Grade 8 Unit Mastery Quiz: Building Fair Comparisons

This quiz is for the Grade 8 topic Building Fair Comparisons in Ratios, Rates and Proportions. It assumes students have already studied the linked records on ratio comparison, proportions, unit rates, and the broader proportional-modeling unit.

Use these linked records for targeted review before or after this quiz:

• rea.m08.number.ratios-rates-proportions.proportional-modeling.unit-mastery-quiz for broader unit coverage
• rea.m08.number.ratios-rates-proportions.ratio-concepts.ratio-comparison.unit-mastery-quiz for direct ratio comparison methods
• rea.m08.number.ratios-rates-proportions.equivalent-ratios.proportion-problems.unit-mastery-quiz for solving proportion situations
• rea.m08.number.ratios-rates-proportions.unit-rates.find-unit-rates.unit-mastery-quiz for unit-rate fluency

Quiz Instructions

• Show all work.
• If two quantities are being compared, make the comparison fair by using the same unit or the same basis.
• Write a sentence to justify each decision question.
• Total: 20 marks

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Questions

1. Better buy by mass (2 marks)

Store A sells 750 g of yogurt for $4.50. Store B sells 1.2 kg of yogurt for $6.84.

Which store offers the better price, and how do you know?

2. Better buy by count (2 marks)

Brand X sells 18 granola bars for $7.92. Brand Y sells 12 granola bars for $5.16.

Which brand is cheaper per bar?

3. Comparing speeds fairly (2 marks)

Lina bikes 9 km in 30 min. Omar bikes 12.6 km in 45 min.

Who is biking faster? Compare fairly using the same time unit.

4. Choosing the right comparison basis (2 marks)

A sports drink advertises: "Bottle A has more electrolytes because it contains 300 mg. Bottle B only contains 260 mg." Bottle A is 750 mL and Bottle B is 500 mL.

Is this a fair comparison? If not, make a fair comparison and decide which bottle has the greater electrolyte concentration.

5. Comparing value in a recipe context (2 marks)

Recipe Mix A uses 5 cups of flour to make 40 cookies. Recipe Mix B uses 7 cups of flour to make 49 cookies.

Which mix is more efficient if the goal is to make the most cookies per cup of flour?

6. Table comparison (2 marks)

Two phone plans charge by data used.

Plan A:

• 2 GB costs $14
• 5 GB costs $35

Plan B:

• 3 GB costs $18
• 6 GB costs $36

Which plan is cheaper per GB? Show how you know.

7. Detecting an unfair claim (2 marks)

A student says, "Team Red is better because they scored 24 points, while Team Blue only scored 21 points." In that game, Team Red took 12 shots and Team Blue took 9 shots.

Explain why the claim may be unfair. Then compare the teams fairly.

8. Equivalent comparison with scaling (2 marks)

A paint mix uses 2 cups of blue paint for every 5 cups of white paint. Another mix uses 6 cups of blue paint for every 14 cups of white paint.

Which mix is more blue-rich? Show a fair comparison.

9. Open response: choose the fair metric (2 marks)

Two summer jobs pay differently.

• Job A pays $84 for 6 hours of work.
• Job B pays $119 for 8.5 hours of work.

Which job pays better? Explain why total pay alone is not enough.

10. Multi-step optimization choice (2 marks)

A school is buying juice boxes for a trip.

• Pack A: 8 boxes for $5.20
• Pack B: 15 boxes for $9.00
• Pack C: 24 boxes for $14.88

Which pack gives the best value per juice box? If the school needs at least 30 boxes and can buy only one pack type, which pack would you recommend based on value?

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Complete Marking Key and Solutions

1. Better buy by mass (2 marks)

Answer: Store B is the better buy.

Solution: Convert to the same unit.

• Store A: 750 g for $4.50
• Store B: 1.2 kg = 1200 g for $6.84

Find cost per 100 g or per gram.

• Store A: $4.50 / 750 = $0.006 per g
• Store B: $6.84 / 1200 = $0.0057 per g

Since $0.0057 per g < $0.006 per g, Store B is cheaper.

Marking:

• 1 mark for converting or setting up equal-unit comparison correctly
• 1 mark for correct conclusion with justification

2. Better buy by count (2 marks)

Answer: Brand Y is cheaper per bar.

Solution:

• Brand X: $7.92 / 18 = $0.44 per bar
• Brand Y: $5.16 / 12 = $0.43 per bar

Since $0.43 < $0.44, Brand Y is cheaper per bar.

Marking:

• 1 mark for correct unit-rate calculations
• 1 mark for correct decision

3. Comparing speeds fairly (2 marks)

Answer: Omar is biking faster.

Solution: Compare in km/h.

• Lina: 9 km in 30 min = 18 km in 60 min = 18 km/h
• Omar: 12.6 km in 45 min

Scale to 60 min: 12.6 / 45 = 0.28 km per min, so in 60 min: 0.28 x 60 = 16.8 km/h

Now compare:

• Lina: 18 km/h
• Omar: 16.8 km/h

So Lina is actually faster.

Marking:

• 1 mark for making the time units the same
• 1 mark for correct conclusion

4. Choosing the right comparison basis (2 marks)

Answer: The advertisement is not fair. Bottle B has the greater concentration.

Solution: Comparing total milligrams is unfair because the bottles have different sizes.

Find mg per mL.

