// Numbas version: exam_results_page_options {"name": "Addition und Subtraktion von Br\u00fcchen", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Addition und Subtraktion von Br\u00fcchen", "tags": [], "metadata": {"description": "

Manipulate fractions in order to add and subtract them. The difficulty escalates through the inclusion of a whole integer and a decimal, which both need to be converted into a fraction before the addition/subtraction can take place.

\n

https://numbas.mathcentre.ac.uk/question/22664/addition-and-subtraction-of-fractions/ by Lauren Richards

\n

Translated to German and Part d) added.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Führen Sie die folgenden Rechnungen durch und geben Sie das Ergebnis als gekürzten Bruch an (in den Teilen a) - c): mit positivem Nenner).

", "advice": "

a)

\n

$\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}+\\frac{\\var{c_coprime}}{\\var{d_coprime}}$

\n

Um Brüche zu addieren oder zu subtrahieren, machen wir sie zuerst gleichnamig, d.h. wir bringen sie durch Erweitern auf einen gemeinsamen Nenner.

\n

Als gemeinsamen Nenner können wir das kleinste gemeinsame Vielfache der beiden Nenner verwenden.

\n

Das kleinste gemeinsame Vielfache von $\\var{b_coprime}$ und $\\var{d_coprime}$ ist $\\var{lcm}.$

\n

Wir erweitern $\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}$ mit $\\displaystyle\\frac{\\var{lcm_b}}{\\var{lcm_b}}$ und erhalten $\\displaystyle\\frac{\\var{alcm_b}}{\\var{lcm}}.$

\n

Wir erweitern $\\displaystyle\\frac{\\var{c_coprime}}{\\var{d_coprime}}$ mit $\\displaystyle\\frac{\\var{lcm_d}}{\\var{lcm_d}}$ und erhalten $\\displaystyle\\frac{\\var{clcm_d}}{\\var{lcm}}.$

\n

Jetzt rechnen wir

\n

$\\displaystyle\\frac{\\var{alcm_b}}{\\var{lcm}}+\\frac{\\var{clcm_d}}{\\var{lcm}}=\\frac{(\\var{alcm_b}+\\var{clcm_d})}{\\var{lcm}}=\\frac{\\var{alcmclcm}}{\\var{lcm}}.$

\n

Zum Schluss schauen wir, ob wir noch kürzen können. Der größte gemeinsame Teiler (ggT) von $\\var{alcmclcm}$ und $\\var{lcm}$ ist $\\var{gcd}.$

\n

Wir kürzen damit und erhalten als Endergebnis $\\displaystyle\\frac{\\var{num}}{\\var{denom}}.$

\n

Der Bruch lässt sich also nicht weiter kürzen und das Endergebnis ist $\\displaystyle\\frac{\\var{num}}{\\var{denom}}$.

\n

\n

b)

\n

$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}-\\frac{\\var{h_coprime}}{\\var{j_coprime}}+2.$

\n

\n

Wir machen wieder die Brüche als erstes gleichnamig, indem wir sie durch Erweitern auf den gemeinsamen Nenner $\\var{lcm2}$ bringen.

\n

Wir erweitern $\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}$ und bekommen $\\displaystyle\\frac{\\var{flcm2_g}}{\\var{lcm2}}$ und erweitern $\\displaystyle\\frac{\\var{h_coprime}}{\\var{j_coprime}}$ zu $\\displaystyle\\frac{\\var{hlcm2_j}}{\\var{lcm2}}.$

\n

Nun haben wir

\n

$\\displaystyle\\frac{\\var{flcm2_g}}{\\var{lcm2}}-\\frac{\\var{hlcm2_j}}{\\var{lcm2}}=\\frac{\\var{flcmhlcm}}{\\var{lcm2}}.$

\n

Jetzt soll noch $2$ addiert werden. Wir schreiben $2$ ebenfalls als Bruchzahl mit Nenner $\\var{lcm2}$:

\n

$\\displaystyle2=2\\bigg(\\frac{\\var{lcm2}}{\\var{lcm2}}\\bigg)=\\frac{\\var{twolcm2}}{\\var{lcm2}}.$

\n

Jetzt rechnen wir weiter:

\n

$\\displaystyle\\frac{\\var{flcmhlcm}}{\\var{lcm2}}+\\frac{\\var{twolcm2}}{\\var{lcm2}}=\\frac{\\var{num2unsim}}{\\var{lcm2}}.$

\n

Zum Schluss schauen wir, ob wir noch kürzen können. Der ggT von $\\var{num2unsim}$ und $\\var{lcm2}$ ist $\\var{gcd2}.$

\n

Wir kürzen damit und erhalten als Endergebnis $\\displaystyle\\simplify{{num2unsim}/{lcm2}}.$

\n

Der Bruch lässt sich also nicht weiter kürzen und wir erhalten als Endergebnis $\\displaystyle\\simplify{{num2unsim}/{lcm2}}$.

