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

# 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.