Interpreting Unit Rates in Context: Extended Worked Examples

Grade 8 focus

This record is for Grade 8 learners studying unit rates in context. It does not repeat the full lesson on what a unit rate is or the broader worked-example set on all unit-rate skills. Use these linked records alongside this one:

• For the main teaching sequence, see rea.m08.number.ratios-rates-proportions.unit-rates.interpret-unit-rates.lesson.
• For a broader set on finding and comparing unit rates, see rea.m08.number.ratios-rates-proportions.unit-rates.worked-examples.
• For common mistakes and repairs, see rea.m08.number.ratios-rates-proportions.unit-rates.misconceptions.
• For independent practice, see rea.m08.number.ratios-rates-proportions.unit-rates.practice-set.

What this record adds

Here the main question is not only "Can you divide?" but also:

• Which quantity is "per 1"?
• What does the answer mean in words?
• Does the question want "dollars per item" or "items per dollar"?
• When comparing two rates, which one is better depends on the context?

Strategy for interpreting a unit rate

For each problem, follow this structure:

1. Name the two quantities and their units.
2. Decide which quantity should be written "per 1".
3. Divide in the correct order.
4. Write the answer with units.
5. Say the meaning in a full sentence.
6. Check whether the interpretation matches the situation.

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Worked Example 1: Reading a price rate

A grocery store sells 5 apples for $4.00. What is the unit rate, and what does it mean?

Step 1: Identify the quantities

• 5 apples
• $4.00

The most useful unit rate here is dollars per apple, because that tells the cost of 1 apple.

Step 2: Divide in the correct order

$4.00 / 5 = $0.80

Step 3: State the unit rate

The unit rate is $0.80 per apple.

Step 4: Interpret it in context

This means each apple costs 80 cents.

Commentary

The important choice is the order of division. If you divide 5 by 4, you get 1.25 apples per dollar, which is a valid rate, but it does not answer the usual shopping question about cost of one item. Interpretation depends on what is useful.

Check

5 apples at $0.80 each cost 5 x $0.80 = $4.00, so the answer is consistent.

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Worked Example 2: Speed in words

A cyclist travels 36 km in 3 hours. Find the unit rate and interpret it.

Step 1: Identify the quantities

• 36 km
• 3 h

For motion, we usually want kilometres per hour.

Step 2: Divide

36 / 3 = 12

Step 3: State the unit rate

The unit rate is 12 km/h.

Step 4: Interpret it in context

This means the cyclist travels 12 kilometres in each hour, on average.

Commentary

The phrase "on average" matters. It does not mean the cyclist moved at exactly 12 km every single hour. It means the total trip works out to 12 km for every 1 hour.

Check

In 3 hours, 12 km/h would give 3 x 12 = 36 km.

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Worked Example 3: Which wording is correct?

A faucet fills 18 litres in 6 minutes. Which interpretation is correct?

• A. 3 litres per minute
• B. 3 minutes per litre
• C. Both are valid, but they mean different things

Step 1: Compute both possible unit rates

18 / 6 = 3, so

• 3 litres per minute

Also, 6 / 18 = 1/3, so

• 1/3 minute per litre

Step 2: Compare with the options

Option A is correct. Option B is not correct because 3 minutes per litre is too large. Option C is false as written, because the reverse rate is 1/3 minute per litre, not 3 minutes per litre.

Final answer

The correct interpretation is A. 3 litres per minute.

Commentary

This problem tests whether you notice that reversing a rate changes both the number and the meaning. The reciprocal rate can be valid, but you must recompute it; you cannot just swap the words.

Check

At 3 litres per minute, in 6 minutes the faucet fills 6 x 3 = 18 litres.

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Worked Example 4: Better deal depends on context

Store A sells 8 granola bars for $6.40. Store B sells 5 granola bars for $3.75. Which store is the better buy?

Step 1: Find cost per bar for Store A

$6.40 / 8 = $0.80 per bar

Step 2: Find cost per bar for Store B

$3.75 / 5 = $0.75 per bar

Step 3: Compare the unit rates

• Store A: $0.80 per bar
• Store B: $0.75 per bar

Since $0.75 is less than $0.80, Store B is cheaper per bar.