\n

\n

c)

\n

$\\displaystyle\\var{k}+\\frac{\\var{l_coprime}}{\\var{m_coprime}}-\\frac{\\var{n_coprime}}{\\var{o_coprime}}.$

\n

Wir müssen die Dezimalzahl in eine Bruchzahl umwandeln und schreiben Sie mit einer entsprechenden Potenz von $10$ im Nenner:

\n

$\\displaystyle\\frac{\\var{k}}{1}\\times\\frac{100}{100}=\\frac{\\var{100k}}{100}.$

\n

An dieser Stelle prüfen wir, ob sich der so entstandene Bruch kürzen lässt. Der ggT von Zähler und Nenner ist $\\var{gcd_k100}.$

\n

Der Bruch lässt sich also nicht kürzen und wir rechnen weiter mit

\n

Wir kürzen dadurch und erhalten

\n

\\[\\simplify{{{100k}}/{100}}\\text{.}\\]

\n

Damit können wir den gegebenen Ausdruck umschreiben als $\\displaystyle\\frac{\\var{k_simp}}{\\var{simp}}+\\frac{\\var{l_coprime}}{\\var{m_coprime}}-\\frac{\\var{n_coprime}}{\\var{o_coprime}}.$

\n

Wir erweitern alle Brüche so, dass wir sie mit Nenner $\\var{gcd3}$ schreiben können.

\n

\\[\\frac{\\var{k_simp}}{\\var{simp}} =\\frac{\\var{k_simp*term1}}{\\var{gcd3}},\\quad \\frac{\\var{l_coprime}}{\\var{m_coprime}}=\\frac{\\var{l_coprime*term2}}{\\var{gcd3}},\\quad\\frac{\\var{n_coprime}}{\\var{o_coprime}}=\\frac{\\var{n_coprime*term3}}{\\var{gcd3}}\\text{.}\\]

\n

Jetzt können wir die Addition durchführen.

\n

\\[\\frac{\\var{k_simp*term1}}{\\var{gcd3}}+\\frac{\\var{l_coprime*term2}}{\\var{gcd3}}-\\frac{\\var{n_coprime*term3}}{\\var{gcd3}}=\\frac{\\var{(k_simp*term1)+(l_coprime*term2)-(n_coprime*term3)}}{\\var{gcd3}}\\text{.}\\]

\n

Zum Abschluss prüfen wir wieder, ob gekürzt werden kann. Der ggT von Zähler und Nenner ist $\\var{gcd_numgcd3}.$

\n

Wir kürzen und erhalten als Ergebnis

\n

Dieser Bruch lässt sich also nicht weiter kürzen, und wir bekommen

\n

\\[\\simplify{{num1}/{gcd3}}\\text{.}\\]

\n

d)

\n

$\\displaystyle{ \\frac{\\var{aad}}{ \\simplify{x-{bd}} } + \\frac{\\var{bbd}}{\\simplify{x-{cd}}} }$

\n

Wir bringen die Brüche auf den gemeinsamen Nenner $(\\simplify{x-{bd}})(\\simplify{x-{cd}})$:

\n

$\\displaystyle{ \\frac{\\var{aad}}{ \\simplify{x-{bd}} } + \\frac{\\var{bbd}}{\\simplify{x-{cd}}}  =   \\frac{\\var{aad} (\\simplify{x-{cd}})  }{ (\\simplify{x-{bd}})(\\simplify{x-{cd}}) } + \\frac{\\var{bbd}(\\simplify{x-{bd}})}{(\\simplify{x-{bd}})(\\simplify{x-{cd}})}   }$

