// Numbas version: exam_results_page_options
{"name": "Calculating area and perimeter using simultaneous equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Calculating area and perimeter using simultaneous equations", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Algebra word problems using area and perimeter.</p>"}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "statement": "<p>Solve the following correct to $1$ decimal place:</p>", "advice": "<p>Have a look at this youtube video to understand simultaneous equations better:&nbsp;<a href=\"https://www.youtube.com/watch?v=Lwto-lQzmec\">Simultaneous Equations</a></p>\n<p><strong>Part 1:</strong></p>\n<p>Width is represented by the variable $x$ so if the width is reduced by $\\var{red1}$ then we can say the new width is $x-\\var{red1}$. So the formula for the area of the new rectangle is given by</p>\n<p>$\\var{lent1} \\times (x - \\var{red1}) = \\var{area1}cm$</p>\n<p>$x - \\var{red1} = \\displaystyle \\frac{\\var{area1}}{\\var{lent1}}$</p>\n<p>$x = \\displaystyle&nbsp;\\frac{\\var{area1}}{\\var{lent1}} +&nbsp;\\var{red1} = \\simplify{{area1/lent1+red1}}$</p>\n<p>The area of the original rectangle is given by $\\var{lent1}x$.</p>\n<p><strong>Part 2:</strong></p>\n<p>We have two equations:</p>\n<p>$(2 \\times length)+&nbsp;(2 \\times width) = \\var{per2}$</p>\n<p>and</p>\n<p>$length = \\var{ratio2} \\times width$</p>\n<p>Substitute the value for the length from the second equation into the first equation and solve to find the width.</p>\n<p>Once you have the value for width, substitute this into the second equation to find the length.</p>\n<p><strong>Part 3:</strong></p>\n<p>Instead of being told that the length is so many times greater than the width, we've been given that it is a percentage greater than the width. But this percentage can be turned into a number. For example, if the length is $50$% greater than the width, then we could multiply the width by $1.5$ to find the length.</p>\n<p>Call the width $w$ and the length $l$.</p>\n<p>For any rectangle, perimeter = $2w + 2l$.</p>\n<p>We also know that $l=\\var{dec}w$. Substitute this value for $l$ into the equation for perimeter and solve for $w$.</p>\n<p></p>\n<p></p>\n<p></p>", "variables": {"per2": {"name": "per2", "description": "", "group": "Ungrouped variables", "definition": "random(32..80#4)", "templateType": "anything"}, "ans11": {"name": "ans11", "description": "", "group": "Ungrouped variables", "definition": "(area1+(red1*lent1))/lent1", "templateType": "anything"}, "ans43": {"name": "ans43", "description": "", "group": "Ungrouped variables", "definition": "ans41*ans42", "templateType": "anything"}, "red1": {"name": "red1", "description": "", "group": "Ungrouped variables", "definition": "random(2..10)", "templateType": "anything"}, "lent1": {"name": "lent1", "description": "", "group": "Ungrouped variables", "definition": "random(15..35)", "templateType": "anything"}, "ans23": {"name": "ans23", "description": "", "group": "Ungrouped variables", "definition": "ans21*ans22", "templateType": "anything"}, "lent3": {"name": "lent3", "description": "", "group": "Ungrouped variables", "definition": "random(15..35#5)", "templateType": "anything"}, "ans31": {"name": "ans31", "description": "", "group": "Ungrouped variables", "definition": "ans32*(1+(lent3/100))", "templateType": "anything"}, "ans41": {"name": "ans41", "description": "", "group": "Ungrouped variables", "definition": "(per4/2)/(1+ratio4)", "templateType": "anything"}, "ans12": {"name": "ans12", "description": "", "group": "Ungrouped variables", "definition": "ans11*lent1", "templateType": "anything"}, "ans21": {"name": "ans21", "description": "", "group": "Ungrouped variables", "definition": "(per2/2)/(1+ratio2)", "templateType": "anything"}, "ratio4": {"name": "ratio4", "description": "", "group": "Ungrouped variables", "definition": "random(1.25..4.75#0.5)", "templateType": "anything"}, "ans22": {"name": "ans22", "description": "", "group": "Ungrouped variables", "definition": "ratio2*ans21", "templateType": "anything"}, "per3": {"name": "per3", "description": "", "group": "Ungrouped variables", "definition": "random(1.2..3.6#0.2)", "templateType": "anything"}, "dec": {"name": "dec", "description": "", "group": "Ungrouped variables", "definition": "lent3/100+1", "templateType": "anything"}, "per4": {"name": "per4", "description": "", "group": "Ungrouped variables", "definition": "random(12.2..24.8#0.2 except[13..24])", "templateType": "anything"}, "area1": {"name": "area1", "description": "", "group": "Ungrouped variables", "definition": "random(402..495#0.5)", "templateType": "anything"}, "ans32": {"name": "ans32", "description": "", "group": "Ungrouped