// Numbas version: exam_results_page_options {"name": "Maths Support: Combining algebraic fractions", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "

13 questions on combining algebraic fractions. An area in which students often need practice.

", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 0.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Combining algebraic fractions 0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "advice": "\n

We have:

\\[\\simplify[std]{{a} + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c}) / (({a2}*x + {d})) = ({a*a2} * x + {a * d + c}) / ( ({a2}*x + {d}))}\\]

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify[std]{{a} + ({c} / ({a2}x + {d}))}\\] as a single algebraic fraction.

Input the fraction here: [[0]].

Input your answer in the form $\\displaystyle \\frac{(ax+b)}{(cx+d)}$ with no other brackets than those shown.

You can click on Show steps for help. You will lose 1 mark if you do so.

", "gaps": [{"type": "jme", "checkingaccuracy": 1e-05, "minlength": {"length": 13.0, "message": "

Input as a single fraction.

", "partialcredit": 0.0}, "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "({a*a2}x+{a*d+c})/({a2}x+{d})", "vsetrange": [10.0, 11.0], "musthave": {"message": "

Input as a single fraction

", "showstrings": false, "strings": [")/"], "partialcredit": 0.0}}], "steps": [{"prompt": "\n

The formula for adding these expressions is :
\\[\\simplify[std]{a + {s1} * (c / d) = (ad + {s1} * bc) / d}\\]

and for this exercise we have $\\simplify{d={a2}x+{d}}$.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Add the following together and express as a single algebraic fraction.

\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "c": {"definition": "random(-9..9 except [0,-a*d])", "name": "c"}, "d": {"definition": "random(-9..9 except [0,a2])", "name": "d"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}, "a2": {"definition": "random(2..9)", "name": "a2"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}}, "metadata": {"notes": "\n \t\t \t\t \t\t

5/08/2012:

\n \t\t \t\t \t\t

Added tags.

\n \t\t \t\t \t\t

Added description.

\n \t\t \t\t \t\t

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.

\n \t\t \t\t \t\t

12/08/2012:

\n \t\t \t\t \t\t

Back to one input of a fraction and trapped input in Forbidden Strings.

\n \t\t \t\t \t\t

Used the except feature of ranges to get non-degenerate examples.

\n \t\t \t\t \t\t

Checked calculation.OK.

\n \t\t \t\t \t\t

Improved display in content areas.

\n \t\t \t\t \n \t\t \n \t\t", "description": "

Express $\\displaystyle a \\pm \\frac{c}{x + d}$ as an algebraic single fraction.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Combining algebraic fractions 1 (Video)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "variables": {"a2": {"definition": "1", "name": "a2", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "a1": {"definition": "1", "name": "a1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "c": {"definition": "random(-9..9 except [0,-a])", "name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "a": {"definition": "random(1..9)", "name": "a", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "b": {"definition": "random(-9..9 except 0)", "name": "b", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "d": {"definition": "random(-9..9 except [0,round(b*a2/a1)])", "name": "d", "description": "", "templateType": "anything", "group": "Ungrouped variables"}}, "statement": "

Add the following two fractions together and express as a single fraction over a common denominator.

", "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "metadata": {"description": "

Express $\\displaystyle \\frac{a}{x + b} \\pm \\frac{c}{x + d}$ as an algebraic single fraction over a common denominator.

Contains a video in Show steps.

", "licence": "Creative Commons Attribution 4.0 International"}, "advice": "\n

The formula for {nb} fractions is :

\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
Hence we have:
\\[\\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}))}\\]

\n ", "preamble": {"js": "", "css": ""}, "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "chain rule", "combining algebraic fractions", "common denominator"], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "s1"], "parts": [{"gaps": [{"variableReplacementStrategy": "originalfirst", "vsetRange": [10, 11], "checkingType": "absdiff", "checkingAccuracy": 1e-05, "showFeedbackIcon": true, "variableReplacements": [], "answer": "({a*a2 + c*a1} * x + {a * d + c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))", "scripts": {}, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "answerSimplification": "std", "failureRate": 1, "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 2, "type": "jme", "showPreview": true, "notallowed": {"strings": [")-", ")+"], "message": "

Input as a single fraction.

", "showStrings": false, "partialCredit": 0}, "checkVariableNames": false, "expectedVariableNames": [], "vsetRangePoints": 5}], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "prompt": "

The formula for {nb} fractions is:

\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.

Note that in your answer you do not need to expand the denominator.

