// Numbas version: finer_feedback_settings
{"name": "Simon's copy of Q5 (Calculating wages using algebraic equations)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p><strong>(a)</strong></p>\n<p>Let $L$ be the wage for 1 labourer, and let $T$ be the wage for 1 tradesman</p>\n<p>Then we can set up two simultaneous equations as follows:</p>\n<p></p>\n<p>$\\var{num41}T+\\var{num42}L=\\var{wages41}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(Equation 1)</p>\n<p>$T + \\var{num43}L=\\var{wages42}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;(Equation 2)</p>\n<p></p>\n<p>Multiplying equation 2 by $\\var{num41}$ gives:</p>\n<p>$\\var{num41}T+\\var{num41*num43}L=\\var{num41*wages42}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(Equation 3)</p>\n<p></p>\n<p>Subtracting equation&nbsp;1 from equation&nbsp;3 eliminates $T$ and gives:</p>\n<p>$\\var{num41*num43}L-\\var{num42}L=\\var{num41*wages42}-\\var{wages41}$</p>\n<p>which we can solve for $L$ as follows:</p>\n<p>$\\var{-num42+num41*num43}L=\\var{-wages41+num41*wages42}$</p>\n<p>$L=\\var{precround((wages41-num41*wages42)/(num42-num41*num43),2)}$</p>\n<p></p>\n<p>We can now find $T$ by substituting this value of $L$ into any of Equations 1-3.</p>\n<p>e.g. Substituting into Equation 2 gives:</p>\n<p>$T + \\var{num43}\\times \\var{precround((wages41-num41*wages42)/(num42-num41*num43),2)} = \\var{wages42}$&nbsp;</p>\n<p>$T= \\var{precround(wages42-num43*(wages41-num41*wages42)/(num42-num41*num43),2)} $&nbsp;</p>\n<p></p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>$\\var{num11}$ tradesmen and&nbsp;$\\var{num12}$ labourers earn&nbsp;$&euro;\\var{wages1}$ between them to do a job. If a tradesman earns&nbsp;$&euro;\\var{num13}$ more than a labourer, calculate the earnings for a tradesman and a labourer.</p>\n<p></p>\n<p>We can set up two simultaneous equations as follows:</p>\n<p>$\\var{num11}T+\\var{num12}L=\\var{wages1}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(Equation 1)</p>\n<p>$T=L+\\var{num13}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(Equation 2)</p>\n<p></p>\n<p>We can replace $T$ in Equation 1 with&nbsp;$L+\\var{num13}$ to obtain:</p>\n<p>$\\var{num11}(L+\\var{num13})+\\var{num12}L=\\var{wages1}$</p>\n<p>Hence:</p>\n<p>$\\var{num11+num12}L+\\var{num11*num13}=\\var{wages1}$</p>\n<p>$\\var{num11+num12}L=\\var{wages1-num11*num13}$</p>\n<p>$L=\\var{precround((wages1-num11*num13)/(num11+num12),2)}$</p>\n<p></p>\n<p>Using Equation 2 we also obtain:</p>\n<p>$T=\\var{precround((wages1-num11*num13)/(num11+num12),2)}+\\var{num13}=\\var{precround(num13+((wages1-num11*num13)/(num11+num12)),2)}$</p>\n<p></p>\n<p></p>\n<p><strong>(c)</strong></p>\n<p>$\\var{num21}$ workmen on a building site earn a total of&nbsp;$&euro;\\var{wages2}$ between them per week. Labourers are paid&nbsp;$&euro;\\var{num22}$ per week and Tradesmen are paid&nbsp;$&euro;\\var{num23}$ per week. How many of each is employed?</p>\n<p>We can set up two simultaneous equations as follows:</p>\n<p>$\\var{num23}T+\\var{num22}L = \\var{wages2}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (Equation 1)</p>\n<p>$T+L = \\var{num21}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(Equation 2)</p>\n<p></p>\n<p>Multiplying equation 2 by $\\var{num23}$ gives:</p>\n<p>$\\var{num23}T+\\var{num23}L=\\var{num21*num23}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;(Equation 3)</p>\n<p></p>\n<p>Subtracting equation&nbsp;1 from equation&nbsp;3 eliminates $T$ and gives:</p>\n<p>$\\var{num23}L-\\var{num22}L=\\var{num21*num23}-\\var{wages2}$</p>\n<p>which we can solve for $L$ as follows:</p>\n<p>$\\var{num23-num22}L=\\var{-wages2+num21*num23}$</p>\n<p>$L=\\var{precround((-wages2+num21*num23)/(num23-num22),0)}$</p>\n<p></p>\n<p>Then substituting this value of $L$ into Equation 2 gives:</p>\n<p>$T+\\var{precround((-wages2+num21*num23)/(num23-num22),0)} = \\var{num21}$</p>\n<p>$T = \\var{num21-precround((-wages2+num21*num23)/(num23-num22),0)}$</p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>", "functions": {}, "statement": "<p>Calculate the following, to the nearest cent!</p>", "preamble": {"js": "", "css": ""}, "name": "Simon's copy of Q5 (Calculating wages using algebraic equations)", "extensions": [], "parts": [{"showCorrectAnswer": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "gaps": [{"mustBeReduced": false, "type": "numberentry", "maxValue": "{ans42}", "variableReplacementStrategy": "originalfirst", "marks": 1, "scripts": {}, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "minValue": "{ans42}", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "showPrecisionHint": false, "unitTests": [], "precision": "2", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "strictPrecision": false}, {"mustBeReduced": false, "type": "numberentry", "maxValue": "{ans41}", "variableReplacementStrategy": "originalfirst", "marks": 1, "scripts": {}, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "minValue": "{ans41}", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "showPrecisionHint": false, "unitTests": [], "precision": "2", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "strictPrecision": false}], "steps": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "unitTests": [], "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "prompt": "<p>Create two simultaneous equations to describe the information given in the question.</p>", "type": "information"}], "type": "gapfill", "customMarkingAlgorithm": "", "prompt": "<p>$\\var{num41}$ tradesmen and&nbsp;$\\var{num42}$ labourers earn&nbsp;$&euro;\\var{wages41}$ per week while 1 tradesman and&nbsp;$\\var{num43}$ labourers earn $&euro;\\var{wages42}$ per week. Find&nbsp;the earnings for 1 tradesman and 1 labourer.