Rate Language and Unit Structure: Graded Practice Set

Grade 8 focus

Use this set after studying [rea.m08.number.ratios-rates-proportions.unit-rates.rate-language.lesson] and [rea.m08.number.ratios-rates-proportions.unit-rates.rate-language.worked-examples]. For broader unit-rate practice, also see [rea.m08.number.ratios-rates-proportions.unit-rates.practice-set]. This record concentrates narrowly on rate language and unit structure: what the order means, what per means, and why reversing a rate changes the meaning.

Core reminder

A rate compares two different kinds of quantities.

• a per b means a for each 1 b
• The first unit belongs to the numerator quantity
• The second unit belongs to the denominator quantity
• Reversing a rate usually creates a different meaning

Example: 60 km/h means 60 kilometres per 1 hour, while 60 h/km means 60 hours per 1 kilometre. Those are not the same statement.

Common traps to watch for

• Mixing up the order of units: pages per minute is not the same as minutes per page.
• Reading per as “times” instead of “for each 1”.
• Comparing rates without checking whether the units are in the same order.
• Giving a numerical answer with the wrong unit label.

Practice Set

A. Read and write rates

1. Write 9 dollars for 3 notebooks in per language in two different ways. Difficulty: easy

2. A runner travels 18 kilometres in 2 hours. Write the rate in words and in slash form. Difficulty: easy

3. The label on a bag says $4.80 per kilogram. a. What quantity is in the numerator? b. What quantity is in the denominator? Difficulty: easy

4. A faucet fills 15 litres in 5 minutes. Write two correct rate statements using the same information. Difficulty: easy

5. A printer produces 24 pages in 3 minutes. Which rate matches the situation? A. 24 min/page B. 8 pages/min C. 3 pages/24 min D. 24 pages/hour Difficulty: easy

6. 72 km/h means: A. 72 hours for each kilometre B. 72 kilometres for each hour C. 72 kilometres for each minute D. 72 kilometres times 1 hour Difficulty: easy

B. Identify meaning and reverse rates

7. A recipe uses 3 cups of flour for 2 batches. a. Write the rate cups per batch. b. Write the reversed rate batches per cup. Difficulty: easy

8. A sign says 5 dollars per movie ticket. Explain in one sentence what this means. Difficulty: easy

9. A cyclist rides at 30 km/h. What does the number 30 tell you in this rate? Difficulty: medium

10. Which statement describes 0.5 h/km? A. Half a kilometre each hour B. Half an hour for each kilometre C. Two kilometres each hour D. Two hours for each kilometre Difficulty: medium

11. A machine packs 48 boxes in 6 minutes. a. Find boxes per minute. b. Find minutes per box. Difficulty: medium

12. Mia says, “If a car goes 80 km/h, then 80 h/km means the same thing.” Is she correct? Briefly explain. Difficulty: medium

13. A babysitter earns $36 in 4 hours. Write: a. dollars per hour b. hours per dollar Difficulty: medium

14. A video downloads 900 megabytes in 3 minutes. Choose the better rate for answering “How much data downloads each minute?” and explain. A. 300 MB/min B. 1/300 min/MB Difficulty: medium

15. A map scale is described as 2 centimetres for every 5 kilometres. Which is the correct interpretation? A. 2 km/cm B. 5 cm/km C. 2 cm/5 km D. 5 km/2 cm only, because the other form is impossible Difficulty: medium

16. A bottle fills at 0.75 litres per second. How many litres is that for each 1 second, and what would the reversed rate mean? Difficulty: medium

C. Choose the correct unit structure in context

17. A student reads 45 pages in 30 minutes. Two classmates describe the situation:

• Kai: 1.5 pages per minute
• Noor: 2/3 minute per page Are they both correct? If so, explain how both can describe the same reading session. Difficulty: hard

