# 6÷2(1+2) = x Solve for x

## 6÷2(1+2) = x Solve for x

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#### rolandson

##### Dr. Stratster
'm still getting x = 1 no matter how I do it.
You're doing what I did....just looking at it and say..."oh yeah... "

Spouse hit me with the tease...
"Put it in the form you have seen before" ...
Trust me, it's sexier in Japanese...

My mistake was order of operations.
First, simplify stuff in parentheses
Second, multiplication and division from left to right

#### buzzword

##### Strat-Talker
Silver Member
I think the answer is that it’s a crappily written math problem being used to annoy people.

I fell for it.

True.

And interesting that entering it into Excel as =6/2(1+2) causes Excel to say it's in error and suggests it should be 6/2*(1+2), it then give a result of 9.

#### Skychurch

##### Strat-Talk Member
Hmmmm HP41C caught my attention. I always wanted one of those. Beautiful programmable calculator and they still command a high price. That’s an engineers machine.

I still have the Ti55 programmable that I could afford in 1979 from my paper route money. Then I got a Ti58C programmable (C is for constant memory, thank goodness). It’s also a worthy programmable.

My Ti55 and Ti58 machines will give me 9.

My circa 1977 RCA CDP1802 powered Cosmac ELF 8-bit computer with 32k RAM / 32k ROM with Basic also gets 9 if I run a simple BASIC program:
10 X = 6/2*(1+2)
20 PRINT X

Maybe I ought to try with an HP 41C emulator. I’ll have to brush up on RPN.

I’ve also got a Sharp PC1251 pocket computer but I’m pretty sure all basic interpreters will give me 9.

The HP-41C is amazing. I think I was in my first year of college. I had been using a Ti-55. I took a summer course called something like "advanced math" which was just a fun course for math geeks who wanted to learn things mathematical outside of the regular curriculum.

The teacher had an HP-41CV. Once I got the hang of RPN I fell in love with that thing. She let me borrow it over a weekend. I was visiting my parents over that weekend and was showing this amazing calculator to my parents.

That Christmas, there was a box with my name on it which contained the usual underwear and that stuff. We laughed and my mother said, "keep looking". At the bottom was an HP-41CV. I freaked out because that was VERY unlike my mother. It's the best gift I have ever received and I still have it today (and it's working).

Being an HP geek, I have simulators for:
HP50g
HP Prime
HP41CX
HP-42S
HP35s
HP12cp
HP-15C
HP 17bII+

They're all free. I have the real hardware for the HP50g, HP Prime, HP41CV and HP-42S.

I guess it's a similar sickness to having all these guitars

Once you get the hang of RPN, there's no turning back. For the expression in question, I did this:

6÷2(1+2) = x Solve for x

6 ENT 2 ENT 1 ENT 2 + X /

Admittedly, I entered the expression how I am seeing it. Others could have done it differently and gotten x = 9.

#### The-Kid

##### Dr. Stratster

No “cheers”, as in “I’m done with you”, but you don’t seem to understand that simple concept either.

BTW: 6÷2(1+2) = 6÷2x3 = 3x3 = 9

Start with understanding the distributive property properly and go from there and you will get the correct answer.
You're doing what I did....just looking at it and say..."oh yeah... "

Spouse hit me with the tease...
"Put it in the form you have seen before" ...
Trust me, it's sexier in Japanese...

My mistake was order of operations.
First, simplify stuff in parentheses
Second, multiplication and division from left to right
Again this ignores the distributive property.

For example simple equation here.

2+2(2+2).

The correct answer is 10 following the distributive property correctly but if you ignore the distributive you get 16 and the wrong answer.

Similarly in this equation in OP ignoring the distributive property gets you the wrong answer.

This is why its important to understand the distributive property whether your doing it on paper or plugging it into a computer or calculator.

