Solved Exercises on Tables and Graphs of Proportional Relationships in Grade 7

Master proportional relationships: constant of proportionality, direct variation, tables, and graphs through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Proportional Relationships in Tables
Exercise 1
Determine if the relationship in the table below is proportional:
x 1 2 3 4
y 3 6 9 12
Definition:

Proportional Relationship: Two quantities are proportional if their ratio is constant. This means y = kx where k is the constant of proportionality.

Method for identifying proportionality 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. The common ratio is the constant of proportionality (k)
Ratios
y/x = 3/1 = 6/2 = 9/3 = 12/4
Result
k = 3
Step 1: Calculate ratios y/x for each pair

Point 1: 3/1 = 3

Point 2: 6/2 = 3

Point 3: 9/3 = 3

Point 4: 12/4 = 3

Step 2: Check if all ratios are equal

All ratios equal 3, so the relationship is proportional

Step 3: Write the equation

Since k = 3, the equation is y = 3x

y = 3x (proportional relationship)
Final answer:

The relationship is proportional with constant of proportionality k = 3. The equation is y = 3x.

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 Graphing Proportional Relationships
Exercise 2
Graph the proportional relationship y = 2.5x and identify its key features.
Definition:

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

Equation
y = 2.5x
Constant
k = 2.5
Slope
m = 2.5
Step 1: Identify the constant of proportionality

In y = 2.5x, the constant of proportionality k = 2.5

Step 2: Create a table of values
x 0 1 2 3 4
y 0 2.5 5 7.5 10
Step 3: Plot the points and draw the line

Plot points (0,0), (1,2.5), (2,5), (3,7.5), (4,10) and connect with a straight line

Linear graph through origin with slope = 2.5
Final answer:

The graph is a straight line passing through the origin with slope 2.5, representing the proportional relationship y = 2.5x.

Applied rules:

• Linear Graph: Proportional relationships graph as straight lines

• Origin Point: All proportional graphs pass through (0,0)

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

↗️
Positive Slope
-Origin-
Passes Through (0,0)
📏
Straight Line
3 Finding Constant of Proportionality
Exercise 3
A car travels 150 miles in 3 hours. If distance varies directly with time, find the constant of proportionality and write the equation.
Definition:

Constant of Proportionality: The ratio between two proportional quantities, often represented as k in the equation y = kx.

Given
Distance = 150 miles, Time = 3 hours
Formula
k = Distance/Time
Result
k = 50 mph
Step 1: Identify the relationship

Distance varies directly with time, so d = kt where k is the constant of proportionality

Step 2: Substitute known values

150 = k × 3

Step 3: Solve for k

k = 150 ÷ 3 = 50

Step 4: Write the equation

d = 50t (where d = distance in miles, t = time in hours)

d = 50t, k = 50 mph
Final answer:

The constant of proportionality is 50 mph, and the equation is d = 50t.

Applied rules:

• Direct Variation Formula: y = kx

• Constant Calculation: k = y/x

• Unit Consistency: Units must match in the calculation

Real-world Application:

The constant of proportionality represents the rate of change - in this case, speed (miles per hour). It tells us how much the dependent variable changes for each unit increase in the independent variable.

Rules and methods, laws,...
\(y = kx\)
Proportional Relationship
Constant of Proportionality
\(k = \frac{y}{x}\)
Ratio of dependent to independent variable
Slope of Graph
\(m = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1}\)
Slope equals constant of proportionality
Direct Variation
\(y \propto x\)
y varies directly with x
Key definitions:

Proportional Relationship: A relationship where the ratio of two variables is constant

Constant of Proportionality: The constant ratio in a proportional relationship (k in y=kx)

Direct Variation: When one variable increases, the other increases at a constant rate

Complete methodology:
  1. Identify Variables: Determine which quantities vary together
  2. Test Proportionality: Check if ratios are constant (tables) or if graph is a straight line through origin
  3. Find Constant: Calculate k = y/x
  4. Write Equation: Express relationship as y = kx
Tip 1: Always check if the graph passes through (0,0) for proportional relationships.
Tip 2: In tables, divide y by x for each pair - if all quotients are equal, the relationship is proportional.
Tip 3: The constant of proportionality has units (e.g., miles per hour, dollars per item).
Tip 4: If x doubles, y should also double in a proportional relationship.
Characteristics: Linear graph through origin, constant ratio, direct variation.
Real Applications: Unit rates, scaling, conversion factors, speed calculations.
Properties of Proportional Relationships:

• Linearity: Graph is always a straight line

• Origin: Graph always passes through point (0,0)

• Constant Rate: Rate of change is constant

• Symmetry: If y is proportional to x, then x is proportional to y

Solution: Exercises 4 to 5
4 Interpreting Proportional Graphs
Exercise 4
The graph shows the cost of apples versus weight. If 4 pounds cost $10, find the equation and predict the cost of 7 pounds.
Definition:

Graph Interpretation: The slope of a proportional relationship graph represents the constant of proportionality.

Given
(4, 10) on the graph
Constant
k = 10/4 = 2.5
Prediction
y = 2.5 × 7 = $17.50
Step 1: Identify the point on the graph

We know that when weight = 4 pounds, cost = $10

Step 2: Find the constant of proportionality

k = cost/weight = $10/4 pounds = $2.50 per pound

Step 3: Write the equation

C = 2.50w (where C = cost in dollars, w = weight in pounds)

Step 4: Make prediction for 7 pounds

C = 2.50 × 7 = $17.50

C = 2.50w, Cost of 7 lbs = $17.50
Final answer:

The equation is C = 2.50w, and 7 pounds of apples cost $17.50.

