# Taking PEMDAS Too Literally: Don't Make This Mistake!

As discussed in Practice With Order of Operations, a common memory device to summarize the order that operations are to be performed is:

Please
Excuse
My
Dear
Aunt
Sally

(PEMDAS)

Unfortunately, people have been known
to take this too literally!
Since the ‘M’ appears
before the ‘D’,
some people think you should do
the multiplications in order from left to right,
and *then* do the divisions in order from left to right.
*This is wrong!!*
*Don't do this!!*

There's a similar problem with the
additions and subtractions.
Since the ‘A’ appears before
the ‘S’,
some people think you should do the
additions in order from left to right,
and *then* do the subtractions
in order from left to right.
*This is wrong!!*
*Don't do this!!*

Multiplication and division
have the *same strength*:
they are done in order, as they appear,
going from left to right.

Similarly, addition and subtraction have the same strength: they are done in order, as they appear, going from left to right.

Here are some counterexamples:

MULTIPLICATION IS *NOT* STRONGER THAN DIVISION!

FAULTY (INCORRECT) PEMDAS:

$$ \begin{align} \cssId{s27}{1\ \cdot\ 2\ \div\ 3\ \cdot\ 4}\ \ &\cssId{s28}{\color{red}{\underset{\text{NO!}}{\overset{?}{=}}} \ \ } \cssId{s29}{(1\cdot 2)\div (3\cdot 4) \ \ }\cr &\ \cssId{s30}{=\ \ \frac{2}{12} \ \ } \cssId{s31}{=\ \ \frac 16} \end{align} $$CORRECT CALCULATION:

$$ \begin{align} \cssId{s33}{1\ \cdot\ 2\ \div\ 3\ \cdot\ 4 \ \ } &\cssId{s34}{=\ \ (1\cdot 2) \div 3 \cdot 4 \ \ }\cr &\cssId{s35}{=\ \ 2\div 3\cdot 4 \ \ }\cr &\cssId{s36}{=\ \ (2\div 3)\cdot 4 \ \ }\cr &\cssId{s37}{=\ \ \frac 23\cdot 4 \ \ } \cssId{s38}{=\ \ \frac 83} \end{align} $$
ADDITION IS *NOT* STRONGER THAN SUBTRACTION!

FAULTY (INCORRECT) PEMDAS:

$$ \begin{align} \cssId{s41}{1\ +\ 2\ -\ 3\ +\ 4\ } &\cssId{s42}{\color{red}{\underset{\text{NO!}}{\overset{?}{=}}}} \cssId{s43}{(1 + 2) - (3 + 4)}\cr &\ \cssId{s44}{=\ \ 3 - 7 \ } \cssId{s45}{=\ -4} \end{align} $$CORRECT CALCULATION:

$$ \begin{align} \cssId{s47}{1\ +\ 2\ -\ 3\ +\ 4 \ \ } &\cssId{s48}{=\ (1 + 2) - 3 + 4 \ \ }\cr &\cssId{s49}{=\ 3 - 3 + 4 \ \ }\cr &\cssId{s50}{=\ (3 - 3) + 4 \ \ }\cr &\cssId{s51}{=\ 0 + 4 \ } \cssId{s52}{=\ 4} \end{align} $$## Practice

Feel free to use a pencil and scrap paper to work these problems. However, do not use your calculator!

Type fractions (as needed) using a slash, e.g. ‘ 1/3 ’.
To be recognized as correct, fractions must be in simplest form.
For example, ‘2/4’ will *not* be recognized as correct, but ‘1/2’ will.