#### CB91710

##### No GAS shortage here
Double Platinum Supporting Member
9 in modern lingo, 1 in classic lingo.

But what about 8 ÷ 2(2 + 2) = ?.Hmmm?
Same thing

8 ÷ 2(2 + 2)
8 ÷ 2(4)
4*4
16 < Correct after 1917

8 ÷ 2(2 + 2)
8 ÷ 2(4)
8 ÷ 8
1 < Correct before 1917

#### fernank017

##### Strat-Talk Member
I was leaning towards it being “ambiguous” at one time, but after searching and finding that the answer “9” appears to be far more prevalent than “1”, it would seem to imply that the consensus agrees more with one than the other, thus making it more “decisive” rather than “ambiguous”.

In fact the poll here (albeit as unscientific as it is) also reflects that sentiment.

And, ultimately, isn’t that how “Standards” are set?.

So, despite the tendency to go either way, the two results can not be considered equal, therefore one can only conclude (if we had to pick one or the other right now) that one is “more accepted” than the other and ambiguous-ness is no longer an absolute truth...

Well if ya want to talk about statistics, having a 55:45 pole isn't exactly "far more prevalent". If I told you that you had a 55% of not having some debilitating illness, would that make you comfortable? The very reason why the vote is pretty close is because the question is ambiguous. If it weren't you'd see a single answer leading closer to 66%~72% (which tends to be the median for most standardized testing). This result is essentially a coinflip.

Please tell me what the 000 means in the context of your post. A G search indicates a connection to the spiritual infinite.

Order of Operations. Sorry for the ambiguity.

#### buzzword

##### Strat-Talker
Silver Member
Basically parenthesis and distribution go hand in hand with order of operations. Its really as simple as that and again stuff learnt in 6th or 7th grade.

You cant just plug in an equation into a calculator willy nilly not knowing and completely understanding this basic math.

If you completely disregard these fundamentals and basics of math then you get 9 easily but since they are a concrete bed of math the answer 9 is just impossible.

Again its just a trick question and folks fall for the trick because most of the time they just dont understand order of operations, how parenthesis work and how distribution works properly.

I'm awfully glad you're not part of the space program.

#### rolandson

##### Dr. Stratster
9 in modern lingo, 1 in classic lingo.

But what about 8 ÷ 2(2 + 2) = ?.Hmmm?
Welllll....The book that I dug out was...
Here, I'll let it speak for itself.

From 1976. Where daughter got it is anyone's guess.

Oh, and ...
16

#### fernank017

##### Strat-Talk Member
I'm awfully glad you're not part of the space program.

It's a good thing physicists evaluate that as 1, then. Because of the conventions they use within their program. There's no universal convention for this.

Y

#### CB91710

##### No GAS shortage here
Double Platinum Supporting Member
For example simple equation here.

2+2(2+2).

The correct answer is 10 following the distributive property correctly but if you ignore the distributive you get 16 and the wrong answer.

Similarly in this equation in OP ignoring the distributive property gets you the wrong answer.
Yes, the answer to that one is 10
2+2(2+2)
2+2(4)
2+8
10

Ignoring the distributive property and simply following PEMDAS/BEDMAS

Perform the operation within the brackets first
2+2(4)
Perform the multiplication next
2+8
Perform the addition from left to right
10

2+2(2+2) calculated as 4(4) is not a violation of only the distributive property, it is also a violation of PEMDAS/BEDMAS.

Like those who are arriving at the conclusion of "1" for the original question, obtaining the result of 16 requires adding a pair of brackets that are neither present in the original offering, nor implied by convention (2+2)(2+2)=16 2+2(2+2)=10

The difference between the two is that following the order of operations, the set of brackets around the first pair is implied by convention. Convention does not imply a set around the right ride.

6÷2(2+1)
6÷2(3)
3(3)

Convention implies brackets around 6÷2 because division and multiplication have equal priority and are thus performed left to right.

#### rolandson

##### Dr. Stratster
This just goes to show that mathematics is a use it or loose it discipline. So is language.

Spouse, born and raised and educated in Japan, before being forced to attend her "safety" college, The University Of Pennsylvania, because she couldn't get in to university in Japan, is forgetting some of her native Japanese.

