I might, also, but the topic of this discussion is how to work with ratios, at the high-school level. Personally, I'm confident that more than 50% of high school students enrolling in college need to revisit fractions. High school students ought not be afforded a pass.

Actually, since the problem

**includes **a fraction, you

**can't** avoid working with fractions in solving it; but you can decide

**how **you want to do so.

Multiplying through by 2, so that you find [imath]2\cdot\frac{3}{2}=3[/imath], is a little easier than the division (though many students have trouble even with this). In fact, I often suggest avoiding division, if only because writing a fraction on top of a fraction is confusing, and risks exactly the error that was made here, even if in principle you know what to do. So, for example, I recommend clearing fractions before solving (which is what this is), which still gives practice working with fractions.

In addition, though, solving by

**completing the square** when there are only two terms is not the best strategy, and it's good to be aware of that, even though practice with both methods is a good idea. I'll often commend a student for any correct work they did while taking the hard way, and then show how they could have avoided it.

But of course you are right that very many students (even in calculus!) struggle with fractions, and need to practice doing all the operations correctly. I've seen it constantly. Both fractions and completing the square are worth learning well; but avoiding them when there are easier and safer methods is also worth learning. (I'd be interested to know whether the instructions for the problem specified completing the square.)