Inverse Proportion (Cambridge (CIE) IGCSE International Maths)

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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

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 $y = \frac{k}{x^{3}}$
- y is inversely proportional to the cube root of x
  - $y \propto \frac{1}{\sqrt[3]{x}}$
  - means that $y = \frac{k}{\sqrt[3]{x}}$
- Each of these would have a different type of graph, depending on the power or root

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 $n$ .

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)

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

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.

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

$\begin{array}{rcl} 60 & = & \frac{150}{\sqrt[3]{n}} \\ \sqrt[3]{n} & = & \frac{150}{60} = 2.5 \\ n & = & 2 . 5^{3} = 15.625 \end{array}$

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

16 people

the (exam) results speak for themselves:

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