What number must be subtracted from the denominator of $\frac{10}{23}$ to make the result $\frac13$? Is it $\frac{10}{23} - \frac{1}x = \frac13$ or something similar?

Given that $\frac{10}{23} - \frac{1}{x} = \frac{1}{3}$ then multiply both sides by $3 \cdot 23 \cdot x$ to obtain $3\cdot 10 \cdot x - 3 \cdot 23 = 23 \cdot x$ which becomes $x = \frac{69}{7}$. This can be seen in the following:
\begin{align}
\frac{10}{23} - \frac{1}{x} &= \frac{10}{23} - \frac{7}{69} = \frac{30}{69} - \frac{7}{69} = \frac{23}{3 \cdot 23} = \frac{1}{3}.
\end{align}
