Lesson BASICS of Operational Order

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BODMAS

This little made up word helps you remember the order that calculations should be done in. Take the following as an example:

7-12/4

How do we do this? There are actually 2 possible approaches, both of which give different answers, but only 1 of them is correct for mathematicians.

The 2 possible solutions are:

1. 7-12 is -5. then -5/4 is -1.25

2. 12/4 is 3. then 7-3 is 4

From a common sense approach, assuming that maths was like English, in that it was done from left to right (like reading), then -1.25 would be the answer. However, for mathematics, the second is the correct answer. This is because of BODMAS, which tells us the order that we must do our operations.

BODMAS stands for:

Brackets
Of
Division
Multiplication
Addition
Subtraction

So, any brackets, then we have to do whatever we can to those terms inside the bracket first, before doing anything else.

"Of" always makes me shrug my shoulders. I have never knowingly done "of", so just assume it is there to make up the word BODMAS rather than having BDMAS.

Division and Multiplication. The order of these 2 does not matter, as these are two opposites. By this i mean, if you multiply 2 numbers, then division is the reverse operation. So, if you wanted, you could do multiplication then division.

Similarly, addition and subtraction are oppositets of each other, so their order does not matter either. Ultimately, we have 3 blocks of work:

1. BRACKETS
2. DIVISION and MULTIPLICATION
3. SUBTRACTION and ADDITION

Now, lets go back to our first example, 7-12/4. It has a subtraction and a division. BODMAS tells us to do the division first (even if it gives a horrible answer - tough). Once we have the answer to that, then we are left with the subtraction.

What if we wanted to get the answer -1.25 though? How could we tell people to do the subtraction before the division? Easy - just write it in brackets, as anything in brackets has to be done first. So, in this case we would have a slightly different looking question: (7-12)/4

So, looking at it, we see the brackets and therefore do the 7-12 first. Then we divide that answer by 4.

One of the best things to remember, is to use brackets more - it makes it less likely to have misunderstandings.

EXAMPLES

1. Calculate -6(6-1)/(3). Well we have a bracket again, so we get (-6x5)/3. Notice I put my own brackets around the -6 and 5, just to make sure that people understand that all this is divisible by 3. It is not needed here at all, since we only have a multiplication and a division left, so it does not matter at all which order we do these in. There are 3 possible ways of calculating this last bit and they should all give the same answer:

a. -6x5 is -30. -30/3 is -10
b. -6/3 is -2. -2x5 is -10
c. 5/3 is 1.66 recurring. 1.66 recurring x -6 is -10.

As you can see, c is the most messy version, so start looking at the questions you have and see if you can do things differently, to make your work easier.

Hopefully, you can see that BODMAS helps us a great deal when we have a term to simplify. Note - it is used in a complicated term to help us simplify it. BODMAS is not used in day-to-day simplifying of equations, such as:

2. Solve {{{3x/2 - 4 = 5}}}

For equations, the way I visualise them is a set of balances: the left hand side EQUALS the right hand side. If we are to do something to one side then we would have to do the same thing to the other side, to keep the balance. So, in this case here, i ultimately want x=the answer, so i have to strip away the 3, the 2 and the 4 from the left hand side.

Looking at it, I have a 3x/2 term and a -4 term. Always start by getting rid of the other terms ie the -4 here. How do we get rid of -4? By adding 4 --> to BOTH sides, to keep the balance in place.

So, we end up with {{{3x/2 - 4 + 4 = 5 + 4}}} which becomes {{{3x/2 = 9}}}. Now we have to get rid of the 3 or the 2. Which one first? It doesn't matter. I would always recommend trying to get rid of the fraction though first, but that is just my preference, so we have "divide by 2". How to get rid of that? By "multiplying by 2".

{{{2*3x/2 = 2*9}}} --> the two 2's on the left cancel each other, to leave 3x = 18.

And now we need to convert the 3x to 1x (or x, as it is lazily written). We have 3*x, so we divide by 3...to both sides: 3x/3 = 18/3. This gives the final answer as x=6.

The full working out for this example is not required for your teachers. All you need to write is:

3x/2 - 4 = 5

3x/2 = 9

3x = 18

x = 6

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EXAMPLES FROM THE WEBSITE

*[problems 26724/14527]