Proportional and nonproportional relationships worksheet: Fill out ... - Free Printable

Let’s go through each problem step by step. We’re looking at whether relationships are proportional or nonproportional, and for some, we need to write ratios.

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What does “proportional” mean?

A relationship is proportional if the ratio between two quantities stays the same no matter how much you change them. That means when you divide one number by the other (like cost ÷ hours), you always get the same answer.

If the ratio changes, it’s nonproportional.

Also, in a proportional relationship, if you start at 0, you should end at 0 — like if you work 0 hours, you earn $0. But sometimes there’s a starting fee (like delivery charge) — that makes it nonproportional.

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## Exercise 1: Number of Hours vs Total Amount Earned

Hours: 1 → $15
Hours: 2 → $30
Hours: 3 → $45
Hours: 4 → $60

Let’s find the ratio: Total Amount Earned ÷ Number of Hours

- 15 ÷ 1 = 15
- 30 ÷ 2 = 15
- 45 ÷ 3 = 15
- 60 ÷ 4 = 15

✔ All ratios are the same → Proportional

Ratios to write in table:
→ 15/1, 30/2, 45/3, 60/4
But usually we simplify or just show the unit rate: $15 per hour

So fill in the "Ratios" row with:
15, 15, 15, 15 (or as fractions: 15/1, 30/2, etc., but since they all reduce to 15, writing 15 is fine)

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## Exercise 2: Number of Packages vs Total Cost

Packages: 1 → $11
Packages: 2 → $20
Packages: 3 → $29
Packages: 4 → $38

Find ratio: Cost ÷ Packages

- 11 ÷ 1 = 11
- 20 ÷ 2 = 10
- 29 ÷ 3 ≈ 9.67
- 38 ÷ 4 = 9.5

✘ Ratios are NOT the same → Nonproportional

Ratios to write:
11, 10, 29/3, 9.5 — but since they’re different, we can leave as decimals or fractions. For simplicity, maybe write:

→ 11/1, 20/2, 29/3, 38/4

But again, since they’re not equal, this shows it’s nonproportional.

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## Exercise 3: Number of Classrooms vs Total Students

Classrooms: 1 → 24 students
Classrooms: 2 → 48 students
Classrooms: 3 → 72 students
Classrooms: 4 → 92 students

Ratio: Students ÷ Classrooms

- 24 ÷ 1 = 24
- 48 ÷ 2 = 24
- 72 ÷ 3 = 24
- 92 ÷ 4 = 23

Wait! 92 ÷ 4 = 23? Let me check: 4 × 23 = 92 → yes.

So first three are 24, last one is 23 → ✘ Not the same → Nonproportional

Ratios:
24, 24, 24, 23

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## Exercises 4–8: Just say “proportional” or “nonproportional”

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Exercise 4:

Hours: 1 → $0.99
Hours: 2 → $1.98
Hours: 3 → $2.97
Hours: 4 → $3.96

Check ratios:

- 0.99 ÷ 1 = 0.99
- 1.98 ÷ 2 = 0.99
- 2.97 ÷ 3 = 0.99
- 3.96 ÷ 4 = 0.99

✔ All same → Proportional

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Exercise 5:

Hours: 1 → $17.25
Hours: 2 → $35.50
Hours: 3 → $50.75
Hours: 4 → $70

Check ratios:

- 17.25 ÷ 1 = 17.25
- 35.50 ÷ 2 = 17.75 ← Wait, that’s different!

Hold on — let me calculate carefully:

35.50 ÷ 2 = 17.75? Actually, 2 × 17.75 = 35.50 → yes.

But 17.25 ≠ 17.75 → already different.

Check next:

50.75 ÷ 3 ≈ 16.916... → even different.

70 ÷ 4 = 17.5

All different → ✘ Nonproportional

Wait — maybe I made a mistake? Let me double-check the numbers.

Actually, let’s see if it’s increasing by a constant amount instead.

From 1 to 2 hours: +$18.25? No:

$35.50 - $17.25 = $18.25
$50.75 - $35.50 = $15.25 → not constant increase either.

But for proportionality, we care about the *ratio*, not the difference.

Since 17.25 ≠ 17.75 ≠ 16.916 ≠ 17.5 → definitely nonproportional

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Exercise 6:

Hours: 1 → 37 pages
Hours: 2 → 73 pages
Hours: 3 → 109 pages
Hours: 4 → 145 pages

Ratio: Pages ÷ Hours

- 37 ÷ 1 = 37
- 73 ÷ 2 = 36.5
- 109 ÷ 3 ≈ 36.333...
- 145 ÷ 4 = 36.25

Not the same → ✘ Nonproportional

Even though it’s close, it’s not exact → so nonproportional.

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Exercise 7:

Lunches: 1 → $2.75
Lunches: 2 → $5.50
Lunches: 3 → $8.25
Lunches: 4 → $11.00

Ratios:

- 2.75 ÷ 1 = 2.75
- 5.50 ÷ 2 = 2.75
- 8.25 ÷ 3 = 2.75
- 11.00 ÷ 4 = 2.75

✔ All same → Proportional

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Exercise 8:

Fred orders pies. Each pie costs $4.50. If less than a dozen (12), there’s a $5 delivery charge.

We need to make a table and decide if cost is proportional to number of pies.

Let’s try small numbers:

Number of Pies | Cost
---------------|-----
1 | 4.50 + 5 = $9.50
2 | 9.00 + 5 = $14.00
3 | 13.50 + 5 = $18.50
...
12 | 12 × 4.50 = $54.00 (no delivery charge)

Now check ratios: Cost ÷ Pies

For 1 pie: 9.50 ÷ 1 = 9.50
For 2 pies: 14.00 ÷ 2 = 7.00
For 3 pies: 18.50 ÷ 3 ≈ 6.17
For 12 pies: 54.00 ÷ 12 = 4.50

Ratios are changing → also, there’s a fixed fee ($5) added only under certain conditions → this breaks proportionality.

Even without calculating, adding a flat fee means it’s not proportional — because if you order 0 pies, cost would be $0, but here if you order 1 pie, you pay extra $5 — so the line doesn’t go through origin.

✔ So, nonproportional

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## Final Answers:

1. Ratios: 15, 15, 15, 15 → Proportional
2. Ratios: 11, 10, ~9.67, 9.5 → Nonproportional
3. Ratios: 24, 24, 24, 23 → Nonproportional
4. Proportional
5. Nonproportional
6. Nonproportional
7. Proportional
8. Nonproportional

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Final Answer:
1. Ratios: 15, 15, 15, 15
2. Ratios: 11, 10, 29/3, 9.5 (or approximately 9.67)
3. Ratios: 24, 24, 24, 23
4. proportional
5. nonproportional
6. nonproportional
7. proportional
8. nonproportional