well quite. you can't do anything with it because it's infinite, you have to approximate it to calculate anything using it, which is where any proof fails.
You can still do calculations on repeating decimals without having to approximate. For example, 0.999r divided by 3 is 0.333r.
Now, since 0.333r is 1/3, you can then see that 0.999r = 1. Tada!
I really have no idea, but I've enjoyed this thread today so here goes a combination of 'seeing' things in words, and what I remember from school, I read this problem as
48 divided by two times nine plus three.
Now, if I was given a reason, I could see it as....
48 divided by two, times nine plus three.
There is no mathematical equivalent of a comma for me to read it like that.
That probably makes NO sense to any of you, but it does to me. And my answer is still 2.
So have I and if you correctly apply BODMAS you get 288..
The issue is of course as others have pointed out, the expression is deliberately ambiguous and therefore BODMAS should be applied literally IMO. In which case it's simply brackets first then left to right.
Those who are getting 2 as the answer are effectively multiplying out the brackets first and are therefore not applying BODMAS correctly.
If we were only multiplying out the brackets first we would get
So have I and if you correctly apply BODMAS you get 288..
The issue is of course as others have pointed out, the expression is deliberately ambiguous and therefore BODMAS should be applied literally IMO. In which case it's simply brackets first then left to right.
Those who are getting 2 as the answer are effectively multiplying out the brackets first and are therefore not applying BODMAS correctly.
They are applying BODMAS correctly if you accept that the 2 immediately next to the bracket is an Other Operation and not a Multiplication.
It is not a straightforward Multiplication as it does not use the standard symbol. It borrows shorthand from algebra and causes confusion because you should not need to know the rules of algebra to do straightforward math.
So have I and if you correctly apply BODMAS you get 288..
The issue is of course as others have pointed out, the expression is deliberately ambiguous and therefore BODMAS should be applied literally IMO. In which case it's simply brackets first then left to right.
Those who are getting 2 as the answer are effectively multiplying out the brackets first and are therefore not applying BODMAS correctly.
We are dealing with the brackets correctly. Those who get 288 are putting a * or x sign in front of the brackets, either implicitly or explicitly.
I am serious, I keep clicking thinking someone will come along an categorically state 'it's......' but oh no...
I can't believe how nuts this is driving me.
OK, I'll do my best to provide an unbiased answer to this.
The question ultimately boils down to this: does a/bc equal:
1) a/(bc)
or:
2) (a/b)c [or, equivalently, (ac)/b]
If you subscribe to the concept of "juxtaposition precedence" (i.e. "the 2 belongs to the (9+3) due to the omission of a multiplication symbol"), you will arrive at the first interpretation, and the answer to the OP is 2.
If you apply the rules of BODMAS and left-to-right precedence which you learned at school, you will arrive at the second interpretation, and the answer to the OP is 288.
Many (I suspect all) of those arguing for an answer of 2 have effectively assumed the existence and validity of "juxtaposition precedence" without having been taught it, or indeed having ever heard of it before this thread. Lacking any significant evidence or education to the contrary, my conclusion is that juxtaposition precedence is (to say the least) very far from standard practice, and I will therefore stick with the conventional rules and the answer of 288.
But I agree that the best answer is "it's a bad question", though I salute the OP for passing it on.
So have I and if you correctly apply BODMAS you get 288..
The issue is of course as others have pointed out, the expression is deliberately ambiguous and therefore BODMAS should be applied literally IMO. In which case it's simply brackets first then left to right.
Those who are getting 2 as the answer are effectively multiplying out the brackets first and are therefore not applying BODMAS correctly.
It depends purely on whether or not you interpret the missing * as associative with the brackets or not. It's poorly written which makes the question ambiguous, and that means it can be 2 or 288 depending on how you read it. simples.
In such a situation I would apply BODMAS literally, that is:
48 ÷ 2 x 12 = 288
There's certainly no reason to multiply the brackets out first as some have done to obtain:
48 ÷ (18+6) = 2
But a division is a fraction also and if it was expressed swapping the division for a fraction line, and treating the numbers either side as the components of the fraction, you would get the latter, as others have said, and laid out better than I care to spend the time doing.
I'd still treat a(b+c) algebraically though and do b+c then multiply by a (or expand) than do b+c then bidmas the rest. It's an entity in itself in those terms.