18. A car mechanic says a repair took 3 hours for 6 brake pads. A customer writes this as 2 brake pads per hour and another writes it as 0.5 hour per brake pad. Which statement answers “How many brake pads were installed each hour?” Which statement answers “How much time was used for each brake pad?” Difficulty: hard

19. A hiker says, “I walked 4 kilometres every hour, so I also walked 4 hours every kilometre.” Identify the error and give the correct reversed rate. Difficulty: hard

20. A juice machine fills 7.2 litres in 9 seconds. a. Find the rate in litres per second. b. Find the reversed rate in seconds per litre. c. State one question each rate could answer. Difficulty: hard

21. A bakery sells 18 muffins for $27. a. Find dollars per muffin. b. Find muffins per dollar. c. Which rate is more useful for answering, “How much does one muffin cost?” Explain. Difficulty: hard

22. A small bus uses 12 litres of fuel to travel 150 kilometres. a. Find litres per kilometre. b. Find kilometres per litre. c. Explain why these two rates are not equal, even though they come from the same trip. Difficulty: hard

23. A worker types 840 words in 12 minutes. Another worker types 75 words per minute. a. Convert the first worker’s rate to words per minute. b. Who types faster? c. Write the slower worker’s rate in minutes per word. Difficulty: hard

24. A water tank drains at 16 litres per minute. A student claims that after reversing, the rate is 16 minutes per litre. a. Explain why the claim is wrong. b. Find the correct reversed rate. c. Interpret the reversed rate in a sentence. Difficulty: hard

Answer Key

1. 9 dollars for 3 notebooks; 3 dollars per notebook; 1/3 notebook per dollar are valid equivalent descriptions, and two correct per statements include 3 dollars per notebook and 1/3 notebook per dollar.
2. 18 kilometres per 2 hours; slash form 18 km/2 h.
3. a. dollars ($4.80) b. kilograms (1 kg).
4. 15 litres per 5 minutes; 3 litres per minute; also valid: 1/3 minute per litre after simplification.
5. B
6. B
7. a. 1.5 cups per batch b. 2/3 batch per cup
8. It means each ticket costs $5.
9. It means 30 kilometres for each 1 hour.
10. B
11. a. 8 boxes/min b. 1/8 min/box
12. No; reversing changes the meaning. The correct reversed rate is 1/80 h/km.
13. a. $9/h b. 1/9 h/$
14. A, because it directly gives data for each minute.
15. C; also the reversed rate 2.5 km/cm can be formed, but 2 cm/5 km is the direct statement given.
16. 0.75 litre for each 1 second; reversed rate means 4/3 seconds per litre.
17. Yes. 1.5 pages/min and 2/3 min/page are reciprocal rates describing the same session.
18. 2 brake pads per hour answers the first question; 0.5 hour per brake pad answers the second.
19. The error is keeping the same number when reversing. The correct reversed rate is 1/4 hour per kilometre.
20. a. 0.8 L/s b. 1.25 s/L c. Example questions vary.
21. a. $1.50 per muffin b. 2/3 muffin per dollar c. dollars per muffin
22. a. 0.08 L/km b. 12.5 km/L c. They compare different “for each 1” units.
23. a. 70 words/min b. Second worker c. 1/70 min/word
24. a. Reversing requires taking the reciprocal, not just switching the words. b. 1/16 min/L c. It means one litre drains in 1/16 of a minute.

Full Solutions for the Hardest Third

17. Reading rate in two forms

Given: 45 pages in 30 minutes

First find pages per minute:

• 45 pages / 30 minutes = 1.5 pages/min

Now find minutes per page:

• 30 minutes / 45 pages = 2/3 minute/page

So both classmates are correct.

Why both work:

• 1.5 pages/min answers: “How many pages are read in 1 minute?”
• 2/3 minute/page answers: “How much time is needed for 1 page?”

These are reciprocal descriptions of the same situation.

