Guided Practice
Learn
In this lesson, you will work through proportional relationship problems with step-by-step guidance. This guided practice reinforces the concepts of constant of proportionality and graphing that you learned in previous lessons.
Key Strategies for Solving Proportional Relationship Problems
- Identify the relationship: Determine if the quantities are directly proportional (y = kx where k is constant).
- Find the constant of proportionality (k): Divide any y-value by its corresponding x-value.
- Write the equation: Express the relationship as y = kx.
- Solve for unknowns: Use the equation to find missing values.
- Verify your answer: Check that your solution maintains the proportional relationship.
Recognizing Proportional Relationships
A relationship is proportional if:
- The ratio y/x is the same for all pairs of values
- The graph passes through the origin (0, 0)
- The graph is a straight line
- When x doubles, y doubles; when x triples, y triples
Examples
Example 1: Finding the Constant of Proportionality from a Table
Problem: The table shows the relationship between hours worked and money earned. Find the constant of proportionality and write an equation.
| Hours (x) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Earnings ($y) | 25 | 50 | 75 | 100 |
Solution:
- Calculate k = y/x for any pair: k = 25/2 = 12.5
- Verify with another pair: k = 50/4 = 12.5 (confirmed)
- Write the equation: y = 12.5x
- The constant of proportionality is $12.50 per hour
Example 2: Using Proportions to Find Missing Values
Problem: If 3 notebooks cost $7.50, how much do 7 notebooks cost?
Solution:
- Find k: k = 7.50/3 = 2.50 (cost per notebook)
- Use the equation: y = 2.50x
- Substitute x = 7: y = 2.50(7) = $17.50
Example 3: Interpreting a Proportional Graph
Problem: A graph shows a straight line through the origin passing through the point (4, 10). Find the constant of proportionality and predict the y-value when x = 6.
Solution:
- Find k using the point (4, 10): k = 10/4 = 2.5
- Write the equation: y = 2.5x
- When x = 6: y = 2.5(6) = 15
Practice
Work through these problems step by step. Show your work and verify your answers.
Problem 1
A recipe calls for 2 cups of flour for every 3 cups of sugar. How much flour is needed for 12 cups of sugar?
Show Solution
k = 2/3 (cups flour per cup sugar)
Flour = (2/3) x 12 = 8 cups of flour
Problem 2
The table shows a proportional relationship. Find the missing value.
| x | 3 | 5 | 8 |
|---|---|---|---|
| y | 18 | 30 | ? |
Show Solution
k = 18/3 = 6
y = 6 x 8 = 48
Problem 3
A car travels at a constant speed. It covers 156 miles in 3 hours. How far will it travel in 5 hours?
Show Solution
k = 156/3 = 52 mph
Distance = 52 x 5 = 260 miles
Problem 4
Determine if this relationship is proportional: When x = 2, y = 6; when x = 4, y = 12; when x = 5, y = 14.
Show Solution
Check ratios: 6/2 = 3, 12/4 = 3, 14/5 = 2.8
Not proportional - the ratios are not all equal.
Problem 5
A map has a scale where 1 inch represents 25 miles. If two cities are 3.5 inches apart on the map, what is the actual distance?
Show Solution
k = 25 miles per inch
Actual distance = 25 x 3.5 = 87.5 miles
Problem 6
If 5 oranges cost $4.00, write an equation relating cost (y) to number of oranges (x). Then find the cost of 12 oranges.
Show Solution
k = 4.00/5 = 0.80
Equation: y = 0.80x
Cost of 12: y = 0.80(12) = $9.60
Problem 7
A line passes through the origin and the point (6, 15). What is the y-coordinate when x = 10?
Show Solution
k = 15/6 = 2.5
y = 2.5 x 10 = 25
Problem 8
A printer prints 24 pages in 4 minutes. How many pages can it print in 15 minutes?
Show Solution
k = 24/4 = 6 pages per minute
Pages = 6 x 15 = 90 pages
Problem 9
The equation y = 4.5x represents a proportional relationship. Complete the table:
| x | 2 | ? | 8 |
|---|---|---|---|
| y | ? | 27 | ? |
Show Solution
When x = 2: y = 4.5(2) = 9
When y = 27: x = 27/4.5 = 6
When x = 8: y = 4.5(8) = 36
Problem 10
Maria earns $84 for 7 hours of work. At this rate, how many hours must she work to earn $180?
Show Solution
k = 84/7 = $12 per hour
Hours = 180/12 = 15 hours
Check Your Understanding
Answer these questions without looking at your notes.
- What must be true about the ratio y/x for a relationship to be proportional?
- If y = kx and k = 7, what is y when x = 9?
- A proportional relationship passes through (0, 0) and (5, 35). What is the equation?
- How can you tell from a graph whether a relationship is proportional?
Show Answers
- The ratio y/x must be constant (the same for all pairs of values).
- y = 7(9) = 63
- k = 35/5 = 7, so y = 7x
- The graph must be a straight line that passes through the origin (0, 0).
Next Steps
- Review any problems where you made errors
- Practice identifying the constant of proportionality quickly
- Move on to Word Problems to apply these skills in context
- Return to this guided practice when preparing for tests