Solved Exercises on Identifying Proportional vs Non-Proportional Relationships in Grade 7

Master identification of proportional vs non-proportional relationships: constant of proportionality, direct variation, tables, graphs, and equations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Table Analysis: Proportional vs Non-Proportional
Exercise 1
Determine if the following tables represent proportional or non-proportional relationships:
Table A
x 1 2 3 4
y 4 8 12 16
Table B
x 1 2 3 4
y 5 9 13 17
Definition:

Proportional Relationship: Two quantities are proportional if their ratio is constant (y/x = k). The equation is y = kx where k is the constant of proportionality.

Non-Proportional Relationship: The ratio y/x is not constant, or the graph doesn't pass through the origin (0,0).

Method for identifying relationships in tables:
  1. Calculate the ratio y/x for each pair of values
  2. Check if all ratios are equal
  3. If ratios are equal, the relationship is proportional
  4. If ratios differ, the relationship is non-proportional
  5. Also check if x=0 gives y=0 (origin test)
Table A
4/1 = 8/2 = 12/3 = 16/4 = 4
Table B
5/1 ≠ 9/2 ≠ 13/3 ≠ 17/4
Step 1: Calculate ratios for Table A

Point 1: 4/1 = 4

Point 2: 8/2 = 4

Point 3: 12/3 = 4

Point 4: 16/4 = 4

Step 2: Calculate ratios for Table B

Point 1: 5/1 = 5

Point 2: 9/2 = 4.5

Point 3: 13/3 ≈ 4.33

Point 4: 17/4 = 4.25

Step 3: Compare ratios

Table A: All ratios equal 4 → Proportional

Table B: Ratios are different → Non-proportional

Table A: Proportional (y = 4x) | Table B: Non-proportional
Final answer:

Table A represents a proportional relationship with equation y = 4x. Table B represents a non-proportional relationship.

Applied rules:

• Constant Ratio Test: In a proportional relationship, y/x = k (constant)

• Direct Variation: y varies directly with x (y = kx)

• Origin Test: The graph passes through (0,0) when both variables are zero

Key Concept:

In a proportional relationship, doubling one quantity results in doubling the other quantity. The relationship maintains the same rate throughout.

2 Graph Analysis: Proportional vs Non-Proportional
Exercise 2
Analyze the following graphs and determine if they represent proportional or non-proportional relationships:
Graph 1: Line through (0,0) and (2,6)
Graph 2: Line through (0,3) and (2,7)
Definition:

Graph of Proportional Relationship: A straight line passing through the origin (0,0) with slope equal to the constant of proportionality.

Graph of Non-Proportional Relationship: Any line that doesn't pass through the origin, or any non-linear curve.

Graph 1
Passes through (0,0) → Proportional
Graph 2
Does not pass through (0,0) → Non-proportional
Slope
m = (6-0)/(2-0) = 3
Step 1: Check if graph passes through origin

Graph 1: Contains point (0,0) → Passes through origin

Graph 2: Contains point (0,3) → Does not pass through origin

Step 2: Analyze linearity

Both graphs are straight lines

Graph 1: Linear + passes through origin = Proportional

Graph 2: Linear + does not pass through origin = Non-proportional

Step 3: Find equations

Graph 1: y = 3x (proportional)

Graph 2: y = 2x + 3 (non-proportional)

Graph 1: Proportional | Graph 2: Non-proportional
Final answer:

Graph 1 represents a proportional relationship (y = 3x). Graph 2 represents a non-proportional relationship (y = 2x + 3).

Applied rules:

• Origin Test: Proportional graphs must pass through (0,0)

• Linearity: Proportional relationships are linear

• Slope-Intercept Form: y = mx + b where b = 0 for proportional relationships

Proportional Characteristics
  • ✓ Passes through origin (0,0)
  • ✓ Linear relationship
  • ✓ Constant rate of change
  • ✓ y = kx format
Non-Proportional Characteristics
  • ✗ Does not pass through origin
  • ✗ May be linear or non-linear
  • ✗ Variable rate of change
  • ✗ y = mx + b (b ≠ 0) format
3 Equation Analysis: Proportional vs Non-Proportional
Exercise 3
Determine if these equations represent proportional or non-proportional relationships:
a) y = 5x
b) y = 3x + 2
c) y = x/4
d) y = 2x - 7
Definition:

Proportional Equation: An equation in the form y = kx where k is a constant and there is no constant term added or subtracted.

Non-Proportional Equation: Any equation that cannot be written in the form y = kx, including those with constant terms.