The following video goes through an example similar to this one.

", "type": "information", "scripts": {}, "showCorrectAnswer": true, "variableReplacements": []}], "type": "gapfill", "stepsPenalty": 1, "prompt": "

Express

\\[\\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\\]

as a single fraction.

Enter the fraction here: [[0]]

Input your answer in the form $\\displaystyle \\frac{(ax+b)}{((cx+d)(ex+f))}$ with no other brackets than those shown.

Click on Show steps if you need help. You will lose one mark if you do so.

"}], "variable_groups": [], "type": "question"}, {"name": "Combining algebraic fractions 3.0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions"], "advice": "

The formula for adding these expressions is:

\\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]

and for this exercise we have $\\simplify{a={b1}}$, $\\simplify{c={c}x+{b2}}$, $\\simplify{d={a2}x+{d}}$.

Hence we have:
\\[\\begin{eqnarray*} \\simplify[std]{{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})} &=& \\simplify[basic,unitFactor]{(({b1}) * ({a2}*x + {d}) + ({c}x+{b2}) ) / ( ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({b1*a2}x+{b1*d})+{c}x+{b2}) / ( ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ( {b1*a2+ c }x+{b1*d+b2}) / (({a2}*x + {d}))}\\end{eqnarray*}\\]

", "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

Express \\[\\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.

Input the fraction here: [[0]].

Click on Show steps to get more information. You will lose one mark if you do so.

\n \n ", "gaps": [{"notallowed": {"message": "

Input as a single fraction.

", "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": "( {c+b1*a2} * x + {b1 * d + b2 })/ ( ({a2}*x + {d}))", "type": "jme"}], "steps": [{"prompt": "\n

Note that:
\\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]

\n \n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Express the following as a single fraction.

\n \n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": 0.0, "name": "a"}, "c": {"definition": "random(1..9 except a)", "name": "c"}, "d": {"definition": "random(-9..9 except [0,round(b2*a2/c)])", "name": "d"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}, "a2": {"definition": 1.0, "name": "a2"}, "b1": {"definition": "random(-5..5 except 0)", "name": "b1"}, "b2": {"definition": "random(-5..5 except 0)", "name": "b2"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}}, "metadata": {"notes": "

18/08/2012:

Added tags.

Added description.

Modified copy of Combining algebraic fractions 3.

Checked calculations.OK.

29/01/2013:

Edited advice so that simplification steps were correct in the solution.

", "description": "

Express $\\displaystyle b+ \\frac{dx+p}{x + q}$ as an algebraic single fraction.

The formula for adding these expressions is:

\\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]

and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={c}x+{b2}}$, $\\simplify{d={a2}x+{d}}$.

Hence we have:
\\[\\begin{eqnarray*} \\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) ) / ( ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+{c}x+{b2}) / ( ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ({a*a2} * x^2 + {a * d +b1*a2+ c }x+{b1*d+b2}) / (({a2}*x + {d}))}\\end{eqnarray*}\\]

\n ", "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.

Input the fraction here: [[0]].

Make sure that you simplify the numerator.

Click on Show steps to get more information. You will lose one mark if you do so.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction. Also make sure that you simplify the numerator so that it is a quadratic.

", "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}*x^2 + {c+b1*a2+a*d} * x + {b1 * d + b2 })/ ( ({a2}*x + {d}))", "type": "jme", "musthave": {"message": "

Input as a single fraction. Also make sure that you simplify the numerator.

", "showstrings": false, "strings": ["^"], "partialcredit": 0.0}}], "steps": [{"prompt": "\n

Note that:
\\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Express the following as a single fraction.

\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "c": {"definition": "random(1..9 except a)", "name": "c"}, "d": {"definition": "random(-9..9 except [0,round(b2*a2/c)])", "name": "d"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}, "a2": {"definition": 1.0, "name": "a2"}, "b1": {"definition": "random(-5..5 except 0)", "name": "b1"}, "b2": {"definition": "random(-5..5 except 0)", "name": "b2"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}}, "metadata": {"notes": "

18/08/2012:

Added tags.

Added description.

Modified copy of Combining algebraic fractions 3.

Checked calculations.OK.

04/02/2013:

Added comments about simplifying the numerator.

", "description": "

Express $\\displaystyle ax+b+ \\frac{dx+p}{x + q}$ as an algebraic single fraction.

The formula for {nb} fractions is :

\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.