\n

Nun addieren wir zu

\n

\\[\\frac{\\simplify[basic]{{aad} (x-{cd})  + {bbd}(x-{bd})}}{(\\simplify{x-{bd}})(\\simplify{x-{cd}})}   = \\frac{ \\var{ad}}{ \\simplify{x^2  - {bd+cd}*x + {bd*cd}}  } \\]

", "rulesets": {}, "extensions": [], "variables": {"d_coprime": {"name": "d_coprime", "group": "Part a", "definition": "d/gcd_cd", "description": "", "templateType": "anything"}, "hlcm2_j": {"name": "hlcm2_j", "group": "Part b", "definition": "h_coprime*lcm2_j", "description": "

PART B

", "templateType": "anything"}, "lcm2": {"name": "lcm2", "group": "Part b", "definition": "lcm(g_coprime,j_coprime)", "description": "

PART B

", "templateType": "anything"}, "denom": {"name": "denom", "group": "Part a", "definition": "lcm/gcd", "description": "

PART A answer for the denominator of part a

", "templateType": "anything"}, "term3": {"name": "term3", "group": "Part c", "definition": "gcd3/o_coprime", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Part a", "definition": "random(1..5)", "description": "

PART A variable a - random number between 1 and 5.

", "templateType": "anything"}, "d": {"name": "d", "group": "Part a", "definition": "random(5..15)", "description": "

PART A variable d - random number between 5 and 15.

", "templateType": "anything"}, "gcd_fg": {"name": "gcd_fg", "group": "Part b", "definition": "gcd(f,g)", "description": "

PART B gcd of first fraction num and denom

", "templateType": "anything"}, "j": {"name": "j", "group": "Part b", "definition": "random(2..10 except h)", "description": "

PART B

", "templateType": "anything"}, "gcd_lm": {"name": "gcd_lm", "group": "Part c", "definition": "gcd(l,m)", "description": "", "templateType": "anything"}, "c_coprimeb_coprime": {"name": "c_coprimeb_coprime", "group": "Part a", "definition": "c_coprime*b_coprime", "description": "

PART A variable c times variable b

", "templateType": "anything"}, "l": {"name": "l", "group": "Part c", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "gcd_ab": {"name": "gcd_ab", "group": "Part a", "definition": "gcd(a,b)", "description": "

PART A simplification of fractions in the question.

", "templateType": "anything"}, "k_simp": {"name": "k_simp", "group": "Part c", "definition": "(100k)/(gcd_k100)", "description": "", "templateType": "anything"}, "o_coprime": {"name": "o_coprime", "group": "Part c", "definition": "o/gcd_no", "description": "", "templateType": "anything"}, "n_coprime": {"name": "n_coprime", "group": "Part c", "definition": "n/gcd_no", "description": "", "templateType": "anything"}, "gcd_k100": {"name": "gcd_k100", "group": "Part c", "definition": "gcd(100k,100)", "description": "", "templateType": "anything"}, "lcm_b": {"name": "lcm_b", "group": "Part a", "definition": "lcm/b_coprime", "description": "

PART A lcm of b and d, divided by b

", "templateType": "anything"}, "num": {"name": "num", "group": "Part a", "definition": "alcmclcm/gcd", "description": "

PART A answer for the numerator input

", "templateType": "anything"}, "b": {"name": "b", "group": "Part a", "definition": "random(5..10 except d)", "description": "

PART A variable b - random number between 5 and 10 and not the same value as d.

", "templateType": "anything"}, "h": {"name": "h", "group": "Part b", "definition": "random(1..10)", "description": "

PART B

", "templateType": "anything"}, "gcd_no": {"name": "gcd_no", "group": "Part c", "definition": "gcd(n,o)", "description": "", "templateType": "anything"}, "f": {"name": "f", "group": "Part b", "definition": "random(1..10)", "description": "

PART B

", "templateType": "anything"}, "gcd_numgcd3": {"name": "gcd_numgcd3", "group": "Part c", "definition": "gcd(num1,gcd3)", "description": "", "templateType": "anything"}, "simp": {"name": "simp", "group": "Part c", "definition": "(100)/(gcd_k100)", "description": "", "templateType": "anything"}, "a_coprimed_coprime": {"name": "a_coprimed_coprime", "group": "Part a", "definition": "a_coprime*d_coprime", "description": "

PART A variable a times variable d

", "templateType": "anything"}, "o": {"name": "o", "group": "Part c", "definition": "random(5..15 except m except n except 7 except 11 except 13)", "description": "", "templateType": "anything"}, "lcm_d": {"name": "lcm_d", "group": "Part a", "definition": "lcm/d_coprime", "description": "

PART A lcm of b and d, divided by d

", "templateType": "anything"}, "gcd2": {"name": "gcd2", "group": "Part b", "definition": "gcd(num2unsim,lcm2)", "description": "

PART B

", "templateType": "anything"}, "c": {"name": "c", "group": "Part a", "definition": "random(1..5)", "description": "

PART A variable c - random number between 1 and 5.

