Calculating Percentages
In this section we learn how to calculate percentages.
By the end of this section we'll know how to calculate things like \(30\%\) of \(70\), or \(8\%\) of \(42\).
To calculate percentages well it's helpful to know how to multiply with fractions as well as know how to simplify fractions, even though
we can get by with the formula taught here.
Calculating Percentages
When we say \(X\%\) this is shorthand writing for \(\frac{X}{100}\) or \(X\div 100\).
So when we write, for example, \(6\%\) this is just a faster way of writing \(\frac{6}{100}\) (which means \(6\div 100\)).
To calculate \(X\%\) of a quantity \(A\), we calculate:
\[\frac{X}{100}\times A = \frac{X\times A}{100}\]
We can also write this:
\[\begin{pmatrix}X\div 100 \end{pmatrix}\times 100\]
The following examples should make this a little clearer.
Example
Calculate \(5\%\) of \(30\).
Solution
Using the first em>formula, we saw above, we em>calculate this percentage/em> as follows: \[\begin{aligned}\frac{5}{100}\times 30 &= \frac{5 \times 30}{100} \\ & = \frac{150}{100} \quad \begin{pmatrix} \text{remember that \(\frac{150}{100}\) means \(150\div 100\)} \end{pmatrix} \\ \frac{5}{100}\times 30 &= 1.5 \end{aligned}\] So \(5\%\) of \(30\) equals \(1.5\).
Example
Calculate \(8\%\) of \(40\).
Solution
Using the second formula we calculate this percentage as follows: \[ \begin{pmatrix}8 \div 100 \end{pmatrix}\times 40 = 0.08 \times 40 = 3.2 \] So \(8\%\) of \(40\) equals \(3.2\).
Example
Calculate \(12\%\) of \(43\).
Solution
To em>calculate this percentage We use the first formula again: \[\begin{aligned} \frac{12}{100} \times 43 &= \frac{12\times 43}{100} \\ & = \frac{516}{100} \\ \frac{12}{100} \times 43 & = 5.16 \end{aligned}\] So \(12\%\) of \(43\) is \(5.16\).
