// Numbas version: exam_results_page_options {"name": "\u00dcbungen (Lektion 9)", "metadata": {"description": "
Arithmettik & Algebra in der Renaissance
", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": "0", "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "questions": [{"name": "Division nach Adam Ries", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "Division durch eine einstellige Zahl gemäß dem 2. Rechenbuch
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "Nach Adam Ries sei die Aufgabe $\\var{c}:\\var{a}$ schriftlich zu berechnen.
", "advice": "Erster Schritt: Wir schreiben unter die 2. Stelle den Divisor, also $\\var{a}$. Wir rechnen $\\var{(c1)*10+c2}:\\var{a}=\\var{b_digits[0]}$, als Rest erhalten wir $\\var{carry1}$, den wir über die 2. Ziffer notieren.
\n\n | {carry1} | \n\n | \n | \n | \n | \n | \n |
{c1} | \n{c2} | \n{c3} | \n{c4} | \n| | \n{b_digits[0]} | \n\n | \n |
\n | {a} | \n\u200b | \n\u200b | \n\n | \n | \n | \n |
Zweiter Schritt: Wir schreiben unter die 3. Stelle den Divisor, also wieder $\\var{a}$. Wir rechnen $\\var{(carry1)*10+c3}:\\var{a}=\\var{b_digits[1]}$, als Rest erhalten wir $\\var{carry2}$, den wir über die 3. Ziffer notieren.
\n\n | {carry1} | \n{carry2} | \n\n | \n | \n | \n | \n |
{c1} | \n{c2} | \n{c3} | \n{c4} | \n| | \n{b_digits[0]} | \n{b_digits[1]} | \n\n |
\n | {a} | \n{a}\u200b | \n\u200b | \n\n | \n | \n | \n |
Dritter Schritt: Wir schreiben unter die letzte Stelle wieder den Divisor, also $\\var{a}$. Wir rechnen $\\var{(carry1*10+floor(mod(c,100)/10)-b_digits[1]*a)*10+c4}:\\var{a}=\\var{b_digits[2]}$. Das geht sich ohne Rest aus, also haben wir die Lösung $\\var{b}$ gefunden.
\n\n | {carry1} | \n{carry2} | \n\n | \n | \n | \n | \n |
{c1} | \n{c2} | \n{c3} | \n{c4} | \n| | \n{b_digits[0]} | \n{b_digits[1]} | \n{b_digits[2]} | \n
\n | {a} | \n{a}\u200b | \n{a}\u200b | \n\n | \n | \n | \n |
Tragen Sie unten die fehlenden Ziffern ein.
\nIn jede Lücke bitte nur genau eine Ziffer eintragen.
\n\n | [[3]] | \n[[4]] | \n\n | \n | \n | \n | \n |
\n{c1}\n | \n\n{c2}\n | \n\n{c3}\n | \n\n{c4}\n | \n\n|\n | \n[[5]] | \n[[6]] | \n[[7]] | \n
\n | [[0]] | \n[[1]]\u200b | \n[[2]]\u200b | \n\n | \n | \n | \n |
Adaptiert von: Katz, V. J. (2009). A history of mathematics: An introduction (3. ed.). Boston: Addison-Wesley, S. 418.
", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "Das folgende Problem stammt (bis auf die Zahlenwerte) von Piero della Francesca (1410/20-1492).
\nDrei Männer gehen eine Geschäftspartnerschaft ein. Der erste investiert {a} Dukaten in das Geschäft, der zweite investiert {b} Dukaten und wir wissen nicht, wie viel der dritte investiert hat.
\nAls Erlös erhalten die drei Partner zusammen {sum} Dukaten, von denen der erste Partner {a2} Dukaten erhält.
", "advice": "Es ist davon auszugehen, dass der Erlös aus dem Geschäft proportional zur in die Partnerschaft investierten Summe ist.
\nDer erste erhält bei einer Investition von $\\var{a}$ Dukaten einen Erlös von $\\var{a2}$ Dukaten.
\nDann muss der zweite bei einer Investition von {b} Dukaten einen Erlös von $\\var{b}\\cdot\\frac{\\var{a2}}{\\var{a}}=\\var{b}\\cdot\\var{a2/a}=\\var{b2}$ Dukaten erhalten.
\nFür den dritten Partner bleiben dann vom Erlös noch $\\var{sum}-\\var{a2}-\\var{b2}=\\var{c2}$ Dukaten übrig. Er hat demnach $\\var{c2}:\\frac{\\var{a2}}{\\var{a}}=\\var{c2}:\\var{a2/a}=\\var{c}$ Dukaten investiert gehabt.
\nIm Original hat das Problem eine nicht-ganzzahlige Lösung, hier wurde als Erlös zur Vereinfachung ein ganzzahliges Vielfaches der Investitionssume gewählt.
", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "50+random(-1,1)*random(1..8)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "80+random(-1,1)*random(3,5,7)", "description": "", "templateType": "anything"}, "sum": {"name": "sum", "group": "Ungrouped variables", "definition": "(a+b+c0)*profit", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "a*profit", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "b*a2/a", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "sum-a2-b2", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "c2*a/a2", "description": "", "templateType": "anything"}, "profit": {"name": "profit", "group": "Ungrouped variables", "definition": "random(3,5,6,7)", "description": "", "templateType": "anything"}, "c0": {"name": "c0", "group": "Ungrouped variables", "definition": "60+random(-1,1)*random(2,4,6,8)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "sum", "a2", "b2", "c2", "c", "profit", "c0"], "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": "Welche Anteile vom Erlös erhalten der zweite und der dritte Partner und wie viel hatte der dritte investiert?
\nDer zweite Mann erhält [[0]] Dukaten vom Erlös.
\nDer dritte Mann erhält [[1]] Dukaten vom Erlös.
\nDer dritte Mann hat [[2]] Dukaten investiert gehabt.
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "b2", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b2", "maxValue": "b2", "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": true, "showPrecisionHint": false, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": true, "customName": "c2", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c2", "maxValue": "c2", "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": true, "showPrecisionHint": false, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": true, "customName": "c", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c", "maxValue": "c", "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": true, "showPrecisionHint": false, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": false, "showresultspage": "oncompletion", "navigatemode": "menu", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "extensions": [], "custom_part_types": [], "resources": []}