Show that $d \geq b+f$.

Question

Let $a, b, c, d, e, f$ be positive integers such that:

$$\dfrac{a}{b}<\dfrac{c}{d}<\dfrac{e}{f}$$

Suppose $af - be = -1$. Show that $d \geq b+f$.

Looked quite simple at first sight...but havent been able to solve this inequality. Have no idea where to start. Need help. Thanks!!

Answer 1

Hint. Since $a,b,c,d$ are positive integers, then $$(a/b) < (c/d)\Rightarrow cb-ad>0\Rightarrow cb-ad\geq 1.$$

Answer 2

Hint: Try to derive that $bf<d$. What can you conclude from there?
(Hint 2: $bf = (b-1)(f-1) + (b+f) - 1$.)