6÷2(1+2) = ?

Just Another Hobby Jogger wrote:

ardour wrote:

The answers here are indicators of how different the education systems are around the world.

In the United States, the most important thing in education is each student having his /her own opinions. And students are told every student's opinion is valuable.

In some other countries, education is about teaching some basic, objective facts.

Except there are no "objective facts" within the conventions of symbolic notation. It's just what people have agreed on over the years, constantly changing, new symbols invented, new syntax becomes standard, etc.

Much like speaking and writing. You and I could barely make sense of Middle English but that doesn't mean they were speaking and writing counter to "objective fact" back in the day. Much like the assertions in this thread that their understanding of order of operations is solely correct...

In the US it's 9 but in Israel it's 1! BTW, its clear as day, today is cloudy where I live!!!

its clear as day wrote:

as a matter of fact wrote:

PEMDAS. There is only one rule......... You go left to right for multiplication or division after you have solved the parentheses and exponents.....

Common mistake. There is no such thing as a left to right rule for multiplication/division (they are the same thing) or addition/subtraction (again they are the same thing) due to the commutative property.

This thread was over when the OP explained this on the 8th comment on the 1st page. The problem is equivalent to 6*(1/2)*(1+2) = 9.

-There is no ambiguity.

-An acronym that doesn't tell the complete story is letting you down if you are getting "1" for an answer.

-The "left to right rule" is not a rule, but a heuristic for often getting the right answer.

-Yes, division is defined via multiplication, and applying this concept makes the problem "clear as day," as the OP said.

How are we still battling over this 100+ posts later.

/thread

That is all your nastiness and narrow-mindedness. Go away.

is it clear as day? wrote:

I think of 2z and 2(z) as 2 x z. However, when I was in algebra class, y/2z would be interpreted as y ÷ (2 x z)

and not y ÷ 2 x z.

So is 2(3) like (2 x 3) or 2 x 3 ?

I think that's where the confusion is.

Correct. 2(3) is of the form 2y, which is held together beneath the division line.

According our AI overlord (MATLAB) it's 9, so I'm gonna say 1

zcvxzcxv wrote:

Correct. 2(3) is of the form 2y, which is held together beneath the division line.

If we go to the arbiter of all things internet, Wikipedia, we find:

“In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression.”

If two people cannot agree on the same convention, they will not agree on the result.

How about the arbiter of the ‘80s, the modern calculator? Seems like Texas Instruments was of two minds:

The TI-82 (gives 1) doesn’t agree with the TI-83 (gives 9).

Some right answers:

- Depends on the conventions of the original writer of the equation

- What is the problem being modeled?

- Use parentheses (or brackets or braces if you prefer)

- Avoid division in inline equations. Write the equation on multiple lines with mathematical fonts and symbols or an equation editor

- Reverse Polish Notation

Texas Instruments explaining the switch:

its clear as day wrote:

I can't believe how much controversy this equation generates. I feel sorry for you if you can't get this.

This isn't math. Mathematicians would never write an expression this way. I am a professional mathematician, and whenever I see someone post these stupid order of operations (PEMDAS) quizzes, it makes me wonder how stupid they are.

matthematician wrote:

its clear as day wrote:

I can't believe how much controversy this equation generates. I feel sorry for you if you can't get this.

This isn't math. Mathematicians would never write an expression this way. I am a professional mathematician, and whenever I see someone post these stupid order of operations (PEMDAS) quizzes, it makes me wonder how stupid they are.

This is what I came to say. I'm not a mathematician but I use maths and formulas everyday. It is clearly ambiguous to anyone who uses maths and the stupid people are those whose only want to deal in certainties.

Brackets first, then order of operation left to right. Answer 9.

Temp Oh wrote:

matthematician wrote:

This isn't math. Mathematicians would never write an expression this way. I am a professional mathematician, and whenever I see someone post these stupid order of operations (PEMDAS) quizzes, it makes me wonder how stupid they are.

This is what I came to say. I'm not a mathematician but I use maths and formulas everyday. It is clearly ambiguous to anyone who uses maths and the stupid people are those whose only want to deal in certainties.

I also use math every day. Its not ambiguous.

When someone starts a basic equation with “six divided by…” that is a big clue that the answer is going to be less than six so long as we’re dealing with positive whole numbers.

