# Which formula?

If no one answers ‘1’, if everyone answers ‘9’, then I shall argue my case for ‘1’ after that, not before.

[H/T Brightside]

## 20 comments for “Which formula?”

1. merv
February 18, 2019 at 18:31

BODMAS says 1

2. February 18, 2019 at 18:57

BODMAS would go this way:

Do the things in brackets first, so we’re left with:

6 divided by 2 times 3.

There is no ‘of’, so next is divide, leaving:

3 times 3, which comes to 9.

BOMDAS goes this way:

Do the things in brackets first, so we’re left with:

6 divided by 2 times 3.

There is no ‘of’, so next is multiply, leaving:

6 divided by 6, which comes to 1.

BODMAS is an error and the reason is this:

‘Of’ means times, i.e. the operation is a multiplication. Why would you go multiply, then divide, then multiply again?

You would do the two multiplications after brackets: ‘of’ and ‘x’, that is the two identical operations, then the divide, then plus, then minus.

The correct method is BOMDAS.

3. Chuckles
February 18, 2019 at 19:49

Close enough for government work.

The question arises because om ambiguities in the way the equation is, or could be, written
The brackets have an implied multiply lurking in there, and similarly we have the subtle difference between a division operation and a fraction.

If you had written it as 6 / 2(1+2) we would have a fraction, and the answer would be 1?
However, as written, I suspect it is properly evaluated as 9?

Similarly, if the latter part had been written 2*(1+2) the brackets would be evaluated first then the ‘equal’ status division and multiplication would be evaluated left to right?

4. woodsy42
February 18, 2019 at 20:22

Brackets first 2+1 =3
multiplication 2*3 = 6
division next, 6 divided by 6
I shall need serious convincing anything different is correct.

• February 18, 2019 at 20:55

Agreed.

• Chuckles
February 18, 2019 at 21:10

“multiplication 2*3 = 6
division next, 6 divided by 6”

No.
Multiplication and division are equal precedence, so they are evaluated left to right.
6÷2×3 = 3 x 3 = 9

• February 18, 2019 at 21:42

No, not right. BOMDAS means just that, as shown, strict order of precedence, times and divide not equal, hence different results.

• woodsy42
February 18, 2019 at 23:01

No Chuckles I don’t agree, it doesn’t matter about the precedence of times and divide or left to right order. The 2 is associated with the bracketed 1+2 – the lack of any sign between them makes them one single expression – so you have to evaluate 2 times 3 before dividing.

• February 18, 2019 at 23:36

That affirms precedence because you opt for that multiply first, as BOMDAS commands. The moment you resolve 2 times 3, as you should, before going further, that is supporting BOMDAS over BODMAS.

Your next action is to look for any ‘of’s or if not, then any times, or if not => proceed to divide and so on.

5. woodsy42
February 19, 2019 at 01:00

This shows that you don’t even need to do the bracket first
https://www.quora.com/Must-we-use-the-BODMAS-rule-all-the-time-when-we-calculate-math. Can I pack in thinking maths and go to bed now?

6. February 19, 2019 at 04:49

I hasten to say, first up, that Woodsy42 was on the right track, this below just explains why.

As demonstrated, BODMAS is wrong, it should never have been taught.

It’s not just unnecessary, it is WRONG and can often lead to the wrong answer.

Let’s go to the test case yet again:

Both methods require two things to be dealt with first:

B and O.

Two things need to be done with the Brackets:

1. All operations within the Brackets need to be resolved;
2. Any multiplier outside the Brackets to the left also needs to be resolved.

Under the egregious BODMAS, here is how it would be resolved:

1. The 1 + 2 inside is resolved to 3;
2. Thus we have left 6 divided by 2(3);
3. As the third operation under BODMAS must be D, the 2(3) cannot yet be resolved, the imperative is to resolve the Division first, thus 6 divided by 2, leaving us with 3(3).
4. Under BODMAS, the next to resolve is the multiplication, giving 9, a wrong answer.

Now, the essential error was that Brackets part two failed to be resolved, the BODMASian left off the resolution of Brackets part two and went over to the Division [which, incidentally, meant going right to left as well].

Under BOMDAS though, as it always was before some clown changed it, this conflict does not arise. Not only is everything within the brackets resolved, but the multiplier 2 outside to the left is also resolved, i.e. 2(3) => 2 x 3 => 6.

This leaves a simple 6 to be divided by 6 = 1. The very fact that the Brackets were even written 2(1 + 2) in the first place indicates that the next step after resolving inside the Brackets MUST be Multiplication to rid the line of its Brackets altogether.

Or to put it another way, the next step AFTER the resolution inside the brackets CANNOT be skipping over to the Division but instead staying to complete the resolution of the Brackets, i.e. a Multiplication operation.

Therefore, Multiplication always trumps Division or put another way, alway precedes it in order of operations.

