 Random Ponderings

September 16, 2020

# The correct way to solve a deliberately misleading math equation

Often people share mathematics equations like the following.

8 ÷ 2(2 + 2) = ?

Every time I have seen a post like this it results in a debate and a great many wrong answers. This is in part due to the fact that this equation is deliberately misleading. Is the answer 10, 1, or 16?

Those who say the answer is 10 mistakenly believe that mathematics is a simple, left-to-right language. It is not.

When reading a mathematics equation you must follow a set of rules known as PPMDAS1 or PEMDAS. PEMDAS is a mnemonic device to help people remember the Order of Operations for a mathematics equation. This table summarizes the Order of Operations for a mathematics equation.

Order of Operations: PEMDAS
Symbols Meaning
P Parenthesis, ()
E Exponents, an
MD Multiplication or Division (Left to right)
AS Addition or Subtraction (Left to right)

Many people who get the wrong answer will quote the PEMDAS rules as evidence to support their wrong answer. Why?

What the people who quote the PEMDAS rule to support their wrong answer forget is that the first rule, parenthesis, only refers to what is inside the parenthesis, not what is inside and around the parenthesis.

Thus, they think the the order of operations is as follows.

8 ÷ 2(2 + 2) = 8 ÷ 2(4) = 8 ÷ 8 = 1

That is not correct.

The a(b) syntax in mathematics is the same as a × (b). This is again due to the fact that the parenthesis rule only says to process what is inside the parenthesis first.

Thus the following equations are equivalent.

8 ÷ 2(2 + 2) = ?

And.

8 ÷ 2 × (2 + 2) = ?

Only the first one is misleading because the × is only implied. Once you add in the implied × it becomes obvious what the answer is as long as you remember to apply the PEMDAS rules.

8 ÷ 2 × (2 + 2) = 8 ÷ 2 × 4 = 16

Note that The New York Times published an article on this very equation, The Math Equation That Tried to Stump the Internet

1. This is how I learned it in sixth grade. I also learned the phrase Pretty Please My Dear Aunt Sally to help me remember it.↩︎