Given $a > b > c > d > 0$ with relatively prime integers. Compare the fractions:

Since all values are positive and $a > b > c > d$:

• $\dfrac{a}{c}$ vs. $\dfrac{a}{b}$: Since $c < b$, we have $\dfrac{a}{c} > \dfrac{a}{b}$

• $\dfrac{a}{c}$ vs. $\dfrac{b}{c}$: Since $a > b$, we have $\dfrac{a}{c} > \dfrac{b}{c}$

• $\dfrac{a}{c}$ vs. $\dfrac{c}{a}$: Since $a > c > 0$, we have $\dfrac{a}{c} > 1 > \dfrac{c}{a}$

• $\dfrac{a}{c}$ vs. $\dfrac{d}{a}$: Since $a > d$ and $c < a$, we have $\dfrac{a}{c} > 1 > \dfrac{d}{a}$

Therefore $\dfrac{a}{c}$ is the greatest.