Which is the correct answer for 48÷2(9+3): 2 or 288?

[JaredJames]

Two engineers here ended up with 2.

Something in my head says x(y) means x lots of y. So thus 48÷2(9+3) is saying 48 divided by two lots of (9+3).

Having it as 48÷2x(9+3) WOULD then equal 288.

Ps my casio calculator agrees with me. She can't be wrong (and i don't like matlab, it butchers fractions).

48÷2(9+3) or 48÷2x(9+3) in a calculator (or python or C or any programming language) gives you 288.

I'd add that 2 is gained only by changing the order of operations.

 

[alcarl]

This is absolutely hilarious.

My 6th grade son came home with this math problem and I posted it on my facebook page. 5 friends posted it on their pages and now it is around the world. My original post was 9pm central time on 4-6-2011. For what it's worth I said 2 and the 2 engineers I work with agree.

All of this inspired me to create this video.

https://www.youtube.com/watch?v=wv19iAncrrQ

 

[JaredJames]

For what it's worth I said 2 and the 2 engineers I work with agree.

So for the record, you're saying all calculators and programming languages are wrong?

I've run this in 4 languages now plus through I don't know how many calculators and I always get 288.

 

[alcarl]

So for the record, you're saying all calculators and programming languages are wrong?

Not all of them, just the ones that say 288 ;)

 

[JaredJames]

Not all of them, just the ones that say 288 ;)

All of them then.

The calculators follow the basic outline as described previously and spit out 288. If you get 2 you are doing it wrong. Have you actually put it into a calculator? Instead of making ridiculous videos like that, why not make a video of you inputting it into a scientific calculator, exactly as it's written, and have it show 2 as the answer?

If you can't do that, then you're argument basically comes down to "all the calculators and computers are wrong".

And that's before we get to programming languages.

 

[alcarl]

Instead of making ridiculous videos like that, why not make a video of you inputting it into a scientific calculator, exactly as it's written, and have it show 2 as the answer?

OK.

https://www.youtube.com/watch?v=gFKGbU6ARQg

Lighten up, this isn't something that's going to alter your life or anything.

The problem is that it's a poorly written equation. When you have juxtaposed values like that, some people will solve that section as a unit before moving on. Tons of debate all over the web, no consensus, no one is likely to change their mind easily. My son did the problem the way his teacher asked him to and now it's just funny watching this seemingly simple question get so much attention.

 

[fierce9]

All of them then.

The calculators follow the basic outline as described previously and spit out 288. If you get 2 you are doing it wrong. Have you actually put it into a calculator? Instead of making ridiculous videos like that, why not make a video of you inputting it into a scientific calculator, exactly as it's written, and have it show 2 as the answer?

If you can't do that, then you're argument basically comes down to "all the calculators and computers are wrong".

And that's before we get to programming languages.

http://epsstore.ti.com/OA_HTML/csks...nge=null&fStartRow=0&fSortBy=2&fSortByOrder=1

my scientific calculator at home also says 2. Do you people really interpret 8/2x as 8/2*x and not 8/(2*x)?

 

[jhae2.718]

main(){
        printf("%d", 48*(9/2+3/2));
}

output:
240

You're doing integer division here, so C returns that 9/2=4 and 3/2=1, giving 48*5=240.

Try using 9.0 and 3.0.

 

[JaredJames]

OK.

Better, now there's a vote the other way.

Lighten up, this isn't something that's going to alter your life or anything.

Suupose this was a drug calculation for the amount of warfarin you need to maintain your life.

If 2 units is correct then 288 will kill you.

That is why the 'alphabet' is not 'ay bee cee...' but 'alpa charlie bravo..' when it matters.

I would suggest you use (lots of) brackets to achieve this aim if there is any possibility of ambiguity. I often find that when I encounter a new program I have to do this on a known test calculation.

It is an issue that really needs addressing.

I completely agree with studiot on this matter. If you want the multiplication done first, put it in brackets.

 

[Mute]

Yeah I understand that, but you re-wrote the problem. Like I stated above though, multiplication by juxtaposition supposedly takes priority before processing other operations. So wouldn't the answer to exactly how it's written really be 2?

According to whom does "multiplication by juxtaposition" take precedence? The standard for order of operations, at least in english-speaking countries, is PEMDAS, which does not include a priority rule for multiplication by juxtaposition. Just because the multiplication sign is implicit doesn't mean the priority of multiplication changes.

Ultimately, the priority rules are just a convention - but PEMDAS is the widely accepted convention. In some contexts (like certain symbolic expressions in academic papers), some other rule might be understood from context (eg., writing $Z = \sum \exp(E_i/kT)$, the kT is understood to be together from a separate context, so one doesn't need to write E/(kT) usually), but without context the default will be PEMDAS, even if "multiplication by juxtaposition" seems more natural.

