Practice with Order of Operations
The order that operations are to be performed (when not clearly identified) is summarized with the following memory device:
Please
Excuse
My
Dear
Aunt
Sally
(PEMDAS)
- Do things inside Parentheses first (using PEMDAS, if needed, inside the parentheses).
- Then do all Exponents, in order as they occur, going from left to right.
- Then do all Multiplications/Divisions (they have equal weight) in order as they occur, going from left to right.
- Finally, do all Additions/Subtractions (they have equal weight) in order as they occur, going from left to right.
In September 2017, this problem was floating around the web:
Here's the solution, with correct order of operations:
$$ \begin{align} &\cssId{s19}{1 + 1 + 1 + 1 + 11 + 1 + 1 + 1 + 11 + 1\times 0 + 1} \cr\cr &\quad\cssId{s20}{= 1 + 1 + 1 + 1 + 11 + 1 + 1 + 1 + 11 + (1\times 0) + 1}\cr &\quad\quad\cssId{s21}{\text{(the multiplication gets done first)}} \cr\cr &\quad\cssId{s22}{= 1 + 1 + 1 + 1 + 11 + 1 + 1 + 1 + 11 + 0 + 1}\cr\cr &\quad\cssId{s23}{= 30} \end{align} $$If this doesn't make sense to you, try the following mental exercise on a shorter (but similar) problem: $$\cssId{s25}{1 + 2 \times 0 + 3}$$
- Replace each plus sign with an (equally-weak) person.
- Replace the multiplication sign with a strong person. Why? Multiplication is ‘stronger than’ addition! And this makes perfectly good sense, since multiplication is ‘super-addition’. For example: $$\cssId{s31}{\,5\times 2 = 2 + 2 + 2 + 2 + 2}$$
- The first (left-most) weak guy is trying to pull together the $\,1\,$ and the $\,2\,,$ to add them.
- The middle (strong) guy is trying to pull together the $\,2\,$ and the $\,0\,,$ to multiply them.
- The right-most weak guy is trying to pull together the $\,0\,$ and the $\,3\,,$ to add them.
- Who wins? Clearly, the strong guy!
1 2 0 3
$$ \begin{align} &\cssId{s37}{1 + 2 \times 0 + 3}\cr &\qquad\cssId{s38}{=\ \ 1 + \overbrace{(2\times 0)}^{\text{strong guy wins}} + 3}\cr\cr &\qquad\cssId{s39}{=\ \ 1 + 0 + 3}\cr\cr &\qquad\cssId{s40}{=\ \ 4} \end{align} $$More Examples
Curious readers may want to explore the optional section: Laurel, Yanny, Cookies, Bananas, and Clocks
Practice
Feel free to use a pencil and scrap paper to work these problems. However, do not use your calculator!