Finding Unit Rates

Grade level: Grade 8 Number

Place in the unit

This lesson is a focused sublesson on how to calculate a unit rate. For the broader meaning of unit rates and how to interpret statements such as "3.5 dollars per kilogram," see rea.m08.number.ratios-rates-proportions.unit-rates.lesson. For larger proportion-solving contexts, see rea.m08.number.ratios-rates-proportions.equivalent-ratios.proportion-problems.overview. After this lesson, use rea.m08.number.ratios-rates-proportions.unit-rates.practice-set for fluency.

Learning goal

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

• recognize when a rate can be rewritten as a unit rate;
• compute a unit rate by dividing correctly;
• state the answer with the correct units;
• check whether the result makes sense in context.

1. Intuition: what a unit rate is

A rate compares two quantities with different units.

Examples:

• 180 kilometres in 3 hours
• $4.50 for 2 kilograms
• 24 students for 6 groups

A unit rate is the same comparison, but written for 1 unit of the second quantity.

Examples:

• kilometres per 1 hour
• dollars per 1 kilogram
• students per 1 group

Visual idea

Imagine 180 km spread evenly across 3 equal 1-hour blocks:

• after 1 hour: ? km
• after 1 hour: ? km
• after 1 hour: ? km

If the trip is at a constant rate, the 180 km is split equally among 3 equal hours. So each 1-hour block gets 180 / 3 = 60 km. The unit rate is 60 km/h.

That is the central idea:

To find a unit rate, split the first quantity equally across the number of second-unit blocks until the second quantity becomes 1.

2. Precise notation

If a rate is

• a units of X in b units of Y,

then the unit rate of X per 1 Y is

• a / b units of X per Y.

Common written forms:

• a/b X per Y
• a:b is a ratio notation, but for unit rates we usually state the division result explicitly
• 60 km/h means 60 kilometres per 1 hour
• $2.25/kg means $2.25 per 1 kilogram

3. Procedure: how to find a unit rate

Use this procedure every time.

1. Identify the two quantities and their units.
2. Decide which quantity should become 1.
3. Divide the first quantity by the second quantity.
4. Write the answer as "___ per 1 ___."
5. Check reasonableness: if the original amount is spread over several equal units, the per-1 amount should usually be smaller than the original first quantity.

Formula pattern

If the rate is a units in b units, then

• unit rate = a / b per 1 unit.

4. Worked examples

Example 1: whole numbers, exact division

A cyclist travels 84 kilometres in 4 hours. Find the unit rate in kilometres per hour.

Step 1. Identify the quantities.

• 84 kilometres
• 4 hours

Step 2. We want kilometres per 1 hour.

Step 3. Divide distance by time.

• 84 / 4 = 21

Step 4. State the unit rate.

• 21 km/h

Interpretation. The cyclist travels 21 kilometres in each hour.

Check. 21 x 4 = 84, so the answer is consistent.

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Example 2: money and mass

A 2 kg bag of rice costs $7.50. Find the unit rate in dollars per kilogram.

Step 1. Identify the quantities.

• $7.50
• 2 kg

Step 2. We want dollars per 1 kilogram.

Step 3. Divide cost by kilograms.

• 7.50 / 2 = 3.75

Step 4. State the unit rate.

• $3.75/kg

Interpretation. Each kilogram costs $3.75.

Check. 3.75 x 2 = 7.50.

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Example 3: decimal result

A car uses 18 litres of fuel to travel 240 kilometres. Find the unit rate in kilometres per litre.

Step 1. Identify the quantities.

• 240 kilometres
• 18 litres

Step 2. We want kilometres per 1 litre.

Step 3. Divide distance by fuel.

• 240 / 18 = 13.333...

Step 4. Decide how to report the answer.

• Exact repeating decimal: 13.333... km/L
• Rounded to the nearest tenth: 13.3 km/L

Step 5. State the unit rate.

