Grade 8 Unit Mastery Quiz: Proportional Modeling and Optimization

Use this quiz as an end-of-unit mastery check for Grade 8 Number: Ratios, Rates and Proportions. For instruction and concept development, study rea.m08.number.ratios-rates-proportions.proportional-modeling.overview, rea.m08.number.ratios-rates-proportions.proportional-modeling.lesson, rea.m08.number.ratios-rates-proportions.proportional-modeling.worked-examples, and rea.m08.number.ratios-rates-proportions.proportional-modeling.practice-set before attempting this assessment.

How this quiz fits with related records

• Use rea.m08.number.ratios-rates-proportions.ratio-concepts.unit-mastery-quiz if a learner still needs a mastery check on ratio meaning and comparison foundations.
• Use rea.m08.algebra.solving-linear-equations.linear-modeling.unit-mastery-quiz after this record when the learner is ready for broader equation modeling beyond purely proportional relationships.
• Use rea.m08.algebra.solving-linear-equations.single-variable-equations.unit-mastery-quiz if solving equations becomes the main barrier rather than building the proportional model itself.

Coverage

This quiz samples the core expectations of the unit:

• recognizing proportional situations;
• finding a constant of proportionality or unit rate;
• writing and using equations of the form y = kx;
• solving missing-value proportion problems in context;
• comparing options fairly;
• optimizing under simple constraints such as budget, package size, or minimum quantity.

Quiz structure

• 10 questions
• 20 marks total
• Suggested time: 35-45 minutes
• Calculators: optional, depending on classroom policy

Mastery recommendation

Recommend mastery at 16/20 or higher (80%), with these conditions:

• the learner must earn at least 6/8 on Questions 7-10 combined, because those questions assess real decision-making and optimization rather than only routine computation;
• the learner should not show a persistent misconception that a lower total price automatically means a better buy, or that a best unit rate automatically settles a constrained packaging problem.

A learner scoring 14-15/20 is close, but should usually revisit the worked examples and practice set before being treated as fully secure.

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Questions

1. Proportional or not? 2 marks

A bike rental shop charges $7 per hour.

1. Is the relationship between hours rented h and total cost C proportional?
2. If yes, write the equation.

2. Constant of proportionality from a table 2 marks

The table shows bags of apples and total cost.

Bags	Cost ($)
2	9
5	22.5
8	36

1. Is the relationship proportional?
2. State the constant of proportionality and what it means.

3. Missing-value proportion 2 marks

A recipe uses 3 cups of oats for 8 granola bars. How many cups of oats are needed for 20 granola bars if the recipe stays proportional?

4. Unit rate and prediction 2 marks

A car travels 144 km on 12 L of fuel.

1. Find the distance per litre.
2. Predict the distance the car can travel on 19 L at the same rate.

5. Write and use a proportional equation 2 marks

A printing shop charges $0.24 per page.

1. Write an equation for total cost C in terms of number of pages p.
2. Find the cost of printing 75 pages.

6. Fair comparison 2 marks

Two cereal boxes are on sale.

• Box A: 750 g for $4.80
• Box B: 1.2 kg for $7.20

Which is the better buy? Show a fair comparison.

7. Best buy under a budget 2 marks

A class has $30 to buy glue sticks.

• Pack P: 8 glue sticks for $4.80
• Pack Q: 15 glue sticks for $8.25

If the class buys only one type of pack and wants the greatest number of glue sticks without going over budget, which pack should it choose, and how many glue sticks can it buy?

8. Minimum quantity with whole packs 2 marks

A team needs at least 50 bottles of water.

• Store A: 12 bottles for $5.40
• Store B: 20 bottles for $9.60

If the team must buy whole packs from only one store, which store gives the lower total cost for meeting the need?

9. Mixed-package optimization 2 marks

A snack table needs at least 28 muffins.

• Tray R: 6 muffins for $4.20
• Tray S: 10 muffins for $6.80

The organizer may buy any combination of trays. Find the least expensive way to get at least 28 muffins.

10. Non-proportional offer vs proportional offer 2 marks

A flyer company offers:

• Offer A: 200 flyers for $18, plus $0.06 per extra flyer beyond 200
• Offer B: $0.15 per flyer for any number of flyers

A club needs 320 flyers.

1. Which offer is cheaper?
2. Is Offer A a proportional relationship for all numbers of flyers? Explain briefly.

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Marking key

Question 1

• 1 mark for identifying the situation as proportional
• 1 mark for C = 7h

Question 2

• 1 mark for correctly identifying the relationship as proportional
• 1 mark for constant of proportionality 4.5, interpreted as $4.50 per bag

Question 3

• 1 mark for correct proportional setup or scaling
• 1 mark for answer 7.5 cups

Question 4

• 1 mark for unit rate 12 km/L
• 1 mark for prediction 228 km

Question 5

• 1 mark for C = 0.24p
• 1 mark for $18.00

Question 6

• 1 mark for correct unit-rate calculations on a common basis
• 1 mark for correct conclusion: Box B is cheaper

Question 7

• 1 mark for testing feasible numbers of packs under $30
• 1 mark for correct choice and quantity: Pack Q, 45 glue sticks

Question 8

• 1 mark for finding required whole-pack counts for both stores
• 1 mark for correct conclusion: Store B, $28.80

Question 9

• 1 mark for valid comparison of feasible combinations
• 1 mark for least-cost answer: 3 Tray S gives 30 muffins for $20.40

Question 10

• 1 mark for correct cost comparison: Offer A $25.20, Offer B $48.00, so A is cheaper
• 1 mark for explaining that Offer A is not proportional for all flyer counts because it includes a fixed first block and then a different rule after 200

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Full solutions

1. Proportional or not?

A constant rate of $7 per hour means total cost is always a fixed multiple of time.

