Finding Unit Rates: Extended Worked Examples

Position in the unit

This record is for Grade 8 learners studying unit rates in Canadian middle-school mathematics. It should be used alongside:

• rea.m08.number.ratios-rates-proportions.unit-rates.worked-examples for a broader set of unit-rate situations
• rea.m08.number.ratios-rates-proportions.unit-rates.rate-applications.overview for the larger lesson arc and applications
• rea.m08.number.ratios-rates-proportions.unit-rates.practice-set for additional independent practice
• rea.m08.number.ratios-rates-proportions.proportional-modeling.worked-examples when unit rates become part of larger decision problems

This record does not re-teach the general idea of a rate. Its purpose is to build accuracy and judgment in finding a unit rate.

Core procedure

To find a unit rate:

1. Identify the two quantities and their units.
2. Decide which quantity should become 1.
3. Divide so that the denominator is 1.
4. State the answer with units in words.
5. Check whether the size of the answer makes sense.

A useful sentence frame is:

unit rate = amount of first quantity / 1 unit of second quantity

Worked Example 1: Simple price per item

Problem. Three notebooks cost $6.75. Find the cost per notebook.

Reasoning choice. The question asks for the price of 1 notebook, so notebooks must become 1.

Work.

$6.75 / 3 = $2.25

So the unit rate is:

$2.25 per notebook

Check. If one notebook costs $2.25, then three notebooks cost 3 x 2.25 = 6.75, which matches.

Commentary. A common mistake is to write 3 / 6.75, which would give notebooks per dollar instead of dollars per notebook. The words in the question tell you which unit belongs after per.

Worked Example 2: Distance per hour

Problem. A cyclist travels 45 km in 3 hours. Find the unit rate.

Reasoning choice. Speed here should be measured in kilometers for each 1 hour.

Work.

45 / 3 = 15

So the unit rate is:

15 km/h

Check. In 3 hours at 15 km/h, the cyclist would travel 15 x 3 = 45 km.

Commentary. For travel contexts, the unit rate often has a familiar combined unit such as km/h or m/s. Writing the units compactly helps prevent confusion.

Worked Example 3: Decimal division in a grocery context

Problem. 1.5 kg of apples costs $4.80. Find the cost per kilogram.

Reasoning choice. Since the question asks for price per kilogram, kilograms must become 1.

Work.

4.80 / 1.5

Multiply both numbers by 10 to clear the decimal:

48 / 15 = 3.2

So the unit rate is:

$3.20 per kg

Check. 1.5 x 3.20 = 4.80.

Commentary. Clearing decimals before dividing is often easier than dividing directly. Also note the final unit: dollars per kilogram, not kilograms per dollar.

Worked Example 4: Fractional amount of time

Problem. A tap fills 7/8 of a container in 5 minutes. How much of the container does it fill in 1 minute?

Reasoning choice. Here the natural unit rate is container per minute.

Work.

(7/8) / 5 = 7/40

So the unit rate is:

7/40 of a container per minute

Check. In 5 minutes:

5 x (7/40) = 35/40 = 7/8

Commentary. Do not force every answer into a decimal if the fraction is clear and exact. Fractions are often the better choice in part-whole settings.

Worked Example 5: Reading a table

Problem. A table shows that 18 pages are read in 30 minutes. Find the unit rate in pages per minute.

Reasoning choice. The question asks for pages in 1 minute.

Work.

18 / 30 = 3 / 5 = 0.6

So the unit rate is:

0.6 pages per minute

It can also be interpreted as 3 pages every 5 minutes.

Check. In 30 minutes:

30 x 0.6 = 18

Commentary. A unit rate can be less than 1. That does not mean it is wrong. It means less than one full page is read each minute, on average.

Worked Example 6: Choosing the useful unit rate

Problem. A car uses 24 L of fuel to travel 300 km. Find a useful unit rate.

Reasoning choice. There are two possible unit rates:

• 300 / 24 = 12.5 km per L
• 24 / 300 = 0.08 L per km

Both are mathematically correct, but the more useful answer depends on the question. Since no single unit is specified, state the more interpretable school-level version first.