Hence we have:
\\[\\begin{eqnarray*}\\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\ &=&\\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}))}\\\\&=&\\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*}\\]

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.

Input the fraction here: [[0]].

Click on Show steps for more information. You will lose one mark if you do so.

\n ", "gaps": [{"notallowed": {"message": "

Input as a single fraction and simplify the numerator.

", "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": "

Simplify the numerator.

", "showstrings": false, "strings": ["^"], "partialcredit": 0.0}}], "steps": [{"prompt": "\n

The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.

Note that in your answer you do not need to expand the denominator.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Add the following two fractions together and express as a single fraction over a common denominator.

\n \n \n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "c": {"definition": "random(-9..9 except [-a,0])", "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": 1.0, "name": "a1"}, "a2": {"definition": 1.0, "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": "

5/08/2012:

Added tags.

Added description.

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.

12/08/2012:

Back to one input of a fraction and trapped input in Forbidden Strings.

Used the except feature of ranges to get non-degenerate examples.

Checked calculation.OK.

Improved display in content areas.

02/02/2013:

Changed variable c so that the coefficient of $x$ in the numerator of the answer is not 0.

Checked calculation and input traps, OK.

", "description": "

Express $\\displaystyle \\frac{ax+b}{x + c} \\pm \\frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator.

The formula for {nb} fractions is :

\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.

Input the fraction here: [[0]].

Make sure that you simplify the numerator and that you input it as a quadratic in $x$.

Click on Show steps for more information. You will lose one mark if you do so.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction.

Input the numerator as a quadratic in $x$.

", "showstrings": false, "strings": ["^"], "partialcredit": 0.0}}], "steps": [{"prompt": "\n

The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.

Note that in your answer you do not need to expand the denominator.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Add the following two fractions together and express as a single fraction over a common denominator.

\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": "

5/08/2012:

Added tags.

Added description.

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.

12/08/2012:

Back to one input of a fraction and trapped input in Forbidden Strings.

Used the except feature of ranges to get non-degenerate examples.

Checked calculation.OK.

Improved display in content areas.

02/02/2013:

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.

Checked calculations again. OK.

", "description": "

Express $\\displaystyle \\frac{ax+b}{cx + d} \\pm \\frac{rx+s}{px + q}$ as an algebraic single fraction over a common denominator.

The formula for {nb} fractions is :

\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{b={a1}*x+{b}}$, $\\simplify{d={a2}*x+{d}}$.
Hence we have:
\\[\\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}))}\\]

Note that the first two fractions are the same as in part a). Hence we immediately have:

\\[\\begin{eqnarray*} \\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}&=&\\simplify[std]{({cofx} * x + {con})/ (({a1}*x + {b}) * ({a2}*x + {d}))+{c1}/({a3}x+{d1})}\\\\&=&\\simplify[std]{((({cofx}*x+{con})({a3}x+{d1})+{c1}*({a1}*x + {b}) * ({a2}*x + {d})))/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\\\&=&\\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}))}\\\\&=&\\simplify[std]{({ans1}x^2+{ans2}x+{ans3})/(({a1}*x + {b}) * ({a2}*x + {d})*({a3}x+{d1}))}\\end{eqnarray*}\\]

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n

Express \\[\\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\\] as a single fraction.

Input the fraction here: [[0]].

Click on Show steps if you need help. You will lose one mark if you do so.

\n ", "gaps": [{"notallowed": {"message": "

Input as a single fraction.

", "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

The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.

Note that in your answer you do not need to expand the denominator.

\n \n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}, {"prompt": "\n

Now add the the following three fractions together to form a single fraction.

$\\displaystyle \\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}=\\;$[[0]]

\n ", "gaps": [{"notallowed": {"message": "

Input as a single fraction.

", "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}], "statement": "\n

In the first part add the two algebraic fractions together and express as a single fraction over a common denominator.

In the second part add the three fractions together and express as a single fraction.

\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": "-a", "name": "c"}, "b": {"definition": "random(-6..6 except 0)", "name": "b"}, "d": {"definition": "random(-6..6 except [0,round(b*a2/a1),b])", "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": "

19/08/2012:

Added tags.

Added description.

Coefficients of $x$ in all three denominators is $1$.

Note that $a=-c$ so that the numerator of the fraction in the first part is an integer.

Checked calculations.OK.

02/02/2013:

Made sure that the numerator of the first part answer is non-zero by changing variable b.