</p>\n<p>Tradesmen: &nbsp;&euro;[[0]]</p>\n<p>Labourer: &nbsp; &nbsp; &euro;[[1]]</p>", "unitTests": [], "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": [], "stepsPenalty": 0}, {"showCorrectAnswer": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "gaps": [{"mustBeReduced": false, "type": "numberentry", "maxValue": "{ans11}", "variableReplacementStrategy": "originalfirst", "marks": 1, "scripts": {}, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "minValue": "{ans11}", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "showPrecisionHint": false, "unitTests": [], "precision": "2", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "strictPrecision": false}, {"mustBeReduced": false, "type": "numberentry", "maxValue": "{ans12}", "variableReplacementStrategy": "originalfirst", "marks": 1, "scripts": {}, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "minValue": "{ans12}", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "showPrecisionHint": false, "unitTests": [], "precision": "2", "customMarkingAlgorithm": "", "mustBeReducedPC": 0, "extendBaseMarkingAlgorithm": true, "precisionPartialCredit": 0, "correctAnswerFraction": false, "showFeedbackIcon": true, "strictPrecision": false}], "type": "gapfill", "customMarkingAlgorithm": "", "prompt": "<p>$\\var{num11}$ tradesmen and&nbsp;$\\var{num12}$ labourers earn&nbsp;$&euro;\\var{wages1}$ between them to do a job. If a tradesman earns&nbsp;$&euro;\\var{num13}$ more than a labourer, calculate the earnings for a tradesman and a labourer.</p>\n<p>Tradesmen: &nbsp;&euro;[[0]]</p>\n<p>Labourer: &nbsp; &nbsp; &euro;[[1]]</p>", "unitTests": [], "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": []}, {"showCorrectAnswer": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "gaps": [{"showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "minValue": "{ans21}", "type": "numberentry", "customMarkingAlgorithm": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{ans21}", "marks": 1, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "allowFractions": false, "showFeedbackIcon": true, "variableReplacements": [], "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "minValue": "{ans22}", "type": "numberentry", "customMarkingAlgorithm": "", "mustBeReduced": false, "unitTests": [], "maxValue": "{ans22}", "marks": 1, "mustBeReducedPC": 0, "correctAnswerFraction": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "allowFractions": false, "showFeedbackIcon": true, "variableReplacements": [], "correctAnswerStyle": "plain"}], "type": "gapfill", "customMarkingAlgorithm": "", "prompt": "<p>$\\var{num21}$ workmen on a building site earn a total of&nbsp;$&euro;\\var{wages2}$ between them per week. Labourers are paid&nbsp;$&euro;\\var{num22}$ per week and Tradesmen are paid&nbsp;$&euro;\\var{num23}$ per week. How many of each is employed?</p>\n<p>[[0]] &nbsp;Labourers</p>\n<p>[[1]] Tradesmen</p>", "unitTests": [], "marks": 0, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacements": []}], "ungrouped_variables": ["num11", "num12", "wages1", "num13", "ans11", "ans12", "ans21", "ans22", "num21", "num22", "num23", "wages2", "num31", "num32", "wages3", "num33", "ans31", "ans32", "num41", "num42", "wages41", "num43", "wages42", "ans41", "ans42"], "variable_groups": [], "variables": {"wages41": {"name": "wages41", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(9000..11000)"}, "ans12": {"name": "ans12", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(wages1-(num11*num13))/(num11+num12)"}, "num13": {"name": "num13", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(71..89 except[80])"}, "num31": {"name": "num31", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(4..11 except [num11,num21])"}, "num21": {"name": "num21", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "ans21+ans22"}, "ans22": {"name": "ans22", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(3..12 except ans21)"}, "wages42": {"name": "wages42", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2405..2795#5)"}, "wages3": {"name": "wages3", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(7005..8995#5)"}, "num43": {"name": "num43", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(3..4)"}, "num22": {"name": "num22", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(500..600#5)"}, "num41": {"name": "num41", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(5..7)"}, "num12": {"name": "num12", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(4..7 except num11)"}, "ans21": {"name": "ans21", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(5..9)"}, "ans11": {"name": "ans11", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "ans12+num13"}, "ans31": {"name": "ans31", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "ans32+num33"}, "ans42": {"name": "ans42", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "wages42-(num43*ans41)"}, "num32": {"name": "num32", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(4..11 except [num11,num21,num31])"}, "ans32": {"name": "ans32", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(wages3-(num31*num33))/(num31+num32)"}, "wages2": {"name": "wages2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(ans21*num22)+(ans22*num23)"}, "num33": {"name": "num33", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(66..84 except[70,75,80])"}, "wages1": {"name": "wages1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(8000..9000)"}, "num42": {"name": "num42", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(7..9 except num41)"}, "num23": {"name": "num23", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(num22+140..800#5)"}, "ans41": {"name": "ans41", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(wages41-(wages42*num41))/(num42 -(num41*num43))"}, "num11": {"name": "num11", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(7..9)"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Calculating wages using algebraic equations</p>\n<p>rebelmaths</p>"}, "rulesets": {}, "type": "question", "contributors": [{"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "TEAME CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/591/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}