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.
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.
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:
Precedence[Divide]
Precedence[Times]
470
400
But in other cases you also have to look at how it groups:
6/2/3
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…
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[1]:= 6 / 2 (1 + 2)
Out[1]= 9
When in doubt, the best way to insure clarity one can write out the formula like this:
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.
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 33
3rd Step 3(3)=3x3= 9.
Perhaps you will change your mind about using the Order of Operations as PEMDAS, after considering the word problem below, while remembering what a “term” is & that a single term (e.g. “2a”) does not need parentheses encasing it to be understood as a single quantity (the product of its factors), and how to divide by a monomial:
In this scenario, I own & run a diner which serves a Breakfast Special from early morning opening until 11 AM. Today, at 10:58 AM, the last of the Breakfast Special customers left the restaurant. But then, at 10:59 AM, two separate groups of a dozen people each come into the diner, wanting to get the Breakfast Special. The 2 separate groups are each seated at their own table. Each member of the 2 groups, consisting of a dozen people apiece, wants eggs. I go into the restaurant kitchen to see how many eggs I have left. When I open the refrigerator, I see that there are 4 full cartons of a dozen eggs each, left on the shelf
Split evenly among the Breakfast Special customers in the diner, how many eggs does each customer get?
The proposition is a single quantity of eggs split evenly among a single quantity of customers:
4 dozen eggs divided by 2 dozen customers
4 dozen ÷ 2 dozen = ?
That same proposition can also be viewed as 4a÷2a when a=12.
In solving this problem, I’d appreciate it if you would please show each mathematical step you take, from beginning to end, so that it can be fully understood how you arrived at the correct answer. If you find more than one mathematical method of solving the monomial division expression in this word problem, please show all of the steps in each alternative.
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I look forward to seeing your methodology to find the correct answer to the word problem.
Thank you for commenting on what I wrote. It’s not that I don’t know how to solve the word problem – I was illustrating a point about how to view juxtaposition.
Obviously, the answer to the question of how many eggs each diner customer gets is 2. The word “dozen” equals 12, so 12 is a factor of both of those terms, 4 dozen & 2 dozen, respectively. Thus, dividing 4 dozen by 2 dozen is written mathematically as:
4(12)/2(12)
…or as the top-and-bottom fraction of 4(12) over 2(12).
That is the same division expression as the monomial division expression of 4a/2a when a=12.
In no Basic Algebra textbook I have ever seen, are young math students instructed to ALWAYS encase a monomial inside parentheses in order to be understood to be a single combined quantity, as illustrated by 4 dozen eggs being understood to be a single combined quantity of 48 individual eggs (4 packs of 12 eggs each) & 2 dozen customers is understood to be a single combined quantity of 24 customers (2 groups of 12 customers each).
There is nothing confusing or ambiguous about the expression 6 ÷ 2(1+2), once one remembers the definition of a term (specifically, a monomial) which holds the value of the PRODUCT of its factors, the rules of the Distributive Law (which directly addresses juxtaposition), and how to divide by a monomial. It is also important to remember, as taught in 5th grade math: that Fraction=Division & therefore, Division=Fraction.
4 dozen divided by 2 dozen is precisely the same expression as the internet division meme from 2011:
48 ÷ 2(9+3)
Factoring out 48 makes the expression…
4(9+3) ÷ 2(9+3)
…or you could say it’s 4 dozen divided by 2 dozen.
In any case, the quotient is 2, not 288 as the PENDAS crowd keeps insisting.
My point is that this should absolutely not be written as 4(12)/2(12). Once one posits “four dozen” of this and “two dozen” of that, these are implicitly grouped (in your terms, “combined quantities”), that is to say, as (4x12) and (2x12). So ungrouping them is kinda a bad idea.
I see the claim “There is nothing confusing or ambiguous about the expression 6 ÷ 2(1+2)…” I would raise the question of whether one should treat any differently the expression 6/2(1+2). If the answer is “no” then I’m hard-pressed to buy that; that expression must evaluate to 9 (I mean, how do you justify first multiplying the 2 by the (1+2). That would be operating from the right, which is to say wrong, end of the expression. I do not think this is taught anywhere.) And i the answer is “yes” then I have to ask why we treat one division sign differently from another. (The answer to that is “we don’t”.)
Getting back to PEMDAS, the rule is that equal precedence goes left to right; this is taught in schools I believe. And it’s the dominant convention nowadays. Not something I would choose to fight, and, in particular, I’d not want to teach algebra in a way that ignores it (even if it is not explicitly stated).