• About 13.3 km/L

Interpretation. For each litre of fuel, the car travels about 13.3 kilometres.

Check. 13.333... x 18 = 240.

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Example 4: choosing the correct direction of division

A bakery makes 96 muffins in 12 trays. Find the unit rate in muffins per tray.

Step 1. Identify the quantities.

• 96 muffins
• 12 trays

Step 2. We want muffins per 1 tray, so trays must become 1.

Step 3. Divide muffins by trays.

• 96 / 12 = 8

Step 4. State the unit rate.

• 8 muffins per tray

Interpretation. Each tray holds 8 muffins.

Important note. If you accidentally divide 12 / 96, you get trays per muffin, which answers a different question.

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Example 5: fraction/decimal context with comparison

Store A sells 750 g of yogurt for $3.60. Find the unit rate in dollars per 100 g and in dollars per kilogram.

Part A: dollars per 100 g

Step 1. Quantities:

• $3.60
• 750 g

Step 2. To get per 100 g, divide both quantities by 7.5 because 750 / 7.5 = 100.

• $3.60 / 7.5 = $0.48
• 750 g / 7.5 = 100 g

So the unitized rate is:

• $0.48 per 100 g

Part B: dollars per kilogram

Since 1000 g = 1 kg, use division directly:

• 3.60 / 0.75 = 4.8

So:

• $4.80/kg

Interpretation. The yogurt costs 48 cents per 100 g, which is the same as $4.80 per kilogram.

Check. 4.80 x 0.75 = 3.60.

5. When division gives the unit rate

There are two equivalent ways to think:

Method A: divide the first quantity by the second

Example: 84 km in 4 h

• 84 / 4 = 21 km/h

Method B: scale the second quantity to 1

Example: 84 km in 4 h

• divide both quantities by 4
• 84 : 4 becomes 21 : 1
• so 21 km per 1 h

Method B often helps learners see why division works.

6. Common misconceptions and repairs

Misconception 1: dividing in the wrong order

Error: using hours / kilometres when the question asks for kilometres per hour.

Repair: Read the words after "per." The quantity before "per" goes on top.

• kilometres per hour = kilometres / hours
• dollars per kilogram = dollars / kilograms

Misconception 2: forgetting the units

Error: writing only 21 instead of 21 km/h.

Repair: Always finish the sentence: "21 what per 1 what?"

Misconception 3: treating any quotient as meaningful without context

Error: computing correctly but not checking whether the result fits the situation.

Repair: Ask, "What does this mean in words?" If you cannot say it clearly, recheck the setup.

Misconception 4: confusing unit rate with total amount

Error: saying $3.75/kg means the whole 2 kg bag costs $3.75.

Repair: A unit rate is the amount for 1 unit, not for the full original amount.

7. Quick checks for reasonableness

Use these after every problem.

• Multiply your unit rate by the original second quantity. You should get back the original first quantity.
• Make sure the units match the question.
• If the second quantity is greater than 1, the per-1 amount is usually smaller than the original first quantity.
• If the second quantity is less than 1, the per-1 amount can be larger.

8. Mini practice with answers

Try these without looking, then check.

1. 45 pages in 3 days. Find pages per day.
   Answer: 45 / 3 = 15, so 15 pages/day.

2. $9.60 for 4 notebooks. Find dollars per notebook.
   Answer: 9.60 / 4 = 2.40, so $2.40 per notebook.

3. 1500 m in 5 minutes. Find metres per minute.
   Answer: 1500 / 5 = 300, so 300 m/min.

4. 8.4 litres for 70 km. Find litres per kilometre.
   Answer: 8.4 / 70 = 0.12, so 0.12 L/km.

9. Summary

A unit rate is a rate written for 1 unit of the second quantity. To find it, divide the first quantity by the second quantity, then write the result with correct units and interpret it in words. The calculation matters, but the meaning and units matter just as much.