Certain Fraction between FractionsIs there always a fraction r s with p q &lt; r s &lt; p + 1 q and s &lt; q for 0 &lt; p &lt; q − 1 ∈ Z and r , s ∈ Z ?

Since 0 &lt; p &lt; q − 1, we have p q &lt; p q − 1 &lt; p + 1 q

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deceptie3j

2022-06-21

Certain Fraction between Fractions
Is there always a fraction

\frac{r}{s}

with

\frac{p}{q} < \frac{r}{s} < \frac{p + 1}{q}

and

s < q

for

0 < p < q - 1 \in Z

and

r, s \in Z

Jaida Sanders

Beginner2022-06-22Added 18 answers

Since

0 < p < q - 1

, we have

\frac{p}{q} < \frac{p}{q - 1} < \frac{p + 1}{q}

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