But a division is a fraction also and if it was expressed swapping the division for a fraction line, and treating the numbers either side as the components of the fraction, you would get the latter, as others have said, and laid out better than I care to spend the time doing.
I'd still treat a(b+c) algebraically though and do b+c then multiply by a (or expand) than do b+c then bidmas the rest. It's an entity in itself in those terms.
Yes that's the whole point, it's written to be ambiguous otherwise there wouldn't be any debate...
OK, I'll do my best to provide an unbiased answer to this.
The question ultimately boils down to this: does a/bc equal:
1) a/(bc)
or:
2) (a/b)c [or, equivalently, (ac)/b]
If you subscribe to the concept of "juxtaposition precedence" (i.e. "the 2 belongs to the (9+3) due to the omission of a multiplication symbol"), you will arrive at the first interpretation, and the answer to the OP is 2.
If you apply the rules of BODMAS and left-to-right precedence which you learned at school, you will arrive at the second interpretation, and the answer to the OP is 288.
Many (I suspect all) of those arguing for an answer of 2 have effectively assumed the existence and validity of "juxtaposition precedence" without having been taught it, or indeed having ever heard of it before this thread. Lacking any significant evidence or education to the contrary, my conclusion is that juxtaposition precedence is (to say the least) very far from standard practice, and I will therefore stick with the conventional rules and the answer of 288.
But I agree that the best answer is "it's a bad question", though I salute the OP for passing it on.
Aw thank you.:)
I am a bit too drunk to read that now, but will read it tomorrow.
OK, I'll do my best to provide an unbiased answer to this.
The question ultimately boils down to this: does a/bc equal:
1) a/(bc)
or:
2) (a/b)c [or, equivalently, (ac)/b]
If you subscribe to the concept of "juxtaposition precedence" (i.e. "the 2 belongs to the (9+3) due to the omission of a multiplication symbol"), you will arrive at the first interpretation, and the answer to the OP is 2.
If you apply the rules of BODMAS and left-to-right precedence which you learned at school, you will arrive at the second interpretation, and the answer to the OP is 288.
Many (I suspect all) of those arguing for an answer of 2 have effectively assumed the existence and validity of "juxtaposition precedence" without having been taught it, or indeed having ever heard of it before this thread. Lacking any significant evidence or education to the contrary, my conclusion is that juxtaposition precedence is (to say the least) very far from standard practice, and I will therefore stick with the conventional rules and the answer of 288.
But I agree that the best answer is "it's a bad question", though I salute the OP for passing it on.
All future posts should simply be referred to this here ^^^^^.
Many (I suspect all) of those arguing for an answer of 2 have effectively assumed the existence and validity of "juxtaposition precedence" without having been taught it, or indeed having ever heard of it before this thread. Lacking any significant evidence or education to the contrary, my conclusion is that juxtaposition precedence is (to say the least) very far from standard practice, and I will therefore stick with the conventional rules and the answer of 288.
The more I think about this, the more I think that this assumed precedence is indeed false, for the reasons I gave earlier.
The simple fact is, the '÷' and 'x' operators do not generally feature in algebra. As I said before, mixing this notation with juxtaposition (the term is something I am aware of -- though I forgot all about the word until this evening) is where the problems come from.
It's a mental block thing, I think. Because we're used to seeing '+' and '-' operators used in algebra, which are of course of lower precedence than multiplication and subtraction, it is easy to 'group' the 2 with the (9+3) and infer a precedence, which may not really exist due to the fact that this isn't something that is generally seen in textbooks.
As I say, I initially fell into the same trap, but quickly realised I was wrong. But the debate here certainly have pause for thought.
Ultimately, the question is duff. It's a curiosity and nothing more. There is no 'correct' answer as such, because the question is invalid. But the "nearest to the truth" answer is probably 288, because the most straightforward way of 'correcting' the equation is to insert a 'x' between the 2 and the bracket.
Else you have to deal with the divide in algebraic terms to match the juxtaposition, and that's when the fun starts.
The B in BIDMAS stands for brackets. The meaning of which is:
"The terms INSIDE the brackets".
NOT
"Any term inside and next to the brackets".
When you come to a stage when there is only operations left on the same level (multiplication and division are on the same level of precedence). You MUST work from left to right.
Comments
Now, since 0.333r is 1/3, you can then see that 0.999r = 1. Tada!
