Greetings Wolfram Alpha,
I am one of the administrators for The Enlightened Consciousness Group on Facebook which has 57,000 members who are fighting over the answer to this seemingly simple equation. The variance in answers and them showing their work has me questioning my own math.
6 ÷ 2(1+2) = ?
The three most common answers on this topic are 9,1, and 7. The logic behind these answers has me questioning my own basic mathematics, so I come to the authority for the answer. The biggest discrepancy is due to the order of operations and that 2 touching the parenthesis.
I was taught to simplify the inside of parenthesis, then work left to right using PEMDAS. Using this method I get 6 ÷ 2(3) = 3(3) = 9
Others have said that you must distribute the 2 before division because it touches the parenthesis. Their math shows 6 ÷ 2(1+2) = 6 ÷ 2 + 4 = 7
Others have said that you evaluate the left and right side of the obelus as independent expressions. Their math shows 6 ÷ 2(1+2) = 6 ÷ 6 = 1
Finally, there was an article written on slate.com written by a math teacher, who asserts that unless specifically stated there is no definitely right way to interpret the equation--and that order of operations is not always appropriate.
That is a classic. The problem is that there is no universal agreement on how to parse expressions with a mix of multiplications and divisions, or with more than one division. Does 1/2a mean 1/(2b) or (1/2)a? It is a matter of convention. Mathematica treats it as (1/2)a, but you may find books where it is meant as 1/(2a), perhaps because this way you save typing parentheses. Usually the ambiguity can be decided from the context, as only one interpretation makes sense. I teach my students to avoid expressions such as a/b/c and always write parentheses to make sure they are not misunderstood. There is a Wikipedia article on the "Order of operations".
The calculation 6 ÷ 2(1+2) = 6 ÷ 2 + 4 = 7 is a gross mistake in my view.
Thank you for your response Gianluca,
For my own edification, what context clues would there be to solve the equation any other way? Which interpretation makes the most sense in this context? 9 or 1? Would a word problem make this easier to understand as well?
Also I very much appreciate any responses in this thread--All of us have access to the same Wikipedia and websites that have greater detail...however for my end users who are obstinate about unreliable sources I wanted to come to the most credible place.
Yes it comes down to whether a space in front of parenthesis in an equation has any significance when parsing a formula. Mathematica doesn't care and even inserts a space to make the formula appear nice. Of course in Mathematice as in many other notations, the multiplication sign can be replaced by a space like 4 5=45 obviously leaving out the space there would be the number 45. with parenthesis the space might not be necessary 2(1+2) = 2 (1+2)=2(1+2), I have not seen the interpretation 2(1+2)=(2*(1+2)), but it is all a matter of agreed convention.
In:= 6 / 2 (1 + 2)
When in doubt, the best way to insure clarity one can write out the formula like this:
In:= 6 / 2*(1 + 2)
In:= 6 / (2*(1 + 2))
It all depends on what they call the precedence of the operators. For Mathematica (Wolfram Language) the Divide has a higher precedence than Times. So it will first do divisions then multiplications:
But in other cases you also have to look at how it groups:
could be 6/(2/3) or (6/2)/3, giving different answers.
have a look here:
To sum up: the author of that equation is just sloppy; you have to assume something in order to solve it, and depending on the conventions... so therefore always add extra parenthesis to rules out those cases...
and similarly things like a^b^c could be (a^b)^c or a^(b^c), so always add parenthesis when there is no clear answer. You can avoid part of it by always doing multiplication first and then division:
a/b c (to be interpreted as (a/b)*c)
should be written as : c*a/b, so it doesn't matter what the order is...
Thank you all!
All things considered would an appropriate answer be "Need more information to solve" since the author gave no context clues to assist?
Yes, without further hints it is not good notation, you need to know the rules of the game. It is like using parenthesis like
we don't know how to deal with that without some extra rules...
Some claim the equation is ambiguous, but the notation is common. It is understood that we apply the Distributive Law in the Parentheses step of PEMDAS. 2(1+2) = (2x1) + (2x2) = 6.
The only number you can divide 6 by to arrive at 9 is 2/3.
2(1+2) != 2/3
2(1+2) == 6
The left to right solvers, by replacing implied multiplication with implicit multiplication, are flipping the parenthetical expression to the inverse.
2÷(1+2) == 2/3
6÷2(1+2) != 6÷2*(1+2)
The Desmos online calculator will return the correct answer when using the keypad.
The Casio fx9860GII SD returns the correct answer to the equation as given 6÷2(1+2) and provides the alternate notation if you were expecting a result of 9.
6÷2(2+1), the answer is actually 9.
For the people who used this principle x/2y, 2y is obviously together in a group. Polynomials is a simpliest form of an equation the doesn't need parentheses. for example, if the problem is written as 6÷[2(2+1)], in algebraic expression it can easily written as 6÷2y or 6/2y with y=2+1.
For the people who said that we should use the distribution, using distribution in 2(2+1) is actually a multiplication method which means you break the rule of PEMDAS that Multiplication and Division are tied in priority which who comes first from left to right should be solve first.
1st step: 6÷2(2+1), Inside the parentheses,
6÷2(3), the parentheses sign is indicate a multiplication, eq. of multiplication signs [x, *, ()]
2nd Step: 6÷2(3), since () is indicate a multiplication, you can rewrite the problem as 6÷2x3, since Division comes first, used Left to Right rule, solve 6÷2=3
3(3) or 3x3 or 3*3
3rd Step 3(3)=3x3= ****9****.