", "templateType": "anything"}, "m": {"name": "m", "group": "Part c", "definition": "random(5..12 except 7 except 11)", "description": "", "templateType": "anything"}, "clcm_d": {"name": "clcm_d", "group": "Part a", "definition": "c_coprime*lcm_d", "description": "

PART A variable c times the lcm of b and d, divided by d

", "templateType": "anything"}, "h_coprime": {"name": "h_coprime", "group": "Part b", "definition": "h/gcd_hj", "description": "

PART B

", "templateType": "anything"}, "twolcm2": {"name": "twolcm2", "group": "Part b", "definition": "2*lcm2", "description": "

PART B

", "templateType": "anything"}, "lcm2_j": {"name": "lcm2_j", "group": "Part b", "definition": "lcm2/j_coprime", "description": "

PART B

", "templateType": "anything"}, "k": {"name": "k", "group": "Part c", "definition": "random(0.01..0.9#0.01)", "description": "", "templateType": "anything"}, "gcd": {"name": "gcd", "group": "Part a", "definition": "gcd(alcmclcm,lcm)", "description": "

PART A greatest common divisor of the variables alcmclcm and lcm

", "templateType": "anything"}, "m_coprime": {"name": "m_coprime", "group": "Part c", "definition": "m/gcd(l,m)", "description": "", "templateType": "anything"}, "j_coprime": {"name": "j_coprime", "group": "Part b", "definition": "j/gcd_hj", "description": "

PART B

", "templateType": "anything"}, "flcmhlcm": {"name": "flcmhlcm", "group": "Part b", "definition": "flcm2_g-hlcm2_j", "description": "

PART B

", "templateType": "anything"}, "term1": {"name": "term1", "group": "Part c", "definition": "gcd3/simp", "description": "", "templateType": "anything"}, "alcm_b": {"name": "alcm_b", "group": "Part a", "definition": "a_coprime*lcm_b", "description": "

PART A variable a times the lcm of b and d, divided by b

", "templateType": "anything"}, "a_coprime": {"name": "a_coprime", "group": "Part a", "definition": "a/gcd_ab", "description": "

PART A

", "templateType": "anything"}, "b_coprime": {"name": "b_coprime", "group": "Part a", "definition": "b/gcd_ab", "description": "

PART A 

", "templateType": "anything"}, "g_coprime": {"name": "g_coprime", "group": "Part b", "definition": "g/gcd_fg", "description": "

PART B g_coprime

", "templateType": "anything"}, "gcd_hj": {"name": "gcd_hj", "group": "Part b", "definition": "gcd(h,j)", "description": "

PART B

", "templateType": "anything"}, "gcd1": {"name": "gcd1", "group": "Part c", "definition": "lcm(simp,m_coprime)", "description": "", "templateType": "anything"}, "g": {"name": "g", "group": "Part b", "definition": "random(2..10 except f except j)", "description": "

PART B

", "templateType": "anything"}, "gcd_cd": {"name": "gcd_cd", "group": "Part a", "definition": "gcd(c,d)", "description": "

PART A 

", "templateType": "anything"}, "lcm": {"name": "lcm", "group": "Part a", "definition": "lcm(b_coprime,d_coprime)", "description": "

PART A lowest common multiple of variable b_coprime and variable d_coprime.

