// Numbas version: exam_results_page_options
{"name": "FY001 In Class Test", "metadata": {"description": "<p>FY001 In Class Test</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 5400, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "In class test", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], []], "questions": [{"name": "Ugur's copy of Algebra: Solving quadratics by factorising", "extensions": [], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Some quadratics are to be solved by factorising</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Solve the quadratic equations by factorising. If there is more than one solution, enter them all separated by a comma.</p>\n<p>-----------------------------------</p>", "advice": "<p>See 5.1 and 5.2 for how to use factorising to solve quadratics.</p>\n<p>See 3.3 and 3.5 for factorising.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "map(num[j][3],j,0..4)", "description": "", "templateType": "anything", "can_override": false}, "solmin": {"name": "solmin", "group": "Ungrouped variables", "definition": "vector([min(-d[0]/c[0],-b[0]/a[0])]+[min(-d[1]/c[1],-b[1]/a[1])]+[min(-d[2]/c[2],-b[2]/a[2])]+[min(-d[3]/c[3],-b[3]/a[3])]+[min(-d[4]/c[4],-b[4]/a[4])])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "map(num[j][1],j,0..4)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "map(num[j][2],j,0..4)", "description": "", "templateType": "anything", "can_override": false}, "solmax": {"name": "solmax", "group": "Ungrouped variables", "definition": "vector([max(-d[0]/c[0],-b[0]/a[0])]+[max(-d[1]/c[1],-b[1]/a[1])]+[max(-d[2]/c[2],-b[2]/a[2])]+[max(-d[3]/c[3],-b[3]/a[3])]+[max(-d[4]/c[4],-b[4]/a[4])])", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "map(num[j][0],j,0..4)", "description": "", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "[[1,random(6..7),1,random(2..5)]] +\nshuffle([\n  [1,random(-4..-6),1,random(-3..3 except 0)],\n  [1,random(-4..-6),random(2..3),random(-1..1 except 0)],\n  [1,random(1,3,5),4,random(1,3,5)*random([1,-1])],\n  [1,random(-4..-6),3,random([-1,1,-2,2])]\n  ])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["num", "a", "b", "c", "d", "solmax", "solmin"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "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": "<p>Factorise $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+ {b[0]*d[0]}}=$ [[0]]</p>\n<p>Hence solve $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+ {b[0]*d[0]}}=0$.</p>\n<p>[[1]]</p>\n<p></p>", "gaps": [{"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[0]}x+{b[0]})({c[0]}x+{d[0]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "[solmin[0],solmax[0]]", "allowFractions": true, "correctAnswerFractions": true, "isSet": false}}], "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": "<p>Factorise $\\simplify{{a[1]*c[1]}x^2+{a[1]*d[1]+b[1]*c[1]}x+ {b[1]*d[1]}}=$ [[0]]</p>\n<p>Hence solve $\\simplify{{a[1]*c[1]}x^2+{a[1]*d[1]+b[1]*c[1]}x+ {b[1]*d[1]}}=0$.</p>\n<p>[[1]]</p>", "gaps": [{"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[1]}x+{b[1]})({c[1]}x+{d[1]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "[solmin[1],solmax[1]]", "allowFractions": true, "correctAnswerFractions": true, "isSet": false}}], "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": "<p>Factorise $\\simplify{{a[2]*c[2]}x^2+{a[2]*d[2]+b[2]*c[2]}x+ {b[2]*d[2]}}=$ [[0]]</p>\n<p>Hence solve $\\simplify{{a[2]*c[2]}x^2+{a[2]*d[2]+b[2]*c[2]}x+ {b[2]*d[2]}}=0$.</p>\n<p>[[1]]</p>", "gaps": [{"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[2]}x+{b[2]})({c[2]}x+{d[2]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "[solmin[2],solmax[2]]", "allowFractions": true, "correctAnswerFractions": true, "isSet": false}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Factorising Quadratic Equations with $x^2$ Coefficients of 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Factorise three quadratic equations of the form $x^2+bx+c$.</p>\n<p>The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Factorise the following quadratic equations.