// Numbas version: finer_feedback_settings
{"name": "Fraction multiplication", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<h4>a)</h4>\n<p>To multiply $\\displaystyle\\frac{\\var{a_coprime}}{\\var{c_coprime}}\\times\\frac{\\var{b_coprime}}{\\var{d_coprime}}$, address the numerators and denominators separately.</p>\n<p>Multiply the numerators across both fractions.</p>\n<p style=\"text-align: center;\">$\\var{a_coprime}\\times\\var{b_coprime}=\\var{ab}$,</p>\n<p>and then multiply the denominators across both fractions.</p>\n<p style=\"text-align: center;\">$\\var{c_coprime}\\times\\var{d_coprime}=\\var{cd}$.</p>\n<p>The values of the multiplied numerators and denominators will be the numerator and denominator of the new fraction: $\\displaystyle\\frac{\\var{ab}}{\\var{cd}}$.</p>\n<p>This answer may need simplifying down, and to do this, find the greatest common divisor in both the numerator and denominator and divide by this number.</p>\n<p>The greatest common divisor of $\\var{ab}$ and $\\var{cd}$ is $\\var{gcd}$.</p>\n<p>By using $\\var{gcd}$ to cancel down the fraction, the final answer is $\\displaystyle\\simplify{{ab}/{cd}}$.</p>\n<p></p>\n<h4>b)</h4>\n<p>To multiply $\\displaystyle\\simplify{{k_coprime}/{j_coprime}}\\times\\var{f}\\frac{\\var{g_coprime}}{\\var{h_coprime}}$, we first need to change the mixed number term into an improper fraction.&nbsp;</p>\n<p>To do this, we need to multiply $(\\var{f}\\times\\var{h_coprime}=\\var{fh})$ and add it to what was already on the numerator of the fraction, $\\var{g_coprime}$.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{(\\var{fh}+\\var{g_coprime})}{\\var{h_coprime}}= \\displaystyle\\frac{\\var{numif}}{\\var{h_coprime}}$.</p>\n<p>Next, we multiply the numerators and denominators across both fractions separately, as done in part <strong>a)</strong>.&nbsp;</p>\n<p style=\"text-align: center;\">$\\var{k_coprime}\\times\\var{numif} = \\var{num}$,</p>\n<p style=\"text-align: center;\">$\\var{j_coprime}\\times\\var{h_coprime}=\\var{denom}$.</p>\n<p>This gives the unsimplified version of the new fraction $\\displaystyle\\frac{\\var{num}}{\\var{denom}}$.</p>\n<p>To simplify, find the greatest common divisor in both the numerator and denominator and divide by this number.&nbsp;</p>\n<p>The greatest common divisor of $\\var{num}$ and $\\var{denom}$ is $\\var{gcdb}$.</p>\n<p>By using $\\var{gcdb}$ to cancel down the fraction, the final answer is $\\displaystyle\\simplify{{num}/{denom}}$.</p>\n<p></p>\n<p></p>\n<h4>c)</h4>\n<p>To square a fraction means to multiply the fraction by itself. To do this, multiply the numerators and denominators across individually.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\bigg(\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\bigg)^2=\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\times\\frac{\\var{l_coprime}}{\\var{m_coprime}}=\\frac{\\var{l_coprime^2}}{\\var{m_coprime^2}}.$</p>\n<p>From this, we should look if it is possible to simplify by finding the highest common divisor of $\\var{l_coprime^2}$ and $\\var{m_coprime^2}.$</p>\n<p>The greatest common divisor is $\\var{gcd_lcmc}$.</p>\n<p><span data-jme-visible=\"gcd_lcmc = 1\">Therefore, it is not possible to simplify this further, and the final answer is</span></p>\n<p><span data-jme-visible=\"gcd_lcmc &gt; 1\">By simplifying with this value, the final answer is</span></p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{l_coprime2}}{\\var{m_coprime2}}$.</p>\n<p></p>\n<h4 style=\"text-align: left;\">d)</h4>\n<p style=\"text-align: left;\">Helen was on holiday for $28$ days and spent $\\displaystyle\\frac{\\var{aa}}{7}$ of her time in Spain.&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{aa}}{7}\\times\\frac{28}{1}=\\frac{\\var{bb}}{7}=\\var{cc}$ days in Spain.&nbsp;</p>\n<p style=\"text-align: left;\">Whilst in Spain, she spends $\\displaystyle\\frac{\\var{dd}}{4}$ of her time in Barcelona.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{dd}}{4}\\times\\frac{\\var{cc}}{1}=\\frac{\\var{ddcc}}{4}=\\var{ee}$ days in Barcelona.&nbsp;</p>\n<h4></h4>", "statement": "<p>Evaluate the following multiplications, giving each fraction in its simplest form.</p>", "variables": {"k": {"name": "k", "group": "Part b", "templateType": "anything", "description": "<p>Random number between 1 and 20</p>", "definition": "random(1..7 except j)"}, "bb": {"name": "bb", "group": "Part d", "templateType": "anything", "description": "", "definition": "28*aa"}, "cc": {"name": "cc", "group": "Part d", "templateType": "anything", "description": "", "definition": "bb/7"}, "g": {"name": "g", "group": "Part b", "templateType": "randrange", "description": "<p>Random number between 1 and 20.</p>", "definition": "random(1..7#1)"}, "cd": {"name": "cd", "group": "Part a", "templateType": "anything", "description": "<p>Variable c times variable d.