// Numbas version: exam_results_page_options {"name": "Brokregning 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": [], "name": "Brokregning 1", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "
Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.
", "rulesets": {}, "parts": [{"stepsPenalty": "0", "prompt": "$\\displaystyle\\frac{\\var{a}}{\\var{b}}+\\frac{\\var{c}}{\\var{b}}=$[[0]]
\n$\\displaystyle\\frac{\\var{d}}{\\var{c}}-\\frac{\\var{a}}{\\var{c}}=$[[1]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Legg sammen tellerne, behold nevner.
\nDisse brøkene har felles nevner, og kan enkelt legges sammen. La oss illustrere det med eksempler (som ikke er like oppgaven):
\n\\[\\frac{2}{3}+\\frac{5}{3}=\\frac{2+5}{3}=\\frac{7}{3}\\]
\nDen samme fremgangsmåten kan brukes for subtraksjon:
\n\\[\\frac{7}{4}-\\frac{3}{4}=\\frac{7-3}{4}=\\frac{4}{4}=1\\]
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "(a+c)/b", "minValue": "(a+c)/b", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": true, "variableReplacements": [], "maxValue": "(d-a)/c", "minValue": "(d-a)/c", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": "0", "prompt": "$\\displaystyle\\simplify{{f}/{g}+{h}/{j}}=$[[0]]
\n$\\displaystyle\\simplify{{h}/{f}-{j}/{g}}=$[[1]]
\n$\\displaystyle \\frac{\\var{a}}{\\var{d}}+\\var{f}=$[[2]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Utvid brøkene slik at de får en felles nevner. Deretter kan du addere eller subtrahere slik som i forrige oppgave.
\nHvis oppgaven var $\\frac{5}{4}+\\frac{3}{8}$ kunne vi utvidet den første brøken til $\\frac{10}{8}$ (ved å multiplisere med 2 i teller og nevner), slik at begge brøkene fikk nevner lik 8. Deretter kunne vi addert brøkene:
\n\\[\\frac{5}{4}+\\frac{3}{8}=\\frac{5\\cdot 2}{4\\cdot 2}+\\frac{3}{8}=\\frac{10}{8}+\\frac{3}{8}=\\frac{13}{8}\\]
\nSom regel må vi utvide begge brøkene for å få en felles nevner, for eksempel når vi skal regne ut $\\frac{5}{4}-\\frac{2}{3}$. Her kan 12 være en felles nevner, og utregningen kan se slik ut:
\n\\[\\frac{5}{4}-\\frac{2}{3}=\\frac{5\\cdot 3}{4\\cdot 3}-\\frac{2\\cdot 4}{3\\cdot 4}=\\frac{15}{12}-\\frac{8}{12}=\\frac{7}{12}\\]
\nEt godt valg for fellesnevneren er minste felles multiplum til de to nevnerne.
\nNB! Husk at hele tall kan skrives som en brøk med nevner lik 1, for eksempel $3=\\frac{3}{1}$.
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\n$\\displaystyle \\frac{1}{\\var{b}}+\\frac{1}{\\var{f}}-\\frac{2}{\\var{b}}\\cdot \\frac{3}{\\var{2*f}}=$[[1]]
\n\n\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Multipliser teller med teller, og nevner med nevner.
\nFor eksempel:
\n\\[\\frac{4}{5}\\cdot \\frac{2}{3}=\\frac{4\\cdot 2}{5 \\cdot 3}=\\frac{8}{15}\\]
\nHusk også regnerekkefølge: multiplikasjon før addisjon/subtraksjon. Eksempel:
\n$\\frac{1}{3}+\\frac{1}{2}-\\frac{1}{4}\\cdot \\frac{2}{3}$ | \n\n $=$ \n | \n$\\frac{1}{3}+\\frac{1}{2}-\\frac{1\\cdot 2}{4 \\cdot 3}$ | \n
\n\n\n | \n\n $=$ \n$=$ \n$=$ \n | \n\n $\\frac{1}{3}+\\frac{1}{2}-\\frac{1}{6}$ \n$\\frac{2}{6}+\\frac{3}{6}-\\frac{1}{6}$ \n$\\frac{4}{6} = \\frac{2}{3}$ \n | \n
(Husk å forkorte hvis mulig)
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\n$\\displaystyle \\var{b}\\cdot(\\frac{1}{\\var{2*b}}+\\frac{1}{\\var{b}}) - \\frac{\\var{2*f}}{\\var{3*h}}:\\frac{2}{\\var{h}}=$[[1]]
\n\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Multipliser med den omvendte brøken.
\nAnta du vil finne $\\frac{3}{7}:\\frac{5}{4}$. Å dividere med $\\frac{5}{4}$ er ekvivalent med å multiplisere med $\\frac{4}{5}$, slik at regnestykket blir:
\n\\[\\frac{3}{7}:\\frac{5}{4}=\\frac{3}{7}\\cdot\\frac{4}{5}=\\frac{3\\cdot 4}{7\\cdot 5}=\\frac{12}{35}\\]
\n\n
Husk også regnerekkefølge: parenteser først, så multiplikasjon/divisjon før addisjon/subtraksjon. Eksempel:
\n$2\\cdot (\\frac{1}{6}+\\frac{2}{3})-\\frac{3}{5}: \\frac{1}{10}$ | \n\n $=$ \n | \n$2\\cdot (\\frac{1}{6}+\\frac{4}{6})-\\frac{3}{5}: \\frac{1}{10}$ | \n
\n\n\n | \n\n $=$ \n$=$ \n$=$ \n\n$=$ \n | \n\n $2\\cdot\\frac{5}{6}-\\frac{3}{5}:\\frac{1}{10}$ \n$\\frac{2\\cdot 5}{6}-\\frac{3}{5}\\cdot \\frac{10}{1}$ \n$\\frac{5}{3}-\\frac{3\\cdot 10}{5\\cdot 1}$ \n$\\frac{2}{3}-6 =-\\frac{16}{3}$ \n | \n
(Husk at $6=\\frac{6}{1}=\\frac{18}{3}$)
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\n\n$\\displaystyle \\frac{\\frac{\\var{d}}{\\var{g}}}{\\var{f}}=$[[1]]
\n\n$\\displaystyle \\frac{\\var{j}}{\\frac{\\var{h}}{\\var{c}}}=$[[2]]
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Her kan man for eksempel betrakte hovedbrøkstreken som divisjon.
\nEksempler:
\n\\[\\frac{7}{\\frac{5}{6}}=7:\\frac{5}{6} =7\\cdot\\frac{6}{5}=\\frac{42}{5}\\]
\n\\[\\frac{\\frac{9}{11}}{\\frac{5}{3}}=\\frac{9}{11}:\\frac{5}{3}=\\frac{9}{11}\\cdot \\frac{3}{5}=\\frac{27}{55}\\]
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