</p>\n<p></p>", "advice": "<p>Quadratic equations&nbsp;of the form</p>\n<p>\\[x^2+bx+c=0\\]</p>\n<p>can be factorised to&nbsp;create&nbsp;an equation of the form</p>\n<p>\\[(x+m)(x+n)=0\\text{.}\\]</p>\n<p>When we expand a factorised quadratic expression we obtain</p>\n<p>\\[(x+m)(x+n)=x^2+(m+n)x+(m \\times n)\\text{.}\\]</p>\n<p>To factorise an equation of the form $x^2+bx+c$, we need to find two numbers which add together to make $b$, and multiply together to make $c$.</p>\n<h4>a)</h4>\n<p>\\[\\simplify{x^2+{v1+v2}x+{v1*v2}=0}\\]</p>\n<p>We need to find two values that add together to make $\\var{v1+v2}$ and multiply together to make $\\var{v1*v2}$.</p>\n<p>\\[\\begin{align}<br/>\\var{v1} \\times \\var{v2}&amp;=\\var{v1*v2}\\\\<br/>\\var{v1}+\\var{v2}&amp;=\\var{v1+v2}\\\\<br/>\\end{align} \\]</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v1})(x+{v2})}=0\\text{.}\\]</p>\n<p></p>\n<h4>b)</h4>\n<p>We can begin factorising by finding factors of $\\var{v3*v4}$ that add together to give $\\var{v3+v4}$.</p>\n<p>\\[\\begin{align}<br/>\\var{v3} \\times \\var{v4}&amp;=\\var{v3*v4}\\\\<br/>\\var{v3}+\\var{v4}&amp;=\\var{v3+v4}\\\\<br/>\\end{align} \\]</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v3})(x+{v4})}=0\\text{.}\\]</p>\n<h4>c)</h4>\n<p>When factorising the quadratic expression</p>\n<p>\\[\\simplify{x^2+{v5*v6}=0}\\]</p>\n<p>we need to find two values that add together to make $0$ and multiply together to make $\\var{v5*v6}$.</p>\n<p>\\begin{align}<br/>\\var{v5} \\times \\var{v6}&amp; = \\var{v5*v6}\\\\<br/>\\simplify[]{ {v5} + {v6}} &amp;= 0 \\\\<br/>\\end{align}</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v5})(x+{v6})}=0\\text{.}\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"v1": {"name": "v1", "group": "Part A ", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "v2": {"name": "v2", "group": "Part A ", "definition": "random(2..6 except v1)", "description": "", "templateType": "anything", "can_override": false}, "v4": {"name": "v4", "group": "Part A ", "definition": "random(1..10 except -v3)", "description": "", "templateType": "anything", "can_override": false}, "v5": {"name": "v5", "group": "Part A ", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "v3": {"name": "v3", "group": "Part A ", "definition": "random(-8..-1)", "description": "", "templateType": "anything", "can_override": false}, "v6": {"name": "v6", "group": "Part A ", "definition": "-v5", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part A ", "variables": ["v1", "v2", "v3", "v4", "v5", "v6"]}], "functions": {}, "preamble": {"js": "question.is_factorised = function(part,penalty) {\n  penalty = penalty || 0;\n  if(part.credit>0) {\n    // Parse the student's answer as a syntax tree\n    var studentTree = Numbas.jme.compile(part.studentAnswer,Numbas.jme.builtinScope);\n\n    // Create the pattern to match against \n    // we just want two sets of brackets, each containing two terms\n    // or one of the brackets might not have a constant term\n    // or for repeated roots, you might write (x+a)^2\n    var rule = Numbas.jme.compile('m_all(m_any(x,x+m_pm(m_number),x^m_number,(x+m_pm(m_number))^m_number))*m_nothing');\n\n    // Check the student's answer matches the pattern. \n    var m = Numbas.jme.display.matchTree(rule,studentTree,true);\n    // If not, take away marks\n    if(!m) {\n      part.multCredit(penalty,'Your answer is not fully factorised.');\n    }\n  }\n}", "css": ""}, "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": "<p>$\\simplify{x^2+{v1+v2}x+{v1*v2}=0}$</p>\n<p>[[0]]&nbsp;$=0$</p>\n<p><em></em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(x+{v1})(x+{v2})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 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{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Use the quadratic formula to calculate values for $x$ in these equations. Input the possible values as $x_1$ and $x_2$, where $x_1&lt;x_2$.</p>\n<p></p>\n<p>Round your answers to two decimals. E.g. if you found 4.1567, round it up to 4.16; and if you found 6.653 round it down to 6.65.