{name} is een grote snoeper. Je bepaalt in wat volgt de kost van alle snoep dat {name} gedurende de laatste week verorberde.
\n{name} at $\\var{a1}$ pakjes koekjes, $\\var{b1}$ doosjes pralines en $\\var{c1}$ doosjes lolly's.
\nJe weet dat een doosje pralines €$\\var{d1}$ meer kost dan een pakje koekjes,
\nen dat een dergelijk doosje pralines $\\var{f1}$ maal zoveel kost als een doosje lolly's .
", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Translation to Dutch of
\"Given a description in words of the costs of some items in terms of an unknown cost, write down an expression for the total cost of a selection of items. Then simplify the expression, and finally evaluate it at a given point.
\nThe word problem is about the costs of sweets in a sweet shop.\"
"}, "parts": [{"useCustomName": false, "extendBaseMarkingAlgorithm": true, "customName": "", "prompt": "Veronderstel dat een pakje koekjes €$x$ kost.
\nSchrijf een uitdrukking neer in termen van $x$ voor de kost van elke snoepsoort (per pakje of doosje):
\nKoekjes: €[[0]]
\nPralines: €[[1]]
\nLolly's: €[[2]]
Schrijf een algebraïsche uitdrukking neer, in termen van $x$ voor de totale kost van het snoepgoed dat {name} verbruikte. Doe dit in de vorm ax+b.
\n€[[0]]
", "variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "sortAnswers": false, "variableReplacements": [], "adaptiveMarkingPenalty": 0, "showCorrectAnswer": true, "marks": 0, "gaps": [{"extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "checkVariableNames": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [{"name": "x", "value": ""}], "checkingType": "absdiff", "failureRate": 1, "showPreview": true, "vsetRange": [0, 1], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "unitTests": [], "useCustomName": false, "customName": "", "answer": "({a1}+{b1}+{c1}/{f1})x+({b1*d1}+{c1*d1}/{f1})", "answerSimplification": "all", "scripts": {}, "checkingAccuracy": 0.001, "variableReplacements": [], "adaptiveMarkingPenalty": 0, "mustmatchpattern": {"message": "Vereenvoudig tot een vorm a*x+b", "partialCredit": 0, "nameToCompare": "", "pattern": "$n*x + $n"}, "marks": 1, "vsetRangePoints": 5, "type": "jme"}], "type": "gapfill", "customMarkingAlgorithm": ""}, {"useCustomName": false, "extendBaseMarkingAlgorithm": true, "customName": "", "prompt": "Nu blijkt dat een pakje koekjes €$\\var{e1}$ kost.
\nGebruik dit om de totale kostprijs te bepalen die {name} gedurende de laatste week spendeerde aan dit snoepgoed.
\n€[[0]]
", "variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "sortAnswers": false, "variableReplacements": [], "adaptiveMarkingPenalty": 0, "showCorrectAnswer": true, "marks": 0, "gaps": [{"extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "checkVariableNames": true, "variableReplacementStrategy": "originalfirst", "valuegenerators": [], "checkingType": "absdiff", "failureRate": 1, "showPreview": true, "vsetRange": [0, 1], "showCorrectAnswer": false, "customMarkingAlgorithm": "", "unitTests": [], "useCustomName": false, "notallowed": {"message": "Don't use brackets
", "partialCredit": 0, "strings": ["(", ")"], "showStrings": true}, "customName": "", "answer": "({a1}+{b1}+{c1}/{f1})*{e1}+({b1*d1}+{c1*d1}/{f1})", "answerSimplification": "all", "scripts": {}, "checkingAccuracy": 0.001, "variableReplacements": [], "adaptiveMarkingPenalty": 0, "marks": 1, "vsetRangePoints": 5, "type": "jme"}], "type": "gapfill", "customMarkingAlgorithm": ""}], "tags": [], "preamble": {"js": "", "css": ""}, "variables": {"d1": {"group": "Number of packets eaten", "definition": "random(1..10)", "name": "d1", "templateType": "anything", "description": "meerkost voor pralines t.o.v. koekjes
"}, "a1": {"group": "Number of packets eaten", "definition": "random(5..10)", "name": "a1", "templateType": "anything", "description": "Aantal pakjes koekjes
"}, "name": {"group": "Ungrouped variables", "definition": "random('Jerry','Jessica')", "name": "name", "templateType": "anything", "description": ""}, "b1": {"group": "Number of packets eaten", "definition": "random(2..10 except a1)", "name": "b1", "templateType": "anything", "description": "Aantal doosjes pralines
"}, "f1": {"group": "Number of packets eaten", "definition": "random(2..5)", "name": "f1", "templateType": "anything", "description": "factor als verhouding pralines t.o.v. lolly's
"}, "c1": {"group": "Number of packets eaten", "definition": "random(2..5)*f1", "name": "c1", "templateType": "anything", "description": "Aantal lolly's.