Say you have 6 minutes to do part of a track workout. Each set is 2 times 2:00 hard, 1:00 jog.

How many sets can you do in the 6:00?

6 / 2(2+1) is a perfectly fine way to write that out. I can do one set of that workout in 6:00.

You’d be an idiot to tell me I could do 9 sets in 6:00.

My challenge is for someone to come up with a word problem that could be expressed in this way that gives an answer of 9. I don’t think it can be done.

The equation is written incorrectly ONLY if you want a value of 9.

The “mathematicians” that are just typing things into calculators are like people who use text-to-speech software and think it gives flawless pronunciation of words.

Use logic, not just your TI-83, and it is clear that the answer is 1.

To great mathematicians like Paul Erdos, the answer to all important problems is either zero or one...in this case one.

Lenny Leonard wrote:

6 / 2(2+1) is a perfectly fine way to write that out.

6 / (2(2+1)) is better.

seattle prattle wrote:

1

But i could be wrong.

I hope thee is an authority at our disposal.

Disclosure: I trust science!

You are wrong. The correct answer is 9

Lenny Leonard wrote:

[quote]Temp Oh wrote:

[quote]matthematician wrote:

[quote]its clear as day wrote:

When someone starts a basic equation with “six divided by…” that is a big clue that the answer is going to be less than six so long as we’re dealing with positive whole numbers.

Say you have 6 minutes to do part of a track workout. Each set is 2 times 2:00 hard, 1:00 jog.

How many sets can you do in the 6:00?

6 / 2(2+1) is a perfectly fine way to write that out. I can do one set of that workout in 6:00.

You’d be an idiot to tell me I could do 9 sets in 6:00.

My challenge is for someone to come up with a word problem that could be expressed in this way that gives an answer of 9. I don’t think it can be done.

.

First of all, it's not about using math everyday. A cashier "uses math" everyday. But people with a degree in math can tell the answer is 9.

Your word problem assumes that there are parentheses around the (2(2+1)), which is why you are getting the intuition you are having. I'm not sure what coming up with a word problem proves. For example, 7^[sin(1.289*(e-3.2))-log(pi+9.2)+sqrt(7-1.92)] is unfathomable to model, but there is a clear answer and it's a valid expression. I don't get to argue about the order of operations just because it's complicated.

But to humor you, let's say that there is a pathway 6 km long, but only half of it is on gravel. You only want to run on gravel and wish to run that portion of the trail hard twice, separated by a recovery interval where you run the distance back. How far did you run?

rekrunner wrote:

How about the arbiter of the ‘80s, the modern calculator? Seems like Texas Instruments was of two minds:

The TI-82 (gives 1) doesn’t agree with the TI-83 (gives 9).

Now that's pretty interesting.

Ambiguous grammars exist in programming languages, like "x += x += 1" in C could be equivalent to x += 1 or x += 2 depending on the compiler.

Lenny Leonard wrote:

When someone starts a basic equation with “six divided by…” that is a big clue that the answer is going to be less than six so long as we’re dealing with positive whole numbers.

You actually typed this?

That's (6/2) * (2+1). You would never write out the problem you just described as 6 / 2(2+1). You just proved my point. It's only ambiguous or confusing if you're trying to make it equal 9.

(2(2+1)) and 2(2+1) are the same thing in this equation. They are one expression. It's like 2x. No one gets up in arms because you don't write that as (2*x).

Rippling Hysteresis wrote:[/b

First of all, it's not about using math everyday. A cashier "uses math" everyday. But people with a degree in math can tell the answer is 9.

To contradict your position, I have two degrees in Math (B.Sc, M. Math) from major Universities that are highly respected. In case it's relevant, I was the top student in almost every math course I've taken (there was a Putnam fellow in a couple classes with me and he was certainly better!), and I scored 100% on four math final exams during my undergrad (on 300/400 level math courses, anyway - I didn't always get my score in 100/200 courses).

I think the expression is poorly written and the interpretation is ambiguous, though I also think that 1 is a better answer, and that's how I would evaluate it. If you have a math degree and think the answer is 9, then my position (that it's ambiguous) has been proven.

Suppose we had twelve xylophones in a music room, and the music teacher told a student to pair the xylophones. Then, we would have 12x ÷ 2x = 6 pairs of xylophones.

We would not have 12x ÷ 2x = 6x^2 pairs of xylophones.