Forms
y = kx vs y = mx + b
Proportional
a) y = 5x, c) y = x/4
Non-proportional
b) y = 3x + 2, d) y = 2x - 7
Step 1: Identify equation forms

a) y = 5x → Only variable term (proportional)

b) y = 3x + 2 → Has constant term (non-proportional)

c) y = x/4 → Only variable term (proportional)

d) y = 2x - 7 → Has constant term (non-proportional)

Step 2: Check for constant terms

Proportional: No constant terms (y = kx)

Non-proportional: Has constant terms (y = mx + b where b ≠ 0)

Step 3: Verify with origin test

a) x=0 → y=5(0)=0 → (0,0) ✓ Proportional

b) x=0 → y=3(0)+2=2 → (0,2) ✗ Non-proportional

Proportional: a, c | Non-proportional: b, d
Final answer:

Proportional relationships: y = 5x and y = x/4. Non-proportional relationships: y = 3x + 2 and y = 2x - 7.

Applied rules:

• Standard Form: y = kx (proportional) vs y = mx + b (non-proportional if b ≠ 0)

• Constant Term Test: Presence of constant term makes relationship non-proportional

• Origin Test: Substituting x=0 should yield y=0 for proportional relationships

Key Concept:

Proportional equations have no constant term added or subtracted. They can be written as y = kx where k is the constant of proportionality.

Identifying Proportional vs Non-Proportional Relationships: Rules and Methods
\(\text{Proportional: } y = kx \quad \text{Non-proportional: } y = mx + b \text{ (} b \neq 0 \text{)}\)
Relationship Forms
Proportional
\(y = kx\)
Direct variation with constant of proportionality
Non-Proportional
\(y = mx + b \text{ (} b \neq 0 \text{)}\)
Linear relationship with y-intercept
Constant of Proportionality
\(k = \frac{y}{x}\)
Ratio of dependent to independent variable
Key definitions:

Proportional Relationship: A relationship where y = kx, with k being a constant and no additional terms

Non-Proportional Relationship: Any relationship that cannot be expressed as y = kx, including those with constant terms

Constant of Proportionality: The constant k in y = kx, representing the rate of change

Complete identification methodology:
  1. Table Analysis: Check if y/x ratios are constant
  2. Graph Analysis: Check if line passes through origin (0,0)
  3. Equation Analysis: Check if equation is in form y = kx (no constant term)
  4. Origin Test: Substitute x = 0, check if y = 0
Tip 1: Always check if the graph passes through (0,0) for proportional relationships.
Tip 2: In equations, if there's anything added or subtracted after the variable term, it's non-proportional.
Tip 3: In tables, divide y by x for each pair - if all quotients are equal, the relationship is proportional.
Tip 4: Proportional relationships show a constant rate of change.
Proportional Characteristics: Linear graph through origin, constant ratio, direct variation.
Non-Proportional Characteristics: Graph doesn't pass through origin, may have variable rate of change.
Tests for Proportionality:

• Table Test: All y/x ratios must be equal

• Graph Test: Must pass through point (0,0)

• Equation Test: Must be in form y = kx with no constant term

• Origin Test: When x = 0, y must equal 0

Solution: Exercises 4 to 5
4 Real-World Application: Shopping Scenarios
Exercise 4
Compare two shopping scenarios:
Scenario A: Apples cost $2 per pound with no membership fee
Scenario B: Apples cost $1.50 per pound with a $5 membership fee
Which represents a proportional relationship?
Definition:

Real-World Proportionality: A relationship where the total cost varies directly with the quantity purchased, without any fixed fees.

Scenario A
Cost = 2pounds
Scenario B
Cost = 1.5pounds + 5
Result
A: Proportional, B: Non-proportional
Step 1: Write equations for each scenario

Scenario A: Total cost = $2 × pounds → C = 2p

Scenario B: Total cost = ($1.50 × pounds) + $5 → C = 1.5p + 5

Step 2: Check for constant terms

Scenario A: C = 2p (no constant term) → Proportional

Scenario B: C = 1.5p + 5 (has constant term) → Non-proportional

Step 3: Verify with origin test

Scenario A: If p = 0, C = 2(0) = $0 → Passes origin test

Scenario B: If p = 0, C = 1.5(0) + 5 = $5 → Fails origin test

Scenario A: Proportional | Scenario B: Non-proportional
Final answer:

Scenario A (apples at $2 per pound with no membership fee) represents a proportional relationship. Scenario B (with membership fee) represents a non-proportional relationship.

Applied rules:

• Fixed Cost Test: Any fixed cost makes relationship non-proportional

• Variable Cost Test: Purely variable costs create proportional relationships

• Origin Test: Zero quantity should yield zero cost for proportional relationships

5 Mixed Representation Analysis
Exercise 5
A relationship is given by: y = 4x. Create a table for x-values 0, 1, 2, 3, and verify that this is proportional by multiple methods.
Definition:

Multiple Representation Verification: Using tables, graphs, and equations together to confirm the nature of a relationship.