We speak it in the home and with our children, but it's mostly she and I, and I am not the native speaker she is.

Same is true with math. How do I know? I at first insisted that the correct solution was 1. But as a 45 year old algebra text demonstrates, the correct solution is 9.

And I hold an MS...
in...
Physics.

Okay, so my thing was imaging and light sciences and the last I saw of algebra was in 1970 except when I asked my son to remind me of the quadratic equation in February 2020 because my brain is swiss cheese but...still.

I should've remembered what I thought I knew...and absent that,

I SHOULD HAVE FUC*KING LOOKED IT UP BEFORE SHOOTING MY MOUTH OFF...as it were.
because I'm stupid.

#### dirocyn

##### Most Honored Senior Member
Sheeezzzzz....!

I am so fu*king embarassed.
Spouse wandered by and I said:
"Quick, what is the solution to...
6÷2(1+2)

She took one brief look and said...
9
I said "you're nuts, it's one".

She said " Says the guy who's hand I held through Diffy Q's..."

She's right. 43 years ago she did hold my hand through that, and Quant.
So I did what every loving spouse of 35 years would do. I set out to prove her as much an imbecile as most of those arguing that the solution is 9...
I went and dug out my daughter's middle school math text.
This is what I found...
View attachment 501759

View attachment 501758

Oh...
Because I'm stupid.

It really is 9
"In this case we need an agreement about the meaning of the expression. Therefore let us agree that operations will be performed as follows..."

The order of operations is not founded on logic or on math itself, but on an agreement. An agreement between whom? Nobody asked ME what the rule should be, they didn't ask you either. It's an agreement proposed by the author of a math textbook for the purpose of use within a class. They could have proposed a different order and the part that's math still works the same. If you change the "agreed" order of operations, you have to write the problems in a different order. That's all.

And then there's the kicker: we don't agree. 45% believe one thing, 55% believe another--this is a nearly even split. Several of us are convinced the answer can only be 1, several are convinced it can only be 9--and the two sides are so dug in and mad about it that they're exchanging rude words like "suck eggs" and "come say that in person." Y'all, it's ambiguous, it was designed to be ambiguous.

Y indeed.

#### The-Kid

##### Dr. Stratster
Yes, the answer to that one is 10
2+2(2+2)
2+2(4)
2+8
10

Ignoring the distributive property and simply following PEMDAS/BEDMAS

Perform the operation within the brackets first
2+2(4)
Perform the multiplication next
2+8
Perform the addition from left to right
10

2+2(2+2) calculated as 4(4) is not a violation of only the distributive property, it is also a violation of PEMDAS/BEDMAS.

Like those who are arriving at the conclusion of "1" for the original question, obtaining the result of 16 requires adding a pair of brackets that are neither present in the original offering, nor implied by convention (2+2)(2+2)=16 2+2(2+2)=10

The difference between the two is that following the order of operations, the set of brackets around the first pair is implied by convention. Convention does not imply a set around the right ride.

6÷2(2+1)
6÷2(3)
3(3)

Convention implies brackets around 6÷2 because division and multiplication have equal priority and are thus performed left to right.
You dont have to imply "invisible" brackets but folks put them there if you dont understand how distribution works and/or as a crutch for longer and bigger equations and also when plugging it into a calculator to have it make the distinction.

Your not ignoring the distributive property by going PEMDAS here your performing Distributive property following into the correct order of operations.

For example another simple equation right........

4÷2(2×8)=x

The correct answer is .125 following distributive property, going into order of operations and again following simple basic maths. But ignoring distributive property your going to get 32 and the wrong answer.

I never knew how important distributive property was to follow through on until calculus. If you ignore distribution and simply plug stuff into a calculator or do it on paper ignoring distribution you keep getting the wrong answer. You would see kids just go insane over this and just not get it.

Its common in regular algebra to get away with it sometimes but definitely not in calculus and its something the teachers drill into the students because its such a big hurtle to get over for a lot of them.

