Refresher questions on topics in algebra that students beginning a maths undergraduate course should be familiar with.
"}, "name": "Common Denominators", "timing": {"timeout": {"action": "none", "message": ""}, "allowPause": true, "timedwarning": {"action": "none", "message": ""}}, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"name": "Combining algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "name": "s1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sgn(c)*random(1..5 except [round(c*d/a2)])", "description": "", "name": "b2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a2"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,-a])", "description": "", "name": "c"}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "name": "nb"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,round(a*b/a1)])", "description": "", "name": "b1"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "description": "", "name": "d"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "b1", "b2", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "gaps": [{"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}))", "musthave": {"message": "Input as a single fraction with the numerator as a quadratic and all terms expanded in the numerator.
", "showStrings": false, "partialCredit": 0, "strings": ["^"]}, "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "showCorrectAnswer": true, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "std", "variableReplacements": [], "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\nNote: you do not need to expand the denominator, but you must enter the numerator as a polynomial in $x$.
\nInput the fraction here: [[0]]
\nClick on Show steps for more information. You will lose one mark if you do so.
", "steps": [{"prompt": "The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (a*d + {s1} * b*c) / (b*d)}\\]
and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\nNote that in your answer you do not need to expand the denominator.
", "scripts": {}, "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "stepsPenalty": 1}], "statement": "Add the following two fractions together and express as a single fraction over a common denominator.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator", "MAS1601", "mas1601"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t \t\t5/08/2012:
\n \t\t \t\t \t\tAdded tags.
\n \t\t \t\t \t\tAdded description.
\n \t\t \t\t \t\tChanged 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\t12/08/2012:
\n \t\t \t\t \t\tBack to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\t \t\tUsed the except feature of ranges to get non-degenerate examples.
\n \t\t \t\t \t\tChecked calculation.OK.
\n \t\t \t\t \t\tImproved display in content areas.
\n \t\t \t\t \n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{ax+b}{x + c} \\pm \\frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator.
"}, "variablesTest": {"condition": "let(\n qa,a*a2+a1*c,\n qb,b*c+a1*b2+b1*a2+a*d,\n qc,b1*d+b2*b,\n roots,[-b/a1,-d/a2],\n \n not (((-qb+sqrt(qb*qb+4*qa*qc))/(2*qa) in roots) or ((-qb-sqrt(qb*qb+4*qa*qc))/(2*qa) in roots))\n)", "maxRuns": "300"}, "advice": "The formula for {nb} fractions is :
\n\\[\\simplify[std]{a / b + {s1} * (c / d) = (a*d + {s1} * b*c) / b*d}\\]
\nand for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\nHence 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*}\\]
Input as a single fraction.
\nAlso make sure that the numerator of your answer is input in the $(ax+b)$ with no brackets other than the ones shown.
", "showStrings": false, "partialCredit": 0, "strings": [")-", ")+"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std1", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\nExpress \\[\\simplify{{a} /((x+{p}) ({a1}x + {b})) + ({c} /( (x+{p})({a2}x + {d})))}\\] as a single fraction.
\nInput the fraction here: [[0]].
\nClick on Show steps if you need help. You will lose one mark of you do so.
\n ", "steps": [{"type": "information", "prompt": "\nNote 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.
\nNote that in your answer you do not need to expand the denominator.
\n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Add the following two fractions together and express as a single fraction over a common denominator.
\nMake sure that your answer has a numerator which you input in form $(ax+b)$ with no brackets other than the ones shown.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator", "MAS1601", "mas1601"], "rulesets": {"std1": ["std", "collectNumbers"], "std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t5/08/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tAdded description.
\n \t\t \t\tChanged 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\t12/08/2012:
\n \t\t \t\tBack to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\tUsed the except feature of ranges to get non-degenerate examples.
\n \t\t \t\tChecked calculation.OK.
\n \t\t \t\tImproved display in content areas.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "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 $(x+r)(px+b)(qx+d)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\nShow steps tells us that a good choice for the denominator of the algebraic fraction we are looking for is $\\simplify{(x+{p})({a1}x+{b})({a2}x+{d})}$.
\nHence 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*}\\]
You must write your answer in the form p/q for integers p and q
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", "showStrings": false, "partialCredit": 0, "strings": ["+", ".", "(", ")", "1-", "2-", "3-", "4-", "5-", "6-", "7-", "8-", "9-"]}, "showpreview": true, "maxlength": {"length": 7, "message": "answer too long
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\\[\\simplify[std]{{a} / {g} + ({s1*b} / {f})}\\]
Input your answer here: [[0]]
No decimal numbers allowed.
\nDo not include brackets in your answer.
\nYou can get help by clicking on Steps. If you do so you will lose 1/2 mark.
", "steps": [{"type": "information", "prompt": "The rule for {action1} fractions is \\[\\simplify{a/b+ {s1}*(c/d)=(a*d+{s1}*b*c)/(b*d)}.\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n{dosomething} the following fractions and reduce the\n \n resulting fraction to lowest form.
Input your answer as a fraction and not\n \n as a decimal.
Putting something here so Loughborough doesn't break.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Add/subtract fractions and reduce to lowest form.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The rule for {action1} fractions is \\[\\simplify{a/b+ {s1}*(c/d)=(a*d+{s1}*b*c)/(b*d)}.\\]
In this case we have:
\\[\\simplify[std,!unitFactor]{{a} / {g} + ({s1*b} / {f}) = ({a} * {f} + {g} * {s1*b}) / ({g} * {f}) ={a*f+s1*g*b}/{g*f}}.\\]
Note that this fraction is in its lowest form as there are no common factors in the denominator and the numerator.
You must write your answer in the form p/q for integers p and q
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", "showStrings": false, "partialCredit": 0, "strings": ["+", ".", "(", ")", "1-", "2-", "3-", "4-", "5-", "6-", "7-", "8-", "9-"]}, "showpreview": true, "maxlength": {"length": 7, "message": "answer too long
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[\\simplify{{g} / ({a} + {s1} * ({b * g} / {f}))}\\]
Input your answer here: [[0]]
Your answer must be of the form a/b for suitable integers a and b. No decimal numbers allowed.
\nDo not include brackets in your answer.
", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Write the following expression as a single fraction in its lowest form:
", "tags": ["checked2015", "Fractions", "fractions", "lowest form", "mas1601", "MAS1601", "simplifying fractions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\nAdded description.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\frac{a} {b + \\frac{c}{d}}$ as a single fraction in the form $\\displaystyle \\frac{p}{q}$ for integers $p$ and $q$.
"}, "advice": "We have:
\\[\\simplify[std]{{g} / ({a} + {s1} * ({b * g} / {f})) = {g} / (({a} * {f} + {s1} * {b * g}) / {f}) ={g} / (({a * f + s1 * b * g}) / {f})= ({f}*{g}) / ({a * f + s1 * b * g}) = ({g * f} / {(a * f + s1 * b * g)})}\\]
Here we use the result that dividing by a fraction $\\frac{a}{b}$ is the same as multiplying by $\\frac{b}{a}$.
The resulting fraction is in lowest form i.e. the top and bottom do not have a common factor.