PEMDAS is taught in elementary school. In 9th grade Basic Algebra, however, a new set of rules is taught which supersede the PEMDAS construct. Included in the superseding rules are defining what a term is (specifically in this case, what a monomial is) & how to divide by a monomial.
The terms “4a” & “2A” are monomials. If you believe that those single terms need to always be encased in parentheses when part of a monomial division expression, please cite specific teaching websites which instruct young algebra students that this is REQUIRED. Here are some teaching sites which say the contrary:
from Mathway’s calculator Algebra Examples:
" 2x÷2x
Rewrite the division as a fraction:
shown as 2x over 2x
Cancel the common factor of 2.
shown as x over x
Cancel the common factor of x.
1 "
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Lumen Learning Courses:
“Dividing Polynomials by a Monomial”
"When there are coefficients attached to the variables, we divide the coefficients and divide the variables.
EXAMPLE
Find the quotient: 56x^5 ÷ 7x^2
Solution
Rewrite as a fraction
shown as 56x^5 over 7x^2
…Answer
56x^5 ÷ 7x^2 = 8x^3 "
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from Cue Math teaching website:
“Practice Questions on Dividing Monomials”
Q.1. Divide. 15a^2b^3 ÷ 5b
Correct answer is shown as 3a^2b^2.
That can only be true if 15a^2b^3 is the entire numerator & 5b is the entire denominator of the top-and-bottom fraction, even though there are no parentheses anywhere in the original horizontally written division statement (using an obelus).
See examples #1, #2 & #3: All are originally written as horizontal expressions using an obelus as the division symbol, which are all immediately rewritten as top-and-bottom fractions with the monomial denominator being the entire term to the right of the division sign – with no parentheses around it.
In example #1, the denominator is 8x^2
In example #2, the denominator is 6x^2
In example #3, the denominator is also 6x^2
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…and there are plenty more examples all over the web of how young algebra students are taught to divide by a monomial, in which the original expression uses an obelus to represent division, and are then instructed to “Rewrite as a fraction,” with the whole monomial term to the right of the obelus as the entire denominator. I can find no examples of young students being told to use only the monomial’s numerical coefficient in the division before multiplying that quotient by the other factor or factors of the monomial.
If you honestly believe that to be incorrect, please show some examples from algebra teaching websites which specifically instruct young students NOT to consider a monomial term such as “4a” or “2a” (without parentheses completely encasing the entire monomial) as having a single value which is the PRODUCT of its factors.
I make no claim about what’s taught in schools. I suspect what you describe is as it was taught even when I took algebra. From a teaching perspective maybe that makes sense, at least if the obelus is regarded as strictly a binary operator.
Here is the catch though. We now have students using calculators and computers in math and also using them post-school in their careers. Computer languages need to follow a grammar, and that grammar needs to be unambiguous. So PEDMAS is back in the picture. I guess one has to choose which convention makes the most sense based on where the question arises (e.g. in an algebra classroom, or in an automated computation).
Best in my view would be to retain the two-dimensional division operation for classroom purposes. I think that handles the ambiguity without sacrificing the integrity (such as it may be) of monomials.
I just showed what is currently being taught on math teaching websites, as to how to divide by a monomial. See Lumen Learning.com, Cue Math.com, Greene Math.com, BYJUS.com teaching website, etc., instructing students to “Rewrite as a fraction,” when they are presented with a linearly written division expression using an obelus as the division symbol. That is because, remembering back to 5th grade math, students are taught that Fraction=Division (and therefore, Division=Fraction). On those teaching websites, there are a number of examples given to current Basic Algebra students, with the solutions, which show this clearly to be the case. Here are a few more examples of this, on math teaching websites to help current Basic Algebra students :
from Teachoo .com:
“Dividing Monomial by Monomial”
"Let’s do some more examples
6x^3 ÷ 3x^2
… = 2x "
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from Cool Math .com:
“Dividing by Monomials”
"Let’s do one
(18x^4 -10x^2 + 6x^7) ÷ 2x^2
Let’s rewrite it like this:
18x^4 -10x^2 + 6x^7
--------- [over] -------- =
2x^2
…
9x^2 - 5 + 3x^5
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from Mometrix Test Preparation:
See example #3 on the video:
8x^5y^3 ÷ x^2y
Note that no parentheses are used in that horizontal division expression.
The eventual answer is shown to be 8x^3y^2…which only works if 8x^5y^3 is the entire numerator & x^2y is the entire denominator of the top-and-bottom fraction.
