// Numbas version: finer_feedback_settings
{"name": "Combining algebraic fractions 5.2", "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 5.2", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "advice": "\n    <p><strong>a)</strong></p>\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 $\\simplify{b={a1}*x+{b}}$, $\\simplify{d={a2}*x+{d}}$.<br/>Hence we have:<br/>\\[\\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d})) = ({a*a2 + c*a1} * x + {a * d + c * b}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\]</p>\n  <p><strong>b)</strong></p>\n  <p>Note that the first two fractions are the same as in part a). Hence we immediately have:</p>\n  <p>\\[\\begin{eqnarray*} \\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}&amp;=&amp;\\simplify[std]{({cofx} * x + {con})/ (({a1}*x + {b}) * ({a2}*x + {d}))+{c1}/({a3}x+{d1})}\\\\&amp;=&amp;\\simplify[std]{((({cofx}*x+{con})({a3}x+{d1})+{c1}*({a1}*x + {b}) * ({a2}*x + {d})))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\\\&amp;=&amp;\\simplify[std]{(({cofx*a3}x^2+{con*a3+d1*cofx}x+{con*d1})+({c1*a1*a2}x^2+{c1*(a1*d+b*a2)}x+{c1*b*d}))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\\\&amp;=&amp;\\simplify[std]{({ans1}x^2+{ans2}x+{ans3})/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\end{eqnarray*}\\]</p>\n    \n    ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "<p>Express \\[\\simplify{{a} / ({a1}x + {b}) + ({c} / ({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 to an expression of the form $ax+b$ for suitable integers $a$ and $b$.</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. Also 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": "({a*a2 +  c*a1} * x + {a * d +  c * b})/ (({a1}*x + {b}) * ({a2}*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 $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.</p>\n          <p>Note that in your answer you do not need to expand the denominator.</p>\n          \n            \n            ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}, {"prompt": "<p>Now add the the following three fractions together to form a single fraction.</p>\n      <p>&nbsp;$\\displaystyle&nbsp;\\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}=\\;$[[0]]</p>\n      <p>Make sure that you simplify the numerator.</p>", "gaps": [{"notallowed": {"message": "<p>Input as a single fraction. Also 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": "({ans1}*x^2+{ans2}*x+{ans3})/(({a1}*x+{b})({a2}x+{d})({a3}x+{d1}))", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\n    <p>In the first part add the two algebraic fractions together and express as a single fraction over a common denominator.</p>\n    <p>In the second part add the three fractions together and express as a single fraction.</p>\n    <p>&nbsp;</p>\n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..6)", "name": "a"}, "cofx": {"definition": "a*a2+c*a1", "name": "cofx"}, "c": {"definition": "random(-6..6 except [0,-a])", "name": "c"}, "b": {"definition": "random(-6..6 except 0)", "name": "b"}, "d": {"definition": "random(-6..6 except [0,round(b*a2/a1)])", "name": "d"}, "ans1": {"definition": "cofx*a3+c1*a1*a2", "name": "ans1"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}, "ans3": {"definition": "con*d1+c1*b*d", "name": "ans3"}, "a1": {"definition": 1.0, "name": "a1"}, "a3": {"definition": 1.0, "name": "a3"}, "a2": {"definition": 1.0, "name": "a2"}, "c1": {"definition": "random(-3..3 except [0,round(-con*d1/b*d)])", "name": "c1"}, "con": {"definition": "a*d+c*b", "name": "con"}, "ans2": {"definition": "con*a3+d1*cofx+c1*(a1*d+b*a2)", "name": "ans2"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}, "d1": {"definition": "random(-6..6 except [0,round(a3*d/a2),round(a3*b/a1)])", "name": "d1"}}, "metadata": {"notes": "<p><strong>19/08/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Added description.</p>\n    <p>Coefficients of $x$ in all three denominators is $1$.</p>\n    <p>Made sure that the numerator in the first part addition has an $x$ term by insisting that $a \\neq -c$.</p>\n    <p>Checked calculations.OK.</p>\n    <p><strong>02/02/2103:</strong></p>\n    <p>Numerator in second part is now non-zero by changing variable c1. Also added comment that the numerator has to be simplified.</p>", "description": "\n        \t\t<p>First part: express as a single fraction: $\\displaystyle \\frac{a}{x + b} +&nbsp; \\frac{c}{x + d},\\; a \\neq -c$.</p>\n        \t\t                <p>Second part: Find $\\displaystyle \\frac{a}{x + b} + \\frac{c}{x + d}+\\frac{r}{x+t}$&nbsp;as a single fraction.</p>\n        \t\t", "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/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}