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# Help with PEMDAS

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What is 48/2(9+3)? There seems to be a lot of debate about this...

Dec 22, 2016

#16
+20216
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True, I did not include WEB2.0 calculator in my list...I just kinda forgot it, but it delivers 288 if you enter this question as written

48/2(9+3)

48/(2)(9+3)     Note the EXTRA parentheses     delivers   ' 2 '     ...with which I am OK.....kinda

Dec 23, 2016

#1
+8918
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What is 48/2(9+3)? There seems to be a lot of debate about this...

$$a(b+c )=(a(b+c))$$

$$Without \ multi-sign, \ this \ is \ one \ thing$$

$$\large 48/2(9+3)=\frac{48}{2(9+3)}=\frac{48}{24}=2$$

!

Dec 22, 2016
edited by asinus  Dec 22, 2016
#2
+5

No, there is no debate about it if you understand what it means:(48/2) x (9+3) =(48/2) x (12/1). Then you multiply the top numbers(numerators) by the bottom numbers(denominators) and you get: 576/2 =288, which is the correct answer!!.

Dec 22, 2016
#3
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No

$$\large x/bc$$     is not     $$\large \frac{x}{b}\times c$$   rather   $$\large\ \frac{x}{bc}$$

so

$$\large 48/2(9+3)=\frac{48}{2(9+3)}=\frac{48}{24}=2$$

!

asinus  Dec 22, 2016
#4
+5

asinus: Look here at Wolfram/Alpha Computing Engine. It is ALMOST the final arbiter on mathematical questions:

http://www.wolframalpha.com/input/?i=48%2F2(9%2B3)%3F

Simplify the following:
(48 (9 + 3))/(2)

(48 (9 + 3))/(2) = (48 (9 + 3))/2:
(48 (9 + 3))/2

Dec 22, 2016
#5
+20216
+5

This shows the importance of parentheses and brackets which are often left out of questions on this forum.  Usually you can discern what was MEANT even if it is asked improperly, but not always.

To me

48/2(9+3) = 48/2   x   (9+3)     = 288

It does NOT equal  48/(2(9+3)) though that is what perhaps was MEANT (but NOT written)

Dec 22, 2016
#6
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What is 48/2(9+3)?

If there is a fractional element (number, variable, parenthesis expression) without a multi-signum, they are a product and are continued as a product.
These elements can not be simultaneously in the numerator and in the denominator.

2(9+3) is clearly a product.

That's why

$$48/2(9+3)=\frac{48}{2(9+3)}=\frac{48}{24}=2$$

!

asinus  Dec 22, 2016
#7
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No one said anything about them being simultaneously in the numerator and denominar.....(9+2) is only in the numerator....

I guess we will just agree to disagree....but go to this website and enter the equation as written:

or this one

https://www.wolframalpha.com/

or this one

http://www.quickmath.com/webMathematica3/quickmath/equations/solve/basic.jsp

or this one

http://www.homeschoolmath.net/worksheets/equation_calculator.php

Dec 23, 2016
#8
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This is an old problem.....here is a good explanantion...but no clear answer as to which is "more correct"

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

I think we should aim for UNambiguity in ALL of our calcs....wouldn't want to be launching in to space with ambiguous calculations used by the engineers !!!

Dec 23, 2016
#9
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I thank all who have thought about it!
Merry Christmas wishes asinus    !

asinus  Dec 23, 2016
#10
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My goodness I have never seen so many any answers for a trivial questions.

I'll add my 2 cents worth  :))

48/2(9+3)

=48/2*12

=24*12

=288

That is why you need to use brackets EVEN if you think the meaning is clear!

Dec 23, 2016
#11
+8918
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Melody,the equation is not unique.
To be clear is not trivial.
A guest mentions this essay. Readable!

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

Merry Christmas for you too.
asinus    !

asinus  Dec 23, 2016
#12
+107426
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ok asinus :)

I'll concede

It is a very good example of why it is so important to use brackets properly - even if the poster thinks they may not really be needed.

Melody  Dec 23, 2016
#13
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Personally, whenever I see something like this, I abstain and move on to the next question.

I can't be bothered to fathom out what the writer intends.

Dec 23, 2016
#14
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Beatng a dead horse .....but....I'm not "conceding" anything here

48/2(9+3)   means  [ using the order of operations ]

48/ 2 * (9 + 3)   =

24 * 12  =

288

For this to equal 2....we would have to have

48 / [ 2 (9 + 3) ]  =

48 / [ 24  ]  =  2

So this........ 48/2(9+3) ....... is clearly  different from this.......  48 / [ 2 (9 + 3) ]

Dec 23, 2016
#15
+1841
+10

Asinus is right, this is debated quite often in academic circles –at least among the science and math majors.

