Mathematics: Is the taught Order of Operations wrong?

There are a number of Maths puzzles doing the rounds on social media and many are based on the following theme:

X ÷ Y(A ± B)

The current one on Facebook is 16 ÷ 4(2+2) with people arguing over whether the answer is 16 or 1. I am writing this article because the “strict and literal use” of the PEMDAS or BODMAS mnemonics does not always give the correct answer.

However, it is possible to use the order of operations to solve to the correct answer of 1 in all situations, even with this deliberately ambiguous mathematical expression.

Using this example, let’s first show the two approaches and then we can look at what went wrong.

Clarification: The mnemonics PEMDAS (US Math) and BODMAS (UK Maths) are actually PE[MD][AS] and BO[DM][AS]. The Multiplication and Division are performed at same time going left to right and Addition and Subtraction are performed at the same time going left to right. This is why the MD in PEMDAS and the DM in BODMAS actually are the same thing. Also, the terminology Parenthesis is the same as Brackets and Exponents is the same as Orders.

Using Strict and Literal PEMDAS

Start with:

16 ÷ 4(2+2)

Calculate inside Parenthesis:

2+2 = 4 so 16 ÷ 4(4)

Incorrectly assuming that 4(4) is exactly the same as 4 x 4:

16 ÷ 4 x 4

Multiply and Divide (from left to right):

16 ÷ 4 = 4 so 4 x 4 = 16

Using PEJMDAS

PEJMDAS as a method was originally taught in schools, before the simplified mnemonics of PEMDAS or BODMAS where even created. The J stands for Juxtaposition and indicates that Multiplication by Juxtaposition (AKA Implied Multiplication or Implicit Multiplication) has a higher precedence in the Order of Operations than standard Multiplication or Division.

Start with:

16 ÷ 4(2+2)

Calculate inside Parenthesis:

2+2 = 4 so 16 ÷ 4(4)

Calculate the Multiplication by Juxtaposition:

4(4) = 16 so 16 ÷ 16

Multiply and Divide (from left to right):

16 ÷ 16 = 1

Which is correct?

The second answer is correct, but there are many who disagree. This all comes down to the assumption that 4(4) = 4 x 4.

While 4(4) equates to 4 x 4, it is not identical to as 4 x 4. Both equate to 16, but they are still a single term in the expression.

In a reply on the post, I was told:

4(4) = (4)4 = (4)(4) = 4 x 4

Which is almost correct, so let’s fix it by correctly expanding the parenthesis/brackets.

Expanding Brackets

Expanding brackets is a method to simplify expressions that are inside brackets by multiplying them out. It means taking the term outside the bracket and multiplying it by each term inside the bracket (including the minus sign, if there is one).

It is no different whether the expression is algebraic (using variables) or not.

Therefore, here is the correction:

4(4) = (4)4 = (4)(4) = (4 x 4)

Yes, the correct expansion of 4(4) is (4 x 4) not 4 x 4. The remaining brackets are important as the signify that this is still a single term in the expression.

Think of it this way. I have moved the 4 from outside the brackets to inside the brackets and multiplied each of the terms inside the brackets.

The brackets are still there around the outside; they don’t suddenly disappear. This is still a single term and must not be split apart.

The brackets can only be removed once the expression is down to a single term inside the brackets and nothing outside.

4(4) = (4 x 4) = (16) = 16

In the same way the correct expansion of

4(2 + 2) = (4 x 2 + 4 x 2) = (8 + 8) = (16) = 16

y(2 + 2) = (y x 2 + y x 2) = (2y + 2y) = (4y) = 4y

and when y = 4,

4y = 4 x 4 = 16

Let’s try the calculation again only using PEMDAS and correctly expanding parenthesis.

Using PEMDAS Correctly

Start with:

16 ÷ 4(2+2)

Calculate inside Parenthesis:

2+2 = 4 so 16 ÷ 4(4)

Expand Parenthesis:

4(4) = (4 x 4) so 16 ÷ (4 x 4)

Calculate inside Parenthesis again:

(4 x 4) = 16

Multiply and Divide (from left to right):

16 ÷ 16 = 1

Start with:

16 ÷ 4(2+2)

Expand Parenthesis:

4(2+2) = (4 x 2 + 4 x 2) so 16 ÷ (4 x 2 + 4 x 2)

Calculate inside Parenthesis:

(4 x 2 + 4 x 2) = (8 + 8) = 16

Multiply and Divide (from left to right):

16 ÷ 16 = 1

Start with and use algebra:

16 ÷ 4(2+2)

Substitute 4 for y where y = 4:

16 ÷ y(2+2)

Expand Parenthesis:

y(2+2) = (y x 2 + y x 2) = (2y + 2y) = (4y)

Substitute y = 4 back:

16 ÷ (4y) = 16 ÷ (4 x 4) = 16 ÷ 16

Multiply and Divide (from left to right):

16 ÷ 16 = 1

Start with and use algebra:

16 ÷ 4(2+2)

Substitute 4 for y where y = 4:

16 ÷ y(2+2)

Calculate inside Parenthesis:

y(2+2) = y(4)

Expand Parenthesis:

y(4) = (y x 4) = (4y)

Substitute y = 4 back:

16 ÷ (4y) = 16 ÷ (4 x 4) = 16 ÷ 16

Multiply and Divide (from left to right):

16 ÷ 16 = 1

You can even take 16 ÷ (4y) and simplify by dividing the nominator and denominator by 4:

16 ÷ (4y) = 16 ÷ 4y = 4 ÷ y

Substitute y = 4 back:

4 ÷ y = 4 ÷ 4

Multiply and Divide (from left to right):

4 ÷ 4 = 1

Using a worded example:

I have 16 apples and there are 4 lines of people waiting to receive apples. Each line is made up of 2 parents and 2 children. How many apples can each person have?

The answer is NOT 16 apples each, unless you can magically turn 16 apples into 256 apples.

The answer is ALWAYS the same. Mathematics is an exact science, equations always have the same solutions, but when the equation could be ambiguous, it should use brackets/parentheses or language to clarify.

This video explains why the calculators (online and physical) might disagree.

The Problem with PEMDAS: Why Calculators Disagree (Direct Link)