What Is 6÷2(1+2) = ? The Correct Answer Explained
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.
What Is 6÷2(1+2) = ?
The Correct Answer Explained
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)
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
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
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outdated usage!)
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
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outdated usage!)
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!)
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:
fraction-9a2-over-3a
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.