</p>", "advice": "<p>The quadratic formula is&nbsp;</p>\n<p>\\[x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]</p>\n<h4>a)</h4>\n<p>From the equation, we can read off values for $a$, $b$ and $c$:</p>\n<p>\\[\\begin{align}<br/>a&amp;=1\\text{,}\\\\<br/>b&amp;=\\var{a+m}\\text{,}\\\\<br/>c&amp;=\\var{a*m} \\text{.}<br/>\\end{align}\\]</p>\n<p>Substituting&nbsp;these values into the quadratic formula,</p>\n<p>\\[x = \\frac {-\\var{a+m}\\pm\\sqrt{\\var{a+m}^2-4\\times \\var{a*m}}}{2}\\text{.}\\]</p>\n<p>Note the $\\pm$ symbol in the formula. This means there are two solutions: one using $+$, the other using $-$.</p>\n<p>The two solutions are</p>\n<p>\\[\\begin{align}<br/>x_1&amp;=\\var{m}\\text{,}\\\\<br/>x_2&amp;=\\var{a}\\text{.}<br/>\\end{align}\\]</p>\n<h4>b)</h4>\n<p><em></em>Note that the right-hand side of the given equation is not zero. We need to&nbsp;rewrite it in the form $ax^2+bx+c=0$:</p>\n<p>\\[\\begin{align}<br/>\\simplify{{a1}x^2+{a2}x+{a3}}&amp;=\\var{a4}\\\\<br/>\\simplify{{a1}x^2+{a2}x+{a3-a4}}&amp;=0\\text{.}<br/>\\end{align}\\]</p>\n<p>Then we can read off values for $a$, $b$ and $c$:</p>\n<p>\\[\\begin{align}<br/>a&amp;=\\var{a1}\\\\<br/>b&amp;=\\var{a2}\\\\<br/>c&amp;=\\var{a3-a4} \\text{.}<br/>\\end{align}\\]</p>\n<p>We can now substitute these values into the quadratic formula:</p>\n<p>\\[x = {\\frac {-\\var{a2}\\pm\\sqrt{\\var{a2}^2-4\\times \\var{a1}\\times \\var{a3-a4}}}{2\\times\\var{a1}}}\\text{.}\\]</p>\n<p>So the two solutions are</p>\n<p>\\[\\begin{align}<br/>x_1&amp;=\\var{dpformat(x1,2)}\\\\<br/>x_2&amp;=\\var{dpformat(x2,2)}\\text{.}<br/>\\end{align}\\]</p>\n<h4>c)</h4>\n<p>We first rearrange our equation&nbsp;into the form $ax^2+bx+c=0$:</p>\n<p>\\[\\begin{align}<br/>\\simplify{{b1}x^2+{b2}x+{b3}}&amp;=0=\\var{b4}x\\\\<br/>\\simplify{{b1}x^2+{b2-b4}x+{b3}}&amp;=0\\text{.}<br/>\\end{align}\\]</p>\n<p>We can then read off the&nbsp;values for $a, b$ and $c$, which are</p>\n<p>\\[\\begin{align}<br/>a&amp;=\\var{b1}\\text{,}\\\\<br/>b&amp;=\\var{b2-b4}\\text{,}\\\\<br/>c&amp;=\\var{b3}\\text{.}<br/>\\end{align}\\]</p>\n<p>Substituting&nbsp;these values into the quadratic formula,</p>\n<p>\\[x = {\\frac {-\\var{b2-b4}\\pm\\sqrt{\\var{b2-b4}^2-4\\times \\var{b1}\\times \\var{b3}}}{2\\times\\var{b1}}},\\]</p>\n<p>we obtain solutions</p>\n<p>\\[\\begin{align}<br/>x_1&amp;=\\var{dpformat(p1,2)}\\text{,}\\\\<br/>x_2&amp;=\\var{dpformat(p2,2)}\\text{.}<br/>\\end{align}\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a3": {"name": "a3", "group": "Ungrouped variables", "definition": "random(-30..4 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(15..20)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(9..a-1)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(2..10 except a1)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "part 2", "definition": "random(-10..4 except 0)^2", "description": "", "templateType": "anything", "can_override": false}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "(-(b2-b4)+((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "random(-30..6 except a3)", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(1..15)", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "(-(a2)-((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "(-(b2-b4)-((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "part 2", "definition": "2c^0.5+a1^2", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": 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"https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>This question aims to test understanding and ability to use the laws of indices.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Using the laws of indices, simplify each expression down to its simplest form. Recall that $a^{0} = 1$ for any number $a$.</p>", "advice": "<h4>a)</h4>\n<p>Here we are using the rule of indices: $a^m \\times a^n = a^{m+n}$.</p>\n<p>Using this rule,&nbsp;</p>\n<p>\\[<br/>\\begin{align}<br/>a^\\var{x} \\times a^\\var{y}\\ &amp;= a^\\simplify[all, !collectNumbers]{{x}+{y}}\\\\<br/>&amp;= a^\\var{x+y}.