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Solution 1.1:5c

From Förberedande kurs i matematik 1

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m (Lösning 1.1:5c moved to Solution 1.1:5c: Robot: moved page)
Current revision (09:21, 22 September 2008) (edit) (undo)
 
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{{NAVCONTENT_START}}
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It is quite easy to see that
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<center> [[Image:1_1_5c-1(2).gif]] </center>
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<center> [[Image:1_1_5c-2(2).gif]] </center>
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{{NAVCONTENT_STOP}}
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<math>\begin{align}
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& \frac{1}{2}=0.5 \\
 +
& \\
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& \frac{2}{3}=2\centerdot \frac{1}{3}=0.666... \\
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& \\
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& \frac{3}{5}=3\centerdot \frac{1}{5}=3\centerdot 0.2=0.6 \\
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\end{align}</math>
 +
 
 +
<math></math>
 +
 
 +
 
 +
which means that
 +
 
 +
 +
<math>{1}/{2<{3}/{5<{2}/{3}\;}\;}\;</math> .
 +
 
 +
Because it is more difficult to evaluate the decimal expansion of
 +
<math>{5}/{8}\;</math>
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and
 +
<math>{21}/{34}\;</math>, we compare instead
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<math>{5}/{8}\;</math>
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and
 +
<math>{21}/{34}\;</math>
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with
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<math>{1}/{2,\ \ {3}/{5}\;}\;</math> and
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<math>{2}/{3}\;</math>
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by rewriting the fractions so that they have a common denominator. We start by comparing
 +
<math>{5}/{8}\;</math>
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with
 +
<math>{1}/{2,\ \ {3}/{5}\;}\;</math>
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and
 +
<math>{2}/{3}\;</math>
 +
 
 +
 
 +
* We have
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<math>\frac{1}{2}=\frac{1\centerdot 4}{2\centerdot 4}=\frac{4}{8}</math>
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and thus
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<math>\frac{1}{2}<\frac{5}{8}</math>
 +
.
 +
 
 +
* Then, we have
 +
<math>\frac{3}{5}=\frac{3\centerdot 8}{5\centerdot 8}=\frac{24}{40}</math>
 +
and
 +
<math>\frac{5}{8}=\frac{5\centerdot 5}{8\centerdot 5}=\frac{25}{40}</math>
 +
, which gives that
 +
<math>\frac{3}{5}<\frac{5}{8}</math>.
 +
 
 +
* Finally,
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<math>\frac{2}{3}=\frac{2\centerdot 8}{3\centerdot 8}=\frac{16}{24}</math>
 +
and
 +
<math>\frac{5}{8}=\frac{5\centerdot 3}{8\centerdot 3}=\frac{15}{24}</math>, and this gives that
 +
<math>\frac{5}{8}<\frac{2}{3}</math>.
 +
 
 +
Thus,
 +
<math>{1}/{2}\;<{3}/{5}\;<{5}/{8}\;<{2}/{3}\;</math>.
 +
 +
When we compare
 +
<math>{21}/{34}\;</math>
 +
with
 +
<math>{1}/{2,\ \ {3}/{5}\;}\;</math>
 +
and
 +
<math>{2}/{3}\;</math>, we obtain:
 +
 
 +
* because
 +
<math>\frac{1}{2}=\frac{1\centerdot 17}{2\centerdot 17}=\frac{17}{34}</math>, so
 +
<math>\frac{1}{2}<\frac{21}{34}</math>
 +
 
 +
 
 +
* furthermore,
 +
<math>\frac{3}{5}=\frac{3\centerdot 34}{5\centerdot 34}=\frac{102}{170}</math>
 +
and
 +
<math>\frac{21}{34}=\frac{21\centerdot 5}{34\centerdot 5}=\frac{105}{170}</math>, i.e.
 +
<math>\frac{3}{5}<\frac{21}{34}</math>.
 +
 
 +
* we have
 +
<math>\frac{5}{8}=\frac{5\centerdot 17}{8\centerdot 17}=\frac{85}{136}</math>
 +
and
 +
<math>\frac{21}{34}=\frac{21\centerdot 4}{34\centerdot 4}=\frac{84}{136}</math>, which gives that
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<math>\frac{21}{34}<\frac{5}{8}</math>.
 +
 
 +
The answer is
 +
<math>{1}/{2}\;<{3}/{5}\;<{21}/{34}\;<{5}/{8}\;<{2}/{3}\;</math>.

Current revision

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 5385.

  • Finally,

32=3828=2416 and 85=8353=2415, and this gives that 8532.

Thus, 12355823.

When we compare 2134 with 12  35 and 23, we obtain:

  • because

21=217117=3417, so 213421


  • furthermore,

53=534334=170102 and 3421=345215=170105, i.e. 533421.

  • we have

85=817517=85136 and 3421=344214=84136, which gives that 342185.

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

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