Direct & Inverse Proportions

• What is direct proportion?

  Direct proportion means that as one variable goes up, the other goes up by the same factor.
• What does it mean when two variables are directly proportional?

  The ratio between the two amounts will always stay the same.
• If x and y are directly proportional, what is the relationship expressed mathematically?

  There will be some value of k such that \( y = kx \).

  
• What type of graph represents a direct proportion relationship?

  The graph is a linear (straight line) graph, with gradient k.
• How can a variable be directly proportional to a function of another variable?

  For example, \( y \) is directly proportional to the square of \( x \) means that \( y = kx^2 \).
• What is the formula for a variable directly proportional to the square root of another variable?


  The formula is \( y = k\sqrt{x} \).
• What is the formula for a variable directly proportional to the cube of another variable?


  The formula is \( y = kx^3 \).
• What are the steps to solve direct proportion questions?

  1. Identify the two variables and set up the formula.
  2. Find k by substituting the values given in the question.
  3. Write the formula for A in terms of B by substituting in your value of k.
  4. Use the formula to find the required quantity.

  
• If A is directly proportional to B, what is the formula used?


  The formula is \( A = kB \).
• How do you find the value of k in a direct proportion problem?

  By substituting the values given in the question into your formula and solving.
• What should you do even if the question doesn’t ask for a formula in direct proportion problems?

  It is always worth working one out and using it in all but the simplest cases.
• In the worked example, what is the relationship between y and x when y is directly proportional to the square of x?

  When \( x = 3 \), \( y = 18 \).

  
• How do you find the value of y when x = 4 in the worked example?

  By using the formula derived from the relationship \( y = kx^2 \) and substituting the values.

  
• What is inverse proportion?

  Inverse proportion means as one variable goes up, the other goes down by the same factor.
• What does it mean when two quantities are inversely proportional?

  One is directly proportional to the reciprocal of the other.
• If x is inversely proportional to y, what is the relationship expressed mathematically?

  There will be some value of k such that \( x = \frac{k}{y} \).

  
• What type of graph represents an inverse proportion relationship?

  The graph will be related to the graph of \( y = \frac{k}{x} \).
• What are the steps to solve inverse proportion questions?

  1. Identify the two variables and set up a formula.
  2. Find k by substituting the values given in the question.
  3. Write the formula for A in terms of B by substituting in your value of k.
  4. Use the formula to find the required quantity.

  
• If A is inversely proportional to B, what is the formula used?


  The formula is \( A = \frac{k}{B} \).
• How do you find the value of k in an inverse proportion problem?

  By substituting the values given in the question into your formula and solving.
• In the worked example, what is the relationship between time and the number of people working on a project?

  The time, \( t \), varies inversely proportional to the number of people, \( n \).
• If 4 people work on a project and it takes 70 hours to complete, how do you write the equation connecting t and n?


  The equation is \( t = \frac{k}{n} \).
• How do you find the minimum number of people needed to complete a project in 18 hours?

  By using the formula and substituting the required time into the equation.
• What is the minimum number of workers required to finish the project in 18 hours?

  16 people is the minimum number of workers required.