# Simple but not so simple mathematical question

30 ÷ 2(2+3)÷ 5

I said 15 but then some argued it to be 0.6. Which is which? Please explain.
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RetiredCommented:

B     Brackets first
O     Orders (ie Powers and Square Roots, etc.)
DM   Division and Multiplication (left-to-right)
Step by step:
B     Brackets first
30 ÷ 2(2+3)÷ 5 ->
DM   Division and Multiplication (left-to-right)
30 ÷ 2 x 5 ÷ 5 ->
15 x 5 ÷ 5 ->
75 ÷ 5
15
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SD-WAN SimplifiedCommented:
Order of operations. First is the () so 2+3 = 5
then is 2*5 = 10
Then go left to right.
30/10=3
3/5=.6
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Commented:
0.6

Use the BODMAS rule for operations
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Author Commented:
Using BODMAS, why not this?

30 ÷ 2(2+3)÷ 5
= 30 ÷ 2 x 5 ÷ 5
= 15 x 5 ÷ 5
= 75 ÷ 5
= 15
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SD-WAN SimplifiedCommented:
Because the 2(2+3) is all together as part of "brackets"
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SD-WAN SimplifiedCommented:
It actually can be rewritten (2*2+2*3)
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Author Commented:
Yes, I think this question is highly debatable. Erm .. who should I believe? A YouTube video from Taiwan revealed models of calculators giving different answers and they claim to say 15 is the right answer.
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RetiredCommented:
If Because the 2(2+3) is all together as part of "brackets" were the case, I think the expression would have to be written as follows: 30 ÷ (2(2+3)) ÷ 5, since 2 multiplies (2+3)...
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Author Commented:
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Information Services ManagerCommented:
Order of operations is not really debatable...lol
Brackets are your means of modifying the order.

30 ÷ 2 * (2+3)  ÷ 5

First order
2 + 3 = 5
Second
2 * (5) = 10
Final
30 / 10 / 5 = 0.6
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Information Services ManagerCommented:
....wait a minute...

Ok ...I see what's going on here.

It is 15.

Parentheses multiplication comes LAST.

Since 2 is appended to the parentheses the division would actually come first.

Therefore...
(2+3) / 5 = 1
2 * 1 = 2
30 / 2 = 15
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RetiredCommented:

@xuserx2000: I agree with you. However, if the BODMAS rule for operations are used, then that limits the debate...;-)

But, in any case, it's an interesting discussion!
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Information Services ManagerCommented:
30 / (2 * (2 + 3)) / 5 = 0.6

but without the parentheses ..it's 15.
30 / 2 * ((2 + 3) / 5)
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RetiredCommented:
and the question is: What is the answer? 30 ÷ 2(2+3)÷ 5, not 30 ÷ (2 * (2 + 3)) ÷ 5  or 30 ÷ 2 * ((2 + 3) ÷ 5)
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Information Services ManagerCommented:

The second equation you posted, is what is inferred by the order of operations, because multiplication with a parenthesed value comes last.  The ((parentheses arithmetic) / value) is inferred as the same operation in the absence of parentheses.
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Commented:
I am positive that the answer is .6

PEMDAS is the correct way to solve this problem
Parenthesis
Exponents
Multiplacation
Division
Subtraction

30 ÷ 2(2+3)÷ 5

Parenthesis first leaves you with

30 ÷ 2(5) ÷5

Simplified to

30 ÷ 10 ÷ 5

Then it is division in the order it appears

30 ÷ 10 = 3

3÷ 5 = .6

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Information Services ManagerCommented:
It's actually a "trick" question, because the multiplication of a parenthesed evaluation is the only exception to the order.  The omission of the brackets for the default order is what is playing head games with everyone here.

2 * (5) -  cannot be the second step unless it was presented as (2 * (2 + 3))
((2 + 3) / 5) is the second step for this reason.

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Information Services ManagerCommented:
""I am positive that the answer is .6"""

....you would be wrong.