• Bottle A: 300 / 750 = 0.4 mg/mL
• Bottle B: 260 / 500 = 0.52 mg/mL

Since 0.52 mg/mL > 0.4 mg/mL, Bottle B has the greater electrolyte concentration.

Marking:

• 1 mark for explaining why the original comparison is unfair
• 1 mark for correct fair comparison and conclusion

5. Comparing value in a recipe context (2 marks)

Answer: Mix A is more efficient.

Solution: Find cookies per cup of flour.

• Mix A: 40 / 5 = 8 cookies per cup
• Mix B: 49 / 7 = 7 cookies per cup

Since 8 > 7, Mix A makes more cookies per cup of flour.

Marking:

• 1 mark for correct unit-rate setup
• 1 mark for correct comparison

6. Table comparison (2 marks)

Answer: Plan B is cheaper per GB.

Solution: Find unit rate from either row in each table.

Plan A:

• $14 / 2 = $7 per GB
• Check: $35 / 5 = $7 per GB

Plan B:

• $18 / 3 = $6 per GB
• Check: $36 / 6 = $6 per GB

Since $6 per GB < $7 per GB, Plan B is cheaper.

Marking:

• 1 mark for correct unit rates
• 1 mark for correct conclusion

7. Detecting an unfair claim (2 marks)

Answer: The claim is unfair because the teams took different numbers of shots. Team Blue performed better per shot.

Solution: Compare points per shot.

• Team Red: 24 / 12 = 2 points per shot
• Team Blue: 21 / 9 = 7/3 = 2.33... points per shot

Even though Team Red scored more total points, Team Blue scored more points per shot, so they were more efficient.

Marking:

• 1 mark for identifying why total points alone is unfair
• 1 mark for correct fair comparison

8. Equivalent comparison with scaling (2 marks)

Answer: The second mix is more blue-rich.

Solution: Compare blue-to-white ratios on the same basis.

First mix:

• 2 blue for 5 white
• Blue per 1 white = 2 / 5 = 0.4

Second mix:

• 6 blue for 14 white
• Blue per 1 white = 6 / 14 = 3 / 7 about 0.4286

Since 0.4286 > 0.4, the second mix has more blue for the same amount of white, so it is more blue-rich.

Marking:

• 1 mark for correct fair comparison setup
• 1 mark for correct conclusion

9. Open response: choose the fair metric (2 marks)

Answer: Job B pays better.

Solution: Total pay alone is not enough because the jobs involve different amounts of time. A fair comparison uses pay per hour.

• Job A: $84 / 6 = $14/h
• Job B: $119 / 8.5 = $14/h

Both jobs pay the same hourly rate.

Check carefully: 8.5 x 14 = 119

So neither job pays better; they are equal at $14 per hour.

Marking:

• 1 mark for identifying hourly pay as the fair metric
• 1 mark for correct calculation and conclusion

10. Multi-step optimization choice (2 marks)

Answer: Pack B gives the best value per juice box, so recommend Pack B.

Solution: Find price per box.

• Pack A: $5.20 / 8 = $0.65 per box
• Pack B: $9.00 / 15 = $0.60 per box
• Pack C: $14.88 / 24 = $0.62 per box

The lowest unit price is Pack B at $0.60 per box.

If the school needs at least 30 boxes and must buy only one pack type, value is still best with Pack B. They would buy 2 packs to get 30 boxes.

• 2 packs of B give 30 boxes for $18.00

So Pack B is the best recommendation based on value.

Marking:

• 1 mark for correct unit-price comparisons
• 1 mark for correct recommendation with context

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Score Interpretation

Recommended mastery threshold

16/20 or higher, with at least:

• 4/6 on questions involving direct unit-rate comparisons: 1, 2, 3, 6, 9, 10
• 3/4 on questions involving fairness of comparison and justification: 4, 5, 7, 8

A student below mastery usually needs one of these repairs:

• Review how to convert quantities to the same unit before comparing.
• Review how to choose the correct basis, such as per item, per hour, per gram, or per shot.
• Review how to explain why a raw total can be misleading when amounts are unequal.

Suggested remediation by error pattern

• If the student makes arithmetic mistakes with unit rates, revisit rea.m08.number.ratios-rates-proportions.unit-rates.find-unit-rates.unit-mastery-quiz.
• If the student compares unmatched ratios incorrectly, revisit rea.m08.number.ratios-rates-proportions.ratio-concepts.ratio-comparison.unit-mastery-quiz.
• If the student struggles with setting up equivalent comparisons, revisit rea.m08.number.ratios-rates-proportions.equivalent-ratios.proportion-problems.unit-mastery-quiz.
• If the student needs broader end-of-unit review, revisit rea.m08.number.ratios-rates-proportions.proportional-modeling.unit-mastery-quiz.

Common Misconceptions to Watch For

• "Bigger total means better." Repair: Compare per 1 unit or per equal-sized unit.

• "I can compare dollars to dollars even if the amounts are different sizes." Repair: Make package size, time, mass, or count the same first.

• "The larger numerator always means the better rate." Repair: A rate depends on both numbers, not just one.

• "If two values look close, estimation is enough." Repair: In fair-comparison problems, compute the actual unit rate before deciding.

Teacher / Self-Study Note

This quiz is intentionally narrow: it assesses whether the learner can build a fair basis for comparison and then use proportional reasoning to make a justified decision. It should be paired with the linked records for reteaching specific subskills rather than used as a replacement for them.