## Task 4 - 040914

It is known about the natural numbers $p$ and $q$ that $0<p<q$.

a) Order the numbers $1, \frac{p}{q}$, and $\frac{q}{p}$ by size! Start with the smallest number!

b) Determine which of the two numbers $\frac{p}{q}$ and $\frac{q}{p}$ is closer to 1!
a) It is $p<q$. Dividing by $q$ (allowed because $Q<0$) yields $\frac{p}{q}<$. Dividing by $p$ results in $1<\frac{q}{p}$, thus

$$
\Rightarrow \frac{p}{q}<1<\frac{q}{p}
$$

b) From $p<q$ we get $p(q-p)<q(q-p)$. Dividing by $pq$ yields

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

This means that the distance from $\frac{p}{q}$ to one is smaller than the distance from $\frac{q}{p}$ to one.