Source:  every calculator on earth.
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Author Commented:
Yes, my original answer was 15 as mentioned in my question.
An unexpected lively debate :)
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Author Commented:
>>Source:  every calculator on earth.

Calculators can even be wrong if you check out this YouTube video.
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Information Services ManagerCommented:
Calculators cannot be wrong.

People who program them can be wrong.
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SD-WAN SimplifiedCommented:
30 ÷ 2(2+3)÷ 5
30 / (2*2 + 2*3) / 5
30 / (4+6) / 5
30 / 10 / 5
3/5
.6
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Information Services ManagerCommented:
It's just that it's a trick because the first order is an "exception" to the rule, and is presented in the last part of the equation.  It's "visually" deceptive.
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Information Services ManagerCommented:
http://www.mathgoodies.com/lessons/vol7/order_operations.html

Note the order of operation, where you Multiply a value by a parenthesed evaluation. -  it comes LAST, not first, therefore the division of the parenthesed evaluation would come FIRST
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RetiredCommented:
@aarontomosky - Aha, I see!
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Information Services ManagerCommented:

That is from Visual Studio.

Dim Int as integer = 30 / 2 * (2 + 3) / 5

Debugging  ...Int = 15
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Commented:
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Commented:
The only debate about this question is which set of rules on the Algebraic Order of Operation you wish to use.

>> Source:  every calculator on earth.
You cannot rely on calculators on earth (but maybe on another planet).

Paste this into your Windows XP PC calculator:
2 + 5*2 =
Windows PC says to do 2+5, which is 7
Then it sees the *, so it gets ready to multiply the 7 by whatever follows, a 2, and 7*2 = 14

However, according to the Rules of Order of Operation, this answer is wrong.
http://en.wikibooks.org/wiki/Algebra/Order_of_Operations
And these are the rules that were taught in the U.S. in public schools as well as in College math/science courses (unless using RPN). I don't know what is taught in private schools.

The correct answer by the standard, most-accepted rules is:
2 + 5*2 = 2 + (5*2) = 2 + 10 = 12
because in the expression 2 + 5 * 2, you can see that the 5 is being vied for by two different binary operators, the + and the *. But according to the rules, in absence of parenthesis, the * attracts the vied for term stronger than the +. This means that the * binds with the 5 leaving the + temporarily stranded. Tough luck, plus.

So, Windows PC calculator does not follow the rules of Algebra. These rules for Order of Operation, although arbitrary and capricious, do provide a systematic and consistent way of communicating with other mathematicians, thereby helping to reduce potential chaos and bitter publications that would ensue with some consensus.
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Commented:
The Standard Order of Operations

Solve expressions in this order:

Please Excuse My Dear Aunt Sally

Parentheses (evaluate what's inside them)
Exponents
Multiplication and/or Division from left to right
Addition and/or Subtraction from left to right
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Information Services ManagerCommented:
...the thing about rules is that there is always an exception.

Seems multiplying a value by a parenthesed equation would come second to the division thereof, in every calculating hardware and software i've tested it on, unless a second set of parentheses are wrapped around it.

I'm not sure why, but it seems to apply only when this is set between two division operations. It may have something to do with a potential divide by zero.

value divide value multiply (value) divide value

It's an excellent brain teaser either way... I would be interested to see how many professional mathematicians would get this wrong.
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Commented:
When applying Please Excuse My Dear Aunt Sally, I don't see any exceptions to this rule.

30 ÷ 2(2+3)÷ 5
30 ÷ 2*(2+3)÷ 5  // show explicitly the implied * binary operator

30 ÷ 2 * (5) ÷ 5     // Please: Parentheses (evaluate what's inside them)

Now, there is a potential conflict with ÷ 2 *. Which binds to the 2 stronger? In this case My Dear doesn't seem to help too much until you read the fine print (i.e., the italicized print:
Multiplication and/or Division from left to right

Multiplication and/or Division have the same binding power to the attacked term. What determines the winner in this case is the rule from left to right. This means that the left operator has a little more attractiveness to the term than the right operator (when both operators have the same binding strength).

