Proportional reasoning worksheet with tables and equations.

Let’s go through each table one by one and determine whether the relationship is proportional. A relationship is proportional if:

- The ratio of y to x (or output to input, etc.) is constant for all pairs (except when x=0, which should give y=0 in a proportional relationship).
- The graph would be a straight line passing through the origin (0,0).
- It can be written as an equation: y = kx, where k is the constant of proportionality.

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1) Table:

| x | y |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |

✔ Proportional? YES

Check ratios:
- 1/1 = 1
- 2/2 = 1
- 3/3 = 1

All ratios are equal → constant of proportionality k = 1

Equation: y = x

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2) Table:

| p | r |
|---|---|
| 0 | 0 |
| 2 | 8 |
| 4 | 16 |
| 5 | 20 |

✔ Proportional? YES

Check ratios (r/p):
- 8/2 = 4
- 16/4 = 4
- 20/5 = 4

Constant ratio → k = 4

Equation: r = 4p

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3) Table:

| x | y |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 9 |
| 4 | 12 |

✔ Proportional? YES

Check ratios (y/x):
- 3/1 = 3
- 6/2 = 3
- 9/3 = 3
- 12/4 = 3

Constant → k = 3

Equation: y = 3x

*(Note: Even though (0,0) isn’t listed, since all other points follow y=3x, and it’s implied that at x=0, y=0, this is still proportional.)*

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4) Table:

| x | y |
|---|-----|
| 2 | 10.5|
| 4 | ? |
| 6 | ? |
| 8 | ? |

We need to check if it’s proportional. Let’s find the constant from the first pair:

y/x = 10.5 / 2 = 5.25

If proportional, then:
- When x=4 → y = 5.25 × 4 = 21
- When x=6 → y = 5.25 × 6 = 31.5
- When x=8 → y = 5.25 × 8 = 42

✔ Proportional? YES (assuming the missing values follow this pattern)

Equation: y = 5.25x
(or as fraction: y = 21/4 x)

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5) Table:

| s | t |
|---|---|
| 2 | 3 |
| 5 | 10|
| 6 | 9 |
| 8 | 16|

Check ratios (t/s):
- 3/2 = 1.5
- 10/5 = 2
- 9/6 = 1.5
- 16/8 = 2

Ratios alternate between 1.5 and 2 — not constant!

✘ Proportional? NO

There’s no single constant of proportionality.

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6) Table:

| Input | Output |
|-------|--------|
| 24 | 3 |
| 72 | 9 |
| 104 | 13 |
| 120 | 15 |

Check Output/Input:
- 3/24 = 0.125
- 9/72 = 0.125
- 13/104 = 0.125
- 15/120 = 0.125

✔ Proportional? YES

Constant ratio = 0.125 = 1/8

Equation: Output = (1/8) × Input
or y = (1/8)x

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7) Table:

| a | b |
|-----|---|
| ⅓ | 3 |
| ⅔ | ? |
| ¾ | 9 |
| ⅚ | ? |

Check ratios (b/a):

First pair: 3 ÷ (1/3) = 3 × 3 = 9

Third pair: 9 ÷ (3/4) = 9 × 4/3 = 12

Not the same → NOT proportional

Wait — let’s double-check:

Is there a constant k such that b = k × a?

From first row: 3 = k × (1/3) → k = 9

Then second row: b = 9 × (2/3) = 6

Third row: b = 9 × (3/4) = 27/4 = 6.75 — but table says 9 → contradiction!

So unless the third row is wrong, it’s not proportional.

But wait — maybe we misread? Let’s test if a × b = constant? That would be inverse, not proportional.

Proportional means b/a = constant.

We have:
- Row 1: b/a = 3 / (1/3) = 9
- Row 3: b/a = 9 / (3/4) = 12 → Not same.

✘ Proportional? NO

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## ✔ Final Answers:

1) Yes — Equation: y = x
2) Yes — Equation: r = 4p
3) Yes — Equation: y = 3x
4) Yes — Equation: y = 5.25x (or y = 21/4 x)
5) No
6) Yes — Equation: Output = (1/8) × Input or y = (1/8)x
7) No

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📝 Bonus Tip: For tables with missing values (like #4), if you’re told to assume proportionality, fill in using the constant ratio from the known pair. In #7, even if we filled in missing values assuming k=9, the given value (9 for a=¾) contradicts that — so it’s definitely not proportional.

Let me know if you want the completed tables too!