Posted August 31, 2016 By Presh Talwalkar
What is 6÷2(1+2) = ?
The problem often generates debate and has millions of comments on Facebook, Twitter, YouTube and other social media sites.
I posted a video with the correct answer.
Keep reading for a text explanation.
The order of operations
The expression can be simplified by the order of operations, often remembered by the acronyms PEMDAS/BODMAS.
First evaluate Parentheses/Brackets, then evaluate Exponents/Orders, then evaluate Multiplication-Division, and finally evaluate Addition-Subtraction.
Everyone is in agreement about the first step: simplify the addition inside of the parentheses.
6÷2(1+2)
= 6÷2(3)
= 6÷2(3)
This is where the debate starts.
The answer is 9
If you type 6÷2(3) into a calculator, Google or WolframAlpha, the input has to be parsed and then computed. All of these will first convert the parentheses into an implied multiplication. The expression becomes the following.
6÷2(3)
= 6÷2×3
= 6÷2×3
According to the order of operations, division and multiplication have the same precedence, so the correct order is to evaluate from left to right. First take 6 and divide it by 2, and then multiply by 3.
6÷2×3
= 3×3
= 9
= 3×3
= 9
This gets to the correct answer of 9.
This is without argument the correct answer of how to evaluate this expression according to current usage.
Some people have a different interpretation. And while it’s not the correct answer today, it would have been regarded as the correct answer 100 years ago.
The other result of 1
Suppose it was 1917 and you saw 6÷2(3) in a textbook. What would you think the author was trying to write?
Historically the symbol ÷ was used to mean you should divide by the entire product on the right of the symbol (see longer explanation below).
Under that interpretation:
6÷2(3)
= 6÷(2(3))
(Important: this is outdated usage!)
= 6÷(2(3))
(Important: this is outdated usage!)
From this stage, the rest of the calculation works by the order of operations. First we evaluate the multiplication inside the parentheses. So we multiply 2 by 3 to get 6. And then we divide 6 by 6.
6÷(2(3))
= 6÷6
= 1
= 6÷6
= 1
This gives the result of 1. This is not the correct answer; rather it is what someone might have interpreted the expression according to old usage.
The symbol ÷ historical use
Textbooks often used ÷ to denote the divisor was the whole expression to the right of the symbol. For example, a textbook would have written:
9a2÷3a
= 3a
(Important: this is outdated usage!)
= 3a
(Important: this is outdated usage!)
This indicates that the divisor is the entire product on the right of the symbol. In other words, the problem is evaluated:
9a2÷3a
= 9a2÷(3a)
(Important: this is outdated usage!)
= 9a2÷(3a)
(Important: this is outdated usage!)
I suspect the custom was out of practical considerations. The in-line expression would have been easier to typeset, and it takes up less space compared to writing a fraction as a numerator over a denominator:
The in-line expression also omits the parentheses of the divisor. This is like how trigonometry books commonly write sin 2θ to mean sin (2θ) because the argument of the function is understood, and writing parentheses every time would be cumbersome.
However, that practice of the division symbol was confusing, and it went against the order of operations. It was something of a well-accepted exception to the rule.
Today this practice is discouraged, and I have never seen a mathematician write an ambiguous expression using the division symbol. Textbooks always have proper parentheses, or they explain what is to be divided. Because mathematical typesetting is much easier today, we almost never see ÷ as a symbol, and instead fractions are written with the numerator vertically above the denominator.
Sources
1. In 2013, Slate explained this problem and provided a bit about the history of the division symbol.
2. The historical usage of ÷ is documented the following journal article from 1917. Read the second and third pages of the article (pages 94 and 95) for the usage of ÷ in evaluating expressions. Notice the author points out this was something of an “exception” to the order of operations.
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