"}, "e1": {"group": "Number of packets eaten", "definition": "random(2..8)", "name": "e1", "templateType": "anything", "description": "kost koekjes
"}, "total": {"group": "Ungrouped variables", "definition": "(a1+b1+c1/f1)*{e1} + b1*d1+c1/f1*d1", "name": "total", "templateType": "anything", "description": "+The total spent.
"}}, "functions": {}, "rulesets": {}, "extensions": ["stats"], "variablesTest": {"condition": "gcd(a1,b1+c1/2)=1", "maxRuns": 100}, "advice": "Volgens de gegevens wordt het aantal pakjes koekjes voorgesteld door de letter $x$.
\nEen doosje pralines kost $€ \\var{d1}$ meer dan een pakje koekjes, i.e. $x+\\var{d1}$.
\nEen doosje pralines kost $\\var{f1}$ maal zoveel als een doosje lolly's,
\nwat impliceert dat een een doosje lolly's als kostprijs ${\\frac{1}{\\var{f1}}}(x+\\var{d1})$ heeft.
\nOm de totale kost te vinden, vermenigvuldig je de supra gevonden uitdrukkingen voor de kostprijs van elk snoepgoed met het aantal verbruikte pakjes of doosjes, en telt deze vervolgens op.
\nZonder vereenvoudiging betekent dit:
\n\\begin{align}
\\text{Totale kost} &= \\simplify[]{{a1}x+{b1}(x+{d1}) + {c1} (1/{f1})*(x+{d1})} \\\\
&= \\simplify[]{{a1}x+{b1}(x+{d1}) + {c1/f1}*(x+{d1})}
\\text{.}
\\end{align}
Werk de haakjes uit:
\n\\begin{align}
\\simplify[]{ {a1}x + {b1}(x+{d1}) + {c1/f1}*(x+{d1})} &= \\simplify[]{ {a1}x + {b1}x + {b1}{d1} + {c1/f1}x + {c1/f1}{d1}} \\\\
&= \\simplify[] { {a1}x + {b1}x + {b1*d1} + {c1/f1}x + {c1/f1*d1} } \\text{.}
\\end{align}
en groepeer gelijksoortige termen voor de optelling:
\n\\begin{align}
\\simplify[] { {a1}x + {b1}x + {b1*d1} + {c1/f1}x + {c1/f1*d1} } &= \\simplify[]{ {a1+b1+c1/f1}x + {b1*d1+c1/f1*d1} } \\text{.}
\\end{align}
Wanneer je weet dat de prijs van een pakje koekjes $€\\var{e1}$ bedraagt, kan je dit voor $x$ substitueren in bovenstaande uitdrukking uit b).
\n\\begin{align}
\\text{Totale kost}&=\\var{a1+b1+c1/f1} \\cdot \\var{e1}+\\var{b1*d1+c1/f1*d1} \\\\
&=\\var{(a1+b1+c1/f1)*{e1}+b1*d1+c1/f1*d1} \\text{.}
\\end{align}
Dus spendeerde {name} voor $€\\var{total}$ aan snoep gedurende de voorbije week.
", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}, {"name": "Paul Verheyen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3610/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}, {"name": "Paul Verheyen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3610/"}]}