Math Problem

[Spike88]

While browsing through Facebook I came across a post that a friend of mine had commented on titled "Let's see whose dumb solve: 6÷2(1+2)". The topic had a thousand some replies with varying answers and explanations.

My first thought is that who ever posted this is an idiot for using 'whose' instead of who's, but that's besides the point.

Back to the math problem. Most people got 9, with a handful getting 1(a few people got 6, 12, 3, etc., these people I worry about).

After doing some research, it seems this question has been asked several times throughout the years, but there isn't a definitive answer. Google Calculator and Wolfram Alpha say the answer is 9, however Myalgebra says the answer is 1.

When I solved it:
6÷2(1+2)
6÷(2+4) Distributive Property/Law
6÷6
1

However, if you don't use the Distributive Property you get:
6÷2(1+2)
6÷2(3)
3(3)
9

Am I misusing the Distributive Property? What's the correct answer here?

Edit: Que "Stupid Americans" response. :O:

[Red October1984]

The distributive property is correct. The answer is one. :up: EDIT: Wait just a second. I think I distributed wrong. Let me solve this.

EDIT2: Yes....One is the answer. I confused myself even though I do this stuff every day...

You would do the same thing for a problem like this:

5x + 4(2x-8x^2)

This would be 5x + 8x - 32x^2 and it would get to 13x-32x^2 for your answer.

Dist Property works for all variables.

[Oberon]

See, now situations like these are the perfect example of why the Second Amendment is still relevant in todays society.

The answer to 6÷2(1+2) is quite obviously, whatever the hell you want it to be if you're holding a gun. Thus making maths irrelevant, and the world a much happier place.

[Red October1984]

See, now situations like these are the perfect example of why the Second Amendment is still relevant in todays society.

The answer to 6÷2(1+2) is quite obviously, whatever the hell you want it to be if you're holding a gun. Thus making maths irrelevant, and the world a much happier place.

'Ngland and 'Merica workin together to solve math.... :sunny:

[Catfish]

I am a definite non-mathematician, but this is easy : 1


Of course, using a gun solves all problems at once :haha:

[Sailor Steve]

I am a definite non-mathematician, but this is easy : 1
I think you're fibbing about either the non-mathematician part or the answer itself.

I don't even know what "distributive property" is.

[edit] Okay, I looked it up and now I'm even more confused. Don't bother trying to explain all that. I'll likely never get it.

[Spike88]

I'll explain it either way. :O:
To put it simply under the Distributive property.
A(B+C) is the same as (AB)+(AC).

You can use it to solve harder multiplication problems easily.

For example if you had to multiply 6 by 23 but didn't have a calculator you could use the distributive property.

6(23) would become (6x20)+(6x3).
6x20=120
6x3=18
So you get 120+18 for 138.

or

5(94) = (5x90)+(5x4)
5x90=450
5x4=20
450+20=470


If you ever multiply large numbers in your head, and break them into smaller numbers, you're probably using it without realizing it, such as when you calculate the tip at a restaurant.

Mind you, this is the simple side to the Property. It gets more complex when you're using it in algebra to solve things like 2(3x+5) and x(2-9).


Edit: I'm at 2,999 Posts. :up:

[Sailor Steve]

If you ever multiply large numbers in your head, and break them into smaller numbers, you're probably using it without realizing it, such as when you calculate the tip at a restaurant.
I recognized that when I looked at your first calculation. Yes, I do that all the time. On the other hand, algebra was my worst subject ever. I was pretty good at geometry, but the others? Not even close.

The funny thing is that my older daughter teaches math, and when doing her post-graduate work at the University of Utah was one of an elite group they called "mathletes".

Edit: I'm at 2,999 Posts. :up:
Well, let's do something about that, shall we? :yep:

[frau kaleun]

I was told there would be no math.

[Sailor Steve]

I was told there would be no math.
They lied. This is math class.

[Spike88]

algebra was my worst subject ever. I was pretty good at geometry, but the others? Not even close.

Funny, my favorite Math Subject was Algebra. I wasn't bad at Geometry, I just didn't really care for it as much as I do for Algebra.

