- Which pairs of numbers gave exactly the same colour, and what do they have in common?
- What happens to the mixture when you add the same amount to both quantities instead of scaling?
Key vocabulary
- Equivalent ratios
- Ratios that describe the same relationship.
- Unit rate
- How much you get for exactly one of something.
- Ratio
- A way of comparing two amounts by division.
Practice questions
0/1 correct
Level 1 · Criterion A
A recipe uses 4 cups of rice for 6 people. Which ratio is equivalent?
MYP criterion tasks
Level 2 · Criterion D
Pack A: 8 pens for 20 QAR. Pack B: 12 pens for 27 QAR. Which is better value? Justify with unit rates.
Reflect & track
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