How To Solve Facebook Math: 6 ÷ 2 (1 + 2)

This is an older video I put out back when I was still doing Tech Chop. Lately, the Facebook math problems have been making their rounds again, so I thought I'd post it here. Check it out:



In the comment section on Youtube, there are a bunch of people still arguing with me over this, and the way I implemented the order of operations. In the video, I reference an article from PurpleMath that says the following:

When you have a bunch of operations of the same rank, you just operate from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ (3 × 4) = 15 ÷ 12, but is rather (15 ÷ 3) × 4 = 5 × 4, because, going from left to right, you get to the division sign first.

So, as mentioned in the video, if you follow the order of operations when solving 6 ÷ 2 (1 + 2), we handle the stuff in parentheses first, which is 1+2 which equals 3.

That now leaves the problem as  6 ÷ 2 (3), which is the same as 6 ÷ 2 x 3. Because everything is the same rank now in the order of operations, we go back to what PurpleMath said, and we solve left to right. The first problem starting from the left is:

6 ÷ 2 = 3

Which leaves us with 3x3 which of course equals 9.

If you want to argue in the comments, fine, but please note that your argument is not with me. It's with PurpleMath and the order of operations.

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EDIT: Okay, TotalMedia in the comments pointed out that PurpleMath actually explains why 9 is not the correct answer on page two. They say:

This next example displays an issue that almost never arises but, when it does, there seems to be no end to the arguing. 

Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.
16 ÷ 2[8 – 3(4 – 2)] + 1
    = 16 ÷ 2[8 – 3(2)] + 1
    = 16 ÷ 2[8 – 6] + 1
    = 16 ÷ 2[2] + 1   (**)
    = 16 ÷ 4 + 1
    = 4 + 1 
    = 5 

The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the [2], rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. 

So, because of how 6 ÷ 2 (1 + 2) is written, with the multiplication not clearly defined like 6 ÷ 2 x (1 + 2), then according to the example above we need to simplify what's in parenthesis first which makes the problem  6 ÷ 2 (3), and since 2 is next to the parenthesis, then it is in essence a multiplication problem that is a part of the parenthesis and must be solved first, and the left-right rule doesn't apply because parenthesis is higher up in the order of operations.

That means that we have to multiply 2(3) which equals 6, and now the problem is 6÷6 which equals 1!

Son of a bitch! That is a tricky problem!