# Order of Operations

What if you were asked to calculate the following:

${4}\hspace{0.33em}{+}\hspace{0.33em}{2}^{2}\times\hspace{0.33em}{7}\hspace{0.33em}{-}\hspace{0.33em}\left({{1}\hspace{0.33em}{+}\hspace{0.33em}{4}^{2}}\right)\hspace{0.33em}\div\hspace{0.33em}{2}$

Well person A could go left to right and do the 4 plus 2 squared to get 8 then multiply by 7 to get 56. then you might think that you should do the operations in the brackets first to get 17 and subtract that from 56 to get 39 then finally divide by 2 to get 19.5.

Person B may first do the operations in the brackets to get 17, then square the 2 to get 4 so that the problem now looked like:

${4}\hspace{0.33em}{+}\hspace{0.33em}{4}\times\hspace{0.33em}{7}\hspace{0.33em}{-}\hspace{0.33em}{17}\hspace{0.33em}\div\hspace{0.33em}{2}$

Then multiply the 4 and 7 to get 28 and divide the 17 by 2 to get 8.5. The problem would now look like:

${4}\hspace{0.33em}{+}\hspace{0.33em}{28}\hspace{0.33em}{-}\hspace{0.33em}{8}{.}{5}$

Then go left to right: add the 4 and 28 to get 32 then subtract the 8.5 to get 23.5. Who is right? Well mathematicians saw that different people would get different answers if there were no rules to dictate how this should be done. So they came up with a rule called order of operations.

This rule is easy to remember with an acronym. Depending on which dialect of maths you use (Australian or American), this acronym is either BODMAS (Australian) or PEDMAS (American). These stand for:

B: Brackets            P: Parentheses
O: Order (Powers)      E: Exponents (Powers)
D: Division            D: Division
M: Multiplication      M: Multiplication
S: Subtraction         S: Subtraction

The main difference being that Americans call brackets, parentheses and Australians call exponents, order. The order of a number is just the exponent or the power of a number.

In actual fact, multiplication and division are equal in order and addition and subtraction are equal. So the acronym could be BOMDAS or BOMDSA but BODMAS is the accepted one. For equal operations, you would proceed left to right. Also, sometimes you get brackets within brackets. You would proceed with the inner most bracket first, and work outwards. By the way, for nested brackets, the convention used so that it is easier to distinguish which end bracket applies to which beginning bracket is {[(     )]}. For my Australian friends these are called brackets “()”, square brackets “[]” and curly brackets “{}”. For my American friends, these are called parentheses, brackets, and braces.

$\left\{{\left[{\left({{4}\hspace{0.33em}{+}\hspace{0.33em}{28}\hspace{0.33em}{-}{9}}\right)\hspace{0.33em}{+}\hspace{0.33em}{7}}\right]\hspace{0.33em}{-}\hspace{0.33em}{8}}\right\}\hspace{0.33em}{=}\hspace{0.33em}\left\{{\left[{{23}\hspace{0.33em}{+}\hspace{0.33em}{7}}\right]\hspace{0.33em}{-}\hspace{0.33em}{8}}\right\}\hspace{0.33em}{=}\hspace{0.33em}\left\{{{30}\hspace{0.33em}{-}\hspace{0.33em}{8}}\right\}\hspace{0.33em}{=}\hspace{0.33em}{22}$

So it turns out, person B did the calculation correctly.

Please try this one and see if you get the answer 28:

$\left[{{\left({{3}\hspace{0.33em}{+}\hspace{0.33em}{4}}\right)}^{2}\hspace{0.33em}{-}\hspace{0.33em}{7}\hspace{0.33em}\times\hspace{0.33em}{3}}\right]$

Posted on Categories Algebra, Pre-VCE