Hope this makes sense and well one cant do simple or advance calculus or even these more simple equations by ignoring the distributive property. I mean you can do it but your going to always get the wrong answer and its going to be a tough class and not a fun time when exams come along.

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#### Skychurch

##### Strat-Talk Member
You dont have to imply "invisible" brackets but folks put them there if you dont understand how distribution works and/or as a crutch for longer and bigger equations and also when plugging it into a calculator to have it make the distinction.

Your not ignoring the distributive property by going PEMDAS here your performing Distributive property following into the correct order of operations.

For example another simple equation right........

4÷2(2×8)=x

The correct answer is .125 following distributive property, going into order of operations and again following simple basic maths. But ignoring distributive property your going to get 32 and the wrong answer.

I never knew how important distributive property was to follow through on until calculus. If you ignore distribution and simply plug stuff into a calculator or do it on paper ignoring distribution you keep getting the wrong answer. You would see kids just go insane over this and just not get it.

Its common in regular algebra to get away with it sometimes but definitely not in calculus and its something the teachers drill into the students because its such a big hurtle to get over for a lot of them.

Hope this makes sense and well one cant do simple or advance calculus or even these more simple equations by ignoring the distributive property. I mean you can do it but your going to always get the wrong answer and its going to be a tough class and not a fun time when exams come along.

0.125 is what I get, too. Not sure how to see it any other way.

#### The-Kid

##### Dr. Stratster
0.125 is what I get, too. Not sure how to see it any other way.

I can see it another way and understand but getting 32 would be the wrong answer because again its ignoring the distributive property.

One, me or even Einstein cant get away with such a fundamental of maths by tossing the distributive property out the door and expect to get any correct results for an equation. Its just not possible.

Whether your dividing and multiplying infinities into each other or doing modest equations like this, one cant take the distributive property for granted.

Otherwise you just keep on getting the wrong answer and the math will still be the math will still be the real math regardless but its something I took for granted and just never understood how important it was until Calc and really not even then until I saw this equation.

In math the distribution property is the equivalent of breathing air into your lungs in the real world. Without it you cant do a damn thing really other than 3rd or 4th grade math at best.

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#### crankmeister

##### Most Honored Senior Member
Interesting article.

The article states,

"If you calculate the problem using this convention, it’s 6 divided by (2(1+2)), which is 1. Typically, though, if the author wanted you to interpret it that way, she would have used parentheses to indicate as much."

I guess that's the part I'm not convinced of ... we don't know who the author is so we don't know for sure how they wanted one to interpret this expression. Maybe they had no freaken clue either
It's postmodern math, where author intent/meaning is abandoned and left to an intersubjective readership.

A clever way of randomizing who does and does not get into X University.

#### rolandson

##### Dr. Stratster
The order of operations was established by operational operators operating operationally on ordered operations

And you have in fact nailed it.

Son walked in a while ago...son who was a math prodigy, hired by a small liberal arts college to teach technical theater* and math, before he'd completed his undergraduate degree...

and I posed the problem to him. He tells me that there is a dispute in acadamia over just this sort of thing. First up, the desire to remove the "÷" symbol from existence. It should never be used.

The correctly written equation would thus be:
6
2(1+2)
There are opposing factions, each demanding sole authority.

And actually his response was

ETA: * That was part of his price...
A) Tenure track
B) He teaches technical theater (lighting, sound, set design) and has control over the performance auditorium
C) in return for which he'd teach undergraduate math as an adjunct professor. Maximum of 2 courses per semester.

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#### The-Kid

##### Dr. Stratster
There are schools where people go to where its "acceptable" to get the answer 9 with such equation such as in the OP..........

My school was where the rich kids went that didnt have good grades to get into UCLA or Stanford and also for the poor kids who just couldnt afford to get into those schools, yet there was still a certain academic standard, quality and care for education.

A proper education and school is so important. I went to the same school Leo Fender went to, pretty funny now that I think about it.

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