// Numbas version: finer_feedback_settings
{"name": "Combining algebraic fractions 6.0", "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 6.0", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator", "quadratic 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 $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x^2+{q}x+{d}}$.<br/>Hence we have:<br/>\\[\\begin{eqnarray*}\\simplify[std]{{a} / ({a1}*x + {b}) + (({c}x+{p}) / ({a2}*x^2+{q}x + {d}))} &amp;=&amp; \\simplify[std]{({a} * ({a2}*x^2 +{q}x+ {d}) + ({c}x+{p}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))}\\\\&amp; =&amp; \\simplify[std,!collectNumbers]{({a*a2} * x^2 + {a *q}x+{a*d}+{a1*c}x^2+{p*a1+b*c}x+{b*p}) / (({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))}\\\\&amp;=&amp;\\simplify[std,!noLeadingMinus]{({co1}x^2+{co2}x+{co3})/(({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))}\\end{eqnarray*}\\]</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "<p>Express \\[\\simplify[std]{{a} / ({a1}x + {b}) + (({c}x+{p}) / ({a2}x^2 +{q}x+ {d}))}\\] as a single fraction.</p>\n      <p>Input the fraction here: [[0]].</p>\n      <p>Do not expand out the denominator.</p>\n      <p>Make sure that you simplify the numerator.</p>\n      <p>Click on Show steps if you need help.You will lose one mark if you do so.</p>\n      <p>&nbsp;</p>", "gaps": [{"notallowed": {"message": "<p>Input as a single fraction. Make sure that you simplify the numerator.</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": "({co1} * x^2 +{co2}*x+ {co3})/ (({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))", "type": "jme"}], "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 $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x^2+{q}x+{d}}$.</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    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..7)", "name": "a"}, "co1": {"definition": "a*a2+a1*c", "name": "co1"}, "c": {"definition": "random(-6..6 except 0)", "name": "c"}, "b": {"definition": "random(-6..6 except 0)", "name": "b"}, "d": {"definition": "r^2+random(1..5)", "name": "d"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}, "co3": {"definition": "a*d+b*p", "name": "co3"}, "a1": {"definition": 1.0, "name": "a1"}, "p": {"definition": "random(-4..4 except 0)", "name": "p"}, "co2": {"definition": "a*q+p*a1+b*c", "name": "co2"}, "q": {"definition": "2*r", "name": "q"}, "r": {"definition": "random(-3..3)", "name": "r"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}, "a2": {"definition": 1.0, "name": "a2"}}, "metadata": {"notes": "<p><strong>19/08/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Added description.</p>\n    <p>&nbsp;Note that the quadratic has no real roots.</p>\n    <p>Looking at Advice, we see that various rules are switched on and off to get a display of the solution which is \"natural\".</p>\n    <p>Checked calculations.OK.</p>\n    <p><strong>02/02/2013:</strong></p>\n    <p>Added homily about simplifying the numerator, both in the part and in the forbidden string warning.</p>", "description": "<p>Express $\\displaystyle \\frac{a}{x + b} + \\frac{cx+d}{x^2 +px+ q}$&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/"}]}