<br/>\\end{align}<br/>\\]</p>\n<h4>b)</h4>\n<p>We are asked to find $\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}$.</p>\n<p>Notice there is a constant in front of each of the terms.</p>\n<p>To do this, write the product out explicitly, as</p>\n<p>\\[\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q} = \\var{c} \\times \\var{d} \\times a^\\var{p} \\times a^\\var{q}.\\]</p>\n<p>We know that $\\var{c} \\times \\var{d} = \\var{c*d}$, and using the rule of indices: $a^\\var{p} \\times a^\\var{q} = a^\\var{p+q}$.</p>\n<p>Therefore:</p>\n<p>\\begin{align}<br/>\\var{c}a^\\var{p} \\times \\var{d}a^\\var{q}&amp;= \\var{c*d} \\times a^\\var{p+q} \\\\<br/>&amp;= \\simplify{{c*d}*a^{p+q}}.<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>Here we are using: $a^m \\div a^n = a^{m-n}$.</p>\n<p>We are asked to simplify the expression, $\\displaystyle\\simplify{{b}*a^{x}/({g}*a^{y})}$.</p>\n<p>To do this, we just have to use the previously mentioned rule of indices. We write this out explicity as</p>\n<p>\\[\\simplify{{b}*a^{x}/({g}*a^{y})} =&nbsp;\\simplify{{b}/{g}} \\times \\simplify{a^{x}/(a^{y})}.\\]</p>\n<p>Using rules of indices,</p>\n<p>\\begin{align} &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\frac{a^\\var{x}}{a^\\var{y}} &amp;= a^\\var{x} \\div a^\\var{y}\\\\<br/>&amp;= a^\\simplify[all, !collectNumbers]{{x}-{y}}\\\\<br/>&amp;= a^\\var{x-y}.<br/>\\end{align}</p>\n<p>Therefore,</p>\n<p>\\begin{align}<br/>\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &amp;=&nbsp;\\simplify{{b}/{g}} \\times \\simplify{a^{{x}-{y}}}\\\\<br/> &amp;=&nbsp;\\simplify{{b}/{g}*a^{x-y}}.<br/>\\end{align}</p>\n<p>Alternatively,&nbsp;</p>\n<p>Using the&nbsp;rule of indices: $a^{-m}&nbsp; = \\displaystyle\\frac{1}{a^{m}}$,&nbsp;we can&nbsp;rewrite the question as:</p>\n<p>\\begin{align}<br/>\\frac{\\var{b}a^\\var{x}}{\\var{g}a^\\var{y}} &amp;=&nbsp;\\simplify{{b}/{g}} \\times \\frac{a^\\var{x}}{a^\\var{y}}\\\\<br/> &amp;=&nbsp;\\simplify{{b}/{g}}&nbsp;\\times a^\\var{x} \\times a^{-\\var{y}}.<br/>\\end{align}</p>\n<p>And then using the rule: $a^m \\times a^n = a^{m+n}$, this becomes:</p>\n<p>\\begin{align}<br/>\\simplify{{b}/{g}}&nbsp;\\times a^\\var{x} \\times a^{-\\var{y}} &amp;=&nbsp;\\simplify{{b}/{g}} \\times a^\\simplify[all,!collectNumbers]{{x}+(-{y})}\\\\<br/> &amp;= \\simplify{{b}/{g}*a^{x-y}}.<br/>\\end{align}</p>\n<h4><strong>d) </strong></h4>\n<p>The question asks us to simplify $(\\simplify{{c}*a^{p}})^{\\var{q}}$.</p>\n<p>To do this we use the rules:</p>\n<p>\\[(a^{m})^{n} = a^{mn},\\]</p>\n<p>\\[(ab)^m = a^mb^m.\\]</p>\n<p>We can&nbsp;then&nbsp;expand the equation as</p>\n<p>\\[(\\simplify{{c}*a^{p}})^{\\var{q}}= \\var{c}^{\\var{q}} \\times (a^{\\var{p}})^{\\var{q}}.\\]</p>\n<p>Then using the rule of indices mentioned previously,</p>\n<p>\\[<br/>\\begin{align}<br/>(\\simplify{{c}*a^{p}})^{\\var{q}}&amp;= \\simplify{{c}^{q}} \\times a^\\var{p*q}\\\\<br/> &amp;= \\simplify{{c}^{q}*a^{p*q}}.<br/>\\end{align}<br/>\\]</p>\n<h4>e)</h4>\n<p>The question asks us to simplify 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"always"}, {"name": "Ugur's copy of Working with standard index form", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "type": "question", "statement": "<p>Working with numbers that are very large or very small can be tricky.</p>\n<p>Standard form allows us to simplify these numbers, using powers of 10.</p>\n<div style=\"text-align: left;\">The standard index form can be defined as</div>\n<div>\\[A&nbsp;\\times 10^n,\\]</div>\n<div>where $1 &le; A &lt; 10$ and $n$&nbsp;is an integer, e.g. $2.26 \\times 10^5$ is a standard form of a number 226000.</div>\n<div>\n<p>&nbsp;</p>\n<p>Write&nbsp;the following in standard index form (for example, for&nbsp;$2.01\\times 10^5$ we would write <code>2.01*10^5</code> in the gap).