// Numbas version: exam_results_page_options
{"name": "Nimm Hundert f\u00fcr Hundert", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Nimm Hundert f\u00fcr Hundert", "tags": [], "metadata": {"description": "<p>Ein&nbsp;sehr bekanntes Problem.</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>Eine Abwandlung der \"100-V&ouml;gel\"-Aufgabe lautet:</p>\n<p>F&uuml;r 100 Sapek sollen 100 {strings[dice[0]][0]} gekauft werden.&nbsp;{strings[dice[0]][1]} kosten&nbsp;{costs[dice[1]][0]} Sapek,&nbsp;{strings[dice[0]][2]} kosten&nbsp;{costs[dice[1]][1]} Sapek und f&uuml;r einen Sapek erh&auml;lt man {costs[dice[1]][2]}&nbsp;{strings[dice[0]][3]}.</p>\n<p></p>\n<p><span style=\"margin: 0px auto; border: 2px dotted #000000; position: absolute; z-index: 2147483647; visibility: hidden; left: 696px; width: 0px; top: 64px; height: 0px;\"></span><span style=\"z-index: 2147483647; position: absolute; visibility: hidden; left: 681px; width: 50px; top: 49px; height: 20px; font-size: 10px; color: black;\"></span></p>", "advice": "<p><strong></strong></p>\n<p>Es seien $x$ die Anzahl der {strings[dice[0]][1]}, $y$ die Anzahl der {strings[dice[0]][1]} und $z$ die&nbsp; Anzahl der {strings[dice[0]][2]}.</p>\n<p>(1) Aus den Anzahlen ermitteln wir dann die Gleichung $x+y+z=100$</p>\n<p>(2) Aus den&nbsp;Kosten zudem&nbsp;die Gleichung $\\var{costs[dice[1]][0]}\\cdot x +&nbsp;\\var{costs[dice[1]][1]}\\cdot y+\\frac{1}{\\var{costs[dice[1]][2]}}\\cdot&nbsp;z=100$</p>\n<p>Durch Einsetzen von (1) in (2) erhalten wir die Gleichung $\\simplify{{costs[dice[1]][0]}*x+{costs[dice[1]][1]}*y+1/{costs[dice[1]][2]}*(100-y-x)=100}$</p>\n<p>Umformen ergibt:&nbsp;$\\simplify{{costs[dice[1]][0]-1/{costs[dice[1]][2]}}*x+{costs[dice[1]][1]-1/{costs[dice[1]][2]}}*y=({100-100/costs[dice[1]][2]})}$&nbsp;bzw.&nbsp;$\\simplify{{costs[dice[1]][2]}*{costs[dice[1]][0]-1/{costs[dice[1]][2]}}*x+{costs[dice[1]][2]}*{costs[dice[1]][1]-1/{costs[dice[1]][2]}}*y=({costs[dice[1]][2]}*{100-100/costs[dice[1]][2]})}$.</p>\n<p>Man w&auml;hlt nun&nbsp;$y$ so, dass sich f&uuml;r $x$ ein ganzzahliger Wert ergibt, der dann auch zu einem ganzzahligen Wert von $z$ f&uuml;hren muss, wobei man noch aufpassen muss, dass $z$ nicht negativ werden darf.</p>\n<p>Eine m&ouml;gliche L&ouml;sung ist $x=\\var{solutions[dice[1]][0]}, y=\\var{solutions[dice[1]][1]}, z=x=\\var{solutions[dice[1]][2]}$.</p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"n": {"name": "n", "group": "Aufgabenteil a)", "definition": "a*b", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Aufgabenteil a)", "definition": "random(3,7,11)^2*5", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Aufgabenteil a)", "definition": "random(7,11,13)", "description": "", "templateType": "anything"}, "costs": {"name": "costs", "group": "Aufgabenteil a)", "definition": "[[5,3,3],[7,3,3],[6,3,3],[5,4,4],[4,3,3]]", "description": "", "templateType": "anything"}, "strings": {"name": "strings", "group": "Aufgabenteil a)", "definition": "[[\"Orangen\",\"Wenzhou-Orangen\", \"gr\u00fcne Orangen\", \"goldene Orangen\"],[\"St\u00fcck Gefl\u00fcgel\",\"H\u00e4hne\", \"Hennen\", \"K\u00fcken\"],[\"St\u00fcck Vieh\",\"Rinder\", \"Milchk\u00fche\", \"K\u00e4lber\"],[\"Schweine\",\"Eber\", \"Sauen\", \"Ferkel\"]]", "description": "", "templateType": "anything"}, "dice": {"name": "dice", "group": "Aufgabenteil a)", "definition": "[random(0,3),random(0,4)]", "description": "", "templateType": "anything"}, "solutions": {"name": "solutions", "group": "Aufgabenteil a)", "definition": "[[4,18,78],[6,6,84],[8,8,84],[15,1,84],[8,14,78]]", "description": "", "templateType": "anything"}, "": {"name": "", "group": "Aufgabenteil a)", "definition": "", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Aufgabenteil a)", "variables": ["n", "a", "b", "costs", "strings", "dice", "solutions", ""]}], "functions": {"euclid_gcd": {"parameters": [["a", "number"], ["b", "number"]], "type": "anything", "language": "javascript", "definition": "  a = +a;\n  b = +b;\n  if (a !