// Numbas version: finer_feedback_settings {"name": "Dezimalzahlen in Bruchzahlen umwandeln", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Dezimalzahlen in Bruchzahlen umwandeln", "tags": [], "metadata": {"description": "

Identify well-known fractional equivalents of decimals. Convert obscure decimals and recurring decimals into fractions.

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https://numbas.mathcentre.ac.uk/question/22784/decimals-to-fractions/ by Lauren Richards

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Translated to German.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Brüche lassen sich als Dezimalzahlen schreiben (und periodische Dezimalzahlen können als Bruchzahlen geschrieben werden). In den folgenden Fragen können Sie üben, Dezimalzahlen in Brüche umzuwandeln.

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Der Überstrich zeigt an, dass sich die Ziffernfolge darunter periodisch wiederholt: $0,\\overline{3} = 0,3333\\dots$.

", "advice": "

a)

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Um eine endliche Dezimalzahl in einen Bruch umzuwandeln, \"erweitern\" wir mit einer geeigneten Zehnerpotenz und kürzen dann den resultierenden Bruch. 

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i)

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$\\var{a}$

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\\[
\\frac{\\var{a}}{1}\\times\\frac{10}{10}=\\frac{\\var{10a}}{10}\\text{.}
\\]

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ii)

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$\\var{b}$

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\\[
\\frac{\\var{b}}{1}\\times\\frac{100}{100}=\\frac{\\var{100b}}{100}=\\simplify{{100b}/{100}}\\text{.}
\\]

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iii)

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\n

$\\var{d}$

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\\[
\\frac{\\var{d}}{1}\\times\\frac{10}{10}=\\frac{\\var{10d}}{10}=\\simplify{{10d}/{10}}\\text{.}
\\]

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iv)

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\n

$0.\\bar{\\var{c}}$

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Um eine periodische Dezimalzahl in eine Bruchzahl umzuschreiben, machen wir den folgenden Ansatz:

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\\[
x=0.\\bar{\\var{c}}\\text{.}
\\]

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Wir multiplizieren beide Seiten mit $10$ und erhalten

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\\[
10x=\\var{c}.\\bar{\\var{c}}\\text{.}
\\]

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Jetzt ziehen wir die erste Gleichung von der zweiten Gleichung ab:

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\\[
\\begin{align}
&&\\var{c}.\\bar{\\var{c}}&={10}x\\\\
-&&{0.\\bar{\\var{c}}}&=x\\\\
&&\\overline{\\qquad} & \\overline{\\qquad}\\\\
&&{\\var{c}}&=9x\\\\
\\\\
&&\\frac{\\var{c}}{9}&=x
\\end{align}
\\]

\n

Zum Schluss kürzen wir $\\displaystyle\\frac{\\var{c}}{9}$ mit $3$ und bekommen $\\simplify{{c}/{9}}$. Also gilt $0.\\bar{\\var{c}}=\\simplify{{c}/{9}}$.

\n

\n

b)

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$\\displaystyle\\var{f}$

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\\[
\\var{f}\\times\\frac{\\var{f1000}}{\\var{f1000}}=\\frac{\\var{f2}}{\\var{f1000}}\\text{.}
\\]

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Wir berechnen den größten gemeinsamen Teiler von Zähler und Nenner; dies ist $\\var{mygcd}$.

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Folglich kann der Bruch nicht weiter gekürzt werden, und die Antwort ist

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Indem wir damit kürzen, erhalten wir

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\\[\\frac{\\var{f3}}{\\var{f4}}.\\]

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c)

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$\\var{h}.\\overline{\\var{j}\\var{k}}.$

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Wir gehen ähnlich vor wie beim letzten Teil von a):

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$x=\\var{h}.\\overline{\\var{j}\\var{k}}.$

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Um in diesem Fall zum Ziel zu kommen, multiplizieren wir mit $100 = 10^2$, weil der periodische Teil die Länge $2$ hat.

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$100x=\\var{h}\\var{j}\\var{k}.\\overline{\\var{j}\\var{k}}.$

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Wir ziehen die erste Gleichung von der zweiten ab und lösen nach $x$ auf.

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\\[
\\begin{align}
&&\\var{h}\\var{j}\\var{k}.\\overline{\\var{j}\\var{k}}&=100x\\\\
-&&\\var{h}.\\overline{\\var{j}\\var{k}}&=x\\\\
&&\\overline{\\qquad} & \\overline{\\qquad} 
\\\\
&&{{\\var{h}}\\var{j}\\var{k-h}}&=99x\\\\
\\\\
&&\\frac{\\var{numerator}}{\\var{g}}&=x\\text{.}\\\\
\\end{align}
\\]

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Zum Schluss prüfen wir noch, ob der Bruch gekürzt werden kann. Der ggT von Zähler und Nenner ist $\\var{gcd1 }$.

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Also lässt sich der Bruch nicht weiter kürzen und wir erhalten

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Indem wir damit kürzen, erhalten wir $\\displaystyle\\simplify{{{numerator}}/{g}}$ und somit 

\n

\\[
\\begin{align}
\\var{h}.\\overline{\\var{j}\\var{k}}=\\simplify{{{numerator}}/{g}}\\text{ als Bruchzahl.}\\\\
\\end{align}
\\]

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Schreiben Sie die folgenden Dezimalzahlen als gekürzte Brüche (mit positivem Nenner).

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i)

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$\\var{a}=$  [[0]] [[1]]

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ii)

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$\\var{b}=$  [[2]] [[3]]

\n

iii)

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$\\var{d}=$  [[6]] [[7]]

\n

iv)

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$0.\\bar{\\var{c}}=$  [[4]] [[5]]

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Schreiben Sie die Dezimalzahl als gekürzten Bruch.

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$\\displaystyle\\var{f} = $  [[0]] [[1]]

\n

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Schreiben Sie die gegebene Dezimalzahl als gekürzten Bruch (mit positivem Nenner).

\n

$\\var{h},\\overline{\\var{j}\\var{k}} = $  [[0]] [[1]]

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