On the matter, my favorite subject was Chemistry for the fact that it's a hybrid between science and Algebra with explosions. I would love to major in Chemistry, but I'm planning to go to school to be a Pharmacist instead.


I was told there would be no math.
You were lied to. :O:


Also, 3,000 posts. :yeah:

[Herr-Berbunch]

Congrats on post #3000 :woot:

But the answer is 9.

[Spike88]

Congrats on post #3000 :woot:

But the answer is 9.

But the Distributive Property says it's 1!

[mookiemookie]

"Let's see whose dumb solve: 6÷2(1+2)".

Obviously the person asking the question is the dumb one. :know:

[Herr-Berbunch]

But the Distributive Property says it's 1!

But I say, with my brain, that it's 9. I'm not using Distributive Property. I can see how you get one, but in my mind it's 9. Always will be. :O:

[Wolferz]

I hates algebra.:shifty:

Please Excuse My Dear Aunt Sally

[NeonSamurai]

I am fairly sure your answer is wrong. The distributive property is all well and good, but it doesn't override the order of operations (plus it also does not change the results, it is a mathematical shortcut).

Order of operations requires you to do brackets first, then multiplication and division (in the order it is written).

So..

6÷2(1+2)
6÷2(3)
3(3)=9

Now you can use the distributive property trick in this equation, but you need to do the division first before you can apply it.

6÷2(1+2)
3(1+2)
(3x1)+(3x2)
3+6=9

[Spike88]

But multiplication by juxtaposition suggests that 2(1+2) is handled before you divide.

If it was 6÷2x(1+2), I could see 9 being the right answer.

[NeonSamurai]

I've never heard of juxtaposition overriding the order of operations, and it is not an accepted convention as far as I am aware of.

Also technically as I said, order of operations requires you handle brackets first, I just wrote the equation to show how you would apply the distributive property (which is something that is only generally used in algebra, and algebraic like expressions).

http://en.wikipedia.org/wiki/Order_of_operations

I also found this...
http://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html

[Penguin]

You're all both correct and incorrect.
The operation needs at least one more pair of brackets to become non-ambivalent.
EDIT: The correct mathemenglish term would be am ambiguous, not ambivalent
So in mathematical correct writing it should either be:

6÷[2(1+2)] =
6÷(2*3) =
6÷6 =
1

or:
(6÷2)(1+2) =
3 (1 +2) =
3 * 3 =
9


:smug:

brackets come first, just as Neon said, multiple brackets = from the inside to the outside


"Let's see whose dumb
:hmmm: :O:

[Platapus]

I
6÷2(1+2)
6÷2(3)
3(3)=9



No.

You must also do the math associated with the brackets or parentheses. next.

It is not 6 divided by 2 times 3 That would be written 6/2*3 in which case you can do any of the math in any order

Six divided by two times 3 equals nine
Six times three divided by two equals nine both are correct

Multiplication and division (which is just a form of multiplication) can be done in any order as long as there are no other operands that need to be completed first.

However, that is not the way the problem was written. The problem was written 6/2(3)

6/2(3) means six divided by the product of two times three. The product of two times three is six. Then, and only then can you work the problem in its order. In this case, six divided by six equals one

[Spike88]

So the distributive property could be used.

[NeonSamurai]

No.

You must also do the math associated with the brackets or parentheses. next.

Show me any reference where that is a rule in mathematics, as I am not aware of any. Some may be naturally inclined to do it that way, but I am not aware of any such convention.

It is not 6 divided by 2 times 3 That would be written 6/2*3 in which case you can do any of the math in any order

Six divided by two times 3 equals nine
Six times three divided by two equals nine both are correct

Multiplication and division (which is just a form of multiplication) can be done in any order as long as there are no other operands that need to be completed first.

However, that is not the way the problem was written. The problem was written 6/2(3)

6/2(3) means six divided by the product of two times three. The product of two times three is six. Then, and only then can you work the problem in its order. In this case, six divided by six equals oneAgain show me a valid reference that says it is a convention to do the mathematics beside the brackets next. The generally accepted convention is that you do the math inside brackets first, and you follow the order of operations in all steps from left to right. Functionally 6/2(3) is the same as 6/2*3, or 6*0.5*3, according to most mathematical references I am aware of.


