Proportional vs non proportional worksheet pdf: Fill out & sign ... - Free Printable

Let's solve each problem step by step, using tables and reasoning to determine whether the relationships are proportional or non-proportional.

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1. Fun Center rents popcorn machines for $20 per hour. In addition to the hourly charge, there is a rental fee of $32. Is the number of hours you rent the popcorn machine proportional to the total cost?

#### Step 1: Fill in the table
- Hourly rate: $20/hour
- Rental fee: $32 (one-time)
- Total cost = $20 × hours + $32

| Hours | 1 | 2 | 3 | 4 |
|-------|-----|-----|-----|-----|
| Cost ($) | 52 | 72 | 92 | 112 |

> Calculation:
> - 1 hr: 20(1) + 32 = 52
> - 2 hrs: 20(2) + 32 = 72
> - 3 hrs: 20(3) + 32 = 92
> - 4 hrs: 20(4) + 32 = 112

#### Step 2: Check proportionality
A relationship is proportional if:
- The ratio of y/x is constant (constant of proportionality).
- The graph passes through the origin (0,0).

But here, when hours = 0, cost = $32 (rental fee), so cost ≠ 0 when hours = 0 → not proportional.

Also, check ratios:
- 52/1 = 52
- 72/2 = 36
- 92/3 ≈ 30.67
- 112/4 = 28 → not constant

✔ Answer: No, the relationship is not proportional because of the fixed rental fee ($32). The total cost includes a constant added amount, so it does not pass through the origin and the ratio is not constant.

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2. Mrs. Govin needs 2 cups of sugar for every cake she makes. Is the number of cakes proportional to the number of cups of sugar?

#### Step 1: Fill in the table
- 1 cake → 2 cups
- 2 cakes → 4 cups
- 3 cakes → 6 cups
- 4 cakes → 8 cups

| Cakes | 1 | 2 | 3 | 4 |
|--------|---|---|---|---|
| Sugar (cups) | 2 | 4 | 6 | 8 |

#### Step 2: Check proportionality
- Ratio of sugar to cakes: 2/1 = 2, 4/2 = 2, 6/3 = 2, 8/4 = 2 → constant
- When cakes = 0, sugar = 0 → passes through origin

✔ Answer: Yes, the relationship is proportional because the ratio of sugar to cakes is constant (2 cups per cake), and it goes through the origin.

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3. CDs cost $11.99 each including tax. Is the number of CDs proportional to the cost?

#### Step 1: Fill in the table

| CDs | 1 | 2 | 3 | 4 |
|------|-------|--------|---------|---------|
| Cost ($) | 11.99 | 23.98 | 35.97 | 47.96 |

> Multiply 11.99 × number of CDs

#### Step 2: Check proportionality
- Ratios: 11.99/1 = 11.99, 23.98/2 = 11.99, 35.97/3 = 11.99, 47.96/4 = 11.99 → constant
- When CDs = 0, cost = 0 → passes through origin

✔ Answer: Yes, the relationship is proportional because the cost per CD is constant and the graph would pass through the origin.

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4. Jean has $280 in her savings account. She deposits $30 each week. Is the amount of money proportional to the number of weeks?

#### Step 1: Fill in the table

| Weeks | 1 | 2 | 3 | 4 |
|--------|-----|-----|-----|-----|
| Savings ($) | 310 | 340 | 370 | 400 |

> Start: $280
> After 1 week: 280 + 30 = 310
> After 2 weeks: 280 + 60 = 340, etc.

#### Step 2: Check proportionality
- At 0 weeks: $280 → not zero
- Ratios: 310/1 = 310, 340/2 = 170, 370/3 ≈ 123.33 → not constant
- Graph would not go through origin due to initial amount

✔ Answer: No, the relationship is not proportional because of the initial $280. This is a linear relationship with a y-intercept, not proportional.

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5. Determine if temperature (Celsius to Fahrenheit) is proportional.

Given:

| Celsius | 0 | 5 | 10 | 15 | 20 |
|---------|----|----|----|----|----|
| Fahrenheit | 32 | 41 | 50 | 59 | 68 |

We know:
F = (9/5)C + 32 → this is not proportional because of the +32.

Check if (F/C) is constant:
- 32/0 → undefined (division by zero)
- 41/5 = 8.2
- 50/10 = 5
- 59/15 ≈ 3.93
- 68/20 = 3.4 → not constant

Also, when C = 0, F = 32 → doesn't pass through origin.

✔ Answer: Not proportional because the relationship includes a constant offset (+32). The graph of F vs C is a straight line that does not pass through the origin.

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6. Popcorn cost:

| Bags of Popcorn | 0 | 1 | 2 | 3 | 4 |
|------------------|---|---|---|---|---|
| Cost ($) | 0 | 4 | 8 | 12 | 16 |

#### Step 1: Check ratios
- 4/1 = 4
- 8/2 = 4
- 12/3 = 4
- 16/4 = 4 → constant ratio
- When bags = 0, cost = 0 → passes through origin

✔ Answer: Proportional — the cost is directly proportional to the number of bags (cost = $4 per bag).

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7. Online DVD rentals: $15/month for unlimited rentals. Is total paid proportional to number of months?

#### Step 1: Create a table

| Month | 1 | 2 | 3 | 4 | ... |
|--------|---|---|---|---|-----|
| Cost ($) | 15 | 30 | 45 | 60 | ... |

Cost = 15 × month

#### Step 2: Check proportionality
- Ratios: 15/1 = 15, 30/2 = 15, 45/3 = 15 → constant
- At 0 months, cost = 0 → passes through origin

✔ Answer: Yes, proportional — the cost increases at a constant rate and starts from zero.

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## ✔ Final Answers Summary:

1. No – Not proportional due to fixed rental fee.
2. Yes – Proportional (2 cups per cake).
3. Yes – Proportional (each CD costs same).
4. No – Not proportional due to initial balance.
5. No – Not proportional (F = (9/5)C + 32, has +32).
6. Yes – Proportional (cost = $4 per bag).
7. Yes – Proportional (cost = $15 per month).

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🔍 How to Graph & Justify:

For problems 5–7, you would plot the points on the coordinate plane.

- A proportional relationship forms a straight line that passes through the origin (0,0).
- A non-proportional relationship may be linear but does not pass through the origin.

> Example: For #5 (temperature), plot (0,32), (5,41), (10,50)... You’ll see a straight line, but it starts at (0,32), not (0,0) → not proportional.

> For #6 and #7, lines go through origin → proportional.

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Let me know if you'd like help sketching the graphs!