As an example, in Canada if you win the lottery you have to answer a "skill testing question" to get your prize money (due to some anti-gambling laws). The questions are always simple order of operations problems. If this were your skill-testing question, answering 2 would lose you your money, as the answer would be written down according to standard PEDMAS.

 

[fraga]

Registered on this fine forum just because of this. :)

Regarding calculators, here is something interesting:

 

[JaredJames]

Registered on this fine forum just because of this. :)

Regarding calculators, here is something interesting:

That's explained in the link given above.

The makers changed things so that juxtaposition took priority.

There is nothing I've seen outside of these makers decision to do so that reinforces this. Certainly nothing in the science world which backs up this decision.

There are a lot of people registering "on this fine forum" just for this.

 

[JaredJames]

You're doing integer division here, so C returns that 9/2=4 and 3/2=1, giving 48*5=240.

Try using 9.0 and 3.0.

It doesn't matter, the rearranging is wrong.

 

[fraga]

There are a lot of people registering "on this fine forum" just for this.

Believe it or not, I initially saw this equation on a bodybuilding forum.
The pictures I posted came from there.

I then proceeded to google the equation.
This forum was on the top of the results.
That's why I registered.

 

[jhae2.718]

If I run the program with the correction, I get 288.

If we have 48/2(9+3), then if we correctly interpret it as
$$\frac{48}{2}(9+3)$$
Then we can distribute the 1/2 through the (9+3) term. E.g. 48/2(9+3)=48/2(12)=48*(12/2)=48*6=288

And (12/2) can be written as (9+3)/2, so I see no problem with writing it that way, other than the fact that writing 12 as the sum of three and nine is cumbersome.

 

[JaredJames]

I can see what's been done. I was running it in Python as a check but with very strict brackets (wrong order) which gave me the same answer.

 

[uart]

I voted for the answer 2 for the following reason. Everywhere in written mathematics (textbooks, papers, exams etc) that I see implied multiplication it is always is given higher precedence than division.

Something like $8 x^2 \div 2x$, for example, invariably means $8 x^2 \div (2x)$. So I've taken to modifying BIDMAS in the following was to also include "implied" multiplication.

BIIDMAS : (brackets, indices, implied multiplication, division, multiplication, addition, subtraction).

 

[JaredJames]

I voted for the answer 2 for the following reason. Everywhere in written mathematics (textbooks, papers, exams etc) that I see implied multiplication it is always is given high precedence than division.

Something like $8 x^2 \divide 2x$, for example, invariably means $8 x^2 \divide (2x)$. So I've taken to modifying BIDMAS in the following was to also include "implied" multiplication.

BIIDMAS : (brackets, indices, implied multiplication, division, multiplication, addition, subtraction).

Well unless that is official notation it's worthless. I've never seen it given precedence.

I'm curious how does 8x²x = 8x²2x?

 

[uart]

Well unless that is official notation it's worthless

I'm not sure what you mean. I'm talking about how written mathematical equations with implied multiplication are invariably interpreted in my experience.

Implied multiplication is where the "times" symbol is not explicitly written but is implied by algebraic convention.

In written mathematics for example $8 x^2 \div 2x$ invariably means $8 x^2 \div (2x)$.

James. I challenge you to find one example in a well written mathematical text or paper where the divide symbol is allow to "break" an implied multiplication. That is an instance where for example $8 x^2 \div 2x$ is written but $(8 x^2 \div 2) \times x$ is meant.

 

[JaredJames]

I'm not sure what you mean. I'm talking about how written mathematical equations with implied multiplication are invariably interpreted in my experience.

$8 x^2 \div 2x$, for example, invariably means $8 x^2 \div (2x)$.

Ah, so there's a divide in there.

Well it's only implied and allows interpretation. If you follow the standard rule on it, the implication is worthless.

The only way to guarantee it is to use brackets - which is something I always do. I'll go overkill on brackets if I need to because I want to make sure everyone knows exactly what I'm doing.

 

[uart]

Ah, so there's a divide in there.

Yes it was always there but a latex error preventing it from displaying properly for the first minute after I posted.

If you follow the standard rule on it, the implication is worthless.

Then please take on my challenge.

James. I challenge you to find one example in a well written mathematical text or paper where the divide symbol is allow to "break" an implied multiplication. That is an instance where for example $8 x^2 \div 2x$ is written but $(8 x^2 \div 2) \times x$ is what is meant.

 

[jhae2.718]

I don't think the $\div$ symbol is used once in any mathematical text I have.

 

[JaredJames]

Then please take on my challenge.

