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{"name": "Fractions/division and multiplication, different ways of presenting the same thing (non-algebraic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Fractions/division and multiplication, different ways of presenting the same thing (non-algebraic)", "tags": [], "metadata": {"description": "<p>Students seem to not realise that $\\frac{a}{b}\\times c=c\\times\\frac{a}{b}=\\frac{a\\times c}{b}=\\frac{c\\times a}{b}=a\\times c \\div b=a\\div b\\times c=c\\div b \\times a \\ne c&nbsp;\\div (b\\times a)\\ldots $ etc. This question is my attempt to help rectify this.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Without the use of a calculator and without actually calculating the values of each answer, which of the following are equal to $\\displaystyle \\var{c}\\times \\frac{\\var{a}}{\\var{b}}$?</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "primes[0]", "description": "", "templateType": "anything", "can_override": false}, "markss": {"name": "markss", "group": "Ungrouped variables", "definition": "[1,1,1,1,1,1,1,1,1,1,-1,-1,-1,-1,-1,1]", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "primes[2]", "description": "", "templateType": "anything", "can_override": false}, "primes": {"name": "primes", "group": "Ungrouped variables", "definition": "shuffle([2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131])", "description": "", "templateType": "anything", "can_override": false}, "choices": {"name": "choices", "group": "Ungrouped variables", "definition": "[\n  '$\\\\displaystyle\\\\frac{\\\\var{a}}{\\\\var{b}}\\\\times\\\\var{c}$',\n    '$\\\\displaystyle\\\\frac{\\\\var{c}}{\\\\var{b}}\\\\times\\\\var{a}$',\n  '$\\\\displaystyle\\\\var{a}\\\\times\\\\frac{\\\\var{c}}{\\\\var{b}}$',\n  '$\\\\displaystyle\\\\frac{\\\\var{c}\\\\times\\\\var{a}}{\\\\var{b}}$',\n  '$\\\\displaystyle\\\\frac{\\\\var{a}\\\\times\\\\var{c}}{\\\\var{b}}$',\n  '$\\\\displaystyle\\\\var{a}\\\\times\\\\var{c}\\\\div\\\\var{b}$',\n    '$\\\\displaystyle(\\\\var{a}\\\\div\\\\var{b})\\\\times\\\\var{c}$',\n  '$\\\\displaystyle\\\\var{c}\\\\div\\\\var{b}\\\\times\\\\var{a}$',\n   '$\\\\displaystyle\\\\frac{\\\\var{a}}{\\\\var{b}}\\\\times\\\\frac{\\\\var{c}}{1}$',\n   '$\\\\displaystyle\\\\var{a}\\\\times\\\\var{c}\\\\times\\\\frac{1}{\\\\var{b}}$',\n  \n  '$\\\\displaystyle\\\\frac{\\\\var{a}}{\\\\var{c}}\\\\times\\\\var{b}$',\n   '$\\\\displaystyle\\\\var{c}\\\\div(\\\\var{b}\\\\times\\\\var{a})$',\n    '$\\\\displaystyle\\\\frac{\\\\var{b}}{\\\\var{c}\\\\times\\\\var{a}}$',\n     '$\\\\displaystyle\\\\frac{\\\\var{a}}{\\\\var{b}}\\\\times\\\\frac{1}{\\\\var{c}}$',\n   '$\\\\displaystyle\\\\var{a}\\\\times\\\\var{c}\\\\div\\\\frac{1}{\\\\var{b}}$',\n  '$\\\\displaystyle\\\\frac{1}{\\\\var{b}}\\\\times\\\\var{a}\\\\times\\\\var{c}$'\n]", "description": "<p>$\\frac{a}{b}\\times c=c\\times\\frac{a}{b}=\\frac{a\\times c}{b}=\\frac{c\\times a}{b}=a\\times c \\div b=a\\div b\\times c=c\\div b \\times a \\ne c&nbsp;\\div (b\\times a)\\ldots $ etc</p>", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "primes[1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["primes", "a", "b", "c", "choices", "markss"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "stepsPenalty": "11", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Recall the following:</p>\n<ul>\n<li>the bar in a fraction is really just division. For example, $\\frac{1}{2}=1\\div 2$.</li>\n<li>the order of operations dictates that if a term contains multiplication and division, then you&nbsp;do the operations from left to right. For example, $4\\div 1&nbsp;\\times 4 = (4\\div 1 )\\times 4=16$.</li>\n<li>the order of multiplication is irrelevant. For example, $3\\times 4=4\\times 3$.</li>\n<li>a whole number can be written as a fraction over 1. For example, $3=\\frac{3}{1}$.</li>\n<li>division&nbsp;by a number is equivalent to multiplication by that numbers reciprocal. For example, $12\\div 3=12\\times \\frac{1}{3}$, and $3 \\div \\frac{2}{3}=3\\times \\frac{3}{2}$.</li>\n<li>multiplication by a number is equivalent to division by that numbers reciprocal. For example, $10\\times \\frac{1}{5}=10\\div 5$.</li>\n</ul>\n<p></p>\n<p>The above gives us (amoung other things)&nbsp;that&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td>$\\displaystyle \\var{c}\\times \\frac{\\var{a}}{\\var{b}}$</td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{c}\\times\\var{a}\\div\\var{b}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{a}}{\\var{b}}\\times\\var{c}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle(\\var{a}\\div\\var{b})\\times\\var{c}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{a}}{\\var{b}}\\times\\frac{\\var{c}}{1}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{a}\\times\\var{c}}{\\var{b}}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{c}\\times\\var{a}}{\\var{b}}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\frac{\\var{c}}{\\var{b}}\\times\\var{a}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{c}\\div\\var{b}\\times\\var{a}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{a}\\times\\frac{\\var{c}}{\\var{b}}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{a}\\times\\var{c}\\div\\var{b}$</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: left;\">$=\\displaystyle\\var{a}\\times\\var{c}\\times\\frac{1}{\\var{b}}$</td>\n</tr>\n</tbody>\n</table>"}], "minMarks": 0, "maxMarks": "0", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": "2", "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": "{choices}", "matrix": "markss"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}