Therefore, BOMDAS is an incorrect method and should NEVER have been taught. I’d like to know which clown thought it up.

Lastly, the tricky ‘O’ in the method. This is ‘Of’ and ‘Of’ is a Multiplier, written differently. It’s for use more with fractions where the Numerator is multiplied, again BEFORE the Division of that Product by the Denominator.

Thus Multiplication trumps Division again, or precedes it.

An example of this might be 3/4 of 12. In this, we were taught to put that as 3/4 x 12/1.

Resolve the Multiplication of the Numerators first => 36, then Divide by 4 x 1 or 4 => 9.

Now I agree that that could have been done, mentally, as a quarter of 12 is 3, times 3 is 9 … and I use that method myself on the boat … and that seems to support Chuckles’s interchangeability of the first two operations, which it does as regards fractions.

But only regarding fractions. For a schoolchild, it’s just far easier to teach the order of precedence M => D => A => S. It always works with such equations.

One last thing – detractors would say that applying that order of precedence renders the top line unwieldy in operations involving fractions – it certainly can do, thus the schoolchild is taught ‘cancelling’ or dividing top and bottom.

Thus 12/16 is far easier to deal with as 3/4 but the opposite is also true: 12/16 + 7/8 is far easier in the head as 12/16 + 14/16. I’m often carrying things like 26/16 in the head to apply to the boat.

Which again supports Chuckles’s interchangeability.

Yes, true, but not in equations requiring BOMDAS.

Incidentally, the boat is 500 inches long, or 500/12 feet.

If someone asks me how long the boat is, I say 500 inches, at which point the other is looking for any nearby brick to apply to my face, thereby resolving all problems.

Did I write ‘One last thing’? One last last thing – I wrote the above between 4.23 a.m. and 5.29 a.m., just thought you’d like to know that.

7. February 19, 2019 at 10:30

Well, I have replaced (as necessary) the implicit product with a “*” and get this.

Python 3.4.2 (default, Sep 26 2018, 01:01:01)
[GCC 4.9.2] on linux
>>> 6/2*(1+2)
9.0
>>>

I’m not totally sure but think that this Pythonic decision is made by treating * and / as of equal precedence and evaluating left to right when there is a choice (ie ambiguity).

>>> 3*6//7
2
>>> 6//7*3
0
>>>

The above, using // (for integer division returning the ‘floor’) seems to support the left to rightness of the Pythonic evaluation with * and // having equal priority.

So, do we prefer ‘artificial intelligence’ when our own is confused?

Best regards

8. opsimath
February 19, 2019 at 11:39

PEMDAS – or PEDMAS (there is no difference) gives the answer as 1.

• February 19, 2019 at 12:34

In this example – yes. PEMDAS by the way is the US version, good to see M precedes D.

9. February 19, 2019 at 11:57

Apologies to Cuffyburgers (and everyone) for missing his second comment (at February 18, 2019 at 21:10) before posting mine.

Certainly I think that equal precedence for divide and multiply and from-left evaluation are the best or a very good way forward. However, I don’t agree that this is what BODMAS necessarily means – it has more than one interpretation – and there-in lies a major problem.

There is much more discussion here on Wikipedia, including the order-of-evaluation issues with several computer programming languages. Also mentioned there are: PEMDAS, BEMDAS, BODMAS and BIDMAS.

My earliest recollections (from FORTRAN IV in late 1971) on computer programming are that one should never trust the compiler’s choice of implicit brackets, but always use one’s own explicitly. Even if the compiler was consistent (under code optimisation etc), it was never clear enough for reliable program maintenance.

Best regards

10. February 19, 2019 at 12:34

None of which, I humbly and respectfully submit to Chuckles, Nigel and Opsimath, with sincere thanks for their angles, alters my basic premise about BODMAS, for which this example sufficed.

Now it’s interesting to note that those coming down on the side of interchangeability of Multiplication and Division do so with a techie background – theirs tend to be techie solutions, whereas we other two come from an old-fashioned maths background where we know they’re not interchangeable.

For example, in the fraction 3/4 divided by 3/8, we were taught to invert the 3/8 and then cancel. That act of inverting in itself is a demonstration of the non-interchangeability.

Kind regards.

11. February 19, 2019 at 13:28

From the Wikipedia article, try evaluating 10 – 3 + 2 according to BODMAS strict application of Add before Subtract (giving 5) or the more ordinary interpretation [which happens to be left to right] (giving 9).

Best regards

• February 19, 2019 at 13:29

I was looking at M and D, not A and S but I’m sure it is fine for those last two operations.

12. Toodles
February 20, 2019 at 09:15

Can’t we just all get along? What is in a number? Would not any number do as well if called by any other number, numeral, whatever? If you have seen one formula you have seen them all, right?All paths, uh formulas lead to the same answer, do they not?

Just trying to bring you all together in a progressive way.

• February 21, 2019 at 05:13

You can be chief mathematician for our land.

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