My thought process: if I had 8÷(4+4) = 1 and factored out a 2, I get 8÷2(2+2) = 1. However, if I say 2(2+2) = 2*(2+2), then 8÷2(2+2) = 8÷2*(2+2) = 1, BUT 8÷2*(2+2) = 16.
Please help I’m feeling dumber by the second
In: Mathematics
It equals 1 because you chose to follow a criteria that gives you 1. If you decided and we all adopted the idea that we’d do the operations from left to right, treating the division symbol as applying to the numbers immediately next to it then it wouldn’t equal 1. But the same way as you can read this text in English because we chose to use these sequence of letters and words as English, we chose the language where the equation equals 1.
This factoring …
8÷(4+4) == 8÷2*(2+2)
is faulty. Taking just 4+4 (without the parens)…
4+4 = 2(2+2) – this is correct, factoring out the 2, creates a NEW set of parens. You must ALSO preserve the original set of parens in the whole equation…
(4+4) = (2*(2+2))…
So the whole thing is …
8÷(2*(2+2)) = 8÷(2*(4)) = 8÷(8) = 1
Yes, the ÷ symbol is often confusing, but the / symbol just means the exact same thing and can be just as confusing.
This is easier…
8 8 8 8
—– = ———- = ———- = ——– = 1
4+4 2*(2+2) 2*(4) 8
For those who keeping spouting PEMDAS, you’re missing something huge with order of operations. The correct interpretation of PEMDAS is:
-Parentheses
-Exponents
-Multiplication AND Division from left to right
-Addiction AND Subtraction from left to right
Lemme say it again: multiplication is not ALWAYS performed before division—it MUST be calculated from left to right. The same applies to addition and subtraction.
If the division symbol were actually a /, then the original equation would be: 8 / 2 * (2 + 2). Following order of operations (which is what is being stressed here, NOT the division symbol vs /)…
Parentheses: 8 / 2 * (2 + 2) = 8 / 2 * (4)
Multiplication/Division from left to right: 8 / 2 * (4) = 4 * (4) = 16
Why is this so complicated for people to understand? I blame Common Core math…
Edits: rewording my statements to better explain myself.
8÷2(2+2) = (8÷2)(2+2) = 16.
If you factor out a 2, you need to put brackets around the 2(2+2) portion, because multiplication and division have the same priority.
You can see a lot of these math lines on Facebook which shows how many people did not pay attention in school.
8÷2(2+2) = 1 is very straight forward.
8÷2(4) = 1
8÷8 = 1
1 = 1
It’s so much easier if you imagine division as a fracture. Just leave the “8” at the top, do the parentheses and multiplication at the bottom and boom 8/8.
The whole American PEDMAS stuff is needlessly confusing.
Nobody uses the notation like that and all these “confusing” examples are just people abusing notation like ÷ .
First thing: Addition and subtraction are two aspects of the same thing and multiplication and division are two aspects of the same thing.
For example 3 -2 +4= 3 +4 -2= 5. Once you know negative numbers (=debt), you can write this as 3 + (-2) + 4. There is only addition, no weird invisible bracket and you can freely reorder things (subtraction is simply adding negative numbers).
Same for multiplication: 6 /3 *2=6 *2 /3=4.
Once you know fraction, you can write this as
6* (1/3) *2 = 6 * 2 * (1/3) =4.
There is only multiplication, no weird invisible bracket and you can freely reorder things (division is just multiplying by a fraction).
Finally, there is one common abbreviation, which is purely for notational convenience. Say you buy 3 coffees for $2 each and 5 donuts for $1 each.
Then we don’t want to write brackets all the time and hence abbreviate:
(3 * 2) + (5 * 1) = 3 * 2 + 5 * 1=11.
Much easier to write, so we say to do multiplication first and then addition.
The issue is the implicit multiplication between the 2 and the 2+2. That is, it’s written as 2(2+2) and not 2*(2+2). By some conventions, this implicit multiplication has a higher priority than other multiplication or division operations, superseding the normal left-to-right order they would be done in.
Why is this the case? Essentially, it’s because if I write something like 1/xy, it’s pretty clear that what I *mean* is 1/(xy), and not (1/x)y. Having it be an explicit rule to give implicit multiplication a higher priority means you can write out quotients of products without needing to use as many brackets or multiplication symbols, thereby making it easier to read.
With this rule, the answer is 1, because you resolve the 2(2+2) first to get 8, and then 8÷8=1. Without this rule, you apply the multiplications and divisions from left to right, meaning the 8÷2 becomes 4, which is then multiplied by the (2+2) to give 16.
Both answers are right, it just depends on which convention you use. Personally, I prefer the implicit multiplication convention, as it results in equations with fewer symbols.
First up, mathematics notation is a language and it’s most helpful to think of it as such. To make this point clear, take an example from a different language such as English. Say you tell someone, “I saw a man with a telescope.” You might mean that you saw a man carrying a telescope. Or you might mean that you looked through a telescope and saw a man. The fact that your actual meaning is unclear to your listener is not a sign that you are a super smart English genius. It’s a sign that you are a barely literate moron.
Same here. The fact that the meaning of the expression 8÷2(2+2) is unclear to you doesn’t mean that you are stupid. But it does tell you a lot about the people asking the question.
So, why is this expression unclear? A small part of it is that the obelus is ambiguous. Most of the time, a-b÷c-d means a-(b÷c)-d. However, occasionally a-b÷c-d means (a-b)÷(c-d). The second meaning is much less common but not entirely unheard of.
By far though, a much greater source of ambiguity is in the implicit multiplication. This is because most of us use implicit multiplication in an ambiguous way most of the time. Consider the expressions 3/2a and the expression 3a/2. Are they equivalent? Most people would say no. Most people would say that 3/2a evaluates to 1.5/a, while 3a/2 evaluates to 1.5×a. But if you apply strict BODMAS, you’d conclude that 3/2a and 3a/2 both evaluate to 1.5×a, ie they are both the same.
The problem is that *most* people sometimes use implicit multiplication to mean high priority multiplication (ie multiplication that must be done before all other multiplication/division) and sometimes use implicit multiplication to mean regular multiplication that is done in the normal left to right order. What we mean by implicit multiplication is usually clear from the context, but in an instance like this question, there is no context.
As a result, it’s not unreasonable for you to conclude that 8÷2(2+2) means 8÷8. This assumes that the implicit multiplication is being used to mean high priority multiplication which is a reasonable assumption because almost everyone uses implicit multiplication this way *some of the time*.
Equally, it’s not unreasonable for you to conclude that 8÷2(2+2) means 4×4. This assumes that the implicit multiplication is being used to mean regular multiplication which is a reasonable assumption because almost everyone also uses implicit multiplication this way *some of the time*.
It really depends on what the person asking the question thinks the answer is.
Oh my god these comments. The answer is 16.
Parentheses first, multiplication and division from left to right.
8 divided by 2 times (2 plus 2) becomes
8 divided by 2 times (4) becomes
4 times (4) becomes
16
According to [cymath](https://www.cymath.com/answer?q=8%232(2%2B2)), the answer is 16. The answer is not 1 unless you interpret the expressions as 8/(2(2+2))
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