Grade 7 Exam > Grade 7 Notes > Math > Cheatsheet: Proportional Relationships
Cheatsheet: Proportional Relationships
1. Understanding Proportional Relationships
1.1 Key Definitions
| Term | Definition |
|---|---|
| Proportional Relationship | A relationship between two quantities where the ratio remains constant; y = kx where k ≠ 0 |
| Constant of Proportionality (k) | The constant ratio in a proportional relationship; k = y/x |
| Ratio | A comparison of two quantities; can be written as a:b, a/b, or a to b |
| Unit Rate | A ratio comparing a quantity to 1 unit of another quantity |
| Origin | The point (0,0) on a coordinate plane; all proportional relationships pass through the origin |
1.2 Characteristics of Proportional Relationships
- The graph is a straight line through the origin (0,0)
- The ratio y/x is constant for all points
- When one quantity is 0, the other is also 0
- The equation is in the form y = kx (no y-intercept other than 0)
- Doubling one quantity doubles the other quantity
2. Identifying Proportional Relationships
2.1 From Tables
| Method | Process |
|---|---|
| Calculate ratios | Find y/x for each row; if all ratios are equal, relationship is proportional |
| Check for (0,0) | If the table includes x = 0, then y must equal 0 for proportional relationship |
2.2 From Graphs
- Graph must be a straight line
- Line must pass through the origin (0,0)
- If either condition is false, relationship is not proportional
2.3 From Equations
| Proportional | Not Proportional |
|---|---|
| y = kx (k is a constant) | y = kx + b where b ≠ 0 |
| y = 5x | y = 5x + 3 |
| y = (2/3)x | y = x² |
2.4 From Verbal Descriptions
- Look for phrases: "per", "each", "for every", "constant rate"
- Check if relationship starts at zero
- Example (proportional): "A car travels 60 miles per hour"
- Example (not proportional): "A phone costs $50 plus $20 per month"
3. Finding the Constant of Proportionality
3.1 Formula
- k = y/x (for any point except the origin)
- k represents the unit rate
- k is the slope of the line
3.2 From Different Representations
| Representation | How to Find k |
|---|---|
| Table | Pick any row, divide y by x: k = y/x |
| Graph | Pick any point (x,y) on the line, calculate k = y/x or count rise/run from origin |
| Equation y = kx | k is the coefficient of x |
| Verbal | Find the unit rate (amount per 1 unit) |
3.3 Example Calculations
- If 3 pounds of apples cost $6, then k = 6/3 = 2 (dollars per pound)
- If point (4, 12) is on the line, then k = 12/4 = 3
- In equation y = 7x, k = 7
4. Writing Equations
4.1 Standard Form
- All proportional relationships: y = kx
- Where y is the dependent variable, x is the independent variable, k is the constant of proportionality
4.2 Steps to Write Equation
- Find the constant of proportionality k using k = y/x
- Substitute k into y = kx
- Simplify if needed
4.3 Example Problems
| Given Information | Equation |
|---|---|
| Point (5, 20) | k = 20/5 = 4; equation: y = 4x |
| 6 gallons cost $18 | k = 18/6 = 3; equation: y = 3x (cost = 3 × gallons) |
| Unit rate of 2.5 | Equation: y = 2.5x |
5. Graphing Proportional Relationships
5.1 Key Steps
- Plot the origin (0,0) first
- Find another point using k (if k = 3, use point (1,3) or any point satisfying y = kx)
- Draw a straight line through both points
- Extend the line in both directions
5.2 Using the Constant of Proportionality
- k represents the slope (rise/run)
- From origin, move right 1 unit, up k units to find point (1, k)
- Example: If k = 2, plot (0,0) and (1,2)
- Example: If k = 1/2, plot (0,0) and (2,1) for easier graphing
5.3 Reading Graphs
- Identify any point (x,y) on the line
- Calculate k = y/x
- Write equation y = kx
- Verify line passes through origin
6. Unit Rates and Comparisons
6.1 Unit Rate Definition
- A ratio with denominator of 1
- Same as the constant of proportionality
- Expressed as "per 1 unit"
6.2 Calculating Unit Rates
| Situation | Calculation |
|---|---|
| $24 for 6 pounds | 24 ÷ 6 = $4 per pound |
| 180 miles in 3 hours | 180 ÷ 3 = 60 miles per hour |
| 12 pencils for $3 | 3 ÷ 12 = $0.25 per pencil OR 12 ÷ 3 = 4 pencils per dollar |
6.3 Comparing Proportional Relationships
- Compare constants of proportionality (k values)
- Higher k means steeper line on graph
- For costs: higher k means more expensive per unit
- For speeds: higher k means faster rate
6.4 Example Comparison
- Store A: 5 pounds for $15 (k = 15/5 = 3)
- Store B: 8 pounds for $20 (k = 20/8 = 2.50)
- Store B has lower unit rate, so better deal
7. Solving Problems
7.1 Problem-Solving Steps
- Identify the two quantities in the relationship
- Determine if relationship is proportional
- Find the constant of proportionality k
- Write equation y = kx
- Substitute known value to find unknown
7.2 Common Problem Types
| Problem Type | Approach |
|---|---|
| Find missing value in table | Calculate k from known row, then use k to find missing value |
| Determine if proportional | Check if all ratios y/x are equal and relationship includes (0,0) |
| Compare unit rates | Calculate k for each option and compare values |
| Write equation from description | Identify unit rate from words, write as y = kx |
7.3 Example Problem
- Problem: If 4 tickets cost $50, how much do 7 tickets cost?
- Find k: k = 50/4 = 12.5
- Equation: y = 12.5x
- Substitute: y = 12.5(7) = 87.5
- Answer: 7 tickets cost $87.50
8. Common Mistakes to Avoid
8.1 Identifying Proportional Relationships
- Mistake: Assuming all linear relationships are proportional
- Remember: Line must pass through origin (0,0)
- y = 2x + 5 is linear but NOT proportional (has y-intercept of 5)
8.2 Finding k
- Mistake: Calculating x/y instead of y/x
- Correct: k = y/x = dependent/independent
- Be consistent with which variable is which
8.3 Graphing
- Mistake: Forgetting to include (0,0)
- Every proportional relationship must pass through origin
- Use at least two points to draw the line
8.4 Writing Equations
- Mistake: Adding extra terms (y = 3x + 2 instead of y = 3x)
- Proportional equations only have form y = kx
- No y-intercept other than 0
9. Key Formulas and Relationships
| Formula/Concept | Description |
|---|---|
| y = kx | Standard form of proportional relationship |
| k = y/x | Formula to find constant of proportionality |
| Unit Rate = k | Constant of proportionality equals unit rate |
| Slope = k | On graph, slope of line equals constant of proportionality |
| (0,0) required | All proportional relationships include the origin |
10. Quick Reference
10.1 Checklist for Proportional Relationships
- ✓ Ratio y/x is constant
- ✓ Graph is straight line through origin
- ✓ Equation has form y = kx
- ✓ No y-intercept (or y-intercept = 0)
- ✓ When x = 0, y = 0
10.2 Converting Between Representations
| From | To |
|---|---|
| Table → Equation | Calculate k = y/x, write y = kx |
| Equation → Graph | Plot (0,0) and (1,k), draw line |
| Graph → Equation | Find k from any point, write y = kx |
| Words → Equation | Find unit rate = k, write y = kx |
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Apr 25, 2026 Last updated
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