I don't think the ÷ symbol is used once in any mathematical text I have.

That was about to be my exact response to that challenge.

It's a non-issue if you use the alternate notation seen pretty much everywhere else.

You can do the division and leave the x term there and not have a problem.

Example:

9 / 2x = 4.5 / x

9x / 2x = 4.5

I've done no multiplication what-so-ever and still have simplified equations (or the answer in the latter case).

 

[uart]

That was about to be my exact response to that challenge.

It's a non-issue if you use the alternate notation seen pretty much everywhere else.

You can do the division and leave the x term there and not have a problem.

Example:

9 / 2x = 4.5 / x

9x / 2x = 4.5

I've done no multiplication what-so-ever and still have simplified equations (or the answer in the latter case).

Utter nonsense! If you do the division first in that example you get,

$9 / 2x = 4.5 x$ instead of the usual interpretation which would be $9 / 2x = 4.5 / x$.

 

[JaredJames]

Utter nonsense! If you do the division first in that example you get,

$9 / 2x = 4.5 x$ instead of the correct $9 / 2x = 4.5 / x$

I put brackets into try and force the multiplication in the calculator.

9/2x = 4.5/x = 18/4x

If you want to simplify the equation, you can just work with the numbers and ignore the x values. Cancel it down in other words.

Not quite what we're going for ey? I see, I've gone the wrong way.

In which case, 9 / 2x = 4.5x, you are correct. But then you've answered your own question of when the division can interrupt the multiplication, surely?

 

[jhae2.718]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
$$9/2x \equiv \frac{9}{2x}$$?

 

[JaredJames]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
$$9/2x \equiv \frac{9}{2x}$$?

Doesn't matter does it? It's reduced the same either way.

 

[jhae2.718]

Well, from what I remember from arithmetic, if we had (9/2)*x it would reduce to 4.5*x, and 9/(2x) would reduce to 4.5/x.

So the 9/2 would always go to 4.5, but the power of x would be either 1 or -1 based on the grouping.

 

[JaredJames]

Well, from what I remember from arithmetic, if we had (9/2)*x it would reduce to 4.5*x, and 9/(2x) would reduce to 4.5/x.

So the 9/2 would always go to 4.5, but the power of x would be either 1 or -1 based on the grouping.

Got me thinking now, think I've got it *** about face.

(Maths isn't my strong point, hence my need to try and follow rules as much as possible - or, well, this happens.)

 

[Borek]

And what about 48÷(9+3)2?

 

[uart]

I'm assuming that that example is if we follow the implied first interpretation, and that in this case
$$9/2x \equiv \frac{9}{2x}$$?

Yes $9 \div 2x \equiv \frac{9}{2x}$. In my experience that is how written mathematics is invariably interpreted.

Also if you have a calculator that can handle implied multiplication (most made in the last few years should allow this) then try typing in something like $12 \div 2\pi$, you'll find that it is interpreted exactly as I say.

BTW. I just checked on my aging "Casio fx-82MS" and

$12 \div 2\pi$ returned 1.909859 and $48 \div 2(9+3)$ returned 2.

In other words, don't just try this on C or MATLAB or python or anything else that doesn't allow algebraic implied multiplication, because it's irrelevant. Try it on a calculator that does allow algebraic implied multiplication if you really want to do a proof by calculator.

 

[jhae2.718]

48/(9+3)*2 = 48/12*2 = 4*2 = 8

 

[uart]

And what about 48÷(9+3)2?

Hey I don't like this notation either Borek but I'm just calling it as I see it commonly interpreted. This is why the fraction notation is preferred by most people as the "fraction bar" provides a well recognized "grouping symbol" and removes any ambiguity.

For that one my old calculator says "syntax error" (it wants either an explicit divide or times symbol after the bracketed expression). But if I was forced to make a call I still say implied multiplication and the answer is still 2, but I really wouldn't use that notation myself.

 

[uart]

48/(9+3)*2 = 48/12*2 = 4*2 = 8

Yep, it certain does if you put an explicit multiplication symbol in there. The whole point of this thread though is about what happens when (and what are the potential ambiguities that can occur when) we use the algebraic implied multiplication in an expression.

 

[Dembadon]

After thought, you might actually be right. For example:
$$a/bc$$

Is it $$\frac{ac}{b}$$ or is it $$\frac{a}{bc}$$?

EDIT: Perhaps the ambiguity of the question is getting to me, and I was correct initially: $$a/bc = \frac{ac}{b}$$

Since parenthesis weren't used around $b$ and $c$, I would interpret $a/bc$ as $\frac{ac}{b}$.

If parenthesis had been used, I would assume $\frac{a}{b}*\frac{1}{c}$.