== a || b !== b) {\n    return [NaN, NaN, NaN];\n  }\n  \n  if (a === Infinity || a === -Infinity || b === Infinity || b === -Infinity) {\n    return [Infinity, Infinity, Infinity];\n  }\n  // Checks if a or b are decimals\n  if ((a % 1 !== 0) || (b % 1 !== 0)) {\n    return false;\n  }\n  var signX = (a < 0) ? -1 : 1,\n    signY = (b < 0) ? -1 : 1,\n    x = 0,\n    y = 1,\n    u = 1,\n    v = 0,\n    q, r, m, n;\n  a = Math.abs(a);\n  b = Math.abs(b);\n\n  while (a !== 0) {\n    q = Math.floor(b / a);\n    r = b % a;\n    m = x - u * q;\n    n = y - v * q;\n    b = a;\n    a = r;\n    x = u;\n    y = v;\n    u = m;\n    v = n;\n  }\n  return [b, signX * x, signY * y];"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "mark:\n  assert(interpreted_answer[0] = floor(interpreted_answer[0]) and\n         interpreted_answer[1] = floor(interpreted_answer[1]) and\n         interpreted_answer[2] = floor(interpreted_answer[2]),\n         fail(\"Bitte nur ganzzahlige L\u00f6sungen!\"));\n  assert(interpreted_answer[0] > 0 and interpreted_answer[1] > 0  and interpreted_answer[2] > 0,\n         fail(\"Keine negativen L\u00f6sungen erlaubt!\"));\n  correctif(interpreted_answer[0] + interpreted_answer[1] + interpreted_answer[2] = 100 and\n          costs[dice[1]][0]*interpreted_answer[0] + costs[dice[1]][1]*interpreted_answer[1] + interpreted_answer[2]/costs[dice[1]][2] = 100)", "extendBaseMarkingAlgorithm": true, "unitTests": [{"variables": [{"name": "n", "value": "3185"}, {"name": "a", "value": "245"}, {"name": "b", "value": "13"}, {"name": "costs", "value": "[ [ 5, 3, 3 ], [ 7, 3, 3 ] ]"}, {"name": "strings", "value": "[ [ \"Orangen\", \"Wenzhou-Orangen\", \"gr\u00fcne Orangen\", \"goldene Orangen\" ], [ \"St\u00fcck Gefl\u00fcgel\", \"H\u00e4hne\", \"Hennen\", \"K\u00fcken\" ] ]"}, {"name": "dice", "value": "[ 0, 1 ]"}, {"name": "solutions", "value": "[ [ 4, 18, 78 ], [ 6, 6, 84 ] ]"}], "answer": {"valid": true, "value": ["3", "4", "5"]}, "notes": [{"name": "mark", "expected": {"value": "", "messages": ["There was an error in this part's marking algorithm. Please report this. Error evaluating variable mark: No definition of 'and' of correct type found."], "warnings": ["There was an error in this part's marking algorithm. Please report this. Error evaluating variable mark: No definition of 'and' of correct type found."], "error": "Error evaluating variable mark: No definition of 'and' of correct type found.", "valid": false, "credit": 0}}]}], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Wie viele&nbsp;{strings[dice[0]][0]} von jeder Sorte soll man kaufen?</p>\n<p>Geben Sie bitte eine m&ouml;gliche <em>ganzzahlige</em> L&ouml;sung an.</p>\n<p>{strings[dice[0]][1]}:&nbsp;[[0]]</p>\n<p>{strings[dice[0]][2]}:&nbsp;[[1]]</p>\n<p>{strings[dice[0]][3]}:&nbsp;[[2]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "a", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "solutions[dice[1]][0]", "maxValue": "solutions[dice[1]][0]", "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": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": true, "customName": "b", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "solutions[dice[1]][1]", "maxValue": "solutions[dice[1]][1]", "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": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": true, "customName": "c", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "solutions[dice[1]][2]", "maxValue": "solutions[dice[1]][2]", "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": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "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/"}]}