18. Matching a rate to a question

Given: 3 hours for 6 brake pads

Find brake pads per hour:

• 6 brake pads / 3 hours = 2 brake pads/hour

Find hours per brake pad:

• 3 hours / 6 brake pads = 0.5 hour/brake pad

Now match each to the question:

• “How many brake pads were installed each hour?” uses 2 brake pads per hour.
• “How much time was used for each brake pad?” uses 0.5 hour per brake pad.

The key idea is that the wording of the question tells you which unit should come first.

19. Reversing a walking rate correctly

Original rate: 4 kilometres every hour = 4 km/h

The hiker’s error:

• He reversed the units but kept the same number.
• That is not valid.

To reverse correctly, compute hours per kilometre:

• 1 hour / 4 kilometres = 1/4 hour/km

So the correct reversed rate is:

• 1/4 hour per kilometre

Interpretation:

• The hiker takes one quarter of an hour to walk each kilometre.

20. Juice machine rate and reversed rate

Given: 7.2 litres in 9 seconds

Part a: litres per second

• 7.2 / 9 = 0.8
• Rate = 0.8 L/s

Part b: seconds per litre

• 9 / 7.2 = 1.25
• Rate = 1.25 s/L

Part c: questions each rate could answer

• 0.8 L/s answers: “How much juice is filled each second?”
• 1.25 s/L answers: “How many seconds does it take to fill 1 litre?”

Same situation, different question, different unit structure.

21. Muffin price in both directions

Given: 18 muffins for $27

Part a: dollars per muffin

• $27 / 18 = $1.50
• Rate = $1.50 per muffin

Part b: muffins per dollar

• 18 / 27 = 2/3
• Rate = 2/3 muffin per dollar

Part c: which is more useful for “How much does one muffin cost?”

• Use dollars per muffin.
• Reason: the question asks for cost of 1 muffin, so dollars must come first.

22. Fuel use in two unit structures

Given: 12 litres for 150 kilometres

Part a: litres per kilometre

• 12 / 150 = 0.08
• Rate = 0.08 L/km

Part b: kilometres per litre

• 150 / 12 = 12.5
• Rate = 12.5 km/L

Part c: why they are not equal

• 0.08 L/km means fuel used for each kilometre.
• 12.5 km/L means distance travelled for each litre.
• The units are reversed, so the meaning changes.
• They are reciprocals, not equal numbers.

23. Comparing typing rates

First worker: 840 words in 12 minutes

Part a: convert to words per minute

• 840 / 12 = 70
• First worker = 70 words/min

Second worker = 75 words/min

Part b: who types faster?

• Compare 70 words/min and 75 words/min
• Since 75 > 70, the second worker types faster.

Part c: slower worker’s rate in minutes per word

• The slower worker is the first worker.
• Use the reciprocal of 70 words/min:
• 1 / 70 min/word

Interpretation:

• The first worker takes 1/70 of a minute for each word.

24. Correcting a mistaken reversed rate

Original rate: 16 litres per minute = 16 L/min

Part a: why 16 min/L is wrong

• The student switched the units but kept the same number.
• Reversing a rate requires taking the reciprocal.
• If 16 litres drain in 1 minute, then 1 litre does not take 16 minutes.

Part b: correct reversed rate

• 1 minute / 16 litres = 1/16 min/L

Part c: interpretation

• 1/16 min/L means one litre drains in one-sixteenth of a minute.
• Since 1 minute = 60 seconds, this is also 60/16 = 3.75 seconds per litre.

Self-check prompts

• Did I read per as “for each 1”?
• Did I keep the units in the same order as the question?
• If I reversed the rate, did I also take the reciprocal?
• Does my final unit label match my numerical answer?

Next links

After this set, move to the broader [rea.m08.number.ratios-rates-proportions.unit-rates.practice-set] for mixed unit-rate fluency, or revisit [rea.m08.number.ratios-rates-proportions.unit-rates.rate-language.worked-examples] if errors are mostly about reversing and interpreting units.