Proportion | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Direct Proportion

What is direct proportion?

• Proportionality describes the relationship between two variables, indicating how they change concerning each other. 
• Direct proportion implies that as one variable increases, the other increases by the same factor. 
  • This ensures that the ratio between the two variables remains constant. 
• If x and y are directly proportional, their values maintain the same ratio, denoted by x : y. 
  • This relationship can be represented by an equation y = kx, where k is a constant value. 
  • Graphically, their relationship forms a linear graph, with a gradient of k.

• In some cases, a variable may be directly proportional to a function of another variable.
• For instance, 
  • y being directly proportional to the square of x is represented as y = kx². 
  • Similarly, y being directly proportional to the square root of x is denoted as y = k√x. 
  • Additionally, y being directly proportional to the cube of x is shown as y = kx³. 
  • Each of these scenarios results in a distinct type of graph, depending on the specific function involved.

How do we deal with direct proportion questions?

Direct proportion questions can be dealt with in the same way:

• STEP 1: Identify the two variables and establish the formula in terms of these variables. For instance, if A is directly proportional to B, the variables are A and B. If A is directly proportional to B, the formula is A = kB. If A is directly proportional to the square of B, then A = kB².
• STEP 2: Determine the constant k by substituting the provided values into the formula and solving the equation.
• STEP 3: Express the formula for A in terms of B by substituting the calculated value of k.
• STEP 4: Utilize the formula to calculate the desired quantity.

Inverse Proportion

What is inverse proportion?

• Inverse proportion means as one variable goes up the other goes down by the same factor
  • If two quantities are inversely proportional, then we can say that one is directly proportional to the reciprocal of the other 
• If x is inversely proportional to y, then
  • will always be the same
  • there will be some value of k such that x = k/y
  • the graph will be related to that graph of y = k/x

How do we deal with inverse proportion questions?

Inverse proportion questions can be approached similarly to direct proportion questions.

• Step 1: Identify the two variables and establish a formula based on those variables.
  • Example: If A is inversely proportional to B, then A and B are the two variables.
  • If A is inversely proportional to B, use the appropriate formula. If A is directly proportional to the square of B, use a different formula.
• Step 2: Determine the constant of proportionality, k, by substituting the given values into the formula and solving.
• Step 3: Express the formula for A in terms of B by substituting the calculated value of k.
• Step 4: Utilize the formula to calculate the desired quantity.