48÷2(9+3) = ?

[Virtuoso]

My wife's working on a math minor for her Chem degree so I'll have to ask her when she get's back but...

..based on my dazed remembrance of what I was taught, I'm saying the answer is 2.

 

[mechanicx]

This equation has multiple threads on the internet lol. While trying to search for the distributive rule's precedence. I ran across another thread on another site and someone posted basically my same argument more eloquently. The question still remains whether or not it is the correct order of precedence.

http://www.letsrun.com/forum/flat_read.php?board=1&thread=3992625&id=3993955

"The 2 is outside the brackets. In fact "48÷2" is outside the brackets, and therefore outside the scope of its influence, in spite of the direct contact of the 2 with the parenthesis. There is no direct contact rule. Associativity, commutativity, distributivity, and whatever other rules or properties you might think of, do not alter the rules of precedence. You can not ignore the preceding terms. Once you perform the calculation inside the brackets, there are no more brackets, and therefore nothing forces the 2 to be resolved with the new non-bracketed expression prematurely. The 2 is connected by multiplication after, and division before. You can not ignore either, but must follow a centuries old convention that binds all mathematicians above the age of 10, worth their salt, to evaluate such infixed expressions from left to right:

Special note: Unary operators, e.g. negative signs, factorials, form part of the number, and must be treated as an integrated number.

48÷2(9+3)=?

Step 1: Perform all groupings, e.g. brackets, parentheses, roots, from left to right
48÷2*12=?

Step 2: Perform all powers (exponents), from left to right (with stacked powers top down)
48÷2*12=?

Step 3: Perform all multiplication and division from left to right
Step 3.1: 24*12=?
Step 3.2: 288=?

Step 4: Perform all addition subtraction from left to right
288=?

Would you like some further help?

The potential problems arising from any "ambiguity" issue was discovered by mathematicians centuries ago, if not longer, and the orders of precedence were agreed by universal convention, and defined by the precedence rule, at the time of creation of the symbolic infix notation for equations. These rules are very precise, leading to only one single interpretation of every equation that gives predictable and repeatable results every time. Mathematics could not have progressed with this kind of basic ambiguity.

Parentheses are not required when it just makes explicit, that which is implicit by universal convention, although it can aid in human understanding."

 

[mechanicx]

Originally Posted By: Warstud
Remember: PEDMAS

P Parentheses first
E Exponents (ie Powers and Square Roots, etc.)
MD Multiplication and Division (left-to-right)
AS Addition and Subtraction (left-to-right)


Exactly so why did you violate it at the end of your solution

.

 

[Warstud]

Originally Posted By: Warstud
Originally Posted By: Warstud
IIRC

48÷2(9+3) =
24 brackets mean x(12)=
or 24x12=288


Wrong #1

Wrong #2
48÷2(9+3) =

48÷2(12)=

48÷2x12=

48÷24=

=2



Hopefully Right
48÷2(9+3) =

48÷2(12)=

24(12)=

24x12=

=288

 

[Virtuoso]

I stand fully corrected. The answer is 288.

Math always did make my head spin. Much more of a language arts kinda guy..

 

[phillyFX4]

if 48 is the number being divided into by a whole #, how can the answer be any higher than 48?

2 x (9+3) is gonna be a whole number right? (24)

the problems answer, as written, should be 2

 

[Al]

Originally Posted By: phillyFX4
if 48 is the number being divided into by a whole #, how can the answer be any higher than 48?

bc its not divided by (9+3)?

http://www.physicsforums.com/showthread.php?t=488334

 

[tom slick]

Originally Posted By: phillyFX4
if 48 is the number being divided into by a whole #, how can the answer be any higher than 48?

2 x (9+3) is gonna be a whole number right? (24)

the problems answer, as written, should be 2


According to the order of operations (PEMDAS) you must divide 48 by 2 before multiplying by (9+3) because regardless of multiplying or dividing you move from left to right.

for any of you that think this form of an equation is irrelevant think about how you'd enter it into a cell in a spreadsheet. If you enter =48/2(9+3) Excel will pop up an error/correction message and correct it to =48/2*(9+3) and solve it at 288.

 

[Reelizmpro]

P in PEMDAS ...Parentheses first. Don't mess with the 48 until you have simplified the expression "2(9+3)". Only way you even touch the 48 first is if it's (48/2)(9+3) explicitly, but it's not. In algebra, when you divide 48/2(9+3) should be typed in MATLAB as it's understood as 48/((2(9+3)) to make the calculation clear. That's where the parentheses are...not around the (48/2).

