Rate Language and Unit Structure: Extended Worked Examples

Position in the unit

This is a Grade 8 worked-examples record for Rate Language and Unit Structure. Use it with, not instead of, the linked lesson record:

• For the core idea of rate language, see rea.m08.number.ratios-rates-proportions.unit-rates.rate-language.lesson.
• For broader unit-rate problem solving, see rea.m08.number.ratios-rates-proportions.unit-rates.worked-examples.
• For applying unit rates in longer contexts, see rea.m08.number.ratios-rates-proportions.unit-rates.rate-applications.lesson.

This record does not reteach the whole lesson. Its job is narrower: to show how the words, the order, and the units control the meaning.

How to read these examples

When you meet a rate, ask four questions in order:

1. What two different quantities are being compared?
2. In what order are they written?
3. What does per mean here?
4. If I invert the rate, how does the meaning change?

A reliable check is: if the rate is a units of A per b units of B, then one unit of B is the input side and A is the output side.

Worked examples

Example 1: Reading a simple rate statement

Problem. A sign says a cyclist travels at 18 kilometres per hour. What do the numbers and units mean?

Solution.

18 kilometres per hour means:

• distance = 18 kilometres
• time = 1 hour
• written as 18 km / 1 h

So in every 1 hour, the cyclist travels 18 km.

Reasoning commentary. The word per means for each or for every. The order matters: this is distance first, time second. The statement answers the question, "How many kilometres for each hour?"

Choice commentary. We write 18 km/h because that compact form preserves the same order as the words.

Check. A rate of 18 h/km would mean something different: it would mean 18 hours for each kilometre, which is clearly not the same situation.

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Example 2: Distinguishing a rate from its reciprocal

Problem. A machine fills bottles at 24 bottles per minute. Which statement matches this rate?

A. It takes 24 minutes to fill 1 bottle. B. It fills 24 bottles in 1 minute. C. It fills 1 bottle in 24 minutes and 24 seconds.

Solution.

24 bottles per minute means 24 bottles / 1 minute.

So the correct statement is:

B. It fills 24 bottles in 1 minute.

Reasoning commentary. The first quantity named is bottles; the second is minutes. So this rate counts output per unit time.

Why A is wrong. 24 minutes per bottle is the reciprocal. It swaps the quantities and changes the meaning.

Why C is wrong. 24 seconds per bottle would be consistent with 24 bottles per minute, because 60 / 24 = 2.5 seconds per bottle, so even that is not correct. This option mixes units incorrectly.

Check by sense-making. A filling machine producing 24 bottles in a minute is fast. A machine taking 24 minutes for one bottle is extremely slow. The context helps catch the inversion error.

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Example 3: Writing a rate from words

Problem. Write each phrase as a rate in symbols.

1. 7 dollars for every 2 notebooks
2. 95 words each minute
3. 3 litres in 5 seconds

Solution.

1. 7 dollars for every 2 notebooks = 7 dollars / 2 notebooks
2. 95 words each minute = 95 words / 1 minute
3. 3 litres in 5 seconds = 3 litres / 5 seconds

Reasoning commentary. Words like for every, each, and in can all signal rate structure. The job is to preserve the original order.

Choice commentary. We do not simplify automatically unless the problem asks for a unit rate. Here the focus is accurate translation, not reduction.

Common pitfall repaired. Some students rewrite 7 dollars for every 2 notebooks as 2/7 notebooks per dollar. That is a valid related rate, but it is not the same statement. Keep the original order first.

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Example 4: Choosing the correct unit rate for the question

Problem. Maya reads 120 pages in 3 hours.

1. Find the rate in pages per hour.
2. Find the rate in hours per page.
3. Which rate is more useful if you want to know how long 1 page takes?

Solution.

1. pages per hour = 120 / 3 = 40

So Maya reads at 40 pages per hour.

2. hours per page = 3 / 120 = 1/40

So Maya takes 1/40 hour per page.

