Grade 7 Proportional Relationship Worksheets 2024 - Free Printable

Let's solve the worksheet step by step and explain each part clearly.

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Problem 1: How much do Dylan and Angela each make per hour?

We are given:
- Dylan makes $336 for 32 hours.
- Angela makes $420 for 42 hours.

To find hourly wage, divide total earnings by number of hours:

Dylan’s hourly wage:
$$
\frac{336}{32} = 10.50
$$
So, Dylan makes $10.50 per hour.

Angela’s hourly wage:
$$
\frac{420}{42} = 10.00
$$
So, Angela makes $10.00 per hour.

✔ Answer:
- Dylan: $10.50/hour
- Angela: $10.00/hour

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Problem 2: Is Dylan’s wage for 25 hours proportional to Amber’s wage for 42 hours? Why or why not?

Wait — there's a problem here. The name Amber is mentioned, but we were only told about Dylan and Angela.

Possibly a typo — maybe it meant Angela instead of Amber?

Assuming it's a typo and they meant Angela, let's proceed with that assumption.

So:
Is Dylan’s wage for 25 hours proportional to Angela’s wage for 42 hours?

But wait — we don't have information about what Dylan earns in 25 hours or what Angela earns in 42 hours directly.

However, we can calculate what each would earn based on their hourly rates.

- Dylan earns $10.50/hour → In 25 hours:
$$
25 \times 10.50 = 262.50
$$

- Angela earns $10.00/hour → In 42 hours:
$$
42 \times 10.00 = 420
$$

Now check if these two wages are proportional — meaning the ratio of money to time should be equal (same rate).

But the question is asking whether Dylan’s wage for 25 hours is proportional to Angela’s wage for 42 hours.

That means: is
$$
\frac{\text{Dylan's 25-hour wage}}{25} = \frac{\text{Angela's 42-hour wage}}{42}
$$

But we already know:
- Dylan’s rate = $10.50/hr
- Angela’s rate = $10.00/hr

These are not equal, so the ratios are not proportional.

Even though both are linear relationships (constant rate), the rates are different, so the two wage amounts are not proportional to each other.

✔ Conclusion:
No, Dylan’s wage for 25 hours is not proportional to Angela’s wage for 42 hours because their hourly rates are different ($10.50 vs. $10.00). For proportions, the unit rates must be equal.

> To determine proportionality between two ratios or rates, the ratios must simplify to the same value (i.e., the unit rates must be equal).

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Table 1 and Table 2: Find the ratio $ \frac{y}{x} $ (Total Cost / Number of Hours)

Let’s compute $ \frac{y}{x} $ for each table.

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#### Table 1:

| Hours (x) | Total Cost (y) | Ratio $ \frac{y}{x} $ |
|----------|----------------|------------------------|
| 1 | $75 | $ \frac{75}{1} = 75 $ |
| 2 | $120 | $ \frac{120}{2} = 60 $ |
| 3 | $165 | $ \frac{165}{3} = 55 $ |
| 4 | $210 | $ \frac{210}{4} = 52.5 $ |
| 5 | $255 | $ \frac{255}{5} = 51 $ |

Ratios: 75, 60, 55, 52.5, 51 → Not constant

So, not proportional

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#### Table 2:

| Hours (x) | Total Cost (y) | Ratio $ \frac{y}{x} $ |
|----------|----------------|------------------------|
| 1 | $45 | $ \frac{45}{1} = 45 $ |
| 2 | $90 | $ \frac{90}{2} = 45 $ |
| 3 | $135 | $ \frac{135}{3} = 45 $ |
| 4 | $180 | $ \frac{180}{4} = 45 $ |
| 5 | $225 | $ \frac{225}{5} = 45 $ |

All ratios = 45 → Constant

✔ So, Table 2 shows a proportional relationship.

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Question 3: Which table shows a proportional relationship?

✔ Answer: Table 2

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Question 4: What makes it a proportional relationship?

A proportional relationship has:
- A constant ratio (unit rate) between $ y $ and $ x $
- The graph passes through the origin (0,0)
- The equation is of the form $ y = kx $, where $ k $ is constant

In this case, Table 2 has a constant ratio of $ \frac{y}{x} = 45 $, so $ y = 45x $

Also, if you extend the pattern back to 0 hours, cost would be $0 → passes through origin.

✔ Conclusion:
To determine proportionality from a table, check if the ratio $ \frac{y}{x} $ is constant for all pairs. If yes, then it's proportional.

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✔ Final Answers Summary:

1. Dylan: $10.50/hour, Angela: $10.00/hour

2. No, Dylan’s wage for 25 hours is not proportional to Angela’s wage for 42 hours because their hourly rates are different.
> To determine proportionality between two ratios or rates, the ratios must simplify to the same value (equal unit rates).

3. Table 2 shows a proportional relationship.

4. A proportional relationship has a constant ratio $ \frac{y}{x} $ across all data points.
> To determine proportionality from a table, check if the ratio $ \frac{y}{x} $ is the same for all rows.

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