Dividing by Fractions in Algebra

Question

You can divide by a fraction by multiplying by its reciprocal (i.e. $\frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c}$). Given the equation $\frac{1}{2}x = 5$, intuitively, you could either multiply both sides by two or divide both sides by $\frac{1}{2}$. Both of these solutions give the same result, $x=10$.

My maths teacher says that, when showing your work, you can only multiply by two. Subjectively, this is simpler. Is this an accepted rule, or is it just personal preference?

P.S. One of my previous teachers did teach to divide by fractions in this case.

Answer 1

That’s totally inaccurate. You can definitely divide by fractions. For example,

$$\begin{align} 0.5x &= 5 \\ \frac{0.5x}{0.5} &= \frac{5}{0.5} \\ x &= 10 \\ \end{align}$$

It’s all the same.

I’d also like to add that “don’t divide by fractions” doesn’t make any sense when you have irrational numbers. For example, if $\pi x=2$, how, are you supposed to solve for $x$? You can’t divide by $\pi$ since it’s not an integer.

Answer 2

From what you mentioned ("multiplying by its reciprocal"), we can see that multiplying by $2$ is dividing by $\frac{1}{2}$.

$\dfrac{x}{\frac{1}{2}}=x\cdot\frac{2}{1}=2x$