Since 1 hour = 60 minutes,

1/40 hour = 60/40 = 1.5 minutes.

So this is also 1.5 minutes per page.

3. If you want to know how long 1 page takes, hours per page or minutes per page is more useful.

Reasoning commentary. Both unit rates are correct, but they answer different questions:

• 40 pages per hour answers "How much reading in one hour?"
• 1.5 minutes per page answers "How much time for one page?"

Choice commentary. The best rate depends on the question, not just the numbers. This is one of the most important habits in rate problems.

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Example 5: Interpreting a price rate carefully

Problem. Apples cost $6 for 4 kilograms.

1. Write the rate in dollars per kilogram.
2. Write the reciprocal rate in kilograms per dollar.
3. Explain what each means.

Solution.

1. dollars per kilogram = 6 / 4 = 1.5

So the price is $1.50 per kilogram.

2. kilograms per dollar = 4 / 6 = 2/3

So the reciprocal rate is 2/3 kilogram per dollar.

3. Meanings:

• $1.50 per kilogram means each 1 kg costs $1.50.
• 2/3 kilogram per dollar means $1 buys 2/3 kg.

Reasoning commentary. These two rates describe the same situation from opposite viewpoints. One tells cost for each kilogram; the other tells amount for each dollar.

Choice commentary. In shopping, dollars per kilogram is usually more useful because stores price items that way and it supports fair comparisons.

Check. The reciprocal relationship is sensible: if each kilogram costs more than $1, then each dollar should buy less than 1 kg.

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Example 6: Matching units to a graph label

Problem. A water tank fills so that after 4 minutes it contains 28 litres of water. A student labels the vertical axis minutes and the horizontal axis litres. Another student labels the vertical axis litres and the horizontal axis minutes.

Which labeling fits the rate statement 7 litres per minute, and why?

Solution.

First find the rate:

28 litres / 4 minutes = 7 litres per minute

This means:

• input: minutes
• output: litres

So the graph should usually show:

• horizontal axis: minutes
• vertical axis: litres

Therefore, the second student has the more natural labeling.

Reasoning commentary. A rate of 7 litres per minute says the amount of water depends on time. Time is the independent quantity, so it is natural on the horizontal axis.

Choice commentary. The rate language guides representation. We are not just computing; we are choosing a structure that matches meaning.

Check. After 1 minute, there should be 7 litres; after 4 minutes, 28 litres. That fits a graph where moving right in time moves up in litres.

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Example 7: Detecting an inversion error in a word problem

Problem. A runner completes 10 kilometres in 50 minutes. Two students write unit rates:

• Student A: 5 minutes per kilometre
• Student B: 5 kilometres per minute

Who is correct?

Solution.

Compute both possible unit rates.

minutes per kilometre = 50 / 10 = 5

So 5 minutes per kilometre is correct.

Now check the other claim:

kilometres per minute = 10 / 50 = 0.2

So the correct distance rate would be 0.2 kilometres per minute, not 5 kilometres per minute.

Therefore, Student A is correct and Student B inverted the quantities incorrectly.

Reasoning commentary. Student B divided in the wrong order. The words in the unit tell you the order of division.

Choice commentary. For a runner, minutes per kilometre is often the more useful rate because it describes pace.

Reality check. 5 km every minute would mean 300 km/h, which is impossible for a human runner. Context catches the mistake.

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Example 8: Building a rate from a two-step description

Problem. A printer uses 3 cartridges to print 1,800 pages.

1. Write a rate in pages per cartridge.
2. Write a rate in cartridges per 100 pages.

Solution.

1. pages per cartridge = 1800 / 3 = 600

So the printer produces 600 pages per cartridge.

2. First find cartridges per page:

3 / 1800 = 1 / 600 cartridge per page.

Now for 100 pages:

100 x (1/600) = 1/6

So the printer uses 1/6 cartridge per 100 pages.

