// Numbas version: exam_results_page_options
{"name": "AE5 - Multiplication of algebraic fractions 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "AE5 - Multiplication of algebraic fractions 2", "tags": [], "metadata": {"description": "<p>Simplifying first is essential in terms of managing expressions that might need factorising.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Expand and simplify $\\displaystyle{\\var{LeftMul}\\times\\var{RightMul}}.$</p>", "advice": "<p>Before we multiply the fractions together first lets check if we can do any cancellation. Notice that $\\var{RightMulBottom}$ has a factor of $\\var{Num}$ so we can cancel this straight away.</p>\n<p>We also have a factor of $x$ in both $\\var{QuadCoeff[0]}x^2+\\var{QuadCoeff[1]}x$ and $\\var{RightMulTop}$ so we're now left with multiplying</p>\n<p>\\[\\frac1{\\var{QuadCoeff[0]}x+\\var{QuadCoeff[1]}}\\times\\frac{\\simplify[all,expandBrackets]{(x+{Lin1Coeff})*({QuadCoeff[0]}x+{QuadCoeff[1]})}}{\\var{Lin2Coeff[0]}x+\\var{Lin2Coeff[1]}}.\\]</p>\n<p>We're not necesserily done with cancellation though! To make sure that a fraction with a quadratic is simplified we have to factorise it to make sure there are no linear factors we can cancel. In this case we have<br/>\\[\\simplify[all,expandBrackets]{(x+{Lin1Coeff})*({QuadCoeff[0]}x+{QuadCoeff[1]})}={(x+\\var{Lin1Coeff})(\\var{QuadCoeff[0]}x+\\var{QuadCoeff[1]})}.\\]</p>\n<p>This gives us one last factor to cancel and then we can finally multiply whats left of each fraction to give us a final answer of</p>\n<p>\\[\\var{ans}.\\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#AlgFrac\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"QuadCoeff": {"name": "QuadCoeff", "group": "Ungrouped variables", "definition": "[random(1..6),random(1..6)]", "description": "", "templateType": "anything", "can_override": false}, "Num": {"name": "Num", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "Lin2Coeff": {"name": "Lin2Coeff", "group": "Ungrouped variables", "definition": "[random(1..6),random(1..6)]", "description": "", "templateType": "anything", "can_override": false}, "Lin1Coeff": {"name": "Lin1Coeff", "group": "Ungrouped variables", "definition": "random(1..6 except Lin2Coeff[1]/Lin2Coeff[0])", "description": "", "templateType": "anything", "can_override": false}, "Lin1": {"name": "Lin1", "group": "Ungrouped variables", "definition": "\"x\"+\"+\"+Lin1Coeff", "description": "", "templateType": "anything", "can_override": false}, "Lin2": {"name": "Lin2", "group": "Ungrouped variables", "definition": "Lin2Coeff[0]+\"x\"+\"+\"+Lin2Coeff[1]", "description": "", "templateType": "anything", "can_override": false}, "Quad": {"name": "Quad", "group": "Ungrouped variables", "definition": "QuadCoeff[0]+\"x^2+\"+QuadCoeff[1]+\"x\"", "description": "", "templateType": "anything", "can_override": false}, "LeftMul": {"name": "LeftMul", "group": "Ungrouped variables", "definition": "expression(Num+\"/(\"+Quad+\")\")", "description": "", "templateType": "anything", "can_override": false}, "RightMulTop": {"name": "RightMulTop", "group": "Ungrouped variables", "definition": "simplify(expression(\"(\"+Quad+\")*(\"+Lin1+\")\"),[\"expandBrackets\",\"all\"])", "description": "", "templateType": "anything", "can_override": false}, "RightMulBottom": {"name": "RightMulBottom", "group": "Ungrouped variables", "definition": "simplify(expression(Num+\"*(\"+Lin2+\")\"),[\"expandBrackets\",\"all\"])", "description": "", "templateType": "anything", "can_override": false}, "RightMul": {"name": "RightMul", "group": "Ungrouped variables", "definition": "expression(\"(\"+string(RightMulTop)+\")\"+\"/\"+\"(\"+string(RightMulBottom)+\")\")", "description": "", "templateType": "anything", "can_override": false}, "Ans": {"name": "Ans", "group": "Ungrouped variables", "definition": "expression(\"(\"+Lin1+\")/(\"+Lin2+\")\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["QuadCoeff", "Num", "Lin2Coeff", "Lin1Coeff", "Lin1", "Lin2", "Quad", "LeftMul", "RightMulTop", "RightMulBottom", "RightMul", "Ans"], "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": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "(`+-$n`?*x+`+-$n`?)/(`+-$n`?*x+`+-$n`?)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}]}]}], "contributors": [{"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}]}