Hello,
so, this problem appeared in my social feed today:

20 ÷ 2 (5+5) = ?

I normally ignore these types of posts. Occasionally, I’d do the problem in my head but always just move on never bothering to reply.

However, this problem has raged into a debate on my feed. I presumed the answer was 100. However a very vocal segment argues that the answer is 1.

Before you chime in, consider this version (solidus substituted for obelus):

20 / 2 (5+5) = ?

Try both in your calculators (app or analog) and see what you get.

So, which is the correct answer? And why? And do we have some issue with obelus vs solidus that may require a tweak to whichever precedence acronym you learned?

So I went over to the other thread, and I didn’t get the answer either. There is a specific question here, that is not answered by BODMAS or PEDMAS (or rather, is ambigious). Does division by obelus vs solidus matter? I don’t believe so, but I’m not a math guy. Or rather, that was the asked question, but the real issue is whether the implicit multiplication of 2(5+5) is done before or after the division.

It seems that it’s not really well defined, according to stack exchange, but math persons can weigh in.

Yes, it’s ambiguous, and deliberately so by its creators. This sort of abuse tends to arouse annoyance in people who are at least vaguely familiar with mathematical notation. It’s created as a “gotcha” by someone who isn’t as smart as he or she thinks he or she is.

As far as legitimate mathematical notation is concerned, the answer is none of the above. Nobody who writes math writes math like that.

So, the new question in this thread, as opposed to all the earlier threads, is whether it makes a difference whether you use the / symbol or the ÷ symbol. I’m pretty sure that we will have 100% agreement that it makes no difference – these symbols are entirely synonymous and interchangeable in this context. And I think we already have a widespread agreement that there is ambiguity in the order of operations involving an explicit division operator, and an implicit (unwritten) multiplication operator.

One new reveal here is that you can’t test the / versus ÷ with very many calculators (as OP suggests we do), since I don’t think there are any that have both of these symbols for you to play with. In my on-screen calculator (gcalctool), the on-screen button shows the ÷ symbol, but if you choose to type it on the physical keyboard, you just type / and it appears on the screen as ÷ anyway.

gcalctool, BTW, doesn’t compute the partial answers as you type the expression like a typical physical calculator and some on-screen calculators do. It displays the entire expression in a text-box as you type it, and only does the calculation when you enter =

The answer I then get is: “Malformed expression”. It doesn’t like the implied multiplication. It insists that I put the explicit multiply operator in there (which is * on the physical keyboard, but × on the on-screen button, and displays as × however you enter it). BTW, the answer I get is 100.

The suggestion in all these threads had been that you ought to write additional
explicit parentheses to clarify the intended order of operations, or else write
an explicit multiplication operator which would have a similar effect: 20 ÷ 2 × (5+5).

Spoken like a computer programmer! (Where you actually would have to write
it like that.)

Actually, no mathematician writes this expression in any of those ways either.
We would write it:

20 20(5+5)
—— (5+5) or: ———————
2 2

using the horizontal line for division with dividend above, divisor below.
Hint: There’s probably a reason why mathematicians devised this notation
for division. Maybe this is it.

Naaah, I’ve seen papers use inline division to save space. In fact, in an old thread someone linked to the rules of a well-known journal which specified that simple fractions should be reduced to one line with the “/” symbol (with multiplication by juxtaposition binding higher than division).

I also commonly use inlining for stacked fractions

2
- + x
3 (2/3) + x
------ = ------------
3 (3/5) + 2
- * 2
5

Even with LaTeX pretty printing stacked fractions can get ugly, especially on smaller fonts.

Two comments:
(1) In this thread (unlike earlier threads), OP asks if it makes a difference whether you use the ÷ symbol versus the / symbol for the division. Okay, so now try the same calculation on your RPN mochine but using the / symbol instead of ÷ and see what you get. I’ll bet you can’t do it! :smack:

(2) Setting that aside, there will still be an argument over whether you just calculated the expression that the OP gave or some different expression. That’s the whole point of all these threads – you inserted an explicit × operation where the OP’s expression only had an implicit multiplication operation, and the whole argument is over the ambiguity of that.

There’s no ambiguity, and it doesn’t matter what symbol is used to represent division. The standard order of operations (which I learned as PEMDAS, which flows off the tongue much better than PEDMAS) applies…and correct answer is 100.

20 / 2 (5+5) - because my keyboard has no obelus key.

First, let’s add in the implied multiplication symbol, just for completeness

20 / 2 * (5 + 5)

Parentheses first

20 / 2 * 10

Multiplication and division operations are done at the same time, left to right, so

20 / 2 = 10
10 * 10 = 100

Of course nobody who actually wants to communicate information would ever write “20 / 2 (5 + 5)” for the very reason that it looks ambiguous. The rules are quite clear on what the answer should be, but we have no way of knowing whether or not the person who wrote this is following the rules. It should be written as

20 / [2 (5 + 5)]

or

(20 / 2) (5 + 5)
BTW, regardless of how this discussion goes, ignorance has been fought here, because despite earning a BA in Mathematics, I never before today learned the words “obelus” or “solidus”.

Are they? Or are multiplications done first, left to right, followed by divisions left to right? I’m sure you’ll find plenty of people who learned each way, and they’ll all say it’s “PEMDAS, just like is taught in schools”.

Yeah, my response would be is to ask the questioner where the expression came from and whop whoever gave it to them upside the head for not being more clear.

This question is a deep as sending out the phrase

“I saw a man on a hill with a telescope.”

and asking the reader to figure out whether I or the man had the telescope.

20÷2×(5+5) is not the same as 20/2×(5+5)
People use to have to use a fraction bar whenever writing equations.
So

20÷2×(5+5) had to be written as

20
_____×(5+5)
2
The symbol ÷ and : are called an obelus in math and are used to make simple ratios to write them in-line fraction form. However people still had to write complex fractions with the fraction bar such as

20

2×(5+5)
So a solidus was created to also write complex fraction in the in-line equation form.

There is actually 13 steps to solving mathematical equations, and not just the 4 steps they teach you up to the 6th grade. Pemdas, which should be pemras to avoid confusion later in math is only 4 steps. Without going into some special functions that you won’t use unless doing extremely difficult math, 7 of the 13 steps are as follows.

Paranthesis
2)exponents and roots
3)juxtaposition (implied multiplication)
4)multiplication and ratio’s
5)addition and subtraction
6)complex fractions
7)return to the beginning and solve the next bracket until the equation is finally in the simplest terms.
Peimrasf. Not exactly easy to roll off the tung as pemras.

There is a mathematical reason to have both symbols. In the early 1900’s a printing press began to use the wrong symbols when printing, the high-school dropouts thought the one symbol looked better than the other symbol and started using them interchangeably.
However to this day, when ever a physicist writes a complex fraction they still only use a solidus, and they use a t-89 or a fx-90 calculator or above to solve the mathematical problems as they are designed to solve equations that have a solidus. While below that they are only capable of simple ratios.