Direct Proportion (Cambridge (CIE) IGCSE Maths) : Revision Note

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Direct Proportion

What is direct proportion?

  • Proportion is a way of talking about how two variables are related to each other

  • Direct proportion means that as one variable goes up the other goes up by the same factor

    • The ratio between the two amounts will always stay the same

  • The symbol proportional to means "proportional to"

    • E.g. y is directly proportional to x, yproportional tox

  • If x and y are directly proportional, then

    • will always be the same

    • there will be some value, k, such that y kx

    • the graph relating x and y is a linear graph, with gradient k

  • k is called the constant of proportionality

Graph showing a directly proportional relationship between x and y.

How do I use direct proportion with powers and roots?

  • Problems may involve a variable being directly proportional to a power or root of another variable

  • For example

    • y is directly proportional to the square of x

      • y proportional to x squared

      • means that y equals k x squared

    • y is directly proportional to the square root of x

      • y proportional to square root of x

      • means that y equals k square root of x

    • y is directly proportional to the cube of x

      • y proportional to x cubed

      • means that y equals k x cubed

    • y is directly proportional to the cube root of x

      • y proportional to cube root of x

      • means that y equals k cube root of x

    • Each of these would have a different type of graph, depending on the power or root

How do I find the equation between two directly proportional variables?

  • Direct proportion questions always have the same process:

    • STEP 1
      Identify the two variables and write down the formula in terms of k

      • E.g. y is directly proportional to x

      • write down the formula y equals k x

    • STEP 2
      Find k by substituting any given values from the question into your formula, then solving to get k

      • E.g. if you are told y = 6 when x = 2

      • then 6 equals k cross times 2 giving k equals 3

    • STEP 3
      Rewrite the formula with the known value of k from above (substitute it in)

      • y equals 3 x

      • This is the equation relating the two variables

    • STEP 4
      Use the equation to answer other parts of the question

      • E.g. find y when x = 10

      • y equals 3 x gives y equals 3 cross times 10 equals 30

Examiner Tips and Tricks

  • Some harder exam questions do not tell you to work out the equation

    • You are expected to do it on your own

Worked Example

It is known that y is directly proportional to the square of x.

When x equals 3, y equals 18.

Find the value of y when x equals 4.

Identify the two variables

y comma space x to the power of 2 space end exponent

We are told this is direct proportion
Write down the formula involving k

y equals k x squared

Find k by substituting in y = 18 when x = 3
Then solve the equation for k

table row 18 equals cell k open parentheses 3 close parentheses squared end cell row 18 equals cell 9 k end cell row cell 18 over 9 end cell equals k row 2 equals k end table

Substitute this value of k back into the formula to get the full equation

y equals 2 x squared

Use this formula to find y when x = 4

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 2 cross times 4 squared end cell row y equals cell 2 cross times 16 end cell row y equals 32 end table

bold italic y bold equals bold 32

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Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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