</p>\n</div>", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"int": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "int", "definition": "random(2..9)"}, "A3dp": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "A3dp", "definition": "random(1..10 #0.001)"}, "B": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "B", "definition": "repeat(random(3..9 #0.01), 4)"}, "small5": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "small5", "definition": "random(0.00001..0.0001 #0.00000001)"}, "A5dp": {"group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "A5dp", "definition": "random(1..10 #0.00001)"}, "ran": 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"musthave": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^2"]}, "variableReplacements": [], "answer": "{A[2]}*10^2", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p style=\"padding-left: 30px;\">$\\var{A[2]*10^2} = $&nbsp; [[0]]</p>", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "answersimplification": "!collectnumbers", "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": "1", "notallowed": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^1"]}, "vsetrange": [0, 1], "musthave": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^", "-1"]}, "variableReplacements": [], "answer": "{A3dp}*10^(-1)", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p style=\"padding-left: 30px;\">$\\var{A3dp*10^(-1)} = $&nbsp; [[0]]</p>", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "answersimplification": "!collectnumbers", "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": "1", "notallowed": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["^-7", "^(-7)"]}, "vsetrange": [0, 1], "musthave": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^7"]}, "variableReplacements": [], "answer": "{A5dp}*10^7", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p style=\"padding-left: 30px;\">$\\var{precround(A5dp*10^7,0)} =$&nbsp; [[0]]</p>", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "answersimplification": "!collectnumbers", "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": "1", "notallowed": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["10^5"]}, "vsetrange": [0, 1], "musthave": {"message": "", "partialCredit": 0, "showStrings": false, "strings": ["*10^", "-5"]}, "variableReplacements": [], "answer": "{{small5}*10^5}*10^(-5)", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p style=\"padding-left: 30px;\">$\\var{small5} = $&nbsp; [[0]]</p>", "type": "gapfill"}], "ungrouped_variables": ["A3dp", "A5dp", "small5", "A", "ran", "B", "int"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Convert a variety of numbers from decimal to&nbsp;standard index form.</p>"}, "preamble": {"css": "", "js": ""}, "advice": "<p>Converting from decimal to a standard form, we are looking for $A&nbsp;\\times 10^n$.</p>\n<p>We need make the first number ($A$) between 1 and 10, so we put the decimal place after the first non-zero digit.</p>\n<p>&nbsp;</p>\n<p><strong>a)</strong></p>\n<p>In $\\var{A[2]*10^2}$, the first non-zero digit is $\\var{siground(A[2] - 0.5, 1)}$ so we get $A = \\var{A[2]}$.</p>\n<p style=\"padding-left: 30px;\">If we moved the decimal place&nbsp;in $\\var{A[2]}$ so it matches our original number $\\var{A[2]*10^2}$, we would go 2&nbsp;places to the right, so $n = 2$.</p>\n<p style=\"padding-left: 30px;\">&nbsp; </p>\n<p><strong>b)</strong></p>\n<p>In $\\var{A3dp*10^(-1)}$, the first non-zero digit is $\\var{siground(A3dp - 0.5, 1)}$ so we get&nbsp;$A = \\var{A3dp}$.</p>\n<p style=\"padding-left: 30px;\">If we moved the decimal place&nbsp;in $\\var{A3dp}$ so it matches our original number $\\var{A3dp*10^(-1)}$, we would go&nbsp;1&nbsp;place to the left, so $n = -1$.</p>\n<p style=\"padding-left: 30px;\">&nbsp; </p>\n<p><strong>c)</strong></p>\n<p>In $\\var{precround(A5dp*10^7,0)}$ the first non-zero digit is $\\var{siground(A5dp - 0.5, 1)}$ so we get $A = \\var{A5dp}$.</p>\n<p style=\"padding-left: 30px;\">If we moved the decimal place&nbsp;in $\\var{A5dp}$ so it matches our original number $\\var{precround(A5dp*10^7,0)}$, we would go&nbsp;7&nbsp;places to the right, so $n = 7$.</p>\n<p style=\"padding-left: 30px;\">&nbsp; </p>\n<h4>d)</h4>\n<p>In $\\var{small5}$ the first non-zero digit is {siground({{small5}*10^5} - 0.5, 1)} so we get $A = \\var{small5*10^5}$.