### The denominator of a rational number is greater than its numerator by $8.$ If $8$ is subtracted from the numerator and $7$ is added to its denominator$,$ the new number becomes $\dfrac { 1 } { 24 }.$ What is the original number$?$

$\dfrac { 9 } { 17 }$
1. Let the numerator be $x.$ According to the question, the denominator will be equal to $x + 8.$ The fraction will be $\dfrac { x } { x+8 } .$
2. The new numerator is $x - 8,$ and the new denominator is $x + 8 + 7 = x + 15.$ The new fraction will become$: \dfrac { x-8 } { x+15 } .$
3. According to the question, $\dfrac { x-8 } { x+15 } = \dfrac { 1 } { 24 }.$ On cross multiplication we get$:$ \begin{align} & 24(x - 8) = x + 15 \\ \implies & 24x - 192 = x + 15 \\ \implies & 23x = 207 \\ \implies & x = 9 \end{align}
4. Hence the original fraction will be equal to $\dfrac { 9 } { 9+8 },$ or $\dfrac { 9 } { 17 }$.