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If you still need to see more of this from other math teaching websites, just say the word & I will provide additional examples of students being instructed to rewrite monomial division expressions as a top-and-bottom fraction, when it was, originally written horizontally with an obelus without parentheses around the divisor (denominator) monomial.
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In teaching your students how to ‘process’ a division expression such as a/b/c, yes, additional parentheses are definitely necessary, to indicate the order of the multiple divisions in the expression. When there is only one division symbol in an expression, however, it is clear that the dividend/numerator is everything to the left of the division symbol (slash or obelus) & the divisor/denominator is everything to the right of the division symbol. See above examples from teaching websites.
In no Basic Algebra textbook are students taught that a numerical coefficient of a term can be separated from its other factor or factors – with or without parentheses surrounding the term. Quite the contrary – students are taught that a monomial term such as “2a” holds a SINGLE VALUE, which is the PRODUCT of “2” & “a.” When a=(2+2), the monomial term 2a equals 8.
As demonstrated on a myriad of math teaching websites showing students how to divide by a monomial, the value of a monomial does not change when a monomial such as “2a” is preceded by a division symbol (slash or obelus). The two in “2a” is not a separate, stand-alone quantity any more than the two in “2 dozen” is a stand-alone quantity. Two dozen equals 24, even when there’s a division symbol immediately to the left of that term. Once again, see all those Basic Algebra teaching websites for confirmation of how to divide by a monomial.
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As for students using calculators, if they are taught the proper way to execute monomial division in the first place, then if they do not get the same result on a calculator, they know it is due to faulty programming. Some manual & online calculators do get it correctly, however. If they get what they understand is an erroneous answer from a calculator program, then they can go back & add more (unnecessary) parentheses.
With the proper knowledge & understanding of how to divide monomials, there is nothing confusing or ambiguous about an expression such as 6 ÷ 2(1+2). 8 ÷ 2(2+2), or 48 ÷ 2(9+3).
This one gives choices and explanations for ordering operations.
These next have the courtesy of rewriting as you go, so at least you know how they parse your input (they all require parens to get the effect shown in the pedagogy sites noted earlier), that is to say, they support PEDMAS at least to the limited extent that I checked.
I wonder how students reconcile this. Maybe they just get used to the difference and expect one convention in class and another online and/or in computer programs.
Here’s how students reconcile poorly programmed calculators & the correct procedure to do monomial division:
LEARN TO EXECUTE MONOMIAL DIVISION PROPERLY, IN THE FIRST PLACE!
When using calculators, it might be a good idea to use additional parentheses for the calculator to interpret the expression correctly, in the event that a calculator programmed was by someone who missed that day in 5th grade math when it was taught that Division=Fraction (and vice versa) & also missed that day in 9th grade Basic Algebra when the teacher went over what a “term” is (specifically, a monomial) & how to divide by a monomial.
Again – once it is understood what a monomial is & the correct procedure to divide by a monomial, there is nothing confusing or ambiguous about an expression such as 6 ÷ 2(1+2). 8 ÷ 2(2+2), or 48 ÷ 2(9+3). It’s monomial division, with the entire term to the right of the division symbol being the denominator/divisor. And the numerator/dividend (the term to the left of the division symbol) can be factored, too. For example:
6 ÷ 2(1+2) =
6=2(1+2)
…making the expression…
2(1+2) ÷ 2(1+2) =
…which is the same statement as…
2a ÷ 2a
…when a=(1+2)…
…which can also correctly be written as the top-and-bottom fraction of 2a over 2a.
That would be nearly every online calculator and I suspect every symbolic programming language. Except those languages are correct, insofar as they require and enforce an unambiguous grammar. Which you will not get from the rules that apply to a quotient of two expressions (I avoid use of “monomials” because it’s not clear that expressions will always be monomials).
Before you insult the people who program those calculators (people like myself), maybe consider the possibility that we are familiar with requirements of unambiguous grammars, and that these requirements go well beyond infix operators from simple arithmetic expressions. Also bear in mind that these operators, division among them, are (often) not restricted to binary. Hence you’ll need to be certain that whatever convention you apply generalizes to three or more arguments.
For what it’s worth, I did not miss those days in class (granted, they may have been different years). Yes, division=fraction. But that alone does not tell a parser where to end a subexpression, one needs more rules for that.
I guess another point to consider is the extent to which it’s worth fighting against prevailing technology. These calculators and programming languages are not going to change just to accommodate the particular interpretation you are providing. It would be good if students know to plan accordingly.