</p>\n<p style=\"padding-left: 30px;\">If we moved the decimal place&nbsp;in $\\var{small5*10^5}$ so it matches our original number $\\var{small5}$, we would go&nbsp;5 places to the left, so $n = -5$.</p>\n<p style=\"padding-left: 30px;\"></p>"}, {"name": "Ugur's copy of Algebraic fractions: operations involving algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Add, subtract, multiply and divide <g class=\"gr_ gr_5 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"5\" data-gr-id=\"5\">alge</g>braic fractions.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate the following and write your answer as a single fraction. Use &nbsp;/ to signify a fraction or division, for example $\\frac{2a-1}{x+3}$ is written (2a-1)/(x+3). Simplify/cancel where possible.</p>", "advice": "<p>Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"primes": {"name": "primes", "group": "numerical fractions", "definition": "shuffle([2,3,5,7,11,13,17])", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "numerical fractions", "definition": "primes[0]", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "numerical fractions", "definition": "primes[3]", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "numerical fractions", "definition": "primes[2]", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "numerical fractions", "definition": "primes[6]", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "numerical fractions", "definition": "primes[5]", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "numerical fractions", "definition": "primes[4]", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "numerical fractions", "definition": "random(primes except [d,g,h])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "numerical fractions", "definition": "primes[1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "numerical fractions", "variables": ["a", "b", "c", "d", "f", "g", "h", "j", "primes"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "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": "<p>$\\displaystyle\\frac{\\var{a}x}{\\var{b}}+\\frac{x+\\var{c}}{\\var{b}}=$&nbsp;[[0]]</p>\n<p>$\\displaystyle\\frac{\\var{d}}{\\var{c}y}-\\frac{\\var{a}}{\\var{c}y}=$&nbsp;[[1]]</p>", "stepsPenalty": "2", "steps": [{"type": "information", "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": "<p><em>Add the tops, leave the bottom the same.</em></p>\n<p></p>\n<p>These fractions have a common denominator (the number on the bottom). This means they are out of the same number of parts and can be compared easily, for example, it is clear $\\frac{2}{3}$ is less than $\\frac{5}{3}$ but not so clear that $\\frac{3}{5}$ is less than $\\frac{2}{3}$.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>Let's say you need to evaluate $\\frac{2}{3}+\\frac{5}{3}$, in words this is 'two thirds plus five thirds', so how many thirds are there in total? Seven&nbsp;thirds!</p>\n<p style=\"text-align: left;\">So we have</p>\n<p style=\"text-align: center;\">\\[\\frac{2}{3}+\\frac{5}{3}=\\frac{2+5}{3}=\\frac{7}{3}\\]</p>\n<p>The same logic is used for subtraction. Suppose you had seven&nbsp;fourths and someone borrowed three fourths, then you are left with&nbsp;four&nbsp;fourths.&nbsp;</p>\n<p>That is</p>\n<p style=\"text-align: center;\">\\[\\frac{7}{4}-\\frac{3}{4}=\\frac{7-3}{4}=\\frac{4}{4}=1\\]</p>\n<p></p>"}], "gaps": [{"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+1}x+{c})/{b}", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"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": "{d-a}/({c}y)", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "y", "value": ""}]}], "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": "<p>$\\displaystyle\\simplify{(a+{f})/{g}+({h}a+1)/{j}}=$&nbsp;[[0]]</p>\n<p>$\\displaystyle\\simplify{(b+{h})/{f}-(b+{j})/{g}}=$&nbsp;[[1]]</p>\n<p>$\\displaystyle \\frac{\\var{a}}{\\var{d}r}+\\var{f}r=$&nbsp;[[2]]</p>", "stepsPenalty": "2", "steps": [{"type": "information", "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": "<p><em>Rewrite the fractions so they have a common denominator. Then perform the addition or subtraction as required.