// Numbas version: finer_feedback_settings
{"name": "Proportionen", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Proportionen", "tags": [], "metadata": {"description": "<p>Zu Euklid Buch 5, Definition 5</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Gegeben sie die folgende Proportion:</p>\n<p style=\"text-align: center;\">$x:\\var{a}=\\var{b}:x$</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 1060px; width: 0px; top: 69px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 1045px; width: 50px; top: 54px; height: 20px; font-size: 10px; color: black;\"></span></p>", "advice": "<p>a) Aus&nbsp;$x:\\var{a}=\\var{b}:x$ folgt unmittelbar $x^2=\\var{square}$ und daraus&nbsp;$\\frac{\\sqrt{\\var{square}}}{\\var{a}}=\\frac{\\var{b}}{\\sqrt{\\var{square}}}\\approx\\var{ratio_approx}$ .</p>\n<p>b) Es ist&nbsp;$m\\cdot x=\\var{m}\\cdot\\sqrt{\\var{square}}\\approx\\var{m*root_approx}$,&nbsp;</p>\n<p>(i) soll $n\\cdot\\var{a}$ gr&ouml;&szlig;er als diese Zahl sein, so muss $n\\cdot \\var{a}&gt;\\var{m*root_approx}\\Leftrightarrow n&gt;\\var{m*root_approx/a}$ sein,</p>\n<p>(ii) soll $n\\cdot\\var{a}$&nbsp;kleiner&nbsp;als diese Zahl sein, so muss $n\\cdot\\var{a}&lt;\\var{m*root_approx}\\Leftrightarrow n&lt;\\var{m*root_approx/a}$ sein,</p>\n<p>die gesuchten nat&uuml;rlichen Zahlen sind daher $\\var{n_min}$ und $\\var{n_max}$.</p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..3)*2", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "(2*random(3..5)+1)", "description": "", "templateType": "anything"}, "square": {"name": "square", "group": "Ungrouped variables", "definition": "a*b", "description": "", "templateType": "anything"}, "root": {"name": "root", "group": "Ungrouped variables", "definition": "sqrt(square)", "description": "", "templateType": "anything"}, "ratio": {"name": "ratio", "group": "Ungrouped variables", "definition": "root/a", "description": "", "templateType": "anything"}, "root_approx": {"name": "root_approx", "group": "Ungrouped variables", "definition": "precround(root,3)", "description": "", "templateType": "anything"}, "ratio_approx": {"name": "ratio_approx", "group": "Ungrouped variables", "definition": "precround(ratio,3)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "precround(1/ratio,1)*10", "description": "", "templateType": "anything"}, "n_max": {"name": "n_max", "group": "Ungrouped variables", "definition": "floor(m*ratio)", "description": "", "templateType": "anything"}, "n_min": {"name": "n_min", "group": "Ungrouped variables", "definition": "ceil(m*ratio)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "square", "root", "ratio", "root_approx", "ratio_approx", "m", "n_max", "n_min"], "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>Ermitteln Sie durch algebraische Umformung:</p>\n<p>i) Den Wert von $x^2=$[[0]]</p>\n<p>ii) Den Wert der Proportion als Dezimalzahl:&nbsp;$\\frac{x}{\\var{a}}=\\frac{\\var{b}}{x}=$&nbsp;[[1]]</p>\n<p><em>Dezimaltrennzeichen ist der Beistrich (,).</em></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "x2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{square}", "maxValue": "{square}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": true, "customName": "prop", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ratio_approx}-0.001", "maxValue": "{ratio_approx}+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}], "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>Laut Eudoxos (gem&auml;&szlig; Elementen des Euklid, Buch V, Definition 5) muss es dann Zahlen $m,n\\in\\mathbb{N}$ geben, so dass</p>\n<p style=\"text-align: center;\">$(1)\\qquad m\\cdot x&lt;n\\cdot\\var{a}\\quad\\wedge\\quad m\\cdot\\var{b}&lt;n\\cdot x$</p>\n<p>bzw.</p>\n<p style=\"text-align: center;\">$(2)\\qquad m\\cdot x&gt;n\\cdot\\var{a}\\quad\\wedge\\quad m\\cdot\\var{b}&gt;n\\cdot x$</p>\n<p>gilt.&nbsp;</p>\n<p>Wir setzen nun $m=\\var{m}$.</p>\n<p>i) Ermitteln Sie die&nbsp;kleinste Zahl $n$, so dass der Fall $(1)$ eintritt: $\\quad n=$[[1]]</p>\n<p>ii) Ermitteln Sie die&nbsp;gr&ouml;&szlig;te Zahl $n$,&nbsp;so dass der Fall $(2)$ eintritt: $\\ \\,\\quad n=$[[0]]</p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 638px; width: 0px; top: 133px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 623px; width: 50px; top: 118px; height: 20px; font-size: 10px; color: black;\"></span></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "n1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{n_max}", "maxValue": "{n_max}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": true, "customName": "n2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{n_min}", "maxValue": "{n_min}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["eu", "plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}]}], "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}