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A common technique for remembering the order of operations is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and these two outrank addition and subtraction (which are together on the bottom rank). When you have a bunch of operations of the same rank, you just operate from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you get to the division first. If you're not sure of this, test it in your calculator, which has been programmed with the Order of Operations hierarchy. For instance, typesetting this into a graphing calculator.
The correct answer is 288. If you dont believe me then go to wolfram alpha and input the equation and it will provide you with 288 as well.
Saw this somewhere else. What is the answer? (There was no Math forum so I placed it here.)
First solution:
Forty-eight divided by two times nine plus three =
48/ 2(9 +3) = 48/ 2(12) = 48/ 24 = 2
Second solution:
Forty-eight divided by two times nine plus three =
48/2 * (9 +3) = 24 * 12 = 288
I can see the confusion. Both are correct depending on the intention of the person creating this equation. I think more parentheses are necessary to clarify the order of operation here.
If the first solution is to be correct, the equation could be better written as "48 / (2(9+3)) =" as one poster indicated. For the second solution to be correct, the equation could be better written as "48/2 * (9+3) = ".
First solution:
Forty-eight divided by two times nine plus three =
48/ 2(9 +3) = 48/ 2(12) = 48/ 24 = 2
Second solution:
Forty-eight divided by two times nine plus three =
48/2 * (9 +3) = 24 * 12 = 288
I can see the confusion. Both are correct depending on the intention of the person creating this equation. I think more parentheses are necessary to clarify the order of operation here.
If the first solution is to be correct, the equation could be better written as "48 / (2(9+3)) =" as one poster indicated. For the second solution to be correct, the equation could be better written as "48/2 * (9+3) = ".
The first solution can not be correct UNLESS it is displayed as such (using the parentheses to encase it as a denominator). The way it is written, you can not get 2 unless you multiply the numerator with the denominator which is a violation of the basic fundamentals of fractions.
As I said, write everything out as a fraction and you see the problem. The only thing that is confusing here is that the "slash" to denote division confuses people num/num*num making them think it is attached to the denominator, but this is an illusion of the manner in which it is displayed due to standard computer characters. Unless otherwise denoted, it is fraction of the whole number in the parentheses.
That is, after you do the addition within the parentheses, it reads as:
forty-eight halves multiplied by twelve. which is to say that the product of forty-eight and twelve is halved.
As I said, write everything in fractions and you can't make the mistake that people are making.
48/2 * (9/1+3/1)
Which to make it easier to see, you separate 48/2's to be:
48/1 * 1/2 * (9/1 +3/1)
48/1 * 1/2 * 12/1
Now combine the fractions.
We get (48*1*12) /(1*2*1) with simple multiplication across like numerators and like denominators and simplify the fraction.
Now you could have left 48/2 and just simplified the parentheses giving you.
48/2 * 12/1 then, you could have used common factors between 2 and 12 to reduce it to 48/1*6/1.
Regardless, you will always come up with 288.
The problem here is that people are forgetting their fractions.
A common technique for remembering the order of operations is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and these two outrank addition and subtraction (which are together on the bottom rank). When you have a bunch of operations of the same rank, you just operate from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you get to the division first. If you're not sure of this, test it in your calculator, which has been programmed with the Order of Operations hierarchy. For instance, typesetting this into a graphing calculator.
The correct answer is 288. If you dont believe me then go to wolfram alpha and input the equation and it will provide you with 288 as well.
Exactly!
This issue really is a problem with people seeing the fraction. I always tell people who have problems with fractions to write everything they are dealing with as a fraction to avoid these problems. The whole number 9 is really 9/1 and by displaying it as such, they never confuse a denominator with a numerator.
As much as I hate to say it, this really is a problem with people and fractions to which they never really properly learned the relationship between them.
In order to get 2 (the wrong answer), you have to do the following.
1/2*(9+3) = 1/2*12=1/24 to which when you multiply to 48/24 = 2.
Again, this is incorrect as you are taking 1/2*12/1 and multiplying a denominator (2) with a numerator (12) to get your 1/24, yet 12 != 1/12 and can not cross multiply as such. You can only multiply across numerators (top) and denominators (bottom), not across numerators and denominators. You can however if you choose find common factors between them and reduce (take out the common factor of each).
So, 2 is the common factor between 2 and 12. There is one two in the (2) denominator and six two's in the (12) numerator that when reduced which gives us:
1/1*6/1= 6, which when multiplied to the numerator 48/1 = 288
Now, what you keep doing is thinking that because 1/2 is written on the same line (a limitation of standard computer fonts) that it implies that the 12 is on the same level as the two. It may "appear" as of such by quick glance, but that is a severe mistake.
What the equation would look like if properly written on a black board or with a program that can use more than a single line to write it is,
48
--- (9+3)
2
This representation is what it is actually saying. Note now the bias of (9+3) appearing to be a denominator is removed because we use the actual full representation of levels to which the equation is.
This shows us really that (9+3) is actual a numerator:
(sorry for the poor spacing, but I can't seem to get some specific BBcodes to work for math or accepting spaces.)
(9+3)
------
1
As you can see, when it appears as such, you no longer mistake it for being a part of the denominator. If we wanted to denote such, we must specifically encase the entire denominator with parentheses:
48/(2*(9+3))
Then it would essentially force it to be considered as a denominator, or rather the numbers would not be 9 and 3, but actually 1/9 and 1/3 which would result in the answer of 2.
So, as I said, the answer is 288 and can never be two unless you denote such explicitly by adding a parentheses to encase the (9+3) in the denominator.
Understand now?
We can use an example of pies to give you a visual representation of the error if you like?
I'm getting a huge kick out of Nomander attempt to prove his way.
There is no trying, I am right here.
Notice how I can provide the proper evidence to show why I am correct and those who claim I am wrong can simply... well... "say" I am. I am right, it is evident as I have shown. People need to show their work and prove their equation. This isn't a subjective discussion.
Though you are welcome to show me otherwise?
Seriously, this is bad for people who are attempting to combat this as it shows their understanding of mathematics is less than grade school level. It is one thing to be ignorant, but another to be arrogant about it as it leads to stupidity.
If people really want to see their error, have them resolve the fractions first and then do the equation. The answer will be... 288.
------------------------------------------
last example:
Algebra.
Let x = 9
48/2*(x+3)=288
Solve and x = 9
48/2*(x+3)=2
Solve and x = -35/12
48/2*(9+3) != 2
Proof that the answer is NOT 2.
I am done, there is no arguing with that.
Last edited by ShadowCaver; 09-25-2011 at 08:41 AM..
it's wrong because after you do the parentheses (9+3), you work left to right. The expression would be better if you added the parentheses, but in algebra they are not necessary.
48÷2(9+3)
48÷2(12)
24(12)
288
if you want 2 the correct expression would be
48÷(2(9+3))
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