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How would I reduce this fraction?

$$\frac{km+kn}{n^2+nm}$$

I think it would be $\frac{2k}{n^2}$ but I am not sure.

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    @Brett, that's not fair! Now everybody else's answers look worse.2012-06-07
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    In general, blind speculation is not a very good problem solving method. Study the solutions well so that you remember a starting point next time (in this case, it was factoring.)2012-06-07
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    In general, when reducing a fraction, factoring is your friend.2012-06-07
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    Since "reducing" is "canceling factors", yes.2012-06-07
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    Why did you think it was $\frac{2k}{n^2}$?2012-06-07
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    @TonyK Sorry, the OP had half uppercase and half lowercase. Lowercase just looked better to me.2012-06-07

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$$\frac{KM+KN}{N^2+NM}=\frac{K(M+N)}{N(N+M)}=\frac{K}{N}$$

As Alex Jordan comments, we can cancel out $M+N$ if and only if $M+N\neq 0$. In this case, given the fact that the denominator is of the form $N(M+N)$ we already know this is a non-zero number, and we can cancel.

On the other hand, if we were given something like $x=y$ then either $x=y=0$ or $x\neq 0$ and then we can divide by $x$ and have $\frac yx=1$.

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    I think it's worth teaching students that the cancellation here was only valid if $M+N$ was not equal to $0$. Equivalently, when $N\neq-M$. So I train my algebra students to write $\frac{K}{N},\quad N\neq -M$.2012-06-07
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    @alex.jordan: I agree. I will add something about that.2012-06-07
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    Well, if $\,n+m=0\,$ then in fact we have $\,\frac{kn+km}{n^2+nm}=\frac{0}{0}\,$ , so *assuming* a legal mathematical expression was given would make the above note reduntant. What is advisable, imo, is that upon solving the exercise, the student should remark that we're doing mathematics *only* under the assumption that $\,n\neq -m\,$2012-06-07
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    @DonAntonio: Yes. This is essentially what I meant, and what I believe alex.jordan meant in his comment. It is worth noting that canceling is equivalent to dividing and one cannot "just cancel a factor". We also do not know how did the OP come by the mathematical expression and whether or not $N+M$ was given in the denominator or "just got there".2012-06-07
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    @DonAntonio I think my point is that it would be wrong to write statements like $\frac{ab}{2b}=\frac{a}{2}$. The RHS makes sense for $b=0$ but the LHS implicitly remains undefined for $b=0$. So the reader should be explicitly told to exclude $b=0$ from the RHS, leaving the expression undefined as it was in the LHS.2012-06-08
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$$\frac{km+kn}{n^2+nm}=\frac{k(m+n)}{n(n+m)}=\frac{k}{n}$$