// Numbas version: exam_results_page_options
{"name": "Aufl\u00f6sen einer Gleichung", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Aufl\u00f6sen einer Gleichung", "tags": [], "metadata": {"description": "<p>Solve for $x$: $\\displaystyle \\frac{px+s}{ax+b} = \\frac{qx+t}{cx+d}$ with $pc=qa$.</p>\n<p>German translation of https://numbas.mathcentre.ac.uk/question/12012/solve-an-equation-in-algebraic-fractions/ by Newcastle University Mathematics and Statistics</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>L&ouml;sen Sie die folgende Gleichung nach $x$ auf.</p>\n<p>Geben Sie Ihre Antwort als Bruchzahl oder als ganze Zahl ein (nicht als Dezimalzahl).</p>", "advice": "<p>Wir multiplizieren mit den beiden Nennern und erhalten:<br/>\\[\\simplify{({p}*x+{s})*({c} * x + {d})=({q}*x+{t})*({a} * x + {b})}\\]<br/>Durch Ausmultiplizieren bekommen wir \\[\\simplify{{p*c}x^2 +{p*d+c*s}x+{s*d}={q*a}x^2 +{q*b+t*a}x+{t*b}}\\] Wir ziehen ${\\var{a*q}}x^2$ von beiden Seiten ab und erhalten \\[\\simplify{{p*d+c*s}x+{s*d}={q*b+t*a}x+{t*b}}\\] Diese lineare Gleichung l&ouml;sen wir auf zu \\[\\simplify{{p*d+c*s-q*b-t*a}x={t*b-s*d}},\\] das bedeutet \\[\\simplify{x={an1}/{an2}}.\\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"q": {"name": "q", "group": "Ungrouped variables", "definition": "p*c/a", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..5 except [p,abs(b)])", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "m*a/g", "description": "", "templateType": "anything"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(-3..3 except r)", "description": "", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "(p*d+c*s-b*q)/a", "description": "", "templateType": "anything"}, "an1": {"name": "an1", "group": "Ungrouped variables", "definition": "b*t-s*d", "description": "", "templateType": "anything"}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "templateType": "anything"}, "an2": {"name": "an2", "group": "Ungrouped variables", "definition": "p*d+s*c-a*t-b*q", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "templateType": "anything"}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "gcd(a,p)", "description": "", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "g", "m", "q", "p", "s", "r", "t", "an2", "an1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "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>\\[\\simplify{({p}*x+{s}) / ({a} * x + {b}) = ({q}*x+{t}) / ({c} * x + {d})}\\]</p>\n<p>$x=\\;$ [[0]]</p>\n<p>F&uuml;r einen Hinweis klicken Sie auf <em>Zeige Tipps</em>. (Daf&uuml;r wird 1 Punkt abgezogen.)</p>", "stepsPenalty": 1, "steps": [{"type": "information", "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>Multiplizieren Sie beide Seiten mit den beiden Nennern:<br/>\\[\\simplify{({p}*x+{s})*({c} * x + {d})=({q}*x+{t})*({a} * x + {b})}\\]<br/>Multiplizieren Sie die Produkte aus zu \\[\\simplify{{p*c}x^2 +{p*d+c*s}x+{s*d}={q*a}x^2 +{q*b+t*a}x+{t*b}}.\\] Ziehen Sie den $x^2$-Term auf beiden Seiten ab. Sie erhalten eine lineare Gleichung in $x$, die Sie noch aufl&ouml;sen m&uuml;ssen.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{an1}/{an2}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.0001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Geben Sie die Antwort als Bruchzahl oder als ganze Zahl ein, nicht als Dezimalzahl.</p>"}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}