# Reciprocals of Reciprocals

**Number Theory**Level 4

\[\large{\begin{cases} \frac{1}{a} - \frac{1}{b} - \frac{1}{c} + \frac{1}{d} &=& \frac{1}{20} \\ a - b - c + d &=& 0 \\ \end{cases} } \]

Let \(a < b < c < d\) be positive integers satisfying the system of equations above. If \( \gcd ( a, b, c , d ) = 1\), then what is the value of \(\dfrac{c}{a} - \dfrac{b}{d}\)?