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>If your question was $\\frac{5}{4}+\\frac{3}{8}$ we could rewrite the first fraction as $\\frac{10}{8}$ (by multiplying the top and bottom by 2) and then both fractions would have a denominator of 8. At this point, we can perform the addition. Our working might&nbsp;look like this:</p>\n<p style=\"text-align: center;\">\\[\\frac{5}{4}+\\frac{3}{8}=\\frac{5\\times 2}{4\\times 2}+\\frac{3}{8}=\\frac{10}{8}+\\frac{3}{8}=\\frac{13}{8}\\]</p>\n<p style=\"text-align: left;\"></p>\n<p></p>\n<hr/>\n<p></p>\n<p style=\"text-align: left;\">Often we need to rewrite both fractions to get a common denominator, for instance, $\\frac{5}{4}-\\frac{2}{3}$. We could multiply the first fraction by 3 on the top and bottom, so that it's denominator was 12, and then multiply the second fraction by 4 on the top and bottom&nbsp;so that it also had a denominator of 12. Then we could perform the subtraction. Our working might look like this:</p>\n<p style=\"text-align: center;\">\\[\\frac{5}{4}-\\frac{2}{3}=\\frac{5\\times 3}{4\\times 3}-\\frac{2\\times 4}{3\\times 4}=\\frac{15}{12}-\\frac{8}{12}=\\frac{7}{12}\\]</p>\n<p style=\"text-align: left;\"></p>\n<p></p>\n<hr/>\n<p></p>\n<p style=\"text-align: left;\">Also, recall that whole numbers are just fractions with a denominator of 1, for example $3=\\frac{3}{1}$.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">In general, the best denominator is the lowest common multiple (LCM) of the two denominators.&nbsp;</p>"}], "gaps": [{"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": "({j+g*h}a+{f*j+g})/{g*j}", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "a", "value": ""}]}, {"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": "({g-f}b+{g*h-f*j})/{f*g}", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "b", "value": ""}]}, {"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}+{f*d}r^2)/({d}r)", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "r", "value": ""}]}], "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": "<p>$\\displaystyle\\frac{m+1}{n+1}\\times \\frac{y}{x}=$&nbsp;[[0]]</p>\n<p>$\\displaystyle -\\frac{\\var{f}+w}{\\var{j}}\\times \\var{d}=$&nbsp;[[1]]</p>\n<p></p>\n<p></p>\n<p></p>", "stepsPenalty": "2", "steps": [{"type": "information", "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": "<p><em>Multiply the tops and the bottoms.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>For example&nbsp;</p>\n<p style=\"text-align: center;\">\\[\\frac{4}{5}\\times \\frac{2}{3}=\\frac{4\\times 2}{5 \\times 3}=\\frac{8}{15}\\]</p>\n<p style=\"text-align: center;\"></p>\n<p></p>\n<hr/>\n<p></p>\n<p style=\"text-align: left;\">Also recall that whole numbers are just fractions with a denominator of 1, for example $7=\\frac{7}{1}$.</p>"}], "gaps": [{"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": "((m+1)*y)/((n+1)*x)", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "m", "value": ""}, {"name": "n", "value": ""}, {"name": "x", "value": ""}, {"name": "y", "value": ""}]}, {"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": "-({d*f}+{d}w)/{j}", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "w", "value": ""}]}], "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": "<p>$\\displaystyle{\\simplify{({f}+{a}x)^2/{h}}}\\div \\simplify{(({f}+{a}x){g})/({j}x)}=$&nbsp;[[0]]</p>\n<p>$\\displaystyle \\frac{\\var{b}q}{\\var{c}q}\\div (\\var{d}+t)=$&nbsp;[[1]]</p>\n<p>$\\displaystyle \\var{j}z\\div \\left(\\frac{\\var{-d}(z+1)^2}{\\var{f}z}\\right)=$&nbsp;[[2]]</p>\n<p></p>", "stepsPenalty": "2", "steps": [{"type": "information", "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": "<p><em>Flip the second fraction and then multiply.</em></p>\n<p></p>\n<hr/>\n<p></p>\n<p>Flipping a fraction is also known as taking the reciprocal of the fraction (or inverting a fraction). Note that a whole number is also a fraction with a denominator of 1, for example, $6=\\frac{6}{1}$.</p>\n<p>How do you find half of a number? You could 'divide it by 2', or you could 'multiply by $\\frac{1}{2}$. Notice that $\\frac{1}{2}$ is the reciprocal of 2. When we divide by a number this is actually the same as multiplying by its reciprocal.