Also that the numerator in the second part is non-zero by redefining c1 so that the constant term is non-zero.

", "description": "\n \t\t

First part: Express $\\displaystyle \\frac{a}{px + b} +\\frac{c}{qx + d},\\;a=-c$. Numerator is an integer.

\n \t\t

Second part: $\\displaystyle \\frac{a}{px + b} +\\frac{c}{qx + d}+ \\frac{r}{sx+t}$ as single fraction

\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Combining algebraic fractions 5.2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "advice": "\n

The formula for {nb} fractions is :

\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

Note that the first two fractions are the same as in part a). Hence we immediately have:

\n \n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\\] as a single fraction.

Input the fraction here: [[0]].

Make sure that you simplify the numerator to an expression of the form $ax+b$ for suitable integers $a$ and $b$.

Click on Show steps if you need help. You will lose one mark if you do so.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction. Also make sure that you simplify the numerator.

", "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

The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.

Note that in your answer you do not need to expand the denominator.

\n \n \n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}, {"prompt": "

Now add the the following three fractions together to form a single fraction.

$\\displaystyle \\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}=\\;$[[0]]

Make sure that you simplify the numerator.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction. Also simplify the numerator.

In the first part add the two algebraic fractions together and express as a single fraction over a common denominator.

In the second part add the three fractions together and express as a single fraction.

\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": "

19/08/2012:

Added tags.

Added description.

Coefficients of $x$ in all three denominators is $1$.

Made sure that the numerator in the first part addition has an $x$ term by insisting that $a \\neq -c$.

Checked calculations.OK.

02/02/2103:

Numerator in second part is now non-zero by changing variable c1. Also added comment that the numerator has to be simplified.

", "description": "\n \t\t

First part: express as a single fraction: $\\displaystyle \\frac{a}{x + b} + \\frac{c}{x + d},\\; a \\neq -c$.

\n \t\t

Second part: Find $\\displaystyle \\frac{a}{x + b} + \\frac{c}{x + d}+\\frac{r}{x+t}$ as a single fraction.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Combining algebraic fractions 5.3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "advice": "\n

The formula for {nb} fractions is :

\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

Note that the first two fractions are the same as in part a). Hence we immediately have:

\n \n \n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\\] as a single fraction.

Input the fraction here: [[0]].

Make sure that you simplify the numerator.

Click on Show steps if you need help. You will lose one mark if you do so.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction. Also make sure that you simplify the numerator.

", "showstrings": false, "strings": [")-", ")+"], "partialcredit": 0.0}, "checkingaccuracy": 1e-05, "vsetrange": [7.0, 8.0], "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

The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.

Note that in your answer you do not need to expand the denominator.

\n \n \n \n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}, {"prompt": "

Now add the the following three fractions together to form a single fraction.

$\\displaystyle \\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))+{c1}/({a3}x+{d1})}=\\;$[[0]]

Make sure that you simplify the numerator.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction. Also make sure that you simplify the numerator.

", "showstrings": false, "strings": [")+", ")-"], "partialcredit": 0.0}, "checkingaccuracy": 1e-05, "vsetrange": [7.0, 8.0], "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}], "statement": "\n

In the first part add the two algebraic fractions together and express as a single fraction over a common denominator.

In the second part add the three fractions together and express as a single fraction.

\n \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*a2/a1])", "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": "random(1..4)", "name": "a1"}, "a3": {"definition": "random(1..4)", "name": "a3"}, "a2": {"definition": "random(1..4)", "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": "

19/08/2012:

Added tags.

Added description.

Checked calculations. OK.

Set accuracy to 0.00001 for abs diff in both parts.

02/02/2013:

Made sure that the numerator in the first part addition has an $x$ term by redefining c.

Similarly for second part by redefining c1.

", "description": "\n \t\t

First part: express as a single fraction: $\\displaystyle \\frac{a}{px + b} + \\frac{c}{qx + d}$.

