Lösung 1.1:5c

Aus Online Mathematik Brückenkurs 1

It is quite easy to see that

21=0532=231=066653=351=302=06

which means that

123523 .

Because it is more difficult to evaluate the decimal expansion of 58 and 2134, we compare instead 58 and 2134 with 12  35 and 23 by rewriting the fractions so that they have a common denominator. We start by comparing 58 with 12  35 and 23

• We have

21=2414=84 and thus 2185 .

• Then, we have

53=5838=4024 and 85=8555=4025 , which gives that \displaystyle \frac{3}{5}<\frac{5}{8}.

• Finally,

\displaystyle \frac{2}{3}=\frac{2\centerdot 8}{3\centerdot 8}=\frac{16}{24} and \displaystyle \frac{5}{8}=\frac{5\centerdot 3}{8\centerdot 3}=\frac{15}{24}, and this gives that \displaystyle \frac{5}{8}<\frac{2}{3}.

Thus, \displaystyle {1}/{2}\;<{3}/{5}\;<{5}/{8}\;<{2}/{3}\;.

When we compare \displaystyle {21}/{34}\; with \displaystyle {1}/{2,\ \ {3}/{5}\;}\; and \displaystyle {2}/{3}\;, we obtain:

• because

\displaystyle \frac{1}{2}=\frac{1\centerdot 17}{2\centerdot 17}=\frac{17}{34}, so \displaystyle \frac{1}{2}<\frac{21}{34}

• furthermore,

\displaystyle \frac{3}{5}=\frac{3\centerdot 34}{5\centerdot 34}=\frac{102}{170} and \displaystyle \frac{21}{34}=\frac{21\centerdot 5}{34\centerdot 5}=\frac{105}{170}, i.e. \displaystyle \frac{3}{5}<\frac{21}{34}.

• we have

\displaystyle \frac{5}{8}=\frac{5\centerdot 17}{8\centerdot 17}=\frac{85}{136} and \displaystyle \frac{21}{34}=\frac{21\centerdot 4}{34\centerdot 4}=\frac{84}{136}, which gives that \displaystyle \frac{21}{34}<\frac{5}{8}.

The answer is \displaystyle {1}/{2}\;<{3}/{5}\;<{21}/{34}\;<{5}/{8}\;<{2}/{3}\;.