</p>\n<p></p>\n<hr/>\n<p></p>\n<p>Suppose you need to evaluate $\\frac{3}{7}\\div\\frac{5}{4}$. Recall this is the same as asking 'how many $\\frac{5}{4}$s are in $\\frac{3}{7}$?', but that doesn't seem to be very helpful here! What is helpful is realising that dividing by $\\frac{5}{4}$ is the same as multiplying by $\\frac{4}{5}$. Our working could&nbsp;look like this</p>\n<p>\\[\\frac{3}{7}\\div\\frac{5}{4}=\\frac{3}{7}\\times\\frac{4}{5}=\\frac{3\\times 4}{7\\times 5}=\\frac{12}{35}\\]</p>\n<p>&nbsp;</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a*j}x^2+{j*f}x)/{g*h}", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b}/({c}({d}+t))", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({j*f}z^2)/({-d}(z+1)^2)", "answerSimplification": "simplifyFractions", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "z", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Algebra VI:  Rearranging equations  to find the value of unknown variable", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Nasir Firoz Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/909/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>This exercise will help you rearrange a linear equation to find the value of a given variable</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Rearrange the following equations to solve for $y$. Enter your answers as a fraction a/b.</p>", "advice": "<p>Use the following steps in both parts to solve for unknown:</p>\n<p>Part a)</p>\n<p>1. Bring the terms of given variable to one side by changing signs.</p>\n<p>Ex- &nbsp;$\\var{a}y - \\var{c}y$ = &nbsp;$\\var{h}y$</p>\n<p>2. Take all the constants to one side by changing signs</p>\n<p>Ex- $\\var{b}- \\var{d}$ = &nbsp;$\\var{w}$</p>\n<p>3. Solving for y, we get-</p>\n<p>$\\ y $= &nbsp;$\\var{f}$</p>\n<p></p>\n<p>Part b)</p>\n<p>1. Bring the terms of fractions of variables &nbsp;to one side by changing signs.</p>\n<p>Ex- &nbsp;$\\frac{\\var{m}}{{y}} - \\frac{\\var{s}}{{y}}&nbsp;= &nbsp;\\frac{\\var{o}}{{y}}$</p>\n<p>2. Take all the constants to one side by changing signs</p>\n<p>Ex- $\\var{t}- \\var{n}$ = &nbsp;$\\var{u}$</p>\n<p>3. Solving for y, we get-</p>\n<p>$\\ y $= &nbsp;$\\var{g}$</p>\n<p></p>\n<p>For further help, check this video-&nbsp;</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<iframe width=\"500\" height=\"375\" src=\"//www.youtube.com/embed/GmMX3-nTWbE\"></iframe></p>", "rulesets": {"std": ["all"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(5..8)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..4 except a)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(10..18)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "precround((o/u),2)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "precround((w/h),2)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "a-c", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..12)", "description": "", "templateType": "anything", "can_override": false}, "o": {"name": "o", "group": "Ungrouped variables", "definition": "m-s", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..12)", "description": "", "templateType": "anything", "can_override": false}, "s": 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"showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\var{a}y + \\var{d}&nbsp;=&nbsp;\\var{b} +\\var{c}y$</p>\n<p>$y=\\;$[[0]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{b-d}/{a-c}", "maxValue": "{b-d}/{a-c}", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle \\frac{\\var{m}}{y}+\\var{n}=\\frac{\\var{s}}{y}+\\var{t}$</p>\n<p>$y=\\;$[[0]]</p>", "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": "{m-s}/{t-n}", 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true, "showanswerstate": false, "allowrevealanswer": false, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": false, "reviewshowexpectedanswer": true, "reviewshowadvice": false, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "extensions": [], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": []}