48 ÷ x(9+3) = 288

**ska** · April 7th 2011, 04:39 PM

Can someone explain why, if 48 ÷ 2(9+3) = 288, then the reason why x doesn't equal 2 in the equation 48 ÷ x(9+3) = 288? Because I got x = 1/72. And I don't remember being taught that equations are ambiguous unless your dealing with absolute value or imaginary numbers, etc... such as in higher level math.

Referring to this:

http://www.mathhelpforum.com/math-he...-a-177168.html

**NOX Andrew** · April 7th 2011, 04:49 PM

If your equation is

$\dfrac{48(9+3)}{x} = 288$ ,

then multiply both sides of the equation by x.

$48(9 + 3) = 288x$

$\iff 48(12) = 288x$

$\iff 576 = 288x$

Divide both sides of the equation by 288.

$\dfrac{576}{288} = x$

$\iff 2 = x$

Did you try to solve the following equation instead?

$\dfrac{48}{x(9+3)} = 288$

This is an example of ambiguity in writing mathematical expressions. It's best to group the entire denominator in parenthesis so that it is clear exactly what is in the denominator if you aren't going to use LaTeX.

**ska** · April 7th 2011, 04:53 PM

I applied the distributive property, because I was taught that you shouldn't separate terms like 12x or x(9+3). So, the second equation.

**NOX Andrew** · April 7th 2011, 04:57 PM

If $\dfrac{48}{x(9+3)} = 288$ is what you meant by 48 ÷ x(9+3) = 288, then your original equation was false.

$\dfrac{48}{2(9+3)} = 288$

$\iff \dfrac{48}{2(12)} = 288$

$\iff \dfrac{48}{24} = 288$

$\iff 2 = 288$

As you can see, the equation must be false because 2 does not equal 288. In fact, the only way the equation is true is if it is written as

$\dfrac{48(9 + 3)}{2} = 288$

In which case, you should similarly write 48 ÷ x(9+3) = 288 as

$\dfrac{48(9 + 3)}{x} = 288$

**ska** · April 7th 2011, 05:11 PM

I meant that, and got 1/72 = x, not 2 = x.

But I was trying to see which equation would normally be assumed when the problem is not specified as to what the denominator is.

e.g. is it 48/[x(9+3)] or (48/x)(9+3).

**TheChaz** · April 7th 2011, 05:23 PM

It is THE SECOND.
This has been asked and answered all over the internets (sic) in the past few hours!

Functionally, there is no ambiguity; when you write 48 ÷ x • (9 + 3) ......(with or without the •)
48 is divided by x first, then the quotient is multiplied by 12.
P
E
M
D
A
S

**NOX Andrew** · April 7th 2011, 05:24 PM

For the equation $\dfrac{48}{x(9 + 3)} = 288$ , $x = \dfrac{1}{72}$ is the correct solution. There isn't any reason $x = 2$ should be a solution to the equation because $\dfrac{48}{2(9 + 3)} \neq 288$ . If the problem were written as $\dfrac{48(9 + 3)}{x} = 288$ (very different from the other way of writing it), then $x = 2$ is is a solution to the equation. I believe this may have caused some confusion, but we have since agreed that 48 ÷ x(9+3) = 288 should be written as $\dfrac{48}{x(9 + 3)} = 288$ (of which x = 2 is not a solution).

In answer to which equation should be assumed if it were ambiguous, I would assume $\dfrac{48}{x} \cdot (9 + 3) = 288 \iff \dfrac{48(9 + 3)}{x} = 288$ . It follows from the order of operations. If I were to evaluate 48 ÷ 12*4, I would divide 48 by 12 (when division and multiplication occur, perform whichever is first as you move from left to right). 48 divided by 12 is 4. Then, I would multiply by 4. 4 * 4 = 16. Therefore, 48 ÷ 12*4 = 16.

**TheChaz** · April 7th 2011, 05:30 PM

Originally Posted by NOX Andrew

...we have since agreed that 48 ÷ x(9+3) = 288 should be written as $\dfrac{48}{x(9 + 3)} = 288$ ..

I didn't agree to that.
See my reference to the TI-83...
http://www.mymathforum.com/viewtopic...t=20148#p79090

**NOX Andrew** · April 7th 2011, 05:39 PM

Sorry, I meant ska and I had since agreed that 48 ÷ x(9+3) = 288 should be written as $\dfrac{48}{x(9 + 3)} = 288$ . I was still composing my post and hadn't realized you had submitted a post of your own. (Personally, I agree with you, but I made an exception once ska clarified which way of writing the equation he had meant.)

**TheChaz** · April 7th 2011, 05:48 PM

I see. Wouldn't it be better to state that $\dfrac{48}{x(9 + 3)} = 288$ should NOT be written as 48 ÷ x(9+3) = 288
but should rather be written as 48 ÷ [x(9 + 3)]

Because while 48 ÷ x(9+3) = 288 may have come from (in a transcribing sense) two different places, it only means one thing - the thing that I have said in 3 threads on 2 forums today

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