Direct Proportion (Cambridge International IGCSE Mathematics)

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 $\propto$ means "proportional to"

  • E.g. y is directly proportional to x, y$\propto$x

• If x and y are directly proportional, then

  • x: y 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

Example of a graph showing direct proportion

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

    • $y\propto x^{2}$
    • means that $y=kx^{2}$

  • y is directly proportional to the square root of x

    • $y\propto\sqrt{x}$
    • means that $y=k\sqrt{x}$

  • y is directly proportional to the cube of x

    • $y\propto x^{3}$
    • means that equation$y=kx^{3}$

  • y is directly proportional to the cube root of x

    • $y \propto \sqrt[3]{x}$
    • means that equation$y = k \sqrt[3]{x}$

  • 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 $y=kx$

  • 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 $6=k\times2$ giving $k=3$

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

    • $y=3x$
    • 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
    • $y=3x$ gives $y=3\times10=30$

Exam Hint 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 $x=3$, $y=18$.

Find the value of $y$ when $x=4$.

Answer:

Identify the two variables

$y,x^{2}$

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

$y=kx^{2}$

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

$\begin{gathered} 18 = k ( 3 )^{2} \\ 18 = 9 k \\ \frac{18}{9} & = k \\ 2 = k \end{gathered}$

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

$y=2x^{2}$

Use this formula to find y when x = 4

$\begin{array}{rcl} y = 2 \times 4^{2} \\ y = 2 \times 16 \\ y = 32 \end{array}$

$y=32$