\n \t\t

Second part: Find $\\displaystyle \\frac{a}{px + b} + \\frac{c}{qx + d}+\\frac{r}{sx+t}$ as a single fraction.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Combining algebraic fractions 6.0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator", "quadratic denominator"], "advice": "\n

The formula for {nb} fractions is :

\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x^2+{q}x+{d}}$.
Hence we have:
\\[\\begin{eqnarray*}\\simplify[std]{{a} / ({a1}*x + {b}) + (({c}x+{p}) / ({a2}*x^2+{q}x + {d}))} &=& \\simplify[std]{({a} * ({a2}*x^2 +{q}x+ {d}) + ({c}x+{p}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))}\\\\& =& \\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}))}\\\\&=&\\simplify[std,!noLeadingMinus]{({co1}x^2+{co2}x+{co3})/(({a1}*x + {b}) * ({a2}*x^2 +{q}x+ {d}))}\\end{eqnarray*}\\]

\n ", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify[std]{{a} / ({a1}x + {b}) + (({c}x+{p}) / ({a2}x^2 +{q}x+ {d}))}\\] as a single fraction.

Input the fraction here: [[0]].

Do not expand out the denominator.

Make sure that you simplify the numerator.

Click on Show steps if you need help.You will lose one mark if you do so.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction. Make sure that you simplify the numerator.

", "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

The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]

and for this exercise we have $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x^2+{q}x+{d}}$.

Note that in your answer you do not need to expand the denominator.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Add the following two fractions together and express as a single fraction over a common denominator.

\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": "

19/08/2012:

Added tags.

Added description.

Note that the quadratic has no real roots.

Looking at Advice, we see that various rules are switched on and off to get a display of the solution which is \"natural\".

Checked calculations.OK.

02/02/2013:

Added homily about simplifying the numerator, both in the part and in the forbidden string warning.

", "description": "

Express $\\displaystyle \\frac{a}{x + b} + \\frac{cx+d}{x^2 +px+ q}$ as an algebraic single fraction over a common denominator.

The formula for {nb} fractions in this case is :

\\[\\simplify[std]{a / b + {s1} * (c / b^2) = (ab+ {s1} * c) / b^2}\\]

and for this exercise we have $\\simplify{b={a1}x+{b}}$.
Hence we have:
\\[\\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a1}*x + {b})^2) = ({a} * ({a1}*x + {b}) + {c} ) / (({a1}*x + {b})^2) = ({a*a1} * x + {a * b + c}) / (({a1}*x + {b})^2 )}\\]

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {b})^2)}\\] as a single fraction.

Input the fraction here: [[0]].

Make sure that you simplify the numerator.

Click on Show steps if you require help. You will lose one mark if you do so.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction and also make sure that you simplify the numerator.

", "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} * x + {a * b + c })/ (({a1}*x + {b})^2)", "type": "jme"}], "steps": [{"prompt": "\n

The formula for {nb} fractions in this case is :
\\[\\simplify[std]{a / b + {s1} * (c / b^2) = (ab + {s1} * c) / b^2}\\]

This is because you can choose the denominator of the single fraction you are trying to find to be an expression which the denominators of the separate fractions both divide into. In this case both divide into $b^2$, so best to choose this.

For this exercise we have $\\simplify{b={a1}x+{b}}$.

Note that in your answer you do not need to expand the denominator.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Add the following two fractions together and express as a single fraction.

\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "c": {"definition": "random(-9..9 except 0)", "name": "c"}, "b": {"definition": "random(-9..9 except 0)", "name": "b"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}, "a1": {"definition": 1.0, "name": "a1"}, "a2": {"definition": "a1", "name": "a2"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}}, "metadata": {"notes": "

19/08/2012:

Added tags.

Added description.

Modified copy of Combining algebraic fractions 1.

Checked calculations.OK.

02/02/2013:

Added instruction to simplify the numerator in the part and also in the forbidden string feedback.

", "description": "

Express $\\displaystyle \\frac{a}{x + b} +\\frac{c}{(x + b)^2}$ as an algebraic single fraction.

Using the information given by Show steps we have:

\\[\\simplify[std]{ {a} / ({a1}x + {b}) + ({c}x+{d}) /( ({a2}x + {b})^2) = ({a} * ({a1}*x + {b}) + {c}*x+{d} ) / (({a1}*x + {b})^2) = ({a*a1+c} * x + {a * b + d}) / (({a1}*x + {b})^2 )}\\]

\n ", "rulesets": {"std": ["all", "fractionNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify[std]{{a} / ({a1}x + {b}) + ({c}x+{d}) /( ({a2}x + {b})^2)}\\] as a single fraction.

Input the fraction here: [[0]].

Make sure that you simplify the numerator.

Click on Show steps if you require help. You will lose one mark if you do so.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction and also make sure that you simplify the numerator.

