Those who use social networks must have already faced the following question: "What is the value of 6÷2(1+2), 1 or 9?" Or, in another version, 48÷2(9+3), 2 or 288?
Whenever it appears published on social networks, there are always dozens of comments, some with arguments that lead to answer 1 and others with arguments that lead to answer 9.
This issue appears to have arisen about six years ago, and since then it has often reappeared. At the time, in Chile, the confusion was such that a university professor of Mathematics had to go on television to explain why only one of the results is correct!
To solve this “problem” we only have to take into account the priority of the operations, which must be done in this order: powers, multiplication and division in the order they appear, addition and subtraction in the order they appear.
If, in the expression, there are parentheses, the operations inside it must be carried out as a matter of priority, and the priorities of the operations described above apply. So the result is 9.
First, do what is inside parentheses, 1+2 = 3, leaving 6÷2×3. Right now we have a division and a multiplication. We know that neither has priority over the other, so it is done in the order they appear. First 6÷2 = 3 and then multiply this result by 3, leaving 3×3 = 9. ÷ 6 2(1 + 2) = ÷ 6 2×3 = 3×3 = 9.
Trying to understand the reason that leads to answer 1, I find two:
Browsing through some forums I “discovered” that one of the arguments that lead to this answer is PEMDAS, which stands for “Parenthesis, Exponents, Multiplication, Division, Addition, Sutraction”, that is, many arrived at answer 1 thinking that PEMDAS tells us the priority by which operations must be performed.
Therefore, first the multiplication would be done and then the division, forgetting that, in the hierarchy of operations, none has priority over the other, carrying out the operations in the order they appear. In this way, you can see why some argue, wrongly, that the answer is 1:
÷ 6 2(1 + 2) = 6÷2 × 3 = 6÷6 = 1
Another argument has to do with the interpretation of the operation a÷bx, which is interpreted as the quotient of a and bx. In this sense, 6÷2×3 would be the quotient between 6 and 2×3, whose result is 1. I understand this interpretation, but it is not correct. The correct one is the product between 6÷2 and 3, whose result is 9.
Often, to avoid this type of “confusion”, mathematicians use fractions or parentheses. For the answer to be 1, the question would have to be written 6÷[2(1+2)].
So, taking into account the priority of the operations, what is the value of 48÷2(9+3), 2 or 288?
Author José Carlos Pereira is a secondary school mathematics teacher; author of school books; responsible for the "If and Only If" column at Clube SPM and for the website Math Success.
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