3.2.2 Solving problems involving proportional relationships

• What happens to the second quantity when the first quantity in an inverse proportional relationship increases?

  It decreases
• What happens to the second quantity when the first quantity in a direct proportional relationship increases?

  It increases
• How many eggs are needed for 6 cakes if 2 cakes require 5 eggs?

  15 eggs
• Steps to solve proportions using cross-multiplication

  1️⃣ Write the proportion

  2️⃣ Cross-multiply

  3️⃣ Simplify

  4️⃣ Solve for the unknown
• What is the x-axis label in the graph?

  Hours
• What is the first step in solving a proportion using cross-multiplication?

  Write the proportion
• What is the equation for direct proportionality?

  $y =$$kx$
• Give an example of direct proportionality.

  More sweets cost more
• Why is it important to simplify and check solutions when solving proportions?

  To ensure accuracy
• Match the proportionality type with its key phrases:

  Direct ↔️ "increases together"

  Inverse ↔️ "one increases, other decreases"
• If 3 apples cost £1.50, how much do 5 apples cost?

  £2.50
• What is the y-axis label in the graph?

  Miles walked
• Match the type of proportionality with its description:

  Direct Proportionality ↔️ Increases together

  Inverse Proportionality ↔️ One increases, the other decreases
• What concept are we applying to real-life situations in this study material?

  Proportional relationship
• Steps to solve the example proportion

  1️⃣ Identify number of apples and cost

  2️⃣ Express known ratio as a fraction

  3️⃣ Set up the proportion

  4️⃣ Use cross-multiplication

  5️⃣ Solve for the unknown cost
• What should you do if substituting the solution into the original proportion results in unequal sides?

  Recheck the solution
• Steps to find the rate of change (speed) from a graph showing miles walked over time.

  1️⃣ Choose two points on the line

  2️⃣ Calculate the distance change (vertical)

  3️⃣ Divide the distance change by the time change (horizontal)
• What phrase indicates direct proportionality?

  "Is directly proportional to"
• Steps to set up proportions using ratio notation

  1️⃣ Identify proportional quantities

  2️⃣ Write down known ratios

  3️⃣ Set up the proportion

  4️⃣ Use cross-multiplication

  5️⃣ Solve for the unknown
• What can you infer about the person's walking pace from the graph?

  • The person walks at a constant pace of 1 mile per hour, as indicated by the linear relationship
  • The slope of the line is 1, meaning they walk 1 mile for every 1 hour
• If the person walked for 6 hours, how many miles would they have walked?

  6 miles
• What is the relationship between the number of hours and the miles walked shown in the graph?

  • The miles walked increases linearly with the number of hours
  • The graph shows a straight line relationship between hours and miles walked
• What should you consider when comparing your solution with expectations in a word problem?

  Real-world logic
• What is the second step in checking the solution of a proportion?

  Verify equality
• What is the final cost of 5 apples in the example?

  £2.50