Applied rules:

• Slope Interpretation: Slope represents the rate of change

• Unit Rate: The constant of proportionality is the unit rate

• Prediction: Use the equation to find unknown values

5 Comparing Proportional Relationships
Exercise 5
Compare two water filling rates: Tank A fills at 8 gallons per minute, Tank B follows the equation y = 6x. Which tank fills faster?
Definition:

Comparison of Rates: Compare constants of proportionality to determine which relationship has a greater rate of change.

Tank A
Rate = 8 gal/min
Tank B
y = 6x → Rate = 6 gal/min
Comparison
8 > 6
Step 1: Identify the rate for Tank A

Tank A fills at 8 gallons per minute, so its rate is 8 gal/min

Step 2: Identify the rate for Tank B

Tank B follows y = 6x, so its rate (constant of proportionality) is 6 gal/min

Step 3: Compare the rates

8 gal/min > 6 gal/min, so Tank A fills faster

Step 4: Verify with specific example

After 10 minutes: Tank A = 8×10 = 80 gal, Tank B = 6×10 = 60 gal

Tank A fills faster (8 gal/min vs 6 gal/min)
Final answer:

Tank A fills faster because its rate of 8 gallons per minute is greater than Tank B's rate of 6 gallons per minute.

Applied rules:

• Rate Comparison: Compare constants of proportionality directly

• Unit Consistency: Ensure rates have the same units

• Verification: Use specific values to confirm comparison

Proportional Relationships Laws, Methods, and Key Concepts
\(y = kx \text{ where } k = \frac{y}{x}\)
Proportional Relationship Formula
Key definitions:

Proportional Relationship: A relationship where y = kx, with k being a constant

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

Direct Variation: When one variable increases, the other increases by the same factor

Complete methodology:
  1. Identify Variables: Determine which quantities vary together
  2. Test Proportionality: Check if ratios are constant (tables) or if graph is a straight line through origin
  3. Find Constant: Calculate k = y/x
  4. Write Equation: Express relationship as y = kx
  5. Make Predictions: Use the equation to find unknown values
Tip 1: Always check if the graph passes through (0,0) for proportional relationships.
Tip 2: In tables, divide y by x for each pair - if all quotients are equal, the relationship is proportional.
Tip 3: The constant of proportionality has units (e.g., miles per hour, dollars per item).
Tip 4: If x doubles, y should also double in a proportional relationship.
Tip 5: The slope of the graph equals the constant of proportionality.
Characteristics: Linear graph through origin, constant ratio, direct variation.
Real Applications: Unit rates, scaling, conversion factors, speed calculations.
Common Errors: Forgetting the origin test, mixing up dependent/independent variables.
Formulas and Properties:

• Proportional Equation: y = kx where k ≠ 0

• Constant Calculation: k = y/x for any point (x,y) on the line

• Graph Property: Straight line passing through origin (0,0)

• Slope Property: Slope = constant of proportionality

• Ratio Property: y₁/x₁ = y₂/x₂ for any two points

Exercise with Visualization: Multiple Proportional Relationships
Exercise 6: Comparing Rates
Compare these proportional relationships:
f₁(x) = 2x (Car A speed)
f₂(x) = 3x (Car B speed)
f₃(x) = 1.5x (Car C speed)

Analysis: The chart shows different proportional relationships with varying rates of change.

  • f₁(x) = 2x: Slower rate (slope = 2)
  • f₂(x) = 3x: Faster rate (slope = 3)
  • f₃(x) = 1.5x: Slowest rate (slope = 1.5)

Questions & Answers

Question: How can I tell if a table shows a proportional relationship just by looking at it? What should I look for specifically?

Answer: Great question! Here are the specific things to look for in a table to identify a proportional relationship:

  1. Calculate Ratios: Divide each y-value by its corresponding x-value (y/x)
  2. Check Consistency: All ratios should be equal
  3. Look for Pattern: When x doubles, y should also double
  4. Origin Test: If x = 0, then y should equal 0

Example: Table with points (1,3), (2,6), (3,9), (4,12)

  • Ratios: 3/1 = 3, 6/2 = 3, 9/3 = 3, 12/4 = 3
  • All ratios equal 3, so it's proportional!

If any ratio is different, the relationship is not proportional.

Question: Why does the graph of a proportional relationship always pass through the origin (0,0)?

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.

This is the fundamental characteristic that distinguishes proportional relationships from other linear relationships (which might have y-intercepts other than 0).

If a line doesn't pass through (0,0), it's linear but not proportional!

Question: I'm confused about the difference between the constant of proportionality and the slope. Are they the same thing?

Answer: Excellent question! For proportional relationships, the constant of proportionality and the slope are actually the same number, but they represent different concepts:

  • Constant of Proportionality (k): The multiplier in the equation y = kx
  • Slope (m): The steepness of the line, calculated as rise/run or (y₂-y₁)/(x₂-x₁)

For proportional relationships: k = m because the equation y = kx is the same as y = mx when the y-intercept is 0.

Example: If y = 4x, then k = 4 (constant) and m = 4 (slope)

Both tell you the same thing: for every 1-unit increase in x, y increases by 4 units.

However, in non-proportional linear relationships (y = mx + b where b ≠ 0), the slope m and the y-intercept b are different values.