Reasoning commentary. The second part is harder because cartridges per 100 pages is not a standard unit rate with denominator 1. We still build it from the same structure.

Choice commentary. We used the reciprocal first because the target wording was cartridges per ..., not pages per ....

Check. If 600 pages need 1 cartridge, then 100 pages should need much less than 1 cartridge. 1/6 fits that expectation.

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Example 9: Comparing two rates with the same quantities in different orders

Problem. Two students describe the same bus trip.

• Student P says the bus travels 60 kilometres per hour.
• Student Q says the bus takes 1 hour per 60 kilometres.

Are these statements equivalent?

Solution.

Student P's rate is:

60 km / 1 h

Student Q's rate is:

1 h / 60 km

These are reciprocals, not identical statements.

Now simplify Student Q's statement:

1 hour per 60 kilometres = 1/60 hour per kilometre

Since 1 hour = 60 minutes,

1/60 hour = 1 minute

So Student Q is saying:

1 minute per kilometre

These two statements are not written the same way, but they are consistent with the same trip because they describe the same motion from opposite viewpoints.

Reasoning commentary. Equivalent situation does not mean identical wording. One rate gives distance for each unit of time; the other gives time for each unit of distance.

Choice commentary. When comparing or solving, decide which form better matches the question being asked.

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Example 10: Challenge problem with mixed interpretation

Problem. A recipe uses 2.5 cups of flour for 20 cookies.

1. Find the rate in cups per cookie.
2. Find the rate in cookies per cup.
3. A student says, "Since there are more cookies than cups, cookies per cup must always be the better rate." Is that correct?

Solution.

1. cups per cookie = 2.5 / 20 = 0.125

So the recipe uses 0.125 cup per cookie.

2. cookies per cup = 20 / 2.5 = 8

So the recipe makes 8 cookies per cup.

3. The student's claim is not correct.

Why not?

• 0.125 cup per cookie is better if you want to know how much flour one cookie needs.
• 8 cookies per cup is better if you want to know how many cookies one cup of flour can make.

The "better" rate depends on the purpose, not on which number is larger.

Reasoning commentary. Larger numbers can look more attractive, but rate choice is about meaning. This is a language-and-structure topic, not a bigger-number topic.

Check. If 1 cup makes 8 cookies, then 2.5 cups should make 20 cookies because 2.5 x 8 = 20. The rates are consistent.

Common misconceptions and repairs

Misconception 1: "Per" does not matter much; I can divide either way.

Repair. The order names the meaning. 12 km/h and 12 h/km are different statements even though they use the same numbers and units.

Misconception 2: The unit rate is always the only useful form.

Repair. Unit rates are powerful, but sometimes a non-unit rate or a reciprocal rate fits the question better. Choose the form that answers the actual question.

Misconception 3: Bigger quotient means better answer.

Repair. The value of a rate comes from interpretation, not size. 40 pages per hour and 1.5 minutes per page are both useful in different contexts.

Misconception 4: Units are decoration.

Repair. Units carry the meaning. If the units do not match the words, the rate is probably wrong.

Quick practice

1. Write 14 dollars for 2 pens as a rate in symbols.
2. A faucet runs at 9 litres per minute. How many litres is that in 1 minute?
3. A walker goes 6 kilometres in 1.5 hours. Find kilometres per hour.
4. A recipe needs 3 cups of rice for 12 servings. Find servings per cup.
5. Explain the difference between 80 words per minute and 80 minutes per word.

Answers

1. 14 dollars / 2 pens
2. 9 litres
3. 6 / 1.5 = 4, so 4 km/h
4. 12 / 3 = 4, so 4 servings per cup
5. 80 words per minute is a fast typing rate; 80 minutes per word is the reciprocal and means extremely slow production of words.

Takeaway

In Grade 8 rate problems, many errors come from language, not arithmetic. If you preserve the order of quantities, read per as for each, and choose units that match the question, your calculations are much more likely to be correct.