// Numbas version: exam_results_page_options
{"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": "<p>Identify well-known fractional equivalents of decimals.&nbsp;Convert obscure decimals and recurring decimals into fractions.</p>\n<p>https://numbas.mathcentre.ac.uk/question/22784/decimals-to-fractions/ by Lauren Richards</p>\n<p>Translated to German.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Br&uuml;che lassen sich als Dezimalzahlen schreiben (und periodische Dezimalzahlen k&ouml;nnen als Bruchzahlen geschrieben werden). In den folgenden Fragen k&ouml;nnen &uuml;ben, Dezimalzahlen in Br&uuml;che umzuwandeln.</p>\n<p>Der &Uuml;berstrich zeigt an, dass sich die Ziffernfolge darunter periodisch wiederholt: $0,\\overline{3} = 0,3333\\dots$.</p>", "advice": "<h4>a)</h4>\n<p>Um eine endliche Dezimalzahl in einen Bruch umzuwandeln, \"erweitern\" wir mit einer geeigneten Zehnerpotenz und k&uuml;rzen dann den resultierenden Bruch.&nbsp; </p>\n<p><em>i)</em></p>\n<p>$\\var{a}$</p>\n<p>\\[<br/>\\frac{\\var{a}}{1}\\times\\frac{10}{10}=\\frac{\\var{10a}}{10}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\"><em>ii)</em></p>\n<p style=\"text-align: left;\">$\\var{b}$</p>\n<p style=\"text-align: left;\">\\[<br/>\\frac{\\var{b}}{1}\\times\\frac{100}{100}=\\frac{\\var{100b}}{100}=\\simplify{{100b}/{100}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\"><em>iii)</em></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">$\\var{d}$</p>\n<p style=\"text-align: left;\">\\[<br/>\\frac{\\var{d}}{1}\\times\\frac{10}{10}=\\frac{\\var{10d}}{10}=\\simplify{{10d}/{10}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\"><em>iv)</em></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">$0.\\bar{\\var{c}}$</p>\n<p style=\"text-align: left;\">Um eine periodische Dezimalzahl in eine Bruchzahl umzuschreiben, machen wir den folgenden Ansatz:</p>\n<p style=\"text-align: left;\">\\[<br/>x=0.\\bar{\\var{c}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\">Wir multiplizieren beide Seiten mit $10$ und erhalten</p>\n<p style=\"text-align: left;\">\\[<br/>10x=\\var{c}.\\bar{\\var{c}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\">Jetzt ziehen wir die erste Gleichung von der zweiten Gleichung ab: </p>\n<p>\\[<br/>\\begin{align}<br/>&amp;&amp;\\var{c}.\\bar{\\var{c}}&amp;={10}x\\\\<br/>-&amp;&amp;{0.\\bar{\\var{c}}}&amp;=x\\\\<br/>&amp;&amp;\\overline{\\qquad} &amp;&nbsp;\\overline{\\qquad}\\\\<br/>&amp;&amp;{\\var{c}}&amp;=9x\\\\<br/>\\\\<br/>&amp;&amp;\\frac{\\var{c}}{9}&amp;=x<br/>\\end{align}<br/>\\]</p>\n<p style=\"text-align: left;\">Zum Schluss k&uuml;rzen wir $\\displaystyle\\frac{\\var{c}}{9}$ mit $3$ und bekommen $\\simplify{{c}/{9}}$. Also gilt $0.\\bar{\\var{c}}=\\simplify{{c}/{9}}$.<br/><br/></p>\n<p style=\"text-align: center;\"></p>\n<h4>b)</h4>\n<p>$\\displaystyle\\var{f}$</p>\n<p style=\"text-align: left;\">\\[<br/>\\var{f}\\times\\frac{\\var{f1000}}{\\var{f1000}}=\\frac{\\var{f2}}{\\var{f1000}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\">Wir berechnen den gr&ouml;&szlig;ten gemeinsamen Teiler von Z&auml;hler und Nenner; dies ist $\\var{mygcd}$.</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"mygcd = 1\">Folglich kann der Bruch nicht weiter gek&uuml;rzt werden, und die Antwort ist<br/></span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"mygcd &gt; 1\">Indem wir damit k&uuml;rzen, erhalten wir<br/></span></p>\n<p style=\"text-align: left;\">\\[\\frac{\\var{f3}}{\\var{f4}}.\\]</p>\n<h4>c)</h4>\n<p style=\"text-align: left;\">$\\var{h}.\\overline{\\var{j}\\var{k}}.$</p>\n<p style=\"text-align: left;\">Wir gehen &auml;hnlich vor wie beim letzten Teil von a):</p>\n<p style=\"text-align: center;\">$x=\\var{h}.\\overline{\\var{j}\\var{k}}.$</p>\n<p><br/>Um in diesem Fall zum Ziel zu kommen, multiplizieren wir mit $100 = 10^2$, weil der periodische Teil die L&auml;nge $2$ hat.</p>\n<p style=\"text-align: center;\">$100x=\\var{h}\\var{j}\\var{k}.\\overline{\\var{j}\\var{k}}.$</p>\n<p></p>\n<p style=\"text-align: left;\">Wir ziehen die erste Gleichung von der zweiten ab und l&ouml;sen nach $x$ auf.</p>\n<p style=\"text-align: left;\">\\[<br/>\\begin{align}<br/>&amp;&amp;\\var{h}\\var{j}\\var{k}.\\overline{\\var{j}\\var{k}}&amp;=100x\\\\<br/>-&amp;&amp;\\var{h}.\\overline{\\var{j}\\var{k}}&amp;=x\\\\<br/>&amp;&amp;\\overline{\\qquad} &amp;&nbsp;\\overline{\\qquad}&nbsp;<br/>\\\\<br/>&amp;&amp;{{\\var{h}}\\var{j}\\var{k-h}}&amp;=99x\\\\<br/>\\\\<br/>&amp;&amp;\\frac{\\var{numerator}}{\\var{g}}&amp;=x\\text{.}\\\\<br/>\\end{align}<br/>\\]</p>\n<p style=\"text-align: left;\">Zum Schluss pr&uuml;fen wir noch, ob der Bruch gek&uuml;rzt werden kann. Der ggT von Z&auml;hler und Nenner ist $\\var{gcd1 }$.