Inverse Proportion (Cambridge International IGCSE Mathematics)

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

Example of a graph showing inverse proportion

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 $y=\frac{30}{2}=15$

Worked Example The time, $t$ hours, it takes to complete a project is inversely proportional to the cube root of the number, $n$, of people working on it.

If 27 people work on the project, it takes 50 hours to complete.

(a) Find an equation connecting $t$ and $n$.

Answer:

Identify the two variables

$t,n$

*We are told this is inverse proportion to the cube root of *$n$ *Write down the formula involving *k

$t = \frac{k}{ \sqrt[3]{n} }$

*Find k by substituting in *$n=27$ and $t=50$ (from the words in the question)

$\begin{gathered} 50 = \frac{k}{ \sqrt[3]{27} } \\ 50 = \frac{k}{3} \\ 50 \times 3 & = k \\ 150 = k \end{gathered}$

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

$t = \frac{150}{ \sqrt[3]{n} }$

(b) Given that the project needs to be completed within 60 hours, find the minimum number of people needed to work on it.

Answer:

Use the formula to find $n$ when $t=60$

$\begin{gathered} 60 = \frac{150}{ \sqrt[3]{n} } \\ \sqrt[3]{n} & = \frac{150}{60} = 2.5 \\ n = 2.5^{3} = 15.625 \end{gathered}$

A sensible answer here is a whole number (as $n$ is the number of people) 15 people would not complete it in time, but 16 would

Final answer: 16 people