Simplicity of a simple problem

$\dfrac{17-\dfrac{1}{18}}{18-\dfrac{1}{17}} \\ = \dfrac{\dfrac{(17\times 18) - 1}{18}}{\dfrac{(17\times 18) - 1}{17}} \\ = \dfrac{\dfrac{1}{18}}{\dfrac{1}{17}}\\ = \dfrac{17}{18}$

$\frac {17 - \frac{1}{18}}{18 - \frac {1}{17}} = \frac { \frac {17 × 18 - 1}{18}}{\frac {18 × 17 - 1}{17}} = \frac {17}{18} × \frac {17 × 18 - 1}{17 × 18 - 1} = \boxed {\frac {17}{18}}$

Let, 17 = x

18 = y

Therefore,

(17-1/18)/(18-1/17)=(x-1/y)/(y-1/x)

→(xy-1/y)*(x/xy-1)

→xy-1 get cancelled out

→x/y

→17/18

$\begin{aligned} \frac{17-\frac{1}{18}}{18-\frac{1}{17}} &= \frac{\frac{305}{18}}{\frac{305}{17}} \\&= \frac{\color{#3D99F6}{305} \cdot 17}{18 \cdot \color{#3D99F6}{305}}\quad\quad{\text{Cancel out}} \\&= \frac{17}{18} \space \square \end{aligned}$

There are many ways to solve for $\frac{17-\frac{1}{18}}{18-\frac{1}{17}}$

$\large \text{ADIOS!!!}$