 

[Reelizmpro]

Originally Posted By: mechanicx
Originally Posted By: Reelizmpro
Look closely...

48/2(9+3)
48/2(12)
2 is actually DISTRIBUTED into parentheses as it's the only way to get rid of parentheses at this point. Which is not the same as 2x12. Technically, it's multiplication through distribution.
48/24
2

another way to do it...

48/2(9+3)
48/(18+6) 2 is Distributed into parentheses but they remain UNTIL the two terms have been added! This takes priority over the DIVIDE symbol.
48/24
2

yet another way to do it...
48/2(9+3)
24/(9+3) expression is simplified by the factor 2. Parentheses still remain!
24/12
2

You must resist the temptation to divide into 48 without first getting rid of the parentheses.

Mechanix, even with the 2 factored out, the parentheses still remain and must be simplified BEFORE dividing.


I completely understand your argument. But again, I'm saying I don't necessarily believe the order of operation states the 2 most be distributed first. Maybe they do maybe they don't but I'd be just taking your word for it. You say the only way to get rid of the parenthesis is by multiplying the 2. But that's not really true since you can just multiply the product of 48/2 by the product in parenthesis and get an answer. Ie, 48/2(12)=24(12)=288. I think 2(12) does equal 2x(12) equals 2x12. I could be wrong but it's a matter of whether or not the order of operation follows the order you are proposing. No one has presented a definitive source other than stating their personal knowledge or belief. So far, we all agree the rules state what's in parenthesis must be factored first, but the rules of distributing the number in front of the parenthsis aren't so certain. Actually you violate that rule in your second example. You seem to be saying the distributive property takes precedence but my understanding of the distributive rule is it just explains how multiplication works and not necessarily the order since the order doesn't really change the product,ie, 2(9+3)=2(12)=18+6=24.


24=24 sure but that's not the point. The point is 2(12) takes precedent over divide and the answer is 2 while 2*12 does NOT and will give you 288.

There's no (48/2) but 48/2(9+3) which implies (and if you type into a calculator...) 48/[2(9+3)]. 2 is a factor of (9+3), it's an expression "2(9+3)". In my examples, I was merely showing different ways to get the answer. My second example of distribution is perfectly within the rules as it's done to help simplify the parentheses. All work done to simplify the parentheses takes priority (which includes multiplication through distribution).

In college I was taught to write out ALL my steps so I know the significance of parentheses and PEMDAS. 48 divided by ...is a trap. If you were to write it on paper as "48" OVER "2(9+3)" would your answer still be 288?

 

[mechanicx]

But the order of operations say solve what's inside parenthesis first ie 2(9+3)=2(12) and not (2*9+2*3) , but that doesn't necessarily mean a number next to the parenthesis must be solved first too. The distributive property just says the result would be the same but does not necesarily say distribution takes precedence.

If we could agree on whether it does then there'd be no debate. That is the question, does implied multipication by proximity of parenthesis take precedence or not? Until both sides can agree on a definitive source for orders of operation the argument is redundant.

 

[Reelizmpro]

Why do you want to group 48/2? The 2 is clearly a factor of (9+3) and can be rewritten as (18+6). Let's say the 2 was never factored out and it was 48 / (18+6). Still getting 288?

 

[Reelizmpro]

Originally Posted By: mechanicx
But the order of operations say solve what's inside parenthesis first ie 2(9+3)=2(12) and not (2*9+2*3) , but that doesn't necessarily mean a number next to the parenthesis must be solved first too. The distributive property just says the result would be the same but does not necesarily say distribution takes precedence.

If we could agree on whether it does then there'd be no debate. That is the question, does implied multipication by proximity of parenthesis take precedence or not? Until both sides can agree on a definitive source for orders of operation the argument is redundant.


2(12) or (18+6) makes no difference. You still distribute the 2 into the parentheses making it 24. How else do you get rid of the parentheses? it takes precedence according to PEMDAS. To answer your question, yes because it's a factor of, that is it's a part of (9+3).

 

[mechanicx]

If you distributed the 2 then there would not be any parenthesis, 2(9+3)=18+6 and not (18+6). The latter is like saying the formula was originally written 48/[2(9+3)]. It was written 48/2(9+3). So if you distribute the 2 why should it not be 48/18+6. And I'm not saying that it would then simplify to 48/24 but to 2.666+6 which would be the wrong solution.

 

[mechanicx]

Originally Posted By: Reelizmpro
Why do you want to group 48/2? The 2 is clearly a factor of (9+3) and can be rewritten as (18+6). Let's say the 2 was never factored out and it was 48 / (18+6). Still getting 288?



