Inverse Proportion

Inverse Proportionvideo

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 

• The symbol, $\propto$, is used to show that one quantity is "directly proportional to the reciprocal of" (inversely proportional to) another quantity

  • "y is inversely proportional to x" is written $y\propto \frac{1}{x}$

• If x and y are inversely proportional then

  • $\frac{1}{x}:y$ will always be the same

  • there will be some value of k such that $y=\frac{k}{x}$

• The graph of $y = \frac{k}{x}$ is shown below

How do I use inverse proportion with powers and roots?

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

• For example

  • y is inversely proportional to the square of x

    • $y\propto \frac{1}{x^2}$

    • means that $y=\frac{k}{x^2}$

  • y is inversely proportional to the square root of x

    • $y\propto \frac{1}{\sqrt{x}}$

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

  • y is inversely proportional to the cube of x

    • $y\propto \frac{1}{x^3}$

    • means that equation$y=\frac{k}{x^3}$

  • y is inversely proportional to the cube root of x

    • $y\propto \frac{1}{\sqrt[3]{x}}$

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

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

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

• Inverse 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 inversely proportional to x

    • write down the formula $y=\frac{k}{x}$

  • 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 = 5 when x = 6

    • then $5=\frac{k}{6}$ giving $k=30$

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

    • $y=\frac{30}{x}$

    • 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 = 2

    • $y=\frac{30}{x}$ gives $y=\frac{30}{2}=15$