Rearranging Equations

In accounting, you will also need to know how to rearrange simple equations to complete balance sheets and find missing values.

We can rearrange equations by using some basic rules of arithmetic. For example, we can rearrange the accounting equation to find a missing value;

A = L + E
can be rearranged to
A – L = E

The key is to remember that whatever you do to one side of the equation, you must do the same to the other. So in the example above, we subtracted L from the right hand side and then also from the left.

Let's say we are trying to work out the contribution margin per unit. This means we are making the contribution margin the 'subject' of the equation, with it on the left side of the equals sign.

Step 1 – Use the equation

$$ \newcommand{\ctext}[1]{\style{font-family:Arial}{\text{#1}}} \ctext{No. of units} = \frac{\left(\ctext{Fixed costs} + \ctext{Target profit}\right)}{\ctext{Contribution margin per unit}} $$

Step 2 – Multiply both sides of the equation by Contribution margin per unit 

$$  \begin{align*} \ctext{No. of units} \times \ctext{Cont. margin per unit} &= \frac{\left( \ctext{Fixed costs} + \ctext{Target profit}\right) \times \ctext{Cont. margin per unit}}{\ctext {Cont. margin per unit}} \cr \ctext{No. of units} \times \ctext{Cont. margin per unit} &=\ctext{Fixed costs}+\ctext{Target profit} \end{align*} $$

Step 3 – Divide each side by Number of units

$$ \begin{align*} \frac{\ctext{No. of units}\times\ctext{Cont. margin per unit}}{\ctext{No. of units}} &=\frac{\left(\ctext{Fixed costs} + \ctext{Target profit}\right)}{\ctext{No. of units}} \cr \ctext{Cont. margin per unit} &= \frac{\left(\ctext{Fixed costs}+\ctext{Target profit}\right)}{\ctext{No. of units}} \end{align*} $$

The same principles used in the above example can be applied to different equations you come across in accounting.

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