Eighth Grade Compare Proportional Relationships and Find Unit ... - Free Printable

Problem Overview:

The task involves comparing proportional relationships and determining which relationship has the larger unit rate. The unit rate is the ratio of the dependent variable to the independent variable when the independent variable is 1. For each problem, we need to calculate the unit rate for both relationships and compare them.

Step-by-Step Solution:

#### Problem 1:
Relationship A: \( y = 1.5x \)
Relationship B: Graph provided

##### Step 1: Calculate the unit rate for Relationship A.
The equation \( y = 1.5x \) is already in the form \( y = mx \), where \( m \) is the unit rate. Here, \( m = 1.5 \).

So, the unit rate for Relationship A is:
\[
\text{Unit Rate (A)} = 1.5
\]

##### Step 2: Calculate the unit rate for Relationship B using the graph.
From the graph:
- When \( x = 2 \), \( y = 3 \).
- The unit rate is calculated as:
\[
\text{Unit Rate (B)} = \frac{y}{x} = \frac{3}{2} = 1.5
\]

##### Step 3: Compare the unit rates.
Both unit rates are equal:
\[
\text{Unit Rate (A)} = \text{Unit Rate (B)} = 1.5
\]

##### Final Answer for Problem 1:
\[
\boxed{=}
\]

---

#### Problem 2:
Relationship A: Graph provided
Relationship B: \( y = 1.75x \)

##### Step 1: Calculate the unit rate for Relationship B.
The equation \( y = 1.75x \) is already in the form \( y = mx \), where \( m \) is the unit rate. Here, \( m = 1.75 \).

So, the unit rate for Relationship B is:
\[
\text{Unit Rate (B)} = 1.75
\]

##### Step 2: Calculate the unit rate for Relationship A using the graph.
From the graph:
- When \( x = 4 \), \( y = 7 \).
- The unit rate is calculated as:
\[
\text{Unit Rate (A)} = \frac{y}{x} = \frac{7}{4} = 1.75
\]

##### Step 3: Compare the unit rates.
Both unit rates are equal:
\[
\text{Unit Rate (A)} = \text{Unit Rate (B)} = 1.75
\]

##### Final Answer for Problem 2:
\[
\boxed{=}
\]

---

#### Problem 3:
Relationship A: Graph provided
Relationship B: \( y = 18x \)

##### Step 1: Calculate the unit rate for Relationship B.
The equation \( y = 18x \) is already in the form \( y = mx \), where \( m \) is the unit rate. Here, \( m = 18 \).

So, the unit rate for Relationship B is:
\[
\text{Unit Rate (B)} = 18
\]

##### Step 2: Calculate the unit rate for Relationship A using the graph.
From the graph:
- When \( x = 2 \), \( y = 9 \).
- The unit rate is calculated as:
\[
\text{Unit Rate (A)} = \frac{y}{x} = \frac{9}{2} = 4.5
\]

##### Step 3: Compare the unit rates.
The unit rate for Relationship B is larger than the unit rate for Relationship A:
\[
\text{Unit Rate (A)} = 4.5 \quad \text{and} \quad \text{Unit Rate (B)} = 18
\]

##### Final Answer for Problem 3:
\[
\boxed{>}
\]

---

#### Problem 4:
Relationship A: Table provided
Relationship B: Graph provided

##### Step 1: Calculate the unit rate for Relationship A using the table.
From the table:
- When \( x = 8 \), \( y = 5 \).
- The unit rate is calculated as:
\[
\text{Unit Rate (A)} = \frac{y}{x} = \frac{5}{8} = 0.625
\]

##### Step 2: Calculate the unit rate for Relationship B using the graph.
From the graph:
- When \( x = 20 \), \( y = 11 \).
- The unit rate is calculated as:
\[
\text{Unit Rate (B)} = \frac{y}{x} = \frac{11}{20} = 0.55
\]

##### Step 3: Compare the unit rates.
The unit rate for Relationship A is larger than the unit rate for Relationship B:
\[
\text{Unit Rate (A)} = 0.625 \quad \text{and} \quad \text{Unit Rate (B)} = 0.55
\]

##### Final Answer for Problem 4:
\[
\boxed{>}
\]

---

Final Answers:

1. \(\boxed{=}\)
2. \(\boxed{=}\)
3. \(\boxed{>}\)
4. \(\boxed{>}\)