Final answer

Store B is the better buy because each bar costs 75 cents instead of 80 cents.

Commentary

The phrase "better buy" means the lower cost for 1 item. In some contexts, "better" could mean something else, but in shopping comparison problems it usually means lower cost per unit.

Check

Difference per bar: $0.80 - $0.75 = $0.05. Store B saves 5 cents per bar.

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Worked Example 5: Interpreting a low rate

A printer produces 45 pages in 15 minutes. What is the unit rate, and what does it tell you about the printer?

Step 1: Choose the usual form

For production, pages per minute is natural.

Step 2: Divide

45 / 15 = 3

Step 3: State the rate

The unit rate is 3 pages per minute.

Step 4: Interpret it

This means the printer produces 3 pages in each minute, on average.

Step 5: Reverse rate if needed

If someone asked for minutes per page instead, then 15 / 45 = 1/3 minute per page

That means each page takes one-third of a minute.

Commentary

Both unit rates are legitimate, but the context usually decides which one is easier to understand. "3 pages per minute" is more natural than "1/3 minute per page" here.

Check

In 15 minutes, 3 pages per minute gives 45 pages.

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Worked Example 6: Unit rate with a decimal result

A car uses 7.5 litres of gasoline to travel 100 km. What is the unit rate in litres per kilometre, and how can you interpret it?

Step 1: Identify the requested form

The problem asks for litres per kilometre.

Step 2: Divide

7.5 / 100 = 0.075

Step 3: State the unit rate

The unit rate is 0.075 L/km.

Step 4: Interpret it

This means the car uses 0.075 litre of gasoline for each kilometre traveled, on average.

Step 5: Make the meaning friendlier

Since 0.075 L is small, some people may prefer a larger comparison unit. For 10 km, the car would use 0.075 x 10 = 0.75 L

So another useful interpretation is: about 0.75 litre every 10 km.

Commentary

A correct unit rate is not always the easiest one to picture. Interpreting a rate sometimes includes rewriting its meaning in a more readable way without changing the mathematics.

Check

0.075 x 100 = 7.5, so the unit rate matches the original data.

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Worked Example 7: When the question asks for the reverse rate

A runner completes 12 km in 50 minutes. How many minutes does the runner take per kilometre?

Step 1: Notice what the question asks

It does not ask for km per minute. It asks for minutes per kilometre.

Step 2: Divide in the required order

50 / 12 = 4.1666...

Step 3: Round reasonably

To the nearest tenth, 4.1666... is about 4.2

Step 4: State the unit rate

The runner takes about 4.2 minutes per kilometre.

Step 5: Interpret it

This means each kilometre takes the runner a little more than 4 minutes, on average.

Commentary

Many students automatically divide distance by time because speed problems often use km/h. Here that would give 0.24 km per minute, which is valid but does not answer the question. Always match the division order to the words asked.

Check

If each kilometre takes about 4.2 minutes, then 12 km would take about 12 x 4.2 = 50.4 minutes, close to 50 minutes because of rounding.

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Worked Example 8: Comparing rates with different units in the story

Machine A fills 24 bottles in 8 minutes. Machine B fills 35 bottles in 10 minutes. Which machine is faster, and what does the comparison mean?

Step 1: Find bottles per minute for Machine A

24 / 8 = 3 bottles per minute

Step 2: Find bottles per minute for Machine B

35 / 10 = 3.5 bottles per minute

Step 3: Compare

Since 3.5 > 3, Machine B fills more bottles each minute.

Final answer

Machine B is faster.

Interpretation

• Machine A fills 3 bottles each minute.
• Machine B fills 3.5 bottles each minute.
• So Machine B fills 0.5 more bottle per minute, on average.

Commentary

When comparing production rates, the larger rate is better because more output in the same time means greater speed. In cost problems, the smaller rate is usually better. Context changes what counts as "better."

Check

In 10 minutes, Machine B produces 10 x 3.5 = 35 bottles, which matches.

