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{"name": "Algebra V: algebraic fractions (simplification)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["n1", "d1", "ans", "a1", "a2", "n2", "p2", "b1", "b2", "b3", "b4", "c1", "c2", "e1", "e2", "e3", "co1", "co2", "f1", "f2", "co3", "co4", "p3", "g1", "g2"], "name": "Algebra V: algebraic fractions (simplification)", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Click 'Try another question like this one' if you need more practice.</p>", "rulesets": {}, "parts": [{"stepsPenalty": "0", "notallowed": {"message": "<p>Please simplify further.</p>", "showStrings": false, "strings": ["^", "x*x"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>$\\simplify{({n1}{a1}x^2+{n1}{a2}x)/({d1}{a1}x+{d1}{a2})}$</p>\n<p></p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>${\\simplify{({n1}{a1}x^2+{n1}{a2}x)/({d1}{a1}x+{d1}{a2})}}$</p>\n<p>Factorise the numerator and denominator so that the binomials in both are the same.</p>\n<p>${\\big(\\frac{\\var{n1}x}{\\var{d1}}\\big)\\big(\\frac{\\var{a1}x+\\var{a2}}{\\var{a1}x+\\var{a2}}\\big)}$</p>\n<p>The binomials cancel, leaving $x$ and its coefficient:</p>\n<p>$\\big({\\simplify{{n1}/{d1}}}\\big)x$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "!basic", "scripts": {}, "answer": "{{n1}/{d1}}*x", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "maxlength": {"length": "0", "message": "", "partialCredit": 0}}, {"stepsPenalty": "0", "notallowed": {"message": "", "showStrings": false, "strings": ["^", "n*n"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>$\\simplify{(({n2}{b1}n^({p2}+1)+{n2}{b2}n^{p2})/({n2}{b3}n^({p2}+1)+{n2}{b4}n^{p2}))}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>${\\simplify{(({n2}{b1}n^({p2}+1)+{n2}{b2}n^{p2})/({n2}{b3}n^({p2}+1)+{n2}{b4}n^{p2}))}}$</p>\n<p>As before, factorise the numerator and denominator. This time, however, you'll notice that the factors themselves are the same.</p>\n<p>$\\big(\\frac{\\var{n2}n}{{\\var{n2}}n}\\big)\\big(\\frac{\\var{b1}n+\\var{b2}}{\\var{b3}n+\\var{b4}}\\big)$</p>\n<p>The factors cancel, leaving:</p>\n<p>$\\big(\\frac{\\var{b1}n+\\var{b2}}{\\var{b3}n+\\var{b4}}\\big)$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "scripts": {}, "answer": "({b1}n+{b2})/({b3}n+{b4})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": "0", "notallowed": {"message": "", "showStrings": false, "strings": ["^", "^2"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>$\\simplify{(x^2+({c1}+{c2})x +{c1}{c2})/(x+{c1})}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>${\\simplify{(x^2+({c1}+{c2})x +{c1}{c2})/(x+{c1})}}$</p>\n<p>Here, the quadratic expression in the numerator needs to be factorised into the product of two binomials.</p>\n<p>$\\frac{({\\simplify{x+{c1}}})({\\simplify{x+{c2}}})}{({\\simplify{x+{c1}}})}$</p>\n<p>You will notice that one of the binomials in the numerator is the same as the denominator, which means that they can be cancelled. This leaves the expression:</p>\n<p>${\\simplify{x+{c2}}}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "scripts": {}, "answer": "x+{c2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": "0", "notallowed": {"message": "", "showStrings": false, "strings": ["^2", "^"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>$\\frac{\\simplify{(n^2+({e1}+{e2})n+{e1}{e2})}}{\\simplify{(n^2+({e1}+{e3})n+{e1}{e3})}}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>$\\frac{{\\simplify{(n^2+({e1}+{e2})n+{e1}{e2})}}}{{\\simplify{(n^2+({e1}+{e3})n+{e1}{e3})}}}$</p>\n<p>This time there is a quadratic expression which needs to be factorised into the products of binomials in both the numerator and denominator.</p>\n<p>$\\frac{({\\simplify{n+{e1}}})({\\simplify{n+{e2}}})}{({\\simplify{n+{e1}}})({\\simplify{n+{e3}}})}$</p>\n<p>The repeated binomials in the numerator and denominator cancel, leaving:</p>\n<p>$\\frac{({\\simplify{n+{e2}}})}{({\\simplify{n+{e3}}})}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "scripts": {}, "answer": "(n+{e2})/(n+{e3})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": "0", "notallowed": {"message": "", "showStrings": false, "strings": ["x"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>$\\simplify{({co1}x+{co1}{f1})/(x+{f2})}\\times \\simplify{({co2}x+{co2}{f2})/(x+{f1})}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>${\\simplify{({co1}x+{co1}{f1})/(x+{f2})}}\\times{\\simplify{({co2}x+{co2}{f2})/(x+{f1})}}$</p>\n<p>For this question, start by factorising each fraction being multiplied.</p>\n<p>$\\big(\\var{co1}\\big)\\big(\\frac{\\simplify{x+{f1}}}{\\simplify{x+{f2}}}\\big)\\times\\big(\\var{co2}\\big)\\big(\\frac{\\simplify{x+{f2}}}{\\simplify{x+{f1}}}\\big)$</p>\n<p>Due to the commmutative nature of multiplication, the factors can be rearranged so that potential simplification becomes easier to spot.