// Numbas version: finer_feedback_settings
{"name": "7d. Adding harder algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "7d. Adding harder algebraic fractions", "tags": [], "metadata": {"description": "<p>(ax+b)/(mx+n)+(cx+d)/(px+q)</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Express the following as a single fraction:</p>\n<p>\\[\\simplify[basic,unitFactor,zeroTerm,zeroFactor]{({a}{L}+{b})/({m}{L}+{n})+({c}{L}+{d})/({p}{L}+{q})}\\]</p>\n<p></p>\n<p><em>Simplify your answer where possible.</em></p>", "advice": "<p>Adding algebraic fractions is the same as adding ordinary fractions. First we need to write both fractions over the same denominator by multiplying the top and bottom of both fractions</p>\n<p>\\[\\simplify[basic,unitFactor,zeroTerm,zeroFactor]{({a}{L}+{b})/({m}{L}+{n})+({c}{L}+{d})/({p}{L}+{q})}=\\simplify[basic,unitFactor,zeroTerm,zeroFactor]{({p}{L}+{q})({a}{L}+{b})/(({p}{L}+{q})({m}{L}+{n}))+({m}{L}+{n})({c}{L}+{d})/(({m}{L}+{n})({p}{L}+{q}))}\\]</p>\n<p>Finally we add the numerators together and simplify to get</p>\n<p>\\[\\simplify[unitFactor,zeroTerm,zeroFactor]{({p}{L}+{q})({a}{L}+{b})/(({p}{L}+{q})({m}{L}+{n}))+({m}{L}+{n})({c}{L}+{d})/(({m}{L}+{n})({p}{L}+{q}))}<br/>=\\simplify[unitFactor,zeroTerm,zeroFactor]{({a*p}{L}^2+{a*q+b*p}{L}+{b*q}+{m*c}{L}^2+{m*d+n*c}{L}+{n*d})/(({m}{L}+{n})({p}{L}+{q}))}\\]<br/>\\[=\\simplify{({a*p}{L}^2+{a*q+b*p}{L}+{b*q}+{m*c}{L}^2+{m*d+n*c}{L}+{n*d})/({m*p}{L}^2+{m*q+n*p}{L}+{n*q})}.\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-8..8 except b)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(0..3 except a)", "description": "", "templateType": "anything", "can_override": false}, "L": {"name": "L", "group": "Ungrouped variables", "definition": "expression(random('x','y','a','w','c','z','n'))", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-3..8 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-3..8 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(-8..8 except d)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(0..3 except c)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["L", "a", "b", "m", "n", "c", "d", "p", "q"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "({a*p}{L}^2+{a*q+b*p}{L}+{b*q}+{m*c}{L}^2+{m*d+n*c}{L}+{n*d})/({m*p}{L}^2+{m*q+n*p}{L}+{n*q})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "?/?", "partialCredit": 0, "message": "", "nameToCompare": "", "warningTime": "submission"}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "resources": []}]}], "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}]}