Proportional Graphs

Grade 7 · ratios · 87 practice problems · read aloud

🔊 Listen to this explanation

Proportional Graphs: Understanding Ratios Visually

📈 What is a Proportional Graph?

A proportional graph shows a relationship where two quantities change at the same constant rate. It's a straight line that always passes through the origin (0,0). This is super useful for seeing real-world relationships, like how much you earn for hours worked, or the cost for pounds of apples.

🧩 How to Solve Problems with Proportional Graphs

  1. Identify the Ratio: Find the constant rate (e.g., 3 cookies per $1).
  2. Plot the Origin: Start by plotting the point (0,0).
  3. Use the Ratio as a Slope: From (0,0), move up and right using your ratio to plot another point (e.g., for 3 cookies/$1, go to (1,3)).
  4. Draw the Line: Connect the points with a straight line through the origin.

🔢 Worked Examples

Example 1: Baking Cookies

A recipe uses 2 cups of flour for every 1 batch of cookies. Let's graph this.

  1. Ratio: 2 cups / 1 batch.
  2. Plot (0,0). Zero batches use zero cups.
  3. From (0,0), move right 1 (batch) and up 2 (cups). Plot (1,2).
  4. Draw a straight line through (0,0) and (1,2).

Example 2: Movie Tickets

Movie tickets cost $12 each.

  1. Ratio: $12 / 1 ticket.
  2. Plot (0,0). Zero tickets cost $0.
  3. From (0,0), move right 1 (ticket) and up 12 (dollars). Plot (1,12).
  4. Draw the line. For 3 tickets, you can see on the graph it's $36.

⚠️ Common Mistakes to Avoid

  • Forgetting the Origin: The line MUST go through (0,0). If it doesn't, it's not proportional.
  • Incorrect Ratio: Mixing up which number goes on the x-axis and which goes on the y-axis. The rate is y divided by x.
  • Drawing a Curved Line: Proportional graphs are always straight lines.

💡 Tips & Tricks

  • The "Origin" Rule: Remember "O" for "Only proportional graphs go through the Origin."
  • Constant Speed Test: If you can say "for every 1 x, there are __ y," it's likely proportional.
  • Check Your Points: Pick any point on your line. Does y/x equal your original ratio? It should!

🎯 How to Practice

  • Find real-life ratios (miles per hour, cost per item) and sketch their graphs.
  • Use online graphing tools to quickly plot points and see the line form.
  • Practice identifying proportional relationships from tables before moving to graphs.

Practice problems

6 of the 87, worked through step by step — try them before opening the answer.

1 y = 12x, x = 9, y = ?

Hint: In a proportional relationship like y = kx, the constant k tells you how many times x is multiplied to get y. Substitute the given x value into the equation.

Show the answer

Answer: 108

  1. The equation is y = 12x.
  2. Substitute x = 9 into the equation: y = 12 × 9.
  3. Multiply: 12 × 9 = 108.
  4. Therefore, y = 108.

The answer is 108.

2 y = 9x, x = 13, y = ?

Hint: Substitute the given value of x into the equation and multiply.

Show the answer

Answer: 117

  1. Start with the equation y = 9x.
  2. Substitute x = 13 into the equation: y = 9 * 13.
  3. Multiply: 9 * 13 = 117.
  4. The final answer is 117.

3 y = 8x, x = 13, y = ?

Hint: Substitute the given x-value into the equation and multiply.

Show the answer

Answer: 104

  1. Start with the equation y = 8x.
  2. Substitute x = 13 into the equation: y = 8 * 13.
  3. Multiply: 8 * 13 = 104.
  4. Therefore, y = 104.

The answer is 104.

4 y = 7x, x = 12, y = ?

Hint: Think about what it means for a relationship to be proportional: the y-value is always the constant of proportionality times the x-value. Substitute the given x into the equation.

Show the answer

Answer: 84

  1. The equation is y = 7x.
  2. Substitute x = 12 into the equation: y = 7 × 12.
  3. Multiply: 7 × 12 = 84.
  4. The final answer is 84.

5 y = 15x, x = 25, y = ?

Hint: Think about how the constant of proportionality relates the two variables in a direct proportion.

Show the answer

Answer: 375

  1. The equation is y = 15x, which is a proportional relationship with constant k = 15.
  2. Substitute x = 25 into the equation: y = 15 × 25.
  3. Multiply: 15 × 25 = 375.
  4. Therefore, y = 375.

The answer is 375.

6 y = 12x, x = 15, y = ?

Hint: Think about how the constant of proportionality relates the two variables. When you multiply the x-value by the constant, you get the y-value.

Show the answer

Answer: 180

  1. The equation is y = 12x, which is a proportional relationship.
  2. Substitute x = 15 into the equation: y = 12 × 15.
  3. Multiply: 12 × 15 = 180.
  4. Therefore, y = 180.

The answer is 180.

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