BODMAS Problems

 BODMAS problems are a recurring challenge throughout the life of a mathematician - especially at school or college.  

BODMAS stands for Brackets, Operators, Division, Multiplication, Addition and Subtraction, and sets out the order in which you must work through a maths problem to get to the correct answer.  Brackets (called Parentheses in the US) always come first, and you have to calculate what's inside them before you do anything else.  Operators are 'powers' or 'indices' and include squares, cubes, square roots and cube roots.  The rest are the normal arithmetic calculations.

For example (3*4)-2 is not the same as 3*(4-2).  And without the brackets, you'd be looking at just 3*4-2 which is ambiguous.  It's not clear which way round you're supposed to do the calculations.  What do you do first?

Mathematicians don't make these rules up to be awkward or difficult.  Mathematicians hate confusion, ambiguity and uncertainty, and therefore they use BODMAS and brackets and all these rules so that when they talk to other mathematicians anywhere in the world, they are entirely clear what they're talking about.  They are even more specific and precise than scientists (in my experience) and will take great care to make sure that they are crystal clear about their calculations.

In my experience, mathematicians who understand BODMAS problems don't get involved in the BODMAS questions that go around on social media, where there are no brackets and a whole flame war kicks off between people who defend their answer to 1+5*3.  These are badly-written problems, written that way on purpose.

Let's take a look at some simple examples of BODMAS problems, and identify some of the possible pitfalls along the way:

a) Let's start with (3*4)-2

The brackets here clearly indicate that we should calculate 3*4 first.  3*4 = 12.  12 -2 = 10.

In fact, the brackets aren't required here - Multiplication always comes before Subtraction.

Let's change the position of the brackets:

3* (4-2)

Without the brackets, we'd do 3*4 first, but with the brackets, we must calculate 4-2 first.

4-2 = 2

3* 2 = 6

So there's a clear distinction between 10 and 6.  We must be careful with our BODMAS problems.

Another one:

b)  4 * (6-2²)

First is Brackets.  We must calculate the contents of the brackets, which is 6-2².
Within these brackets, we must do the Operator first.  Operators are squares, square roots, and any other powers.  In this question, the operator is the 2 squared.

2² = 4
6-4 = 2

So the contents of the Brackets comes to 2.

4*2 = 8

Let's compare this with the same calculation without the brackets, and use BODMAS to find out the value of 4 * 6-2²

BODMAS says that we do the Operators first (there are no Brackets in this question), and we know that 2² = 4

This gives us:  4 * 6 - 4 

Next, we do the multiplication, so 4 *6 = 24
And finally, the subtraction, 24 -4 = 20

So now we have a difference between
 4 * (6-2²) = 8
4 * 6 - 4 = 20 

Clearly we're going to have to be careful!

c) This time, let's do a calculation with algebra:

x²  + (2x * 6x) + (x -z)

Remembering to do our brackets first, we get:

x²  + 12x² + x - z

Operators: we can't do anything further with these at this stage; all that remains now are Addition and Subtraction.  Addition comes first:

x²  + 12x² + x - z = 13x²  + x - z

And we can't simplify this any further.

Why does BODMAS cause problems?

BODMAS expressions, like 3* (3+5+6) can occur even in simple everyday maths situations.

For example, how many wheels are there in total with on four four-wheeled cars if each car is carrying a caravan which has two wheels?

A car has four wheels; a caravan has two wheels.  That's six wheels.  And there are four cars and caravans, so that's 4 * 6 = 24 wheels altogether.

However, if you misunderstood or miscalculated, you might work like this:

There are four cars, each with four wheels.  That's sixteen wheels.  And two wheels for the caravan makes 18 wheels.

4 * (4+2) is very different from (4*4) + 2.  And that's why we have BODMAS - to help us avoid problems and misunderstandings, in cases from cars and caravans to planets and stars!