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So given: $\frac{2}{15x^2} + \frac{3}{5x}$ LCD: $15x^2$ Therefore you change the second term to a equal term with the LCD as its denominator by multiplying by $3x$: $\frac2{15x^2} + \frac{9x}{15x^2}$

For an answer of $\frac{2 + 9x}{15x^2}$ (correct)

My question is why can you not multiply the whole expression to begin with by the LCD? Why is the following not allowed:

$15x^2 (\frac{2}{15x^2} + \frac{3}{5x}) = 2 + 9x $(incorrect)

I know you can multiply the whole expression by LCD in complex rational expressions like:

$$\frac{2(1/2+1/2)}{2(1/2+1/2)}=\frac{1+1}{1+1}=\frac{2}{2}=1$$

( Assume you do not know a/a=1 ; I know its an easy example but Im using this just to show how it works in another scenario and ask why it works in one and not the first)

Thanks for helping me out with an easy problem, that I should really know by now...

Ted
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Sphygmomanometer
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    Because you are not multiplying by 1, you are multiplying by $15x^2$, and not $\frac{15x^2}{15x^2}$ – Jose M Serra Oct 29 '17 at 23:45
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    You can multiply an equation by the LCD to get rid of denominators, and the result will be a logically equivalent equation (with caveats for restricted values). You can also multiply an expression by an LCD to get rid of denominators, but then the resulting expression may not be equivalent to the first one. For instance, if you multiply $\frac{1}{2}+\frac{1}{2}$ by $2$ you get $1+1$, and these are inequivalent. It's not at all clear to me what you're trying to say when you multiplied $\frac{1}{2}+\frac{1}{2}$ by $2$; please edit your question to be more readable. – anon Oct 29 '17 at 23:46
  • @Sphygmomanometer Please verify that the edits did not change your intended meaning. – Ted Oct 30 '17 at 00:03
  • Thank you anon and Funky, I figured it out. When getting rid of the denominator I was changing the expression. To anon: in the now cleared up example, thanks Ted, I found the LCD was 2 and multiplied everything by 2 to get rid of denominators. Why is this allowed? Isnt 1/2 being changed to 1? I understand the answer is correct but it is different from the first scenario. – Sphygmomanometer Oct 30 '17 at 00:12
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    In your example with $1/2$s you are not multiplying the fraction by $2$, you are multiplying it by $\frac{2}{2}$, which is just $1$. When you multiply an expression by $1$, you do not change what it is or represents. When you multiply an expression by something other than $1$, that can change it. In particular, when you multiply an expression by $15x^2$, that changes it to something else. – anon Oct 30 '17 at 01:04
  • I understand the changing part but in the 1/2s I am multiplying the fraction by [distributing] 2 (LCD) into 1/2 and 1/2 to cancel out the denominators so I am left with 1 + 1/ 1+ 1. – Sphygmomanometer Oct 30 '17 at 02:45

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