Eli5, people are saying that pemdas is absolute and that 1000 / 5(4-2) = m could be either 100 or 400 because the variable m isn’t defined. Could you all elaborate on what that means?
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If you can’t tell what the order of operations is supposed to be, the equation is improperly formatted.
All these meme equations trying to obfuscate the correct order with weird application of parentheses and “/“ symbols are poorly structured specifically to confuse.
Don’t overthink it.
This, and many, many similar problems online, are rage/engagement bait. They put the operations together in a way where it’s ambiguous what order you are meant to solve the problem in.
“The variable ‘m’ isn’t defined” doesn’t mean anything here. The issue is that one interpretation is (1000/5) * (4-2), and another interpretation is 1000 / (5*(4-2)), and those result in an answer of 400 or 100, respectively.
I was an English major so I could be way off, but I believe that is incorrect. It can only be 400. Let’s write out the operators to make this more clear.
>1000 **÷** 5 **x** (4-2) = m
First, calculate the parentheses.
>1000 ÷ 5 x **2** = m
Then, division/multiplication. These you *calculate left to right.*
>**200** x 2 = m
>**400** = m.
1000 / 5(4-2) is not a well written equation. And that’s on purpose. You see these all over the place in different variations. There is no “right” answer and they are designed to be ambiguous on purpose just to create internet drama.
The ambiguity lies in the use of the “/” symbol. If it is meant to represent division, then m = 400.
But “/” is also used to represent fractions in which case a valid interpretation is that this is 1000 *over* 5(4-2), in which case m = 100.
Either of those are possible interpretations of that equation.
These people failed third grade math. In this, 1000/5(2) is what you get after the first step. From there, multiplication and division are on the same plane, so to speak, and you do them left-to-right. So the answer *is* 400.
Either way, nobody writes equations like this to solve any real-world problems. 1000/5 would be expressed as a fraction, and the correct solution would be obvious
When you multiply something you can omit the multiplication sign. This is called implied multiplication.
For example `2×y` can be written as `2y`. Mathematicians do anything to write less symbols.
This implied multiplication is one half of the problem. The other half is the way we write divisions on one line.
If you write “proper” math you’d write divisions like this: https://i.imgur.com/AwjnAmM.gif
That expression is perfectly clear with no room for ambiquity. But when you try to write it on one line you get this: `1/2y`
And `1/2y` is ambiquous. If you do the implied multiplication first it means `1/(2×y)` but otherwise it means `(1/2)×y`.
Since mathematicians hate writing extra symbols they silently have agreed that `1/2y` means `1/(2×y)`. If you instead meant `(1/2)×y` you would just write `y/2`. Now both expressions can be written with so few symbols and mathematicians around the world are happy.
This is an unwritten rule that implied multiplication is done first. But not everyone follows that rule so you shouldn’t rely on it too much.
For example Texas instruments says this https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/11773:
> Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.
> This order of precedence was changed for the TI-83 family, TI-84 Plus family, TI-89 family, TI-92 Plus, Voyage™ 200 and the TI-Nspire™ Family. Implied and explicit multiplication is given the same priority.
Additionally the ISO-80000-2 standard says this:
> [Either[multiplication sign] can be omitted if no misunderstanding is possible](https://i.imgur.com/OwIi5Pq.png)
In the end the expression `1000/5(4-2)` is intentionally made so that misunerstanding will occur. It is not properly written. You can slap the person who presented it and demand that they rewrite it so that no misunderstanding is possible.
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I’d like to also point out that any and all problems with “division is done before multiplication” or “multiplication is done before division” rise from the way we write math on one line.
The single line division and multiplication combination hides what number multiplies what and that leads to confusion.
For example `2/3*5`. Without reading further quickly say what number is the `5` multiplying.
If you said that `5` multiplies `3` you are wrong.
If you write it “properly” you immediately see that `5` in fact multiplies `2`.
If you write the “proper” way as shown above you can do multiplications and divisions in any order you want and always get the exact same result. When done properly division and multiplication **always** have same “priority”.
The problem here is that when you stick two things next to each other, we mean to multiply them. 4y means 4×y. When they’re just stuck to each other, it’s called implicit multiplication. When it’s written out, it’s called explicit multiplication. We sort of treat it as a single object.
But this can lead to ambiguity. With BODMAS/PEMDAS/etc, multiplication and division happen from left to right. But should the implicit stuff skip the queue and come first? I think so, and Casio agrees with me. Others, including Texas Instruments, disagree.
The thing is, it doesn’t matter. No self-respecting mathematician or scientist would ever write something in a way where that made a difference. They’d write it much clearer, using actual fractions (instead of ÷) or more brackets!
The only thing that is absolute about PEMDAS is that it is absolutely useless for anything but the most basic of expressions. I think we can all agree that 2+3×4 is 14, not 20. But that’s about it.
As others have said, most of the PEMDAS stuff out there on facebook and the like are deliberately ambiguous to try to draw you in by anger and annoyance.
Specifically, the M and D in PEMDAS are sometimes supposed to be “M and D whichever is first” and sometimes “all the M’s then all the D’s” and there is no absolute rule about which. Combine this with the fact you have multiple ways of representing multiplication (x, *, a dot, a gap, two things next to each other without a gap) and division (/, ÷, a horizontal line with stuff above and below, little fractions like ½). Different people apply different rules depending on which of these are used. Well written things make it easy to tell from context what should be done. Badly written things need clarification. Memes need to be ignored.
Your best best is to claim that you asked your professor zirself, and ze said the answer is definitely <fill in some number that it definitely can’t be, like 250>. Make sure you use ze rather than he or she. Then, and this is the lost important bit, walk away. Don’t engage.
There are some good answers here explaining why it’s ambiguous, but I just want to add that in any real-life situation you would have some more context to help you understand what the intended meaning is.
It’s perfectly normal in mathematical and scientific communication, including school textbooks, to have expressions that would be ambiguous if you removed all context. But there will almost always be some text explaining what the numbers and symbols are, and how the expression was obtained, and usually this will make it very obvious how to resolve the ambiguity. With memes like this we’re just given a bunch of random numbers and symbols with no explanation of what any of them represent. Not only does this make it impossible to be sure what the intended interpretation was, it also means it doesn’t really matter. I’ve no idea what m even means, so why should I care whether it’s 100 or 400?
It’s bad, ambiguous notation that’s confusing on purpose in order to create social media flame wars.
People who write math for real would write either (1,000 / 5)(4 – 2) or 1,000 / [5(4 – 2)] or just use a horizontal line instead of a slash to represent division.
As written in the OP, it’s unclear if the bunching together of 5(4-2) is supposed to imply parentheses, or if it’s just inconsistent formatting.
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