Equation
y = 4x
Table
(0,0), (1,4), (2,8), (3,12)
Verification
All tests confirm proportionality
Step 1: Create the table
x 0 1 2 3
y 0 4 8 12
Step 2: Test with ratio method

For each point: y/x = 4/1 = 8/2 = 12/3 = 4 (constant ratio)

Step 3: Test with origin method

When x = 0, y = 4(0) = 0 → Passes through origin

Step 4: Test with equation form

Equation is y = 4x (form y = kx with no constant term)

All tests confirm: y = 4x is proportional
Final answer:

The relationship y = 4x is proportional as confirmed by all three methods: constant ratios, origin test, and equation form.

Applied rules:

• Consistency Check: All representation methods should agree

• Multiple Testing: Use several methods to verify relationships

• Origin Verification: Essential test for proportional relationships

Comprehensive Summary: Proportional vs Non-Proportional Relationships
\(\text{Proportional: } y = kx \text{ | Non-proportional: } y = mx + b \text{ (} b \neq 0 \text{)}\)
Key Formula Distinction
Core Definitions:

Proportional Relationship: A relationship between two variables where the ratio of one variable to the other is constant. This means y = kx where k is the constant of proportionality.

Non-Proportional Relationship: A relationship where the ratio between variables is not constant, or the relationship cannot be expressed in the form y = kx.

Constant of Proportionality (k): The constant ratio in a proportional relationship, calculated as k = y/x.

Identification Methods:
  1. Table Method: Check if all y/x ratios are equal
  2. Graph Method: Check if the graph passes through origin (0,0)
  3. Equation Method: Check if equation is in form y = kx (no constant term)
  4. Origin Test: Substitute x = 0, check if y = 0
Quick Check: If an equation has a number added or subtracted (like y = 3x + 5), it's non-proportional.
Pattern Recognition: In proportional relationships, doubling x results in doubling y.
Visual Cue: Proportional graphs always go through the point (0,0).
Rate Consistency: Proportional relationships have a constant rate of change.
Proportional Examples: y = 2x, y = 0.5x, y = (3/4)x, cost = $2.50 × items
Non-Proportional Examples: y = 2x + 3, y = x - 5, y = 0.5x + 10
Key Rules and Properties:

• Constant Ratio: In proportional relationships, y₁/x₁ = y₂/x₂ for any two points

• Origin Requirement: All proportional relationships pass through (0,0)

• Linear Nature: Proportional relationships are always linear

• No Constant Term: Proportional equations have no additive constant

• Direct Variation: As one variable increases, the other increases by a constant factor

Proportional Relationships
  • ✓ Equation: y = kx
  • ✓ Graph passes through (0,0)
  • ✓ Constant rate of change
  • ✓ y/x ratio is constant
  • ✓ Direct variation
Non-Proportional Relationships
  • ✗ Equation: y = mx + b (b ≠ 0)
  • ✗ Graph does not pass through (0,0)
  • ✗ May have variable rate of change
  • ✗ y/x ratio is not constant
  • ✗ Not direct variation

Questions & Answers

Question: How can I quickly tell if an equation is proportional without doing calculations? What's the fastest way?

Answer: Great question! Here's the fastest way to identify proportional equations:

  1. Look for constant terms: If you see any number added or subtracted (like "+5" or "-3"), it's non-proportional
  2. Check the form: Only equations in the form y = kx (with just the variable term) are proportional
  3. Examples:
    • Proportional: y = 3x, y = 0.5x, y = (2/3)x
    • Non-proportional: y = 3x + 5, y = 0.5x - 2, y = x + 10

Memory Tip: "No extra numbers allowed!" Proportional equations can only have the variable term multiplied by a constant.

This quick visual check works because any added or subtracted number prevents the relationship from passing through the origin (0,0).

Question: Why does the graph of a proportional relationship always pass through the origin (0,0)? What happens if it doesn't?

Answer: This happens because of the mathematical definition of proportional relationships. Let's break it down:

  1. Equation Form: Proportional relationships have the form y = kx
  2. Substitute Zero: When x = 0, y = k(0) = 0
  3. Point Verification: This gives us the point (0,0)

Real-world example: If you buy 0 items, the cost is $0. If you drive for 0 hours, you travel 0 miles.

If a graph doesn't pass through (0,0), it's still linear but it's NOT proportional. This usually indicates there's a fixed starting value or fee involved.

For example, if you have to pay a $5 membership fee regardless of purchases, the relationship becomes non-proportional.

Question: I'm still confused about why some relationships are called "linear" but not "proportional." Can you explain the difference?

Answer: Excellent question! There's an important distinction here:

  • Linear Relationship: Any relationship whose graph is a straight line (y = mx + b)
  • Proportional Relationship: A special type of linear relationship where b = 0 (y = mx)

Analogy: Think of squares and rectangles. All squares are rectangles, but not all rectangles are squares. Similarly, all proportional relationships are linear, but not all linear relationships are proportional.

Examples:

  • Linear AND Proportional: y = 3x (passes through origin)
  • Linear BUT Not Proportional: y = 3x + 5 (doesn't pass through origin)

The key difference is whether the relationship passes through the origin (0,0).