Work.

300 / 24 = 12.5

So one useful unit rate is:

12.5 km per L

Check. 24 x 12.5 = 300.

Commentary. This example matters because many errors come from assuming there is only one correct unit rate. Often there are two reciprocal rates; the context decides which one is more useful.

Worked Example 7: Comparing two deals by finding unit rates

Problem. Which is the better buy for granola bars?

• Pack A: 8 bars for $5.20
• Pack B: 12 bars for $7.08

Reasoning choice. To compare fairly, both prices must be converted to the same unit: dollars per bar.

Work.

Pack A:

5.20 / 8 = 0.65

So Pack A costs $0.65 per bar.

Pack B:

7.08 / 12 = 0.59

So Pack B costs $0.59 per bar.

Since 0.59 < 0.65, Pack B is the better buy.

Check. A lower cost per one bar means a better value.

Commentary. Students sometimes compare total prices only, which is invalid because the numbers of bars are different. Unit rates create a fair comparison.

Worked Example 8: Finding a unit rate after converting units

Problem. A runner covers 1500 m in 6 minutes. Find the speed in meters per second.

Reasoning choice. The target unit is meters per second, so time must be converted first.

Work.

6 minutes = 360 seconds

Now divide distance by time:

1500 / 360 = 25 / 6 = 4.166...

So the unit rate is about:

4.17 m/s

Check. 4.17 x 360 is about 1501.2, close because the speed was rounded.

Commentary. The key choice here is to convert before dividing. If you divide by 6 first, you get meters per minute, not meters per second.

Worked Example 9: Working backward to verify a claimed unit rate

Problem. A sign says that 10 cans cost $8.50, so the unit price is $0.80 per can. Is the sign correct?

Reasoning choice. First compute the actual unit rate, then compare it with the claim.

Work.

8.50 / 10 = 0.85

The actual unit rate is:

$0.85 per can

So the sign is not correct.

Check. If the price were $0.80 per can, then 10 cans would cost $8.00, not $8.50.

Commentary. This shows why unit rates are not just calculations; they are also a way to check claims and labels.

Worked Example 10: Multi-step challenge with area and cost

Problem. A rectangular patio is 4.5 m by 3.2 m. Paving stones cost $28.80 per square meter. Find the total cost.

Reasoning choice. The given unit rate is cost per 1 m^2, so first find the area in square meters.

Work.

Area:

4.5 x 3.2 = 14.4 m^2

Now apply the unit rate:

14.4 x 28.80 = 414.72

So the total cost is:

$414.72

Check. Since 14.4 is a little more than 14, and each square meter costs about $29, an estimate is 14 x 29 = 406, which is close.

Commentary. The challenge is recognizing that the unit rate cannot be used until the quantity is expressed in the matching unit. Here the matching unit is square meters, not meters.

Common misconceptions and repairs

• Mistake: Dividing in the wrong order. Repair: Read the question aloud using the word per. If it asks for dollars per notebook, compute dollars / notebooks.
• Mistake: Ignoring units after calculating. Repair: Write units at every step. Numbers without units are easy to misread.
• Mistake: Assuming a unit rate must be greater than 1. Repair: A unit rate can be less than 1, such as 0.6 pages per minute.
• Mistake: Comparing totals instead of unit rates. Repair: Convert both options to the same per 1 quantity.
• Mistake: Skipping needed conversions. Repair: Match the denominator unit in the question before dividing.

Quick practice

Find each unit rate.

1. 9 pencils cost $4.50. What is the cost per pencil?
2. A hiker walks 16 km in 4 h. What is the speed in km/h?
3. 2.4 L of juice fills 6 bottles equally. How many liters per bottle?
4. A printer produces 135 pages in 15 minutes. How many pages per minute?

Answers

1. $0.50 per pencil
2. 4 km/h
3. 0.4 L per bottle
4. 9 pages per minute

What to study next

After you can reliably find a unit rate, the next step is to interpret what that unit rate means in context, compare rates efficiently, and use unit rates inside larger proportional decisions.