# Reducing a fraction?

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. – TonyK Jun 7 '12 at 18:21
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.) – rschwieb Jun 7 '12 at 18:43
In general, when reducing a fraction, factoring is your friend. – Cameron Buie Jun 7 '12 at 19:01
Since "reducing" is "canceling factors", yes. – The Chaz 2.0 Jun 7 '12 at 19:10
@TonyK Sorry, the OP had half uppercase and half lowercase. Lowercase just looked better to me. – Brett Frankel Jun 7 '12 at 19:48

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

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 $\frac{K}{N},\quad N\neq -M$. – alex.jordan Jun 7 '12 at 19:27
@alex.jordan: I agree. I will add something about that. – Asaf Karagila Jun 7 '12 at 19:29
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 $\,n\neq -m\,$ – DonAntonio Jun 7 '12 at 21:28
@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". – Asaf Karagila Jun 7 '12 at 21:29
@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. – alex.jordan Jun 8 '12 at 0:01

$$\frac{km+kn}{n^2+nm}=\frac{k(m+n)}{n(n+m)}=\frac{k}{n}$$

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