So the relationship is proportional.

Equation: C = 7h

Answer: Yes, proportional; C = 7h.

2. Constant of proportionality from a table

Check cost per bag:

• 9 / 2 = 4.5
• 22.5 / 5 = 4.5
• 36 / 8 = 4.5

The ratio cost / bags stays constant, so the relationship is proportional.

The constant of proportionality is 4.5. This means $4.50 per bag.

Answer: Yes; constant of proportionality 4.5, meaning $4.50 for each bag.

3. Missing-value proportion

3 cups make 8 bars.

Scale from 8 bars to 20 bars: 20 / 8 = 2.5

So multiply the oats by 2.5: 3 x 2.5 = 7.5

Answer: 7.5 cups.

4. Unit rate and prediction

Distance per litre: 144 / 12 = 12

So the car travels 12 km/L.

For 19 L: 12 x 19 = 228

Answer: 12 km/L and 228 km.

5. Write and use a proportional equation

If the shop charges $0.24 per page, then cost equals rate times number of pages.

Equation: C = 0.24p

For 75 pages: C = 0.24(75) = 18

Answer: C = 0.24p; cost is $18.00.

6. Fair comparison

Convert 1.2 kg to grams: 1.2 kg = 1200 g

Find price per gram:

• Box A: 4.80 / 750 = 0.0064 dollars per gram
• Box B: 7.20 / 1200 = 0.006 dollars per gram

Since 0.006 < 0.0064, Box B has the lower unit price.

Equivalent price per 100 g:

• Box A: $0.64 per 100 g
• Box B: $0.60 per 100 g

Answer: Box B is the better buy.

7. Best buy under a budget

Check whole packs under $30.

Pack P: 30 / 4.80 = 6.25, so at most 6 packs 6 x 8 = 48 glue sticks Cost: 6 x 4.80 = 28.80

Pack Q: 30 / 8.25 = 3.63..., so at most 3 packs 3 x 15 = 45 glue sticks Cost: 3 x 8.25 = 24.75

Compare quantities:

• Pack P gives 48
• Pack Q gives 45

Even though Q has a better unit price (8.25 / 15 = 0.55 each vs 4.80 / 8 = 0.60 each), the budget and whole-pack restriction make P the better choice here.

Answer: Choose Pack P; the class can buy 48 glue sticks.

8. Minimum quantity with whole packs

Need at least 50 bottles.

Store A: 50 / 12 = 4.16..., so 5 packs are needed Bottles: 5 x 12 = 60 Cost: 5 x 5.40 = 27.00

Store B: 50 / 20 = 2.5, so 3 packs are needed Bottles: 3 x 20 = 60 Cost: 3 x 9.60 = 28.80

Compare total costs:

• Store A: $27.00
• Store B: $28.80

Answer: Store A gives the lower total cost.

9. Mixed-package optimization

Need at least 28 muffins.

Unit prices:

• Tray R: 4.20 / 6 = 0.70 per muffin
• Tray S: 6.80 / 10 = 0.68 per muffin

Tray S has the lower unit price, but check combinations because only whole trays can be bought.

Test reasonable combinations:

• 5R = 30 muffins, cost 5 x 4.20 = 21.00
• 3S = 30 muffins, cost 3 x 6.80 = 20.40
• 2S + 2R = 32 muffins, cost 13.60 + 8.40 = 22.00
• 1S + 3R = 28 muffins, cost 6.80 + 12.60 = 19.40

The smallest cost among valid options is $19.40.

Answer: Buy 1 Tray S and 3 Tray R for 28 muffins at $19.40.

10. Non-proportional offer vs proportional offer

Need 320 flyers.

Offer A:

• First 200 flyers cost $18
• Extra flyers needed: 320 - 200 = 120
• Extra cost: 120 x 0.06 = 7.20
• Total: 18 + 7.20 = 25.20

Offer B: 320 x 0.15 = 48.00

Compare:

• Offer A: $25.20
• Offer B: $48.00

So Offer A is cheaper.

Is Offer A proportional for all flyer counts? No. A proportional relationship must have one constant multiplier and be expressible as y = kx for the whole situation. Offer A uses one rule up to 200 flyers and then another rule after 200, so it is not proportional for all numbers of flyers.

Answer: Offer A is cheaper; Offer A is not proportional for all flyer counts.

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Common errors to watch for when marking

• Treating a better unit rate as an automatic final answer without checking budget or whole-pack constraints.
• Comparing total prices directly when package sizes differ.
• Forgetting to convert units, such as kilograms to grams.
• Using a proportional model for a piecewise pricing rule with a starting block or fixed fee.
• Rounding too early in close comparisons.

Interpretation guide

• 18-20/20: strong unit mastery
• 16-17/20: recommended mastery threshold met
• 14-15/20: near mastery; targeted review recommended
• 0-13/20: reteaching and guided practice recommended before reassessment