48÷2(9+3) = ????

[Z-man]

If we start with:

48÷2(9+3) = n

I think we all agree that this can be turned into:

48÷2(12) = n

(So far, so good).

Here is where the paths diverge. What is the 2 'linked to' first? The 48? or the 12?

I assert that the parenthesis serve to prioritize the 9+3 part, and thus, do not prioritize the 2(12) piece. Additionally, since all of the functions are now 'on the same plane' (ie multiplication) the PEDMAS rule dictates that from here on out, the equation must be solved from LEFT to RIGHT. Therefore, the equation can be re-written thusly:

48÷2*12 = n

Applying the left to right rule, the equation becomes:

24 * 12 =n

Solve for n:

24 * 12 = 288.

In my opinion, the only way the result could be 2 is if the equation were written like this:

48÷(2(9+3)) = n

With the addition of the 2nd set of parenthesis, the 2 and (9+3) must be reduced before considering the 48. But without the 2nd set of parenthesis, there is no reason to mutliply the 2 with (9+3) before considering the 48.

Unless someone can prove, using references that 2(9+3) means that after the equation within the forumla is solved, then the 2 MUST be applied to the result, then I contend that by the laws PEDMAS, the answer must be 288. Unless you can show me how 2(9 + 3) is not the same as 2 * (9+3), then I cannot accept any other answer than 288.

Well, maybe 42. (Zaphod & Mr. Dent would be proud)

(Note: this is the same argument I used in the beginning of this thread...)
-Z

[Geary]

48÷2(9+3) = 48÷[2x(9+3)] = 2

The 2 alongside the parenthesis without an "x" sign means that it is "connected to" that parenthesis, and MUST multiply whatever total is within the parenthesis. This is not open to interpretation. It is a basic rule.

For the result to be 288, the equation would be written :

48÷2x(9+3) or equivalent.

[Z-man]

Originally Posted by Geary

48÷2(9+3) = 48÷[2x(9+3)] = 2

The 2 alongside the parenthesis without an "x" sign means that it is "connected to" that parenthesis, and MUST multiply whatever total is within the parenthesis. This is not open to interpretation. It is a basic rule.

Please support the above statement with a link to a mathematical rule that supports this. (as in school, wikipedia is NOT a valid reference! )

-Z

[DARISC]

Originally Posted by winders

...the guy that builds a ton of successful race cars...

Well, I call BS on that!!!

The minimum amount of race cars, greater than 1, that a guy can build that weigh a two thousand pound ton = 2.

Reducing the car scale to 1:18, the maximum amount of race cars that a guy can build that weigh a two thousand pound ton = 288.

THereford I am biased against trusting that guy's brake bias for a Porsche, which is too heavy.

[svandamme]

Originally Posted by Geary

48÷2(9+3) = 48÷[2x(9+3)] = 2

The 2 alongside the parenthesis without an "x" sign means that it is "connected to" that parenthesis, and MUST multiply whatever total is within the parenthesis. This is not open to interpretation. It is a basic rule.

For the result to be 288, the equation would be written :

48÷2x(9+3) or equivalent.

48÷2(9+3) IS equivalent to 48÷2x(9+3)

the lack of an X does not mean it's connected any different then if there is an x

[T77911S]

when i first did it i thought it was 2
but i remember thinking something was not right when i multiplied 2 times 12. i should have gone left to right.

288 is correct

[vash]

288...i got bored and entered the equation into my calculator.

my gut said "2"..hahaha

[Geary]

Originally Posted by svandamme

48÷2(9+3) IS equivalent to 48÷2x(9+3)

the lack of an X does not mean it's connected any different then if there is an x

2( ) means 2-of whatever is within the parenthesis. The 2 cannot be involved in any other operation in the equation. It is strictly multiplication of whatever total is within the parenthesis .. and ONLY because it lack the "x" which would separate it from the parenthesis.

While I don't have any math book on hand to prove ANY operation's sequence, I know for certain that this is a hard rule we were taught .. over & over through repetition. (Our teacher, Peter Cole, was an actual mathematician who only taught so that he could have a more flexible surfing schedule )

[Z-man]

Originally Posted by Geary

2( ) means 2-of whatever is within the parenthesis. The 2 cannot be involved in any other operation in the equation. It is strictly multiplication of whatever total is within the parenthesis .. and ONLY because it lack the "x" which would separate it from the parenthesis.

While I don't have any math book on hand to prove ANY operation's sequence, I know for certain that this is a hard rule we were taught .. over & over through repetition. (Our teacher, Peter Cole, was an actual mathematician who only taught so that he could have a more flexible surfing schedule )

Well, for now, it is your word -- post up a reference to your rule, and then we an discuss further...