", "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} * x + {a * b + d })/ (({a1}*x + {b})^2)", "type": "jme"}], "steps": [{"prompt": "\n

The formula for {nb} fractions in this case is :
\\[\\simplify[std]{a / b + {s1} * (c / b^2) = (ab + {s1} * c) / b^2}\\]

This is because you can choose the denominator of the single fraction to be an expression which the denominators of the separate fractions both divide into. In this case both divide into $b^2$, so best to choose this.

For this exercise we have $\\simplify{b={a1}x+{b}}$.

Note that in your answer you do not need to expand the denominator.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Add the following two fractions together and express as a single fraction.

\n \n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "c": {"definition": "random(-9..9 except 0)", "name": "c"}, "b": {"definition": "chcp(a1,1,9,random(-9..9 except 0))", "name": "b"}, "d": {"definition": "random(-6..6 except[0,round(b*c/a2)])", "name": "d"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}, "a1": {"definition": "random(1..8 except 0)", "name": "a1"}, "a2": {"definition": "a1", "name": "a2"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}}, "metadata": {"notes": "

19/08/2012:

Added tags.

Added description.

Modified copy of Combining algebraic fractions 1.

Introduced function chcp to make sure that the denominator was not of the form (ax+b) where a and b have a common factor.

Checked calculations.OK.

02/02/2013:

Added instructions to simplify the numerator.

", "description": "

Express $\\displaystyle \\frac{a}{x + b} +\\frac{cx+d}{(x + b)^2}$ as an algebraic single fraction.

Show steps tells us that a good choice for the denominator of the algebraic fraction we are looking for is $\\simplify{(x+{p})({a1}x+{b})({a2}x+{d})}$.

Hence we have:
\\[\\begin{eqnarray*} \\simplify{{a} /((x+{p}) ({a1}x + {b})) + ({c} /( (x+{p})({a2}x + {d})))} &=& \\simplify[std]{({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / ((x+{p})({a1}*x + {b}) * ({a2}*x + {d})) }\\\\&=& \\simplify{({a*a2 + c*a1} * x + {a * d + c * b}) / ((x+{p})({a1}*x + {b}) * ({a2}*x + {d}))}\\end{eqnarray*}\\]

\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"], "std1": ["std", "collectNumbers"]}, "parts": [{"stepspenalty": 1.0, "prompt": "

Express \\[\\simplify{{a} /((x+{p}) ({a1}x + {b})) + ({c} /( (x+{p})({a2}x + {d})))}\\] as a single fraction.

Input the fraction here: [[0]].

Make sure that you simplify the numerator.

Click on Show steps if you need help. You will lose one mark of you do so.

", "gaps": [{"notallowed": {"message": "

Input as a single fraction and make sure that you simplify the numerator.

", "showstrings": false, "strings": [")-", ")+"], "partialcredit": 0.0}, "checkingaccuracy": 1e-05, "vsetrange": [10.0, 11.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std1", "marks": 2.0, "answer": "({a*a2 + c*a1} * x + {a * d + c * b})/ ((x+{p})({a1}*x + {b}) * ({a2}*x + {d}))", "type": "jme"}], "steps": [{"prompt": "\n

Note that the denominators both have the factor $\\simplify{x+{p}}$ hence we see that a common denominator is $\\simplify{(x+{p})({a1}x+{b})({a2}x+{d})}$ as both denominators, $\\simplify{(x+{p})({a1}x+{b})}$ and $\\simplify{(x+{p})({a2}x+{d})}$, divide into it.

Note that in your answer you do not need to expand the denominator.

\n ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n

Add the following two fractions together and express as a single fraction over a common denominator.

\n \n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "c": {"definition": "random(-9..9 except 0)", "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(1..8)", "name": "a1"}, "p": {"definition": "random(-9..9 except [round(b/a1),round(d/a2)])", "name": "p"}, "a2": {"definition": "random(1..8)", "name": "a2"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}}, "metadata": {"notes": "

5/08/2012:

Added tags.

Added description.

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.

12/08/2012:

Back to one input of a fraction and trapped input in Forbidden Strings.

Used the except feature of ranges to get non-degenerate examples.

Checked calculation.OK.

Improved display in content areas.

02/02/2013:

Added instructions to simplify the numerator.

", "description": "

Express $\\displaystyle \\frac{a}{(x+r)(px + b)} + \\frac{c}{(x+r)(qx + d)}$ as an algebraic single fraction over a common denominator. The question asks for a solution which has denominator $(x+r)(px+b)(qx+d)$.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}