EDIT: The correct mathemenglish term would be am ambiguous, not ambivalent

Yes.

Anyhow that link I posted before goes into the whole problem in great detail, and was written by a mathematics professor at Berkeley
http://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html

[Red October1984]

Ah! TOO MUCH MATH! I have enough of it every day and this is just too much for my already dead brain to handle....


Soon....

We get out May 14th.... :) :woot:

[Platapus]

So the distributive property could be used.


Yes it could, but it did not have to be used.

Let's walk through it

In the expression

6÷2(1+2)

we can translate that into

Six divided by the product two times one plus the product of two times two

Or in mathspeak: 6/((2*1) + (2*2)) Notice the double parenthesis This is called a second level parenthesis. This is important. (see the bottom of my post)

which reduces down to

6/(2+4) Notice that I got rid of the two first level parenthesis, but I still have to deal with that important second level one that remains.

Which reduces further to
6/6 which equals 1

With a statement as simple as 2(1+2) one can use several mathematical operations to calculate the solution.

Two times three Simply adding the numbers inside the parenthesis and then multiplying by the number just prior to the parenthesis

or

Using the distributive process

(Two times one) plus (two times two)

which reduces down to

two plus four

Which equals six

Same expression, same answer, two different mathematical processes. If the expression were not so simple, for example if it contained algebraic expressions, then the distributive process would probably be more appropriate.

What about a mistake?

Suppose I made a translation mistake and wrote the expression

Six divided by the product two times one plus the product of two times two
as:

6/(2*1) + (2*2) Forgetting the second level parenthesis. Does this change the expression?

Let's walk through it

Six divided by two plus four

6/2+4 The next rule is that I must handle multiplication/Division before addition and subtraction. Therefore, this expression reduces to"

3+4 = 7

Yes it does change the expression. A lot!

The moral of the story is that if you have expressions with parentheses, everything within the parenthesis must be calculated first. Even if you choose to re-state individual expressions in the parenthetical expression. That's why that second level parenthesis that encompassed both sub levels of two parenthetical expressions was important.

Otherwise the first expression whose correct answer is 1 would become the second expression whose correct answer is 7

1 seldom equals 7. :D

[Platapus]

Show me any reference where that is a rule in mathematics, as I am not aware of any.


The order of operations in math are


Parentheses and Brackets -- Simplify the inside of parentheses and brackets before you deal with the exponent (if any) of the set of parentheses or remove the parentheses.
Exponents -- Simplify the exponent of a number or of a set of parentheses before you multiply, divide, add, or subtract it.
Multiplication and Division -- Simplify multiplication and division in the order that they appear from left to right.
Addition and Subtraction -- Simplify addition and subtraction in the order that they appear from left to right.


http://www.algebrahelp.com/lessons/simplifying/oops/

Does that help?

[Armistead]

9 is the correct answer.

[mapuc]

I'm sorry but this thread and the answer my friends and their friends give on Facebook, when such a math problem is at hand you get several solution to the problem

But what many of these people forget and wich Platapus correctly pointed out

There are some rules that you have to follow. So simple is it.

Markus

[NeonSamurai]

The order of operations in math are

Parentheses and Brackets -- Simplify the inside of parentheses and brackets before you deal with the exponent (if any) of the set of parentheses or remove the parentheses.
Exponents -- Simplify the exponent of a number or of a set of parentheses before you multiply, divide, add, or subtract it.
Multiplication and Division -- Simplify multiplication and division in the order that they appear from left to right.
Addition and Subtraction -- Simplify addition and subtraction in the order that they appear from left to right.

http://www.algebrahelp.com/lessons/simplifying/oops/

Does that help?

Nope because it doesn't say what you are saying anywhere in there. It does however highlight what I have been saying quite nicely. It says simplify what is inside the brackets, then deal with any exponents outside (none in this case) and remove the brackets. After that it says do the multiplication and the division in the order they appear from left to right.

So following those instructions...