Like the majority, I get '2'.
48 divided by two times nine plus three.
Now, if I was given a reason, I could see it as....
48 divided by two, times nine plus three.
There is no mathematical equivalent of a comma for me to read it like that.
That probably makes NO sense to any of you, but it does to me. And my answer is still 2.
If we were only multiplying out the brackets first we would get
48 ÷ 18 + 6 which gives 8.666666r
..... and this is where I'm completely out of my depth..... or else I'm too old
It is not a straightforward Multiplication as it does not use the standard symbol. It borrows shorthand from algebra and causes confusion because you should not need to know the rules of algebra to do straightforward math.
OK, I'll do my best to provide an unbiased answer to this.
The question ultimately boils down to this: does a/bc equal:
1) a/(bc)
or:
2) (a/b)c [or, equivalently, (ac)/b]
If you subscribe to the concept of "juxtaposition precedence" (i.e. "the 2 belongs to the (9+3) due to the omission of a multiplication symbol"), you will arrive at the first interpretation, and the answer to the OP is 2.
If you apply the rules of BODMAS and left-to-right precedence which you learned at school, you will arrive at the second interpretation, and the answer to the OP is 288.
Many (I suspect all) of those arguing for an answer of 2 have effectively assumed the existence and validity of "juxtaposition precedence" without having been taught it, or indeed having ever heard of it before this thread. Lacking any significant evidence or education to the contrary, my conclusion is that juxtaposition precedence is (to say the least) very far from standard practice, and I will therefore stick with the conventional rules and the answer of 288.
But I agree that the best answer is "it's a bad question", though I salute the OP for passing it on.
It depends purely on whether or not you interpret the missing * as associative with the brackets or not. It's poorly written which makes the question ambiguous, and that means it can be 2 or 288 depending on how you read it. simples.
48÷2(9+3)
Becomes 48÷2(12)
Multiplication and subtraction are equal so we go left to right
48÷2=24
24 x 12 is 288
If the sum was 48÷(2(9+3)) then the answer would be 2.
Indeed, and that is the issue in a nutshell...
In such a situation I would apply BODMAS literally, that is:
48 ÷ 2 x 12 = 288
There's certainly no reason to multiply the brackets out first as some have done to obtain:
48 ÷ (18 + 6) = 2
But a division is a fraction also and if it was expressed swapping the division for a fraction line, and treating the numbers either side as the components of the fraction, you would get the latter, as others have said, and laid out better than I care to spend the time doing.
I'd still treat a(b+c) algebraically though and do b+c then multiply by a (or expand) than do b+c then bidmas the rest. It's an entity in itself in those terms.
Aw thank you.:)
I am a bit too drunk to read that now, but will read it tomorrow.
All future posts should simply be referred to this here ^^^^^.
If you believe a ÷ bc is a ÷ b * c then the answer is 288
The more I think about this, the more I think that this assumed precedence is indeed false, for the reasons I gave earlier.
The simple fact is, the '÷' and 'x' operators do not generally feature in algebra. As I said before, mixing this notation with juxtaposition (the term is something I am aware of -- though I forgot all about the word until this evening) is where the problems come from.
It's a mental block thing, I think. Because we're used to seeing '+' and '-' operators used in algebra, which are of course of lower precedence than multiplication and subtraction, it is easy to 'group' the 2 with the (9+3) and infer a precedence, which may not really exist due to the fact that this isn't something that is generally seen in textbooks.
As I say, I initially fell into the same trap, but quickly realised I was wrong. But the debate here certainly have pause for thought.
Ultimately, the question is duff. It's a curiosity and nothing more. There is no 'correct' answer as such, because the question is invalid. But the "nearest to the truth" answer is probably 288, because the most straightforward way of 'correcting' the equation is to insert a 'x' between the 2 and the bracket.
Else you have to deal with the divide in algebraic terms to match the juxtaposition, and that's when the fun starts.
48÷2(9+3)
48÷2(12) aka 48÷2*12 (brackets)
24(12) = 24*12 (can't do indices so move to division)
24(12) = 288 (multiplication)
BIDMAS.
In algebraic notation - the multiplication is implied by the lack of specified mathematical operator between 2 and (
Therefore 48/2*(9+3) is the same as 48/2(9+3)
Most people say 288 because it is 288.
Edit: Yes poor terminology on my part.