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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!!

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marked as duplicate by Martin R, Rohan, mrp, Antonios-Alexandros Robotis, kingW3 Feb 20 '17 at 16:59

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

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  • $\begingroup$ I am still not able to prove it...I tried a lot. How to proceed after this? $\endgroup$ – SirXYZ Nov 6 '16 at 10:32
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Hint: Try to derive that $bf<d$. What can you conclude from there?
(Hint 2: $bf = (b-1)(f-1) + (b+f) - 1$.)

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