So the potential conflict is resolved:
÷ 2 *
The ÷ will bind to the 2, not because it is stronger than the * (they have equal binding strength), but because it is to the left of the 2.

Now the expression can be written as:
(30 ÷ 2) * (5) ÷ 5 =  (15) * (5) ÷ 5

Here again we have a * and a ÷ trying to bind to the middle term (5). Again, they have equal binding strength, so the left to right arbiting rule comes into play. This time it is the * that binds to the middle term because it is to the left of the (5).

Now the expression can be written as:
(15) * (5) ÷ 5 = (15 * 5) ÷ 5 = (3) ÷ 5 = 15
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Commented:
So, knowing all this, what is the value of  81/36/4 ?

Is it 81/9 = 9?
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Commented:
A professional mathematician would not get this wrong. Our operators are ambiguous unless a convention for the order of evaluation is agreed upon. And it is

(1) Inner most parenthesis
(2) Functions
(3) x and  ÷
(4) + and -

Given 2 entities of the same order of operation  then proceed left to right. ( eg 16/4*2 = 4*2 = 8 )

So

30 ÷ 2*(2+3)÷ 5   (inner most parenthesis)

30 ÷ 2*5 ÷ 5      (all same order proceed left to right)

15*5÷5

75 ÷ 5

15
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Commented:
Anyone want to take a stab at 81/36/4 ?

Is it 81/9 = 9?
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Commented:
81/36/4

= 2.25/4

= 0.5625

ie not 9

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Commented:
>> ie not 9
because ÷ and ÷ are both vying for the middle term, 36, to avoid any ambiguity, the left to right rule is the arbiter.

Then the implied parenthesis in the expression, 81/36/4, is (81/36)/4 does not equal 9.

Likewise, 81*36*4 becomes (81*36)*4. If you thought that the answer was 81*(36*4), well, you would be right, but maybe for the wrong reason.

In fact, there is more than one arithmetic rule to remember. It turns out that a*(b*c) = (a*b)*c.

But, as you can see a/(b/c) != (a/b)/c all of the time - they are the same, sometimes; one example, if a = b = c = 1.
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Commented:
There are more than one standards.
http://mathforum.org/library/drmath/view/57021.html
and writing it as
30 ÷ 2(2+3)÷ 5
rather than as
30 / 2 * (2+3) / 5
could suggest that we need not limit ourselves to standards used by programming languages.
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RetiredCommented:
I found out (about a year ago) that the order of operations in mathematics has changed since I went to high school in the sixties! ;-)
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RetiredCommented:
I learned in parentheses first, powers and square Roots, etc. second, addition and subtraction third and multiplication and division fourth. Always from left to right!

Apparently, in Quebec (Canada) this is no longer the case! But I can't remember the new order...
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Commented:
Well, I'll be. I never heard of the "implicit multiplication first" rule before. Not sure what country it is popular in. Never saw it in HS or Universities (or anywhere except here, right now, on EE).
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Commented:
>>I found out (about a year ago) that the order of operations in mathematics has changed since I went to high school in the sixties! ;-)

That'll be why the economy is in such a mess if economists are using two different ways to calculate!

I was taught to resolve everything to do with the parentheses first (i.e. 2(2+3) = 10 ) & when left with multiplication and division there was no specific priority so read from left to right.  My electronic Casio friend seems to do the same (damn those Japanese programmers :)).

It's certainly a good job my career doesn't involve any high level math (or does it ....?)

Thanks for the debate.
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Commented:
I suggest that if there can be this much debate about a small sum that it is written incorrectly. The omission of a couple of brackets has created an ambiguity. Written language is meant to convey ideas and in this case it has not done the job.
If you were trying to show the sum a÷b where a=3 and b=3÷2, the result is 3÷(3÷2), if you write 3÷3÷2 there may be a rule to tell you how to interpret it, but you need to be very sure that everybody reading your work has the same interpretation.
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