Finding Unit Rates — Grade 8 Unit Overview

Where This Unit Fits

This lesson-sized unit belongs to Grade 8 Number: Ratios, Rates and Proportions, inside the larger unit Unit Rates and Rate Interpretation. It focuses on the calculation step: taking a rate or ratio and rewriting it for 1 unit of one quantity.

For the broader context, see:

• Unit Rates and Rate Interpretation
• Ratios, Rates and Proportions

What You Will Be Able To Do After This Unit

By the end of this unit, you should be able to:

• identify when a comparison is a rate because it compares unlike quantities
• compute a unit rate by dividing both quantities so one quantity becomes 1
• write unit rates with correct language and notation, such as 5 dollars per notebook, 60 km/h, or 18 words per minute
• decide which quantity should be made 1, depending on the question being asked
• explain the meaning of a unit rate in context, not just calculate it
• compare two situations fairly by converting both to unit rates with the same units
• check whether an answer is reasonable using estimation and mental math

Prerequisite Skills

Before starting this unit, you should already be comfortable with ideas from these earlier records:

• Ratio Concepts and Comparison You need to understand ratio language, equivalent ratios, and how ratios compare relationships rather than just totals.
• Ratios, Rates and Proportions You should recognize that rates compare quantities with different units and that proportional reasoning is about preserving multiplicative relationships.
• Unit Rates and Rate Interpretation This parent unit introduces why unit rates matter and how they support comparison and decision-making.

You also need basic fluency with:

• division of whole numbers and decimals
• simplifying fractions
• interpreting units such as dollars, hours, kilometres, litres, and items

Core Idea

A unit rate is a rate written so that one of the quantities is exactly 1.

Examples:

• 12 dollars for 3 notebooks becomes 4 dollars per 1 notebook
• 180 kilometres in 3 hours becomes 60 kilometres per 1 hour
• 750 mL for 5 servings becomes 150 mL per 1 serving

Why do this? Because comparing "for 1" is fair and easy. If two deals use different package sizes, or two trips take different times, rewriting each situation for 1 unit lets you compare them directly.

Intuition Before Procedure

Think of a unit rate as asking:

• "How much for one?"
• "How far in one hour?"
• "How many in one minute?"

If 3 notebooks cost 12 dollars, then one notebook must cost less than 12 dollars, and because 12 split equally among 3 is 4, the cost is 4 dollars per notebook.

This is not a new relationship. It is the same relationship rewritten in a more useful form.

Procedure: How To Find a Unit Rate

Method 1: Divide directly

1. Identify the rate and the quantity you want to make 1.
2. Divide the other quantity by that quantity.
3. Write the answer with per 1 or standard rate notation.

Example: 24 dollars for 6 pens

• Make pens become 1.
• 24 ÷ 6 = 4
• Unit rate: 4 dollars per pen

Method 2: Scale both quantities by the same factor

1. Write the rate as a fraction or pair.
2. Divide both parts by the same number so one part becomes 1.
3. Keep the units attached.

Example: 150 km in 2 hours

• Divide both numbers by 2
• 150 ÷ 2 = 75, 2 ÷ 2 = 1
• Unit rate: 75 km per hour

These two methods are really the same idea.

Fully Worked Examples

Example 1: Cost per item

A pack of 8 granola bars costs 10.40 dollars. Find the unit rate.

We want cost for 1 granola bar.

10.40 ÷ 8 = 1.30

So the unit rate is 1.30 dollars per granola bar.

Reasonableness check:

• 8 items for a little over 10 dollars means 1 item should cost a little over 1 dollar.
• 1.30 makes sense.

Example 2: Speed

A cyclist travels 45 km in 1.5 hours. Find the unit rate.

We want distance for 1 hour.

45 ÷ 1.5 = 30

So the unit rate is 30 km/h.

Interpretation: At this rate, the cyclist travels 30 kilometres in each hour.

Example 3: Rate with a fraction answer

3 kilograms of apples cost 8 dollars. Find the cost per kilogram.

We want cost for 1 kilogram.

8 ÷ 3 = 2.666...

So the exact unit rate is 8/3 dollars per kilogram.

As money, this is usually written as about 2.67 dollars per kilogram.

Important point: Some unit rates are terminating decimals, some repeat, and some are best left as fractions until rounding is needed.

Common Misconceptions and Repairs

Misconception 1: Dividing the wrong way

A learner may compute 3 ÷ 12 instead of 12 ÷ 3 for 12 dollars for 3 notebooks.

Repair: Ask, "What am I finding for one?" If the question is cost per notebook, divide cost by number of notebooks.

Misconception 2: Dropping the units

A learner writes 4 instead of 4 dollars per notebook.

Repair: Always say the answer as a sentence: "Each notebook costs 4 dollars." Units carry the meaning.

Misconception 3: Thinking unit rate means the answer must be 1

A learner may think a unit rate should equal 1.

Repair: Only one of the quantities becomes 1. The other quantity is usually not 1.

Misconception 4: Mixing up ratio and rate

A learner may treat 3 red : 5 blue like a rate.

Repair: A rate compares quantities with different units. A colour-to-colour comparison is a ratio, not a rate.

Suggested Order of Study

1. Review ratio meaning and equivalent ratios in Ratio Concepts and Comparison.
2. Review what makes a comparison a rate in Unit Rates and Rate Interpretation.
3. Start with whole-number unit rates, such as cost per item or kilometres per hour.
4. Move to decimal and fractional rates, including money and measurement contexts.
5. Practise choosing which quantity should become 1 based on the question.
6. Use unit rates to compare two options fairly.
7. Continue into interpretation and decision-making, then later into broader applications in Proportional Modeling and Optimization.

Study Advice for a Self-Taught Learner

When you practise, say each answer aloud in words, not just numbers. For example, say "2.5 litres per minute" instead of only writing 2.5. This helps prevent reversing the division and keeps the units meaningful.

A good self-check is: if I multiply my unit rate back by the original number of units, do I recover the original amount?

Example: 4 dollars per notebook × 3 notebooks = 12 dollars

If that works, your unit rate is likely correct.

Quick Practice

Practice 1

15 oranges cost 9 dollars. What is the cost per orange?

Answer: 9 ÷ 15 = 0.60 So the unit rate is 0.60 dollars per orange.

Practice 2

A car travels 210 km in 3 hours. What is the speed?

Answer: 210 ÷ 3 = 70 So the unit rate is 70 km/h.

Practice 3

2.4 litres of juice are poured equally into 6 bottles. How much juice is in each bottle?

Answer: 2.4 ÷ 6 = 0.4 So the unit rate is 0.4 litres per bottle.

What Comes Next

Once you can reliably find a unit rate, the next step is to interpret it, compare unit rates across options, and use it to make decisions. That moves this skill from calculation to modelling and problem-solving.