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Simple but not so simple mathematical question

Posted on 2011-04-22
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Last Modified: 2012-05-11
What is the answer?

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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Question by:hongjun
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Expert Comment

by:Aaron Tomosky
ID: 35448294
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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Expert Comment

by:☠ MASQ ☠
ID: 35448346
0.6

Use the BODMAS rule for operations
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Author Comment

by:hongjun
ID: 35448474
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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by:Aaron Tomosky
ID: 35448588
Because the 2(2+3) is all together as part of "brackets"
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Expert Comment

by:Aaron Tomosky
ID: 35448597
It actually can be rewritten (2*2+2*3)
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Accepted Solution

by:
Paul Sauvé earned 800 total points
ID: 35448645

B     Brackets first
O     Orders (ie Powers and Square Roots, etc.)
DM   Division and Multiplication (left-to-right)
AS   Addition and Subtraction (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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Author Comment

by:hongjun
ID: 35448677
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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Expert Comment

by:Paul Sauvé
ID: 35448690
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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Expert Comment

by:Ron Malmstead
ID: 35448719
Order of operations is not really debatable...lol
Multiplication, addition, division, subtraction
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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Assisted Solution

by:Ron Malmstead
Ron Malmstead earned 400 total points
ID: 35448766
....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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Expert Comment

by:Paul Sauvé
ID: 35448769

@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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Expert Comment

by:Ron Malmstead
ID: 35448788
30 / (2 * (2 + 3)) / 5 = 0.6

but without the parentheses ..it's 15.
30 / 2 * ((2 + 3) / 5)
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Expert Comment

by:Paul Sauvé
ID: 35448833
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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Expert Comment

by:Ron Malmstead
ID: 35448877
The answer is 15.

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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Expert Comment

by:imonfireDAMMIT
ID: 35448911
I am positive that the answer is .6

PEMDAS is the correct way to solve this problem
Parenthesis
Exponents
Multiplacation
Division
Addition
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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Expert Comment

by:Ron Malmstead
ID: 35448915
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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Expert Comment

by:Ron Malmstead
ID: 35448926
""I am positive that the answer is .6"""

....you would be wrong.

Source:  every calculator on earth.
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Author Comment

by:hongjun
ID: 35448932
Yes, my original answer was 15 as mentioned in my question.
An unexpected lively debate :)
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Author Comment

by:hongjun
ID: 35448937
>>Source:  every calculator on earth.

Calculators can even be wrong if you check out this YouTube video.
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Expert Comment

by:Ron Malmstead
ID: 35448954
Calculators cannot be wrong.

People who program them can be wrong.
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Expert Comment

by:Aaron Tomosky
ID: 35448959
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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Expert Comment

by:Ron Malmstead
ID: 35448964
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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by:Ron Malmstead
ID: 35448991
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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Expert Comment

by:Paul Sauvé
ID: 35449045
@aarontomosky - Aha, I see!
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Expert Comment

by:Ron Malmstead
ID: 35449085
order of operations - Visual Studio evaluation
That is from Visual Studio.

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


Debugging  ...Int = 15
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Expert Comment

by:phoffric
ID: 35449205
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by:phoffric
ID: 35449418
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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Expert Comment

by:phoffric
ID: 35449456
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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Expert Comment

by:Ron Malmstead
ID: 35449516
...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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Assisted Solution

by:phoffric
phoffric earned 400 total points
ID: 35449738
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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Expert Comment

by:phoffric
ID: 35449756
So, knowing all this, what is the value of  81/36/4 ?

Is it 81/9 = 9?
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Assisted Solution

by:GwynforWeb
GwynforWeb earned 400 total points
ID: 35449816
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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Expert Comment

by:phoffric
ID: 35450094
Anyone want to take a stab at 81/36/4 ?

Is it 81/9 = 9?
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Expert Comment

by:GwynforWeb
ID: 35450163
81/36/4

= 2.25/4

= 0.5625

ie not 9

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Expert Comment

by:phoffric
ID: 35450222
>> 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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Expert Comment

by:ozo
ID: 35451070
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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Expert Comment

by:Paul Sauvé
ID: 35451143
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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by:Paul Sauvé
ID: 35451164
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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Expert Comment

by:phoffric
ID: 35451381
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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Expert Comment

by:☠ MASQ ☠
ID: 35452208
>>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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LVL 17
ID: 35454805
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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