More intuitive solution to simplifying complex fraction?

Question

My problem is this:
$$\frac{3 - \frac{1}{x}}{\frac{1}{3x} - 1}$$

This simplifies to $-3$. So to solve this you must get everything with a denominator of $3x$ for each term in the complex fraction. Is there a more intuitive way to solve this problem? Currently, there are three major steps.

1. Multiply each term in the expression to get a common denominator of $3x$ in each term and simplify
2. Then, we can rearrange the denominator:
3. Factor out $(3x-1)$ and simplify.

Answer 1

Note that $$\frac{3-\frac{1}{x}}{\frac{1}{3x}-1}=\frac{3x-1}{\frac{1}{3}-x}=\frac{9x-3}{1-3x}$$

Answer 2

I would express it as find the common denominator of the smaller fractions. Here it is $3x$, so multiply by $\frac {3x}{3x}$ $$\frac {3x}{3x}\cdot \dfrac{3 - \frac{1}{x}}{\frac{1}{3x} - 1}=\frac {9x-3}{1-3x}=-3$$

Answer 3

I would set

$$y=\frac1{3x}\implies \dfrac{3 - \frac{1}{x}}{\frac{1}{3x} - 1}=\dfrac{3 - 3y}{y - 1}=-3\cdot \dfrac{y-1}{y - 1}=-3$$

Answer 4

Well, it wouldn't work in general, but I notice that I can reverse the order of subtraction in the denominator by taking (factoring) out a -1. Then the denominator looks a lot "closer" to the numerator. In fact 3 times the denominator is the numerator, but if I throw in the 3 in the denominator to make the fraction unity, I have to balance with a 3 up above. That times the -1 I took out is -3. The combined observations and result probably take less than 15 seconds.

My description's sloppy wrt mathematical terminology, but on many equations like that it's useful to me to see whether I can do that kind of mental simplification before brute-forcing a well-defined process like we're taught (which we know will give us the correct answer!) If the stakes are high, you can even double check your answer. Test and homework questions are often contrived to result in a simple answer, and sometimes they're constructed such that a quick observation or simplification at the start will great speed up the solution process. Worst case you crunch it through the long and tedious way that you're familiar with...

Answer 5

I would try to make the numerator look similar to the denominator as follows: $$\frac {3 - \frac 1 x} {\frac 1 {3x} - 1} = - \frac {3 - \frac 1 x} {1 - \frac 1 {3 x}} = - 3 \cdot \frac {1 - \frac 1 {3x}} {1 - \frac 1 {3x}} = -3.$$ By the way, this is quite similar to both gcbound's and gimusi's answers.