Or - I can counter with "Our teacher, Dr. Martin Brubaker was an actual mathemetician who only taught so that he could torture us college students)



-Z-man.

[dad911]

Originally Posted by Geary

.....
The 2 alongside the parenthesis without an "x" sign means that it is "connected to" that parenthesis, and MUST multiply whatever total is within the parenthesis. This is not open to interpretation. It is a basic rule.
.....

Please provide source for that "basic rule".

So is 1/2(x) equal to .5x or 1/(2x) ??

By 'left to right' convention it is .5x, by your 'basic rule' it is the latter.

[masraum]

Order of arithmetic operations; in particular, the 48/2(9+3) question.

George Mark Bergman's Home Page

George M. Bergman | Department of Mathematics at University of California Berkeley

[Bill Verburg]

Sad to say but not everyone teaching math has a complete grasp of all it's nuances

the pertinent property that is causing all the issues is one of the most basic, so basic that many just dash through it w/o truly appreciating its meaning

once again multiplication is associative as is addition, one can place parenthesis wherever one chooses thus altering the basic left to right rule, or leave the parenthesis off retaining the basic left to right rule, the result does not change, ie you have consistent results no matter how it's done.

(ab)c = a(bc) = abc and (a+b)+c = (a+b)+c = a+b+c

division and subtraction do not have the associative property, you can not place a parentheses affecting any division or subtraction in any expression that alters the rule left to right in any way.

(a/b)c is not equal to a/(bc) and (ab)/c is not equal to a(b/c)

In any expression containing division or subtraction you have to go in strict left to right order. If you do not then you get different results as we have seen here in this thread.

I demonstrated by alternate methods ie rationalizing, that strict left to right will give consistent results



additionally the Parenthesis part of PEMDAS is apparently misunderstood. It says to do any operation inside a parenthesis if you can, in fact it is not necessary to do the inside of the parenthesis first as in this problem you can use the distributive property

additionally one of the more important basic properties of multiplying the coefficient outside a parentheses by the inside is that it clears the parenthesis. ie once you multiply the coefficient by the parenthesis the parenthesis is gone and cannot be reused

using the distributive property and going strictly left to right where the division is the first thing gives the following





we go no further here because another property has been violated, when a term outside the parentheses is multiplied by the parenthesis it clears the parenthesis from the expression if the parentheses is cleared as it ought to you get



another inconsistent result

[svandamme]

Originally Posted by Geary

2( ) means 2-of whatever is within the parenthesis. The 2 cannot be involved in any other operation in the equation. It is strictly multiplication of whatever total is within the parenthesis .. and ONLY because it lack the "x" which would separate it from the parenthesis.

While I don't have any math book on hand to prove ANY operation's sequence, I know for certain that this is a hard rule we were taught .. over & over through repetition. (Our teacher, Peter Cole, was an actual mathematician who only taught so that he could have a more flexible surfing schedule )

Well my teacher told me that multiplication could be indicated in many ways

it could be 2xY, 2*Y or 2.Y or 2Y, all those are the same thing...

Hence what Bergman says
"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,"

So he clearly confirms that the symbol could be used there. Once it's (9+3) has been worked down to .... 12



so 2x(9+3) = 2*(9+3)= 2.(9+3) = 2(9+3) It's 2 times whatever follows

It only works that way if it's algebra , pure numbers math, would be confusing

2 12 or 2 . 12 would not make sense, so the symbol x is required



But that wasn't the case here.. it was 2(9+3)

Not (2(9+3))

The only way you can reduce 2(9+3), is by first taking out the 9+3, making a 12 out of it

At that point the () serves no purpose anymore, andit becomes 2*12

Which is fine

Except there's still a 48/ in front of it, which has to be dealt with first.

48 / 2 * 12 is and always will be 288


And the above post, of a maths professor calling it ambigious.. He's obviously making a case for using parenthesis whenever possible so there can't be any confusion by those who don't know the rules...

And although he did not actually take sides as to the outcome having to be 2 or 288..

he did in fact confirm that the 2(9+3) = 2x12 or 2x(9+3)

And by doing so , he actually confirms that it the outcome ought to be 288

[flatbutt]

entering this =48/2*(9+3) into Excel yields 288. Does that help?

[winders]

This is not about the associative property. It's not about the actual mathematics. It's not about anything other than convention. The reason this is a problem is that some people use a convention that states implicit multiplication (juxtaposition) has precedence over division or explicit multiplication. That's it. That's the whole enchilada.

