// Numbas version: finer_feedback_settings
{"name": "Combining algebraic fractions 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Combining algebraic fractions 4", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "advice": "\n  <p>The formula for {nb} fractions is :</p>\n  <p>\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]</p>\n  <p>and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.</p>\n  <p>Hence we have:<br/>\\[\\begin{eqnarray*}\\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &amp;=&amp; \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\ &amp;=&amp;\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+({a1*c}x^2+{b*c+a1*b2}x+{b*b2})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\&amp;=&amp;\\simplify[std]{ ({a*a2 + c*a1} * x^2 + {a * d +a1*b2+b1*a2+ c * b}x+{b1*d+b*b2}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\end{eqnarray*}\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "<p>Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.</p>\n        <p>Input the fraction here: [[0]].</p>\n        <p>Make sure that you simplify the numerator and that you input it as a quadratic in $x$.</p>\n        <p>&nbsp;Click on Show steps for more information. You will lose one mark if you do so.</p>", "gaps": [{"notallowed": {"message": "<p>Input as a single fraction.</p>", "showstrings": false, "strings": [")-", ")+"], "partialcredit": 0.0}, "checkingaccuracy": 1e-05, "vsetrange": [10.0, 11.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "({a*a2+a1*c}*x^2 + {b*c+a1*b2+b1*a2+a*d} * x + {b1 * d +  b2 * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))", "type": "jme", "musthave": {"message": "<p>Input the numerator as a quadratic in $x$.</p>", "showstrings": false, "strings": ["^"], "partialcredit": 0.0}}], "steps": [{"prompt": "\n          <p>The formula for {nb} fractions is :<br/>\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]</p>\n          <p>and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.</p>\n          <p>Note that in your answer you do not need to expand the denominator.</p>\n          ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "extensions": [], "statement": "\n    <p>Add the following two fractions together and express as a single fraction over a common denominator.</p>\n    <p>&nbsp;</p>\n    \n  \n  \n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "c": {"definition": "random(-9..9 except [0,round(-a*a2/a1)])", "name": "c"}, "b": {"definition": "random(-9..9 except 0)", "name": "b"}, "d": {"definition": "random(-9..9 except [0,round(b*a2/a1)])", "name": "d"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}, "a1": {"definition": "random(2..5)", "name": "a1"}, "a2": {"definition": "random(2..5)", "name": "a2"}, "b1": {"definition": "random(-5..5 except [0,round(a*b/a1)])", "name": "b1"}, "b2": {"definition": "sgn(c)*random(1..5 except [round(c*d/a2)])", "name": "b2"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}}, "metadata": {"notes": "<p><strong>5/08/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Added description.</p>\n    <p>Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.</p>\n    <p><strong>12/08/2012:</strong></p>\n    <p>Back to one input of a fraction and trapped input in Forbidden Strings.</p>\n    <p>Used the except feature of ranges to get non-degenerate examples.</p>\n    <p>Checked calculation.OK.</p>\n    <p>Improved display in content areas.</p>\n    <p><strong>02/02/2013:</strong></p>\n    <p>Added requirement that the numerator is input as a quadratic. This is following on from changing variable c so that the coefficient of $x^2$ is non zero.</p>\n    <p>Checked calculations again. OK.</p>", "description": "<p>Express $\\displaystyle \\frac{ax+b}{cx + d} \\pm&nbsp; \\frac{rx+s}{px + q}$&nbsp;as an algebraic single fraction over a common denominator.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "resources": []}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}