variables", "definition": "(per3/2)/(2+(lent3/100))", "templateType": "anything"}, "ratio2": {"name": "ratio2", "description": "", "group": "Ungrouped variables", "definition": "random(1.5..4.5#0.5)", "templateType": "anything"}, "ans42": {"name": "ans42", "description": "", "group": "Ungrouped variables", "definition": "ratio4*ans41", "templateType": "anything"}}, "extensions": [], "ungrouped_variables": ["lent1", "red1", "area1", "ans11", "ans12", "per2", "ratio2", "ans21", "ans22", "ans23", "lent3", "per3", "ans31", "ans32", "ratio4", "per4", "ans41", "ans42", "ans43", "dec"], "tags": [], "functions": {}, "rulesets": {"std": []}, "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"showCorrectAnswer": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "precisionType": "dp", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precision": "1", "maxValue": "{ans11}", "strictPrecision": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "{ans11}", "showFeedbackIcon": true, "precisionPartialCredit": 0}, {"variableReplacementStrategy": "originalfirst", "precisionType": "dp", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precision": "1", "maxValue": "{ans12}", "strictPrecision": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "{ans12}", "showFeedbackIcon": true, "precisionPartialCredit": 0}], "type": "gapfill", "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "type": "information", "marks": 0, "variableReplacements": [], "prompt": "<p>First sketch the original rectangle with the information given. Next draw the second rectangle with the new width. How can you represent this new width using $x$? Now use the information given to find $x$.</p>", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}}], "prompt": "<p>A rectangle has a length of $\\var{lent1} cm$ and a width of&nbsp;$x cm$. If the width was reduced by&nbsp;$\\var{red1} cm$ but the length remained the same, the area would be&nbsp;$\\var{area1} cm^2$. Find the original width and area of this rectangle.</p>\n<p>Width: [[0]]$cm$</p>\n<p>Area: [[1]]$cm^2$</p>", "stepsPenalty": 0, "showFeedbackIcon": true, "scripts": {}}, {"showCorrectAnswer": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "precisionType": "dp", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precision": "1", "maxValue": "{ans21}", "strictPrecision": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "{ans21}", "showFeedbackIcon": true, "precisionPartialCredit": "0"}, {"variableReplacementStrategy": "originalfirst", "precisionType": "dp", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precision": "1", "maxValue": "{ans22}", "strictPrecision": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "{ans22}", "showFeedbackIcon": true, "precisionPartialCredit": 0}, {"variableReplacementStrategy": "originalfirst", "precisionType": "dp", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precision": "1", "maxValue": "{ans23}", "strictPrecision": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "{ans23}", "showFeedbackIcon": true, "precisionPartialCredit": 0}], "type": "gapfill", "marks": 0, "variableReplacements": [], "prompt": "<p>Find the width, length and area of a rectangular room which has a perimeter of $\\var{per2} m$, if its length is $\\var{ratio2}$ times its width.</p>\n<p>Width:&nbsp;[[0]]$m$</p>\n<p>Length: [[1]]$m$</p>\n<p>Area:&nbsp;[[2]]$m^2$</p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}}, {"showCorrectAnswer": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "precisionType": "dp", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precision": "1", "maxValue": "{ans31}", "strictPrecision": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "{ans31}", "showFeedbackIcon": true, "precisionPartialCredit": 0}, {"variableReplacementStrategy": "originalfirst", "precisionType": "dp", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "precision": "1", "maxValue": "{ans32}", "strictPrecision": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "showPrecisionHint": false, "minValue": "{ans32}", "showFeedbackIcon": true, "precisionPartialCredit": 0}], "type": "gapfill", "marks": 0, "variableReplacements": [], "prompt": "<p>The&nbsp;length of a rectangle is $\\var{lent3}$% greater than its width. If the perimeter of the rectangle&nbsp;is $\\var{per3} m$, find the length and width.</p>\n<p>Length: [[0]]$m$</p>\n<p>Width: [[1]]$m$</p>\n<p></p>", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}}], "type": "question", "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}]}]}], "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}]}