When BIDMAS goes bad

A reader asks:

"When you're working out an expression, why do you sometimes divide after you multiply, when the BIDMAS rules say D comes before M?"

This is exactly the reason I don't like BIDMAS - because it suggests something that simply isn't true (that division is before multiplication and addition before subtraction).

What is BIDMAS?

BIDMAS is a well-intentioned memory-aid to help work out the order you work things out in: it stands for Brackets, Indices, Divide, Multiply, Add, Subtract, and it's very nearly very useful.

How it works: you compute anything in a bracket before you do anything else. You then work out any indices (things like $3^2$) you can. After that, it gets messy (and this is the problem): divide and multiply are really the same thing in reverse, so they have the same precedence - you do them from left to right, after you've done the indices, but before you do the add/subtracts. Although BIDMAS looks like D comes before M, they could just as easily be the other way around.

Similarly, add and subtract happen from left to right as well.

If you're going to use BIDMAS, I recommend writing it like this:
B
I
(DM)
(AS)

Some examples

Let's start with an easy(ish) one:

$3 - 4 + 7$

If you do that the way BIDMAS seems to suggest, you get the wrong answer: you think "A comes before S, so I work out $4+7 =11$ and then take it away from $3$ to get $-8$."

Our survey said nut-uh.

You start by doing $3-4$, because the minus is to the left of the plus. You get $-1 + 7 = 6$, the correct answer.

Now, I'm writing this first thing on a Monday morning, and my coffee hasn't kicked in properly yet, so I've not been able to find an example where dividing before multiplying gives the wrong answer. The order you do things in can make a difference, though. Consider:

$6 \div 3 \times 2$

The right way is to say "$6 \div 3 = 2$, then multiply that by $2$ to get $4$."

The wrong way - our poor American cousins who remember PEMDAS might fall into this trap - would be to say "Multiply first! $3 \times 2 = 6$ - then $6 \div 6 = 1$." Wrong answer.

But WHY?!

Add and subtract are opposites - but they're the same sort of thing: counting up and counting down, if you like. You can even see subtraction as 'adding a negative number' - which sounds more complicated, but is actually easier once you get your head around it.

Similarly, multiply and divide are opposites of the same sort: adding repeatedly and taking away repeatedly. Dividing by 3 (say) is just the same as multiplying by $\frac{1}{3}$ - in fact, it's very unusual for a mathematician to use the $\div$ symbol; we generally prefer fractions.

So, add and subtract are on the same BIDMAS level because they're really the same thing! Similarly, a divide is really a multiply in disguise.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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7 comments on “When BIDMAS goes bad

  • BParkEd

    I’ve had a heated debate with my (science teacher) aunt over this. Luckily her (GCSE student) son agreed with me. Phew.

    I’ve also had arguments with maths teacher colleagues on the same subject, which is worrying. One of them even pulled rank on the “I got a higher degree classification than you” front, and continues, as far as I’m aware, to get it wrong (and teach it wrong) to this day. Scary stuff.

  • Mitch

    While I agree with what you have written, the order in which you do addition and subtraction doesn’t matter if it is done correctly.

    In the example you gave the value of the 4 isn’t just 4, its negative 4…

    -4+7=+3
    then
    3+3=6

    • Colin

      I half-agree! I definitely teach my students to think of the operations as attached to the numbers – but I don’t agree that the “-4” in 3 – 4 + 7 directly represents the number -4 (unless there’s a + before it). It can be mangled into that, and that’s a nice way to look at it, but this statement explicitly uses – as a binary operator, meaning “subtract 4 from 3” – not as a unary operator (meaning negative 4), which wouldn’t make sense in the context.

      • M Melia-Cochran

        Please could you explain 10 divided by 2 multiplied by 3?

        • Colin

          If you see something like $10 \div 2 \times 3$, you should respond “write the bloody thing properly.” Hope that helps.

          A calculator will (probably) tell you it’s 15, but it’s reasonable to interpret it as $\frac{10}{2} \times 3$ or as $\frac{10}{2\times 3}$, so we don’t write it with a divide sign, we use fractions.

          Hope that helps :o)

  • Tony

    What if expressed as 10/2(2+1) =?
    Does it become 10/2(3) =
    Is now 10/2 x 3
    Or 10/6 does th brackets not mean they need to be first multiplied or not?

    • Colin

      This falls under “Write the bloody thing properly.” Like “I hit the man with the umbrella”, it’s ambiguous as written, and should be clarified.

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