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:
$$
\begin{alignat}{2}
1 + 1 + \,&1 +\, 1 + 11 + 1 + 1 + 1 + 11 + 1\times 0 + 1&& \cr
&= 1 + 1 + 1 + 1 + 11 + 1 + 1 + 1 + 11 + (1\times 0) + 1 &\qquad&\text{(the multiplication gets done first)} \cr
&= 1 + 1 + 1 + 1 + 11 + 1 + 1 + 1 + 11 + 0 + 1\cr
&= 30
\end{alignat}
$$
If this doesn't make sense to you, try the following mental exercise on a shorter (but similar) problem: $1 + 2 \times 0 + 3$
- 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, $\,5\times 2 = 2 + 2 + 2 + 2 + 2\,$.
- 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
$$1 + 2 \times 0 + 3
\ \ =\ \ 1 + \overbrace{(2\times 0)}^{\text{strong guy wins}} + 3
\ \ =\ \ 1 + 0 + 3 \ \ =\ \ 4$$
MORE EXAMPLES:
[beautiful math coming... please be patient]
$-1 + 3\times 5 - 2 = -1 + (3\times 5) - 2 = 12$
[beautiful math coming... please be patient]
$2 - 10\div 5 + 3 = 2 - \frac{10}{5} + 3 = 3$
Feel free to use a pencil and scrap paper to work these problems.
However, do not use your calculator!