6/2(1+2) = 6/2(3)= 6/2*3 = 3*3 = 9

[Platapus]

I could be wrong. I am by no means a math guy. :D

[NeonSamurai]

I could be wrong. I am by no means a math guy. :D

I think the main issue, as was pointed out, is that how it is written is ambiguous, which is why there is so much disagreement over it as you can make arguments for doing it either way. Math should be written clearly.

As professor Bergman wrote for the similar equation 48/2(9+3) Depending on whether one interprets the expression as (48/2)(9+3) or as 48/(2(9+3)) one gets 288 or 2. There is no standard convention as to which of these two ways the expression should be interpreted, so, in fact, 48/2(9+3) is ambiguous. To render it unambiguous, one should write it either as (48/2)(9+3) or 48/(2(9+3)). This applies, in general, to any expression of the form a/bc : one needs to insert parentheses to show whether one means (a/b)c or a/(bc).

It is interesting that in the 48/2(9+3) problem, the last element was written 9+3 rather than 12. If the latter had been used, it would have been necessary to insert a multiplication sign, 48/2×12, and I would guess that a large majority of people would have then made the interpretation (48/2)×12. Perhaps we will never know where this puzzle originated; perhaps it was cunningly designed so that one interpretation would seem as likely as the other; or perhaps it came up as a real expression that someone happened to write down, not thinking of it as ambiguous, but that other people did have trouble with.

He also had this to say about juxtaposition
Finally, the convention in algebra of denoting multiplication by juxtaposition (putting symbols side by side), without any multiplication symbol between them, has the effect that one sees something like ab as a single unit, so that it is natural to interpret ab+c or a+bc as a sum in which one of the summands is the product ab or bc. Without that typographic convention, the order-of-operations convention might never have evolved. When one has numbers rather than letters, one can't use juxtaposition, since it would give the appearance of a single decimal number, so one must insert a symbol such as ×, and there is less natural reason for interpreting 2 × 3 + 4 as (2 × 3) + 4 rather than 2 × (3 + 4), but I suppose that we do so by extension of the convention that arose in the algebraic context. Likewise, because addition and subtraction constitute one "family" of operations, and multiplication and division another, and perhaps also because the slant "/" doesn't seem to separate two expressions as much as a + or − does, we are ready to read a/b+c etc. as involving division before addition. But when it comes to a/bc, where the operations belong to the same family, the left-to-right order suggests doing the division first, while the "unseparated letters" notation suggests doing the multiplication first; so neither choice is obvious.

http://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html

[Wolferz]

This is proof positive that over engineering the plumbing just clogs the pipes that much quicker.:hmmm:

Simple math and arithmetic will suffice in most situations.

Letters are for spelling and punctuation is for grammar, not math.

Algebra was always a less than desirable subject for me. I flunked it on purpose:-?

[TorpX]

My TI-85 give an answer of 1. This is because it gives a higher priority to implied multiplication over stated multiplication.

Though it pains me to disagree with my trusty TI, I have to go along with Neon Samurai; I know of no rule that states implied multiplication comes first. Apparently, later models make no distinction. When I was in math class, we used to write out fractions with long horizontal lines, so there was no ambiguity.

Hmmm.... I just checked my old Radio Shack calculator; it gives the same answer as the TI - 85.

[Platapus]

This problem bothered me a lot. So I decided to consult with some of my PhD co-workers. With out exaggeration, we have some of the smartest people in the country working at my company. Why they hired me is another question.

I hunted down a couple of these 50 pound brains and gave them the problem

6/2(2+1)

The told me that the answer is 9

So I was wrong. :wah:

But, they said that they could understand my mistake. This was, they told me, an example of a poorly written expression. It is the math version of bad grammar. Math is supposed to be unambiguous and this expression has ambiguity.

The problem is the use of the symbol "/" to indicate division. When writing out expressions on a single line, the use of "/" can cause confusion and that is why it is usually defined by parenthetical sub-expressions. The problem with

6/2(2+1)

is that it can be interpreted in two ways


6
--------- = 1
2(2+1)

Or


6
---- (2+1) =9
2

Using the strict definitions of order of operation, the answer is 9. But they also said that a very common interpretation would be the first one with the answer of 1.

Which is why this particular expression is common on the Internets Tubes -- It is written to be ambiguous.