</p>\n<p>$\\big(\\var{co1}\\times\\var{co2}\\big)\\Big(\\frac{(\\simplify{x+{f1}})(\\simplify{x+{f2}})}{(\\simplify{x+{f2}})(\\simplify{x+{f1}})}\\Big)$</p>\n<p>The binomial expressions in the fraction all cancel, leaving the answer as the product of the factorised coefficients:</p>\n<p>$\\var{co1}\\times\\var{co2}=\\simplify{{co1}*{co2}}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{co1}*{co2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": "0", "vsetrangepoints": 5, "prompt": "<p>$\\simplify{{co3}/(x^({p3}+1) + {g1}x^{p3})}\\div \\simplify{{co4}/({g2}x+{g2}{g1})}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>$\\simplify{{co3}/(x^({p3}+1) + {g1}x^{p3})}\\div \\simplify{{co4}/({g2}x+{g2}{g1})}$</p>\n<p>When the divisor is a fraction, it's often useful to flip the numerator and denominator and change the operation to multiplication. This 'flipping' action is more commonly referred to as taking the reciprocal and it holds that dividing by a number or expression is the same as multiplying by its reciprocal.</p>\n<p>$\\simplify{{co3}/(x^({p3}+1) + {g1}x^{p3})}\\times \\simplify{({g2}x+{g2}{g1})/{co4}}$</p>\n<p>Each fraction should then be separately factorised.</p>\n<p>$\\big(\\frac{\\var{co3}}{x^{\\var{p3}}}\\big)\\Big(\\frac{1}{\\simplify{x+{g1}}}\\Big)\\times\\big(\\frac{\\var{g2}}{\\var{co4}}\\big)\\Big(\\frac{\\simplify{x+{g1}}}{1}\\Big)$</p>\n<p>Rearrange the factors to help spot simplification possibilities.</p>\n<p>$\\big(\\frac{\\var{co3}}{x^{\\var{p3}}}\\big)\\big(\\frac{\\var{g2}}{\\var{co4}}\\big)\\times\\Big(\\frac{1}{\\simplify{x+{g1}}}\\Big)\\Big(\\frac{\\simplify{x+{g1}}}{1}\\Big)$</p>\n<p>When multiplied, the final two fractions simplify and the binomials cancel.</p>\n<p>$\\big(\\frac{\\var{co3}}{x^{\\var{p3}}}\\big)\\big(\\frac{\\var{g2}}{\\var{co4}}\\big)\\times\\Big(\\frac{\\simplify{x+{g1}}}{\\simplify{x+{g1}}}\\Big)=\\big(\\frac{\\var{co3}}{x^{\\var{p3}}}\\big)\\big(\\frac{\\var{g2}}{\\var{co4}}\\big)\\times1$</p>\n<p>This leaves the answer as the simplification of the remaining fractions multiplied.</p>\n<p>$\\big(\\frac{\\var{co3}}{x^{\\var{p3}}}\\big)\\big(\\frac{\\var{g2}}{\\var{co4}}\\big)=\\simplify{({co3}*{g2})/({co4}*x^{p3})}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "({co3}*{g2})/({co4}*x^{p3})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "extensions": [], "statement": "<p><strong>Simplify the following algebraic expressions.</strong></p>\n<p><strong><em>Note:</em></strong> <em>Although the question may accept coefficients in their decimal forms, it would be more appropriate to keep them in their most simplified fraction forms.</em></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"f1": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "f2": {"definition": "random(-5..5 except f1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "ans": {"definition": "n1/d1", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "b4": {"definition": "random(1..9 except b3)", "templateType": "anything", "group": "Ungrouped variables", "name": "b4", "description": ""}, "b1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random(1..9 except b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "b3": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "d1": {"definition": "random(1..9 except n1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "co1": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "co1", "description": ""}, "co3": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "co3", "description": ""}, "co2": {"definition": "random(2..5 except co1)", "templateType": "anything", "group": "Ungrouped variables", "name": "co2", "description": ""}, "co4": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "co4", "description": ""}, "g2": {"definition": "random(-5..5 except 0 except g1)", "templateType": "anything", "group": "Ungrouped variables", "name": "g2", "description": ""}, "g1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "g1", "description": ""}, "a1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "random(1..9 except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "c2": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}, "c1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "e1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "e1", "description": ""}, "e3": {"definition": "random(-5..5 except 0 except e2)", "templateType": "anything", "group": "Ungrouped variables", "name": "e3", "description": ""}, "e2": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "e2", "description": ""}, "p2": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "p2", "description": ""}, "p3": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "p3", "description": ""}, "n1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}}, "metadata": {"description": "<p>A question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}]}]}], "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}]}