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.
\nhttps://numbas.mathcentre.ac.uk/question/22664/addition-and-subtraction-of-fractions/ by Lauren Richards
\nTranslated 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": "$\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}+\\frac{\\var{c_coprime}}{\\var{d_coprime}}$
\nUm Brüche zu addieren oder zu subtrahieren, machen wir sie zuerst gleichnamig, d.h. wir bringen sie durch Erweitern auf einen gemeinsamen Nenner.
\nAls gemeinsamen Nenner können wir das kleinste gemeinsame Vielfache der beiden Nenner verwenden.
\nDas kleinste gemeinsame Vielfache von $\\var{b_coprime}$ und $\\var{d_coprime}$ ist $\\var{lcm}.$
\nWir erweitern $\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}$ mit $\\displaystyle\\frac{\\var{lcm_b}}{\\var{lcm_b}}$ und erhalten $\\displaystyle\\frac{\\var{alcm_b}}{\\var{lcm}}.$
\nWir erweitern $\\displaystyle\\frac{\\var{c_coprime}}{\\var{d_coprime}}$ mit $\\displaystyle\\frac{\\var{lcm_d}}{\\var{lcm_d}}$ und erhalten $\\displaystyle\\frac{\\var{clcm_d}}{\\var{lcm}}.$
\nJetzt rechnen wir
\n$\\displaystyle\\frac{\\var{alcm_b}}{\\var{lcm}}+\\frac{\\var{clcm_d}}{\\var{lcm}}=\\frac{(\\var{alcm_b}+\\var{clcm_d})}{\\var{lcm}}=\\frac{\\var{alcmclcm}}{\\var{lcm}}.$
\nZum Schluss schauen wir, ob wir noch kürzen können. Der größte gemeinsame Teiler (ggT) von $\\var{alcmclcm}$ und $\\var{lcm}$ ist $\\var{gcd}.$
\nWir kürzen damit und erhalten als Endergebnis $\\displaystyle\\frac{\\var{num}}{\\var{denom}}.$
\nDer Bruch lässt sich also nicht weiter kürzen und das Endergebnis ist $\\displaystyle\\frac{\\var{num}}{\\var{denom}}$.
\n\n$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}-\\frac{\\var{h_coprime}}{\\var{j_coprime}}+2.$
\n\nWir machen wieder die Brüche als erstes gleichnamig, indem wir sie durch Erweitern auf den gemeinsamen Nenner $\\var{lcm2}$ bringen.
\nWir 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 $\\displaystyle\\frac{\\var{hlcm2_j}}{\\var{lcm2}}.$
\nNun haben wir
\n$\\displaystyle\\frac{\\var{flcm2_g}}{\\var{lcm2}}-\\frac{\\var{hlcm2_j}}{\\var{lcm2}}=\\frac{\\var{flcmhlcm}}{\\var{lcm2}}.$
\nJetzt soll noch $2$ addiert werden. Wir schreiben $2$ ebenfalls als Bruchzahl mit Nenner $\\var{lcm2}$:
\n$\\displaystyle2=2\\bigg(\\frac{\\var{lcm2}}{\\var{lcm2}}\\bigg)=\\frac{\\var{twolcm2}}{\\var{lcm2}}.$
\nJetzt rechnen wir weiter:
\n$\\displaystyle\\frac{\\var{flcmhlcm}}{\\var{lcm2}}+\\frac{\\var{twolcm2}}{\\var{lcm2}}=\\frac{\\var{num2unsim}}{\\var{lcm2}}.$
\nZum Schluss schauen wir, ob wir noch kürzen können. Der ggT von $\\var{num2unsim}$ und $\\var{lcm2}$ ist $\\var{gcd2}.$
\nWir kürzen damit und erhalten als Endergebnis $\\displaystyle\\simplify{{num2unsim}/{lcm2}}.$
\nDer Bruch lässt sich also nicht weiter kürzen und wir erhalten als Endergebnis $\\displaystyle\\simplify{{num2unsim}/{lcm2}}$.
\n\n$\\displaystyle\\var{k}+\\frac{\\var{l_coprime}}{\\var{m_coprime}}-\\frac{\\var{n_coprime}}{\\var{o_coprime}}.$
\nWir müssen die Dezimalzahl in eine Bruchzahl umwandeln und schreiben Sie mit einer entsprechenden Potenz von $10$ im Nenner:
\n$\\displaystyle\\frac{\\var{k}}{1}\\times\\frac{100}{100}=\\frac{\\var{100k}}{100}.$
\nAn dieser Stelle prüfen wir, ob sich der so entstandene Bruch kürzen lässt. Der ggT von Zähler und Nenner ist $\\var{gcd_k100}.$
\nDer Bruch lässt sich also nicht kürzen und wir rechnen weiter mit
Wir kürzen dadurch und erhalten
\\[\\simplify{{{100k}}/{100}}\\text{.}\\]
\nDamit können wir den gegebenen Ausdruck umschreiben als $\\displaystyle\\frac{\\var{k_simp}}{\\var{simp}}+\\frac{\\var{l_coprime}}{\\var{m_coprime}}-\\frac{\\var{n_coprime}}{\\var{o_coprime}}.$
\nWir 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{.}\\]
\nJetzt 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{.}\\]
\nZum Abschluss prüfen wir wieder, ob gekürzt werden kann. Der ggT von Zähler und Nenner ist $\\var{gcd_numgcd3}.$
\nWir kürzen und erhalten als Ergebnis
\nDieser Bruch lässt sich also nicht weiter kürzen, und wir bekommen
\n\\[\\simplify{{num1}/{gcd3}}\\text{.}\\]
\nd)
\n$\\displaystyle{ \\frac{\\var{aad}}{ \\simplify{x-{bd}} } + \\frac{\\var{bbd}}{\\simplify{x-{cd}}} }$
\nWir bringen die Brüche auf den gemeinsamen Nenner $(\\simplify{x-{bd}})(\\simplify{x-{cd}})$:
\n$\\displaystyle{ \\frac{\\var{aad}}{ \\simplify{x-{bd}} } + \\frac{\\var{bbd}}{\\simplify{x-{cd}}} = \\frac{\\var{aad} (\\simplify{x-{cd}}) }{ (\\simplify{x-{bd}})(\\simplify{x-{cd}}) } + \\frac{\\var{bbd}(\\simplify{x-{bd}})}{(\\simplify{x-{bd}})(\\simplify{x-{cd}})} }$
\nNun 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}} } \\]
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", "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
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", "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.
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", "templateType": "anything"}, "h_coprime": {"name": "h_coprime", "group": "Part b", "definition": "h/gcd_hj", "description": "PART B
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", "templateType": "anything"}, "lcm2_j": {"name": "lcm2_j", "group": "Part b", "definition": "lcm2/j_coprime", "description": "PART B
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", "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
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", "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.
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", "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
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", "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": "$\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}+\\frac{\\var{c_coprime}}{\\var{d_coprime}}=$
$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}-\\frac{\\var{h_coprime}}{\\var{j_coprime}}+2=$
$\\displaystyle \\var{k}+\\frac{\\var{l}}{\\var{m}}-\\frac{\\var{n}}{\\var{o}}=$
Schreiben Sie Ihr Ergebnis so, dass der Koeffizient von $x^2$ im Nenner positiv ist.
\n$\\displaystyle{ \\frac{\\var{aad}}{ \\simplify{x-{bd}} } + \\frac{\\var{bbd}}{\\simplify{x-{cd}}} }$ =