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Worked Example 9: Multi-step interpretation in a buying decision

Juice Brand X costs $3.60 for 1.5 L. Juice Brand Y costs $4.20 for 2 L. Which brand gives more juice for each dollar?

Step 1: Read the question carefully

The question asks for juice per dollar, not dollars per litre.

Step 2: Find litres per dollar for Brand X

1.5 / 3.60 = 0.4166... So Brand X gives about 0.417 L per dollar.

Step 3: Find litres per dollar for Brand Y

2 / 4.20 = 0.47619... So Brand Y gives about 0.476 L per dollar.

Step 4: Compare

Since 0.476 > 0.417, Brand Y gives more juice for each dollar.

Final answer

Brand Y is the better value for this question because each dollar buys more juice.

Commentary

This is a classic interpretation trap. If you instead computed dollars per litre, you could still solve the problem, but you would compare in the opposite direction: lower dollars per litre would be better. Since the question asks for juice per dollar, larger is better.

Check by reverse calculation

Brand X dollars per litre: 3.60 / 1.5 = 2.4 dollars per litre Brand Y dollars per litre: 4.20 / 2 = 2.1 dollars per litre Brand Y is still better, which confirms the decision.

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Worked Example 10: Challenge problem with choosing the meaningful interpretation

A water tank is being drained. After 4 minutes, 28 litres have been removed.

1. Find the unit rate in litres per minute.
2. Find the unit rate in minutes per litre.
3. Explain which unit rate is more useful if the question is "How quickly is the tank being emptied?"

Part 1: Litres per minute

28 / 4 = 7

So the rate is 7 L/min.

Interpretation

The tank is losing 7 litres each minute.

Part 2: Minutes per litre

4 / 28 = 1/7

So the rate is 1/7 min/L.

Interpretation

It takes 1/7 of a minute to remove 1 litre.

Part 3: Which is more useful?

For the question "How quickly is the tank being emptied?", 7 L/min is more useful because it directly describes the amount removed each minute. It is easier to picture and compare.

Commentary

This challenge shows that more than one unit rate can be correct mathematically. Interpreting in context means choosing the form that communicates the situation most clearly. In draining, filling, driving, printing, and earning contexts, the most useful version is usually the one people naturally ask about.

Check

If 7 litres are removed each minute, then after 4 minutes the amount removed is 4 x 7 = 28 litres.

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Patterns to notice across the examples

• The numbers alone are not enough; the units decide the meaning.
• Reversing a rate changes both the wording and the number.
• In cost contexts, smaller cost per unit is usually better.
• In speed or production contexts, larger output per unit time is usually better.
• A unit rate may be correct mathematically but awkward to interpret.
• Always end with a sentence that answers the real-world question.

Common interpretation errors to watch for

For fuller discussion, see rea.m08.number.ratios-rates-proportions.unit-rates.misconceptions. The most important quick checks are:

• Wrong order of division: Ask, "Per 1 what?"
• Swapped wording without recomputing: Reciprocal rates must be recalculated.
• Comparing the wrong direction: Lower is better for cost per item; higher is better for items per dollar or distance per hour.
• Ignoring context: A rate can be numerically correct but not answer the question asked.

Short practice check with answers

Try these after the worked examples.

1. A baker makes 18 muffins in 3 batches. What is the unit rate? Answer: 6 muffins per batch.

2. A streaming plan costs $12 for 4 months. Interpret the unit rate. Answer: $3 per month, meaning each month costs $3.

3. A bus travels 150 km in 2.5 h. Which is the better interpretation: 60 km/h or 0.0167 h/km? Answer: 60 km/h is the more useful interpretation for travel speed.

4. A box of 9 pens costs $5.85. Another box of 12 pens costs $8.04. Which is the better buy? Answer: First box: $0.65 per pen. Second box: $0.67 per pen. The 9-pen box is slightly cheaper per pen.

Closing note

This record develops the interpretation skill specifically. Once you can compute a unit rate, the next level is deciding what it says, whether it is the right form for the question, and how to use it to compare choices in context.