Addition and subtraction

Inverse operations to check calculations

Inverse operations to check calculations

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Tutor: Toby


Inverse operations to check calculations

​​In a nutshell

Addition and subtraction are inverse operations to each other. By re-expressing a calculation, you can check your answer with the inverse operation.

Inverse operation

To be an inverse operation means to undo what the operation did. If you add five, this can always be undone by subtracting five. Hence the inverse operation of addition is subtraction. Similarly, if you subtract two, you can always undo this by adding two. Therefore addition is the inverse operation of subtraction.

Example 1

Some number plus six equals fourteen. What is the number?

First express this as a mathematical equation. Since you don't yet know the number, call it xx. So:


To find xx, find the inverse of what has been applied to xx, and apply it to 1414. The inverse of adding 66 is subtracting 66. Hence the following is also true:


This gives 88, so


Checking answers

Beyond being used to solve problems like the example above, inverse operations are helpful when checking answers.

Example 2

Calculate 744974-49.

67144925\begin {array} { c c c}& ^6\cancel7 & ^14 \\- & 4 & 9 \\ \hline& 2& 5\\ \hline\end {array}​​

So 7449=2574-49=25. Now you can check this is correct using the inverse operation. Initially, you subtracted 4949 from 7474 to find 2525. The inverse is to add 4949 to 2525. You can now check if this does give 7474:

215+4974\begin {array} { c c c}& \overset{1}2 & 5 \\+ & 4 & 9 \\ \hline& 7 & 4 \\ \hline\end {array}​​

As expected, you get 7474, so you can be confident that 7449=2574-49=25.

Example 3

Calculate 119+15119+15.​​

Start with the five units:

+1+1+1+1+1118119120121122123124\begin{array}{cccccccccccccc}&&&+1&&+1&&+1&&+1&&+1&&& \\& &&\curvearrowright &&\curvearrowright &&\curvearrowright&&\curvearrowright &&\curvearrowright &&\\118 &&\underline{119}&&120&&121&&122&&123&&\underline{124}&& \\|&&|&&|&&|&&|&&|&&| \\ \hline\end{array} ​​

Now count up the one ten:

+1094104114124134144154\begin{array}{cccccccccccccc}&&&&&&&+10&&&&& \\& &&&& &&\curvearrowright&& &&&&\\94&&104&&114&&\underline{124}&&\underline{134}&&144&&154&& \\|&&|&&|&&|&&|&&|&&| \\ \hline\end{array}​​

To check this answer, find the inverse of adding 1515, and apply this inverse to 134134. The inverse is subtracting 1515. Start with the five units:

11111128129130131132133134\begin{array}{cccccccccccccc}&&&-1&&-1&&-1&&-1&&-1&&& \\& &&\curvearrowleft &&\curvearrowleft &&\curvearrowleft&&\curvearrowleft &&\curvearrowleft &&\\128 &&\underline{129}&&130&&131&&132&&133&&\underline{134}&& \\|&&|&&|&&|&&|&&|&&| \\ \hline\end{array} ​​

Then subtract the one ten:

108999109119129139149\begin{array}{cccccccccccccc}&&&&&&&-10&&&&& \\& &&&& &&\curvearrowleft&& &&&&\\89&&99&&109&&\underline{119}&&\underline{129}&&139&&149&& \\|&&|&&|&&|&&|&&|&&| \\ \hline\end{array}​​

See that by inverting the process, you have returned to 119119, hence you can be more confident that 119+15=134119+15=\underline{134}.

Note: You need not stick to just the column method, or just the number line method. Use whichever you prefer, unless it is stated to use one particular method.

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