Proportional Relationships (Edexcel IAL Maths)

Spec point — 2.1

Proportional relationships

What are proportional relationships?

• Proportional relationships describe a proportional connection between two variables

• This can happen in two ways

  • Direct proportion $y=kx$

    • one variable increases or decreases the other does the same

  • Inverse proportion $y=\frac{k}{x}$

    • one variable increases the other decreases and vice versa

• Proportional relationships use the symbol $\propto$ which means is proportional to

• Both direct and inverse proportion can be represented graphically

  • Direct proportion creates a linear graph where k is the gradient
  • Inverse proportion creates a reciprocal graph

What is direct proportion?

• $y\propto x$ means y is proportional to x

• y increases as x does, k determines the rate (gradient)

• by changing this to the equation $y=kx$ we can substitute in given values and solve to find k

  • Note that this means the ratio of x and y is constant k = y / x

What is inverse proportion?

• $y\propto\frac{1}{x}$ means y is proportional to $\frac{1}{x}$ or y is inversely proportional to x

• y decreases as x increases and vice versa, k determines the rate

• by changing this to the equation $y=\frac{k}{x}$ we can substitute in given values and solve to find k

  • Note that this means the product of x and y is constant k = xy

• Set up your proportional relationship using $\propto$ then change to = k

• Be clear about what y is proportional to …

  • “… the square of x” (x²)
  • “… x plus four” (x + 4)

• Calculate or deduce the value of k from the information given or a graph

• Once you've found k sub it back in to your original proportion equation

• You can now find any values using this proportional relationship

• y = mx + c rearranges to y – c = mx so (y - c) is directly proportional to x

• Proportional relationships are often used in modelling

Worked Example