That is not a widely held convention but it is widely held enough to render the original equation ambiguous. Ambiguity is why the mathematics community debates the equation so ferociously and why the absolute consensus is that it should be written more clearly. Before you say it is not ambiguous, don't. It is, period. It's ambiguous for the same reason 1/2x is ambiguous. Because the intent of the author, the convention they follow, is unclear. Some authors will mean x/2 while others might mean 1/(2x).

You can say there is nothing to debate. But, clearly that is completely wrong as the debate rages on all over the Internet...

Scott

[Bill Verburg]

Originally Posted by Geary

48÷2(9+3) = 48÷[2x(9+3)] = 2

....

That is a false equality

the only rule that allows the addition of a parenthesis that was not explicitly written in the original problem is the the associative property of multiplication or the associative property of addition. The reason is that doing so can alter the order of operations and thus the result, as we have seen here.

division and subtraction do not have the associative property so you cannot add a parenthesis anywhere that will alter the left to right rule, this rule was violated above when [] was added to the original problem.

[winders]

Originally Posted by Bill Verburg

That is a false equality

That depends solely on what conventions you are using. Geary is using a convention that states implicit multiplication (juxtaposition) has precedence over division or explicit multiplication.

Using that convention, this is true:

48÷2(9+3) = 48÷[2x(9+3)] = 2

Again, this is not about the associative property.

Scott

[Bill Verburg]

Originally Posted by winders

That depends solely on what conventions you are using. Geary is using a convention that states implicit multiplication (juxtaposition) has precedence over division or explicit multiplication.

Using that convention, this is true:

48÷2(9+3) = 48÷[2x(9+3)] = 2

Again, this is not about the associative property.

Scott

unfortunately that is not true

the basic properties are the foundation of all mathematics, it's when people make up rules that confusion and inconsistent results pop up

I suggest that you go rear a basic math text

the associative property is the only one that allows the addition of a parentheses that was not explicitly written in the problem

it is usually given as abc =(ab)c =a(bc) or a+b+c = (a+b)+c = a+(b+c)

it allows the introduction of a parenthesis so as to alter the order of operations, it is not allowed when division and subtraction are involved. This is the only way a parentheses can be added to an expression.

[DARISC]

Originally Posted by winders

This is not about the associative property. It's not about the actual mathematics. It's not about anything other than convention.

Scott

OK Scott, NOW I truly think I see the root cause why your butting up against Bill Verburg's attempts to enlighten you. You are misunderstanding the meaning of, or misusing the word 'convention'.

Why I say this is because you state that:

"The reason this is a problem is that some people use a convention that states implicit multiplication (juxtaposition) has precedence over division or explicit multiplication."

You equate a convention with a rule of math - they are not the same.

Let me paraphrase this statement:

"The reason this is a problem is that some people use a convention that uses a rule that implicit multiplication (juxtaposition) has precedence over division or explicit multiplication."

And, assuming that you agree thai is a reasonable paraphrasing, I think you'll agree that this whole dispute is about a rule that implicit multiplication has precedence over division or explicit multiplication.

Cite where that rule can be found or you have no argument. I don't believe you can. As Bill Verburg posted:

"'There is a convention that says you do implicit multiplication before division' is a rule only in your world, it is not a convention recognized anywhere else except in the minds of others that don't understand math."

[winders]

Originally Posted by Bill Verburg

unfortunately that is not true

the basic properties are the foundation of all mathematics, it's when people make up rules that confusion and inconsistent results pop up

I suggest that you go rear a basic math text

the associative property is the only one that allows the addition of a parentheses that was not explicitly written in the problem

it is usually given as abc =(ab)c =a(bc) or a+b+c = (a+b)+c = a+(b+c)

it allows the introduction of a parenthesis so as to alter the order of operations, it is not allowed when division and subtraction are involved. This is the only way a parentheses can be added to an expression.

Bill, Bill, Bill......

Mathematical convention - AoPSWiki

Math 1010 on-line

A "convention" is not mathematics. It's not a basic property of mathematics. Solving from left to right is a convention. Tomorrow, the notation could be changed to right to left and the mathematics would still work. We use infix notation today. That is a convention. Sometime in the future, it could be decided that we will be using postfix (Reverse Polish) notation. The mathematics would still be the same.

If the mathematics community decided that implicit multiplication (juxtaposition) has precedence over division or explicit multiplication, only the notation would change. The mathematics would be unaffected.

Don't go confusing conventions with properties of mathematics or the foundations of mathematics.

Scott