", "templateType": "anything"}, "flcm2_g": {"name": "flcm2_g", "group": "Part b", "definition": "f_coprime*lcm2_g", "description": "

PART B

", "templateType": "anything"}, "lcm2_g": {"name": "lcm2_g", "group": "Part b", "definition": "lcm2/g_coprime", "description": "

PART B

", "templateType": "anything"}, "term2": {"name": "term2", "group": "Part c", "definition": "gcd3/m_coprime", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Part c", "definition": "random(1..5)", "description": "", "templateType": "anything"}, "alcmclcm": {"name": "alcmclcm", "group": "Part a", "definition": "alcm_b+clcm_d", "description": "

PART A 

", "templateType": "anything"}, "gcd3": {"name": "gcd3", "group": "Part c", "definition": "lcm(gcd1,o_coprime)", "description": "", "templateType": "anything"}, "f_coprime": {"name": "f_coprime", "group": "Part b", "definition": "f/gcd_fg", "description": "

PART B

", "templateType": "anything"}, "num1": {"name": "num1", "group": "Part c", "definition": "(k_simp*term1)+(l_coprime*term2)-(n_coprime*term3)", "description": "", "templateType": "anything"}, "c_coprime": {"name": "c_coprime", "group": "Part a", "definition": "c/gcd_cd", "description": "", "templateType": "anything"}, "num2unsim": {"name": "num2unsim", "group": "Part b", "definition": "flcmhlcm+twolcm2", "description": "

PART B

", "templateType": "anything"}, "l_coprime": {"name": "l_coprime", "group": "Part c", "definition": "l/gcd_lm", "description": "", "templateType": "anything"}, "ad": {"name": "ad", "group": "Part d", "definition": "-aad*cd - bbd*bd", "description": "", "templateType": "anything"}, "bd": {"name": "bd", "group": "Part d", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "cd": {"name": "cd", "group": "Part d", "definition": "random(-5..7)", "description": "", "templateType": "anything"}, "aad": {"name": "aad", "group": "Part d", "definition": "random(-3,-2,-1,1,2,3,4,5)", "description": "", "templateType": "anything"}, "bbd": {"name": "bbd", "group": "Part d", "definition": "-aad", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "!(ad = 0) && !(bd = cd)", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part a", "variables": ["a", "a_coprime", "b", "b_coprime", "gcd_ab", "c", "c_coprime", "d", "d_coprime", "gcd_cd", "lcm", "a_coprimed_coprime", "c_coprimeb_coprime", "lcm_b", "lcm_d", "alcm_b", "clcm_d", "alcmclcm", "gcd", "num", "denom"]}, {"name": "Part b", "variables": ["f", "f_coprime", "g", "g_coprime", "gcd_fg", "h", "h_coprime", "j", "j_coprime", "gcd_hj", "lcm2", "lcm2_g", "flcm2_g", "lcm2_j", "hlcm2_j", "flcmhlcm", "twolcm2", "num2unsim", "gcd2"]}, {"name": "Part c", "variables": ["k", "gcd_k100", "k_simp", "simp", "l", "l_coprime", "m", "m_coprime", "gcd_lm", "n", "n_coprime", "o", "o_coprime", "gcd_no", "gcd1", "gcd3", "term1", "term2", "term3", "num1", "gcd_numgcd3"]}, {"name": "Part d", "variables": ["ad", "bd", "cd", "aad", "bbd"]}], "functions": {}, "preamble": {"js": "", "css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > 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": "

$\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}+\\frac{\\var{c_coprime}}{\\var{d_coprime}}=$ [[0]] [[1]]

", "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": "num", "maxValue": "num", "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": "denom", "maxValue": "denom", "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": "

$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}-\\frac{\\var{h_coprime}}{\\var{j_coprime}}+2=$  [[0]] [[1]]

", "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": "num2unsim/gcd2", "maxValue": "num2unsim/gcd2", "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": "lcm2/gcd2", "maxValue": "lcm2/gcd2", "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": "

$\\displaystyle \\var{k}+\\frac{\\var{l}}{\\var{m}}-\\frac{\\var{n}}{\\var{o}}=$ [[0]] [[1]] .

", "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": "num1/gcd_numgcd3", "maxValue": "num1/gcd_numgcd3", "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": "gcd3/gcd_numgcd3", "maxValue": "gcd3/gcd_numgcd3", "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": "

Schreiben Sie Ihr Ergebnis so, dass der Koeffizient von $x^2$ im Nenner positiv ist.

\n

$\\displaystyle{ \\frac{\\var{aad}}{ \\simplify{x-{bd}} } + \\frac{\\var{bbd}}{\\simplify{x-{cd}}} }$ = [[0]][[1]]

", "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": "{ad}", "maxValue": "{ad}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x^2 - ({bd+cd})*x + {bd*cd}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Ulrich G\u00f6rtz", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7603/"}]}