Back in 2003, the world learned that a cute, vegetarian panda could turn into a violent gun-toting creature thanks to nothing more than a misplaced comma. Thanks to the breakout success of the book Eats, Shoots and Leaves, talented literary editor Lynn Truss, taught people the importance of proper punctuation.
I think of this book every time I see that stupid math problem on Facebook. You know the one, it looks something like this: 6 ÷ 3(1+2).
Millions of people have quibbled over the “correct” solution to the problem with a “my way or the highway” certainty, based on nothing more than the confidence that their third grade teacher had a full grasp of the world of mathematics.
The PEMDAS Principle
At play here is a mathematical convention known as PEMDAS which suggests a certain order be followed when calculating mathematical equations. The order is as follows: Parentheses, Exponents, Multiplication, Division, Addition and Subtraction. Except for many Americans, remembering even those operator names is too complicated, so PEMDAS has generally been “shortened” to the longer mnemonic “Please Excuse My Dear Aunt Sally.”
Although PEMDAS can and does serve as a guide to solving this generic math problem, the larger issue at play is that this is a badly written equation and without context (or a better set of defining parentheses), there will never be a wholly correct answer.
Yes, yes, I already hear you arguing that math (and PEMDAS) demands that multiplication comes before division.
Math vs. Logic
Well first dear, gentle reader, let’s clearly define our terms.
While people will argue profusely over the “math” behind this internet problem, the real issue isn’t math at all. Rather PEMDAS is a definition of the “logic” of which order the math should be applied.
buHere, I don’t mean logic in a Captain Spock, “I find humans to be illogical” kind of way (although sometimes I believe that’s also true) but rather in the true Oxford English definition.
For the purposes of this story…
Math is defined as “the science of numbers and their operations” and
Logic is defined as “a system or set of principles underlying the arrangements of elements”
So first we must understand the logic (order) of the equation before we can apply the math (operators) behind it.
Get it? No?
Cartons vs. Pastries
Okay, let’s take a look at this problem visually..
For this story, Dear Aunt Sally is a pastry-making queen. Before defining the underlying problem, we know two things to be true.
6 ÷ 2(1 + 2)
Aunt Sally starts out with six cartons
🥡🥡🥡🥡🥡🥡
6 ÷ 2(1 + 2)
And in one pastry baking session (the session being the container defined by the parentheses) Aunt Sally makes three pastries.
🍪🍩🍩
In this example, I am also working under the assumption that since 1+2 are separately defined within the parentheses and not immediately identified as the number 3, that there is some differentiation between the two. (Yes, there goes the idea that 2 plus 1 always equals three) While we know Aunt Sally has made three pastry items, we’re going to assume one of them is a cookie and two of them are donuts..
What Happens to Mr. In Between?
This is where the world goes crazy!
The entire metaverse is arguing over what happens to the number 2.
6 ÷ 2(1 + 2)
6 ÷ 2 * (1 + 2)
6 ÷ 2 * (3)
Just like a shifting comma can change the very nature of our beloved panda, the shifting placement of the number 2 defines the very question that defines this equation. .
Either the number two defines the number of pastries made:
6 ÷ (2 * 3)
Or it defines the number of cartons: used:
(6 ÷ 2) * 3
Example One: The Answer is One
If you’re of the camp that believes the correct answer is one, then for you the number two defines what’s happening to the baked goods.
6 ÷ 2(1 + 2)
6 ÷ 2 * (1 + 2)
6 ÷ 2 * (3)
6 ÷ 2 * 3
6 ÷ 6
In this case, the number two tells us Aunt Sally has baked twice. Or times two. Or x2. Which means she has baked a total of six pastries.
2 * (1 + 2)
2 x (🍪🍩🍩) OR
🍪🍩🍩🍪🍩🍩
6 ÷ 6
Ultimately that turns this equation into a question of division.
So here we know Aunt Sally has started with six cartons and she has baked six pastries. The clock is ticking as six customers are lining up at her door and she needs to know how many pastries she can evenly fit (divide) into each of her six cartons.
Fortunately the math is pretty easy.
🥡🥡🥡🥡🥡🥡 / 🍪🍩🍩🍪🍩🍩
🥡=🍪, 🥡=🍪, 🥡=🍩, 🥡=🍩, 🥡=🍩, 🥡=🍩
So the answer here is one.
Six pastry items divided into six cartons means one pastry item goes in each carton. Phew, six customers are satisfied with their orders, thanks for your help!
Example Two: The Answer is Nine
For those who assert the correct answer is nine, we are now first shifting emphasis onto the number of cartons rather than the number of pastries.
6 ÷ 2(1 + 2)
6 ÷ 2 * (1 + 2)
3 * (1 + 2)
3 * (3)
9
6 ÷ 2 * (1 + 2)
Oh no! Aunt Sally dropped half (or “ / 2” or “ ÷ 2”) of her cartons on the floor. Oh dear, she only has three remaining.
🥡🥡🥡🥡🥡🥡 / 2
🥡🥡🥡
But don’t worry, since this is now an equation of multiplication, she really only needs to know how many pastries she needs to make.
6 ÷ 2 * (1 + 2)
While Aunt Sally only has three cartons, in this example each carton holds three (times 3 or x3 or *3) pastry items. So the question now shifts to how many pastry items must Aunt Sally make in total to fill her three cartons.
3 * (1 + 2)
1🥡 has 🍪🍩🍩
Times Three
🥡 has 🍪🍩🍩 +
🥡 has 🍪🍩🍩 +
🥡 has 🍪🍩🍩
OR
3🥡 have 3 (🍪🍩🍩)
OR
🥡🥡🥡= 🍪🍩🍩🍪🍩🍩🍪🍩🍩
Here, the correct answer is 9.
.
Three cartons holding three pastry items each, means Aunt Sally will need to bake 9 pastries to complete the order. Apparently these were much bigger cartons than the ones she used in example one.
The Power of Storytelling
I hear you. You see the pastries, you see the cartons, but still PEMDAS must rule all! There’s been lots of great writing on this topic, but if you’d like to explore the subject in greater detail, I recommend reading this Slate story by math teacher Tara Haelle.
The larger point of my article, however, is that often math and logic don’t matter nearly as much as the story behind the numbers.
No matter if you believe the two in this example should modify the number of cartons or the number of pastries, the addition of the story shifts the sentiment of the piece. The idea that Aunt Sally can net better profit margins despite the loss of half her cartons can have marketers shifting their language towards Example 2, whether or not the math and logic always back that up.