I’ll add another two cents worth.

A fraction is ostensibly the same as division.  Most will agree that all of these are the same thing the same as

$$3 \div 5 \\ \dfrac {3}{5}\\ 3/5 \\$$

Most will also know that order of operations (PEMDAS) says division take precedence over addition. Meaning 5+20/4+1 = 11 (not 5)

But

$$\dfrac {5+20}{4+1} = 5$$

There are no parentheses here, and addition now appears to takes precedence over division --both before and after the division  to give the "correct" answer of 5.

(Take note, guest #7, you listed several calculators but left out this site’s calculator)

These are comments from my favorite troll:

If you paste this  48/(2)(9+3)  into the site calculator, it returns 2 as the solution.

This is isn’t a bug. The parentheses cause the value to be treated as a variable. Implicit multiplication of variables takes precedents over division – a noted exception, dating back to the late 1960s, to the normal convention of mathematical hierarchy . Herr Massow’s calculator is the only one I know of that that does this, and it’s probably because it allows the use of variables.

Of course, if you put the explicit multiplication operator in: 48/(2)*(9+3) or leave the parentheses off the 2 then it returns 288,  the normal solution for numeric values.

This calc does have one hierarchal fault. “Stacked powers” are right-associated. The calc resolves them from the left.

4^3^2 returns (4^3)^2 = 4096 (It shows the order at the top).

With the correct hierarchy, it is 262144.

It’s always a good idea to know the quirks and faults of the tools you might use.  I drove an old car once that ran great, but the breaks didn’t work so well –I had to drag my foot to help it stop. It sure wore out my shoes faster than normal. :)

Dec 23, 2016
edited by GingerAle  Dec 23, 2016
#16
+20216
+5

True, I did not include WEB2.0 calculator in my list...I just kinda forgot it, but it delivers 288 if you enter this question as written

48/2(9+3)

48/(2)(9+3)     Note the EXTRA parentheses     delivers   ' 2 '     ...with which I am OK.....kinda

ElectricPavlov  Dec 23, 2016
#17
+107426
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Thanks Ginger, you have made some very good points.

I did not realize that the correct (by convention) answer for 48/(2)(9+3) is 2 but I can see why this is the case.

However, I do not accept that tthe following does not have brackets.  It is like saying that 6x does not have a multiply sign. The multiply sign is implied just as the brackets are implied in your example.

$$\dfrac {5+20}{4+1} = 5$$

this simpy means

(5+20)/(4+1)=5

The same as square root is done before addition but

$$\sqrt{21+4}=\sqrt{25}=5$$

the brackets are implied.

Melody  Dec 24, 2016
#18
+1841
+5

I may have created more ambiguity with my post.

I notice Asinus has started a discussion on this topic on the German forum.  And there’s EP’s reference to this post, here:    http://web2.0calc.com/questions/help_57875#r1

The “this is not a bug” comment may have implied that 48/(2)(9+3) =2 is true because of the parentheses.  It is not.  I’m sure the “not a bug” had to do with the nature of how Web2.0calc’s complier interprets variables. Not that this is a true statement.

This sentence implies it should not do that:

“Herr Massow’s calculator is the only one I know of that that does this, and it’s probably because it allows the use of variables.”

I understood this because of his previous comments not directly related to this post, I learned the only way to concatenate variables on the web2.0 calculator is to place parentheses around them: (a)(b), else “ab” will be treated as a variable named “ab” instead of implicit multiplication of “a” and “b”.  Using this a(b) will cause the calc to treat it as function “a” with argument “b”.

You can place parentheses around numbers in other calculators, but it returns 288.  At lot of calculators will solve for variables x and y, but few, if any, allow a value to be directly assigned to a variable.  So, it just explains why web2.0calc  treats literal numbers as variables (and gives a wrong answer).

This statement  . . .

“Of course, if you put the explicit multiplication operator in: 48/(2)*(9+3) or leave the parentheses off the 2 then it returns 288,  the normal solution for numeric values.”

implies that 2 is an abnormal result and is clear that 288 is the normal result for numeric values.

This statement …

“Implicit multiplication of variables takes precedents over division – a noted exception, dating back to the late 1960s, to the normal convention of mathematical hierarchy.”

makes it is clear that the multiplication of variables takes precedents over division.  But the example above has no variables in it.  I suppose, if pressed, he would agree it is in fact a bug, because there are no variables in the parenthetical operators.

---------------------------------

I have several instructables and comments by our troll relating to the forum’s calculator -- including a humorous one starting with, “I wish Herr Massow would stick to neutering his pets instead of his calculator. . . .”

This comment preceded a lament that all the constants except for pi and e are gone.

“ . . .  Now I have to manually enter the constants to calculate how hyper-polarized I am. What a bummer!”