</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd1 = 1\">Also l&auml;sst sich der Bruch nicht weiter k&uuml;rzen und wir erhalten<br/></span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd1 &gt; 1\">Indem wir damit k&uuml;rzen, erhalten wir $\\displaystyle\\simplify{{{numerator}}/{g}}$ und somit&nbsp; <br/></span></p>\n<p style=\"text-align: left;\">\\[<br/>\\begin{align}<br/>\\var{h}.\\overline{\\var{j}\\var{k}}=\\simplify{{{numerator}}/{g}}\\text{ als Bruchzahl.}\\\\<br/>\\end{align}<br/>\\]</p>", "rulesets": {}, "extensions": [], "variables": {"answer": {"name": "answer", "group": "Part a", "definition": "round(10c-c)", "description": "", "templateType": "anything"}, "cround": {"name": "cround", "group": "Part a", "definition": "c/3", "description": "", "templateType": "anything"}, "g": {"name": "g", "group": "Part b", "definition": "99", "description": "", "templateType": "anything"}, "j": {"name": "j", "group": "Part b", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "g_coprime": {"name": "g_coprime", "group": "Part b", "definition": "g/gcd1", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Part a", "definition": "random(0.2,0.4,0.6,0.8)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Part a", "definition": "random(0.1,0.3,0.7,0.9)", "description": "", "templateType": "anything"}, "gcd1": {"name": "gcd1", "group": "Part b", "definition": "gcd(numerator,g)", "description": "", "templateType": "anything"}, "h": {"name": "h", "group": "Part b", "definition": "random(1..5)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Part a", "definition": "random(0.25,0.75)", "description": "", "templateType": "anything"}, "f4": {"name": "f4", "group": "Part b", "definition": "f1000/gcd(f2,f1000)", "description": "", "templateType": "anything"}, "d_coprime": {"name": "d_coprime", "group": "Part a", "definition": "10d/gcd(10d,10)", "description": "", "templateType": "anything"}, "f1000": {"name": "f1000", "group": "Part b", "definition": "1000", "description": "", "templateType": "anything"}, "numerator_coprime": {"name": "numerator_coprime", "group": "Part b", "definition": "numerator/gcd1", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Part a", "definition": "random(3,6)", "description": "", "templateType": "anything"}, "k": {"name": "k", "group": "Part b", "definition": "random(h..9 except j except h)", "description": "", "templateType": "anything"}, "f2": {"name": "f2", "group": "Part b", "definition": "precround(f1000*f,0)", "description": "", "templateType": "anything"}, "f3": {"name": "f3", "group": "Part b", "definition": "f2/gcd(f2,f1000)", "description": "", "templateType": "anything"}, "numerator": {"name": "numerator", "group": "Part b", "definition": "h*100+j*10+k - h", "description": "", "templateType": 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numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Schreiben Sie die folgenden Dezimalzahlen als gek&uuml;rzte Br&uuml;che (mit positivem Nenner).</p>\n<p><em>i)</em></p>\n<p>$\\var{a}=$<em>&nbsp;</em><fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>\n<p><em>ii)</em></p>\n<p><em></em>$\\var{b}=$&nbsp;<fraction> <numerator>[[2]]</numerator> <denominator>[[3]]</denominator> </fraction></p>\n<p><em>iii)</em></p>\n<p>$\\var{d}=$&nbsp;<fraction> <numerator>[[6]]</numerator> <denominator>[[7]]</denominator> </fraction></p>\n<p><em>iv)</em></p>\n<p><em></em>$0.\\bar{\\var{c}}=$&nbsp;<fraction> <numerator>[[4]]</numerator> <denominator>[[5]]</denominator></fraction></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a*10", "maxValue": "a*10", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "10", "maxValue": "10", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b_coprime", "maxValue": "b_coprime", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "4", "maxValue": "4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "cround", "maxValue": "cround", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "3", "maxValue": "3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "d_coprime", "maxValue": "d_coprime", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "5", "maxValue": "5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Schreiben Sie die Dezimalzahl als gek&uuml;rzten Bruch. </p>\n<p>$\\displaystyle\\var{f} = $&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>\n<p><em></em></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "f3", "maxValue": "f3", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "f4", "maxValue": "f4", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Schreiben Sie die gegebene Dezimalzahl als gek&uuml;rzten Bruch (mit positivem Nenner).</p>\n<p>$\\var{h},\\overline{\\var{j}\\var{k}} = $&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "numerator_coprime", "maxValue": "numerator_coprime", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "g_coprime", "maxValue": "g_coprime", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], 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