What is the correct answer to this infamously ambiguous arithemetic problem 6÷2(1+2)?

Question

I am going to make the case that $6÷2(1+2) = 1$. Many people often quote the PEMDAS rule and get the answer $9$ but look at this way:

$$ 6 \div 2(1+2)= \frac{6}{2(1+2)}$$

Algebraically, we can treat the division operator(obelus or solidus) as a fraction. Everything to the left of the operator is in the numerator and everything to the right is in the denominator. So then you apply the usual PEMDAS rule to simplify the denominator:

$$\frac{6}{2(1+2)}=\frac{6}{2(3)}=\frac{6}{2\ast 3}=\frac{6}{6}=1$$

A lot of engineering types just work left to right and end up with $9$ because they don't appreciate how division lacks 2 properties of multiplication: commutativity and distributivity. But if there is a good mathematical reason why I'm wrong, I'd like to hear it!

https://math.stackexchange.com/questions/1212662/8-4-4-2-what-is-answer/1212674 — Kenta S, Mar 24 '21 at 03:11
@Lt.Commander.Data A lot of this rule has reasonable justifications. For example, you would want $3x-2x = x$ instead of $3x-2x =3(x-2)x$, so you would do multiplication first before subtraction. — Some Guy, Mar 24 '21 at 03:56

score 5 · Accepted Answer · answered Mar 24 '21 at 03:12

Actually, no, you cannot treat $\div$ as a fraction line, it should be treated as dividing the thing that comes right after, not ALL the things that come right after. There is no really good mathematical reason you are wrong, that's just it, $\div$ doesn't work like a fraction line. Thus, by PEMDAS, you should actually get

$6 \div 2(1+2) = 6 \div2(3) =3(3) = 9$

If you're still confused, try the mnemonic PEMoDAoS, which stands for parantheses, exponents, (multiplication or division), (addition or subtraction). Multiplication and division have the same priority, so you have to evaluate from left to right.

Hope this helps

Yes it does help. This is a just a question of convention. – Mr X Mar 24 '21 at 16:35 — Mr X, Mar 24 '21 at 16:35

score 2 · Answer 2 · answered Mar 24 '21 at 03:35

The 'ambiguity' here comes from the fact that we use parenthesis as a multiplication operator as well. We can agree that $2(1+2) = 2 \times (1+2)$
The problem is then: $ 6 \div 2 \times (1+2)$

By convention, multiplication and division have the same hierarchy and when this happens, we always go from left to right. Conventions are followed to avoid ambiguous situations such as this.

If you mean to use the division operator as you did, then you're not following the usual convention. That is, you're using the same symbols differently than what most mathematicians would interpret it.

Obviously, if you're using symbols differently, then anything can happen.

How calculators evaluate the input all depends on how it was programmed, hence why there is the 'ambiguity'

score 1 · Answer 3 · answered Mar 24 '21 at 03:13

In order to get $\frac{6}{ 2(1+2)}$, we would need the original problem to have: $$6 \div (2 (1+2)).$$

Because there are no extra parentheses, we cannot put the $1+2$ under the fraction bar as well. You MUST work from left to right and get $6 \div 2 \cdot (1+2) = 3 \cdot (1+2) = 9.$

What is the correct answer to this infamously ambiguous arithemetic problem 6÷2(1+2)?

3 Answers3