someone Wrote:I'm applying the Law of Distribution of Multiplication Over Addition. In truth, the very first step in solving that is distributing that two through the parenthesis, so it becomes:
48 ÷ 2( 9 + 3) = 48 ÷ ( 2 * 9 + 2 * 3 ) = 48 ÷ ( 18 + 6 ) = 48 ÷ ( 24) = 48 ÷ 24 = 2
I do indeed have a degree for this sort of thing, I wasn't taught this law until Abstract Algebra, so I can see how you may be confused if all you've had is up to College Algebra; however, do believe me when I say 2 is the correct answer. Because there is no explicit multiplication between the 2 and (9+3), that 2 is associated to the parenthesis. As in johnfarson's example, 4x ÷ 2x, the coefficients are associated to the variable, thus having higher precedence than the strict left-to-right multiplication/division. It's the same with the parenthesis. If you were to, however, explicitly write the equation as he did in his second example 4 * x ÷ 2 * x, then you dissasociate the coefficients and it would indeed be 2x². It's confusing, I know, but association is an important concept that really needs to be taught at a lower level than college.

johnfarson Wrote:When there is multiplication with no explicit symbol, like in distribution into parenthesis it has priority over other multiplication and division. Lets take this for example:

4x ÷ 2x = 2

No one will argue the answer is 2. But if you strictly go left to right, the answer you would get is:

4 * x ÷ 2 * x = 2x²

Which is obviously wrong. Therefore, when there is no multiplication symbol it is implied that the entire expression between operators will be multiplied before any other operations are performed. In other words, it's not really 4x ÷ 2x, but (4x) ÷ (2x). The parenthesis are implied because there is no multiplication symbol between the number and variable.

Now lets see what happens if you substitute x = (9 + 3), and perform the calculation the wrong way:

4 * (9 + 3) ÷ 2 * (9 +3) = 288

288 is obviously wrong. We already know 4x ÷ 2x = 2. Note that 2x² = 288 when x=12. So we know the equation should have been:

(4 * (9 + 3)) ÷ (2 * (9 +3)) = 2

Which is probably how this originally popped up. Some math student made the error, and either he or his teacher posted it on the internet. He first did the calculation for 4x:

(4 * (9 +3)) = 48

Then he substituted 2(9 +3) in for 2x, but did not add the implied parenthesis and typed this into his calculator:

48 ÷ 2(9 + 3) = 288

He should have entered:

48 ÷ (2 * (9 +3)) = 2

The equation in OP is not the original equation. It simply showed a students error midway through solving the problem. And how he came to get an answer of 288, when the correct answer was 2.