What I'm saying is even if we accepted that the 2 is factored out you should still solve inside the parenthesis first so 2(12) means the same thing as 2*12 and neither would take precedence over working left to right and dividing first. Of course I could be wrong but I'd just be taking your word for it.

 

[kender]

Originally Posted By: mechanicx
If you distributed the 2 then there would not be any parenthesis, 2(9+3)=18+6 and not (18+6). The latter is like saying the formula was originally written 48/[2(9+3)]. It was written 48/2(9+3). So if you distribute the 2 why should it not be 48/18+6. And I'm not saying that it would then simplify to 48/24 but to 2.666+6 which would be the wrong solution.


if you distribute the 2 first,2(9+3)=(18+6),the parenthesis remain until you add (18+6)=24. If you add the (9+3) first inside the parenthesis (9+3)=(12), the parenthesis remain until you multiply by 2....2(12)=24.

 

[mechanicx]

Originally Posted By: kender
if you distribute the 2 first,2(9+3)=(18+6),the parenthesis remain until you add (18+6)=24.


OK by what rule says they parenthesis remain? Once the 2 is distributed there is no good reason why they should remain.


Quote:
If you add the (9+3) first inside the parenthesis (9+3)=(12), the parenthesis remain until you multiply by 2....2(12)=24.


The parenthesis remain because they also signify multiplication. If you distribute the 2 first you no longer need to signify multiplication. If I add the (9+3) first that doesn't mean I have to multiply the 2 next and not divide the 48. Also what rule says I can't do this 2(12)=2*12?

 

[Reelizmpro]

Originally Posted By: mechanicx
Originally Posted By: kender
if you distribute the 2 first,2(9+3)=(18+6),the parenthesis remain until you add (18+6)=24.


OK by what rule says they parenthesis remain? Once the 2 is distributed there is no good reason why they should remain.


Quote:
If you add the (9+3) first inside the parenthesis (9+3)=(12), the parenthesis remain until you multiply by 2....2(12)=24.


The parenthesis remain because they also signify multiplication. If you distribute the 2 first you no longer need to signify multiplication. If I add the (9+3) first that doesn't mean I have to multiply the 2 next and not divide the 48. Also what rule says I can't do this 2(12)=2*12?



Hmmm the distributive rule of algebra? The reason they remain is because it shows you need to add 9+3 before anything else...pretty important step. The problem is sloppy math. Take it step by step. Write the problem on paper and show each and every step. Get rid of the parentheses in the proper manner. The parentheses remain until the expression 2(9+3) = 24.

Please understand the difference...
2(12) shows multiplication through distribution. Gets priority.
2*12 shows simple multiplication only.
note: there are no multiplication signs in this problem so don't add any.

 

[hate2work]

Somehow I missed this thread until this morning. Since my math skills are rather lacking, I asked someone I know that has multiple degrees in math and computer science, my brother Jeff. Below is his answer, and I'm willing to bet he's correct. His response is below. My comments are preceded with an >

=================================================================

Hi Don,

Jeff here.

>Here's the equation.... 48÷2(9+3) = ?

>The answer depends on the order of operations, right?

Yep.

>IIRC, it's parentheses first, then divide, then multiply...288...right?

Yep.

The “conventional” order of operations is :

1.) evaluate anything in parentheses first. When there are multiple parentheses,

evalluate them inside out and then left to right

Then

2.) if an exponent appears in the expression, it’s next

then

3.) multiplication and division (same priority)


and lastly

4.) addition and subtraction (same priority)

and furthermore, within each category, expressions are evaluated left to right.

So, 48/2(9+3) = 48/2(12) = 24(12) = 288.

The 48 gets divided by 2 first , THEN multiplied by 12, because of the “left to right” rule for operations within the same priority.

Another example :

5/10/15 (5 divided by 10 divided by 15)

Is the answer 1/30 or 7.5 ?

Left to right, it’s 5/10/15 = 1/2/15 = 1/30

Not 5/10/15 = 5/2/3 = 7.5 (right to left).

Or another example

3 + 5 * 10

It’s 53, not 80 (multiplication/division priority over addition/subtraction).

Jeff.

 

[Reelizmpro]

Originally Posted By: hate2work


So, 48/2(9+3) = 48/2(12) = 24(12) = 288.


48/2(12) = 48/24 = 2
LOL...Parentheses first. Ask him to check his work with pen/paper...no calculators.