</p>", "definition": "c_coprime*d_coprime"}, "a": {"name": "a", "group": "Part a", "templateType": "anything", "description": "<p>Random number from 1 to 12.</p>", "definition": "random(2..12 except c)"}, "d": {"name": "d", "group": "Part a", "templateType": "randrange", "description": "<p>Random number from 1 to 12.</p>", "definition": "random(2..12#1)"}, "l": {"name": "l", "group": "Part c", "templateType": "anything", "description": "", "definition": "random(1..12)"}, "numif": {"name": "numif", "group": "Part b", "templateType": "anything", "description": "<p>Numerator of the improper fraction converted from a mixed number.</p>", "definition": "(f*h_coprime)+g_coprime"}, "gcd_gh": {"name": "gcd_gh", "group": "Part b", "templateType": "anything", "description": "", "definition": "gcd(g,h)"}, "fh": {"name": "fh", "group": "Part b", "templateType": "anything", "description": "<p>Variable f times variable h</p>", "definition": "f*h_coprime"}, "g_coprime": {"name": "g_coprime", "group": "Part b", "templateType": "anything", "description": "", "definition": "g/gcd_gh"}, "j_coprime": {"name": "j_coprime", "group": "Part b", "templateType": "anything", "description": "", "definition": "j/gcd_kj"}, "gcd_kj": {"name": "gcd_kj", "group": "Part b", "templateType": "anything", "description": "", "definition": "gcd(k,j)"}, "f": {"name": "f", "group": "Part b", "templateType": "randrange", "description": "<p>Random number between 1 and 4 - integer part of the mixed number.</p>", "definition": "random(1..4#1)"}, "c_coprime": {"name": "c_coprime", "group": "Part a", "templateType": "anything", "description": "", "definition": "c/gcd_ac"}, "gcd": {"name": "gcd", "group": "Part a", "templateType": "anything", "description": "", "definition": "gcd(ab,cd)"}, "b": {"name": "b", "group": "Part a", "templateType": "randrange", "description": "<p>Random number from 1 to 12.</p>", "definition": "random(2..12#1)"}, "d_coprime": {"name": "d_coprime", "group": "Part a", "templateType": "anything", "description": "", "definition": "d/gcd_bd"}, "ddcc": {"name": "ddcc", "group": "Part d", "templateType": "anything", "description": "", "definition": "dd*cc"}, "gcdb": {"name": "gcdb", "group": "Part b", "templateType": "anything", "description": "", "definition": "gcd(num,denom)"}, "gcd_ac": {"name": "gcd_ac", "group": "Part a", "templateType": "anything", "description": "<p>PART A</p>", "definition": "gcd(a,c)"}, "denom": {"name": "denom", "group": "Part b", "templateType": "anything", "description": "<p>Denominator of new fraction.</p>", "definition": "j_coprime*(h_coprime/gcda)"}, "l_coprime": {"name": "l_coprime", "group": "Part c", "templateType": "anything", "description": "", "definition": "l/gcd_lm"}, "m": {"name": "m", "group": "Part c", "templateType": "anything", "description": "", "definition": "random(1..12 except l)"}, "a_coprime": {"name": "a_coprime", "group": "Part a", "templateType": "anything", "description": "", "definition": "a/gcd_ac"}, "h": {"name": "h", "group": "Part b", "templateType": "randrange", "description": "<p>Random number between 1 and 20.</p>", "definition": "random(7..10#1)"}, "num": {"name": "num", "group": "Part b", "templateType": "anything", "description": "<p>Numerator of gap 0</p>", "definition": "k_coprime*{numif/gcda}"}, "m_coprime": {"name": "m_coprime", "group": "Part c", "templateType": "anything", "description": "", "definition": "m/gcd_lm"}, "aa": {"name": "aa", "group": "Part d", "templateType": "anything", "description": "", "definition": "random(1..6)"}, "gcda": {"name": "gcda", "group": "Part b", "templateType": "anything", "description": "<p>gcd of the numerator of the improper fraction</p>", "definition": "gcd({numif},{h_coprime})"}, "h_coprime": {"name": "h_coprime", "group": "Part b", "templateType": "anything", "description": "", "definition": "h/gcd_gh"}, "ee": {"name": "ee", "group": "Part d", "templateType": "anything", "description": "", "definition": "ddcc/4"}, "c": {"name": "c", "group": "Part a", "templateType": "anything", "description": "<p>Random number from 1 to 12.</p>", "definition": "random(3,5,7,11)"}, "b_coprime": {"name": "b_coprime", "group": "Part a", "templateType": "anything", "description": "", "definition": "b/gcd_bd"}, "l_coprime2": {"name": "l_coprime2", "group": "Part c", "templateType": "anything", "description": "", "definition": "l_coprime^2/gcd_lcmc"}, "k_coprime": {"name": "k_coprime", "group": "Part b", "templateType": "anything", "description": "", "definition": "k/gcd_kj"}, "j": {"name": "j", "group": "Part b", "templateType": "anything", "description": "<p>Random number between 1 and 20</p>", "definition": "Random(3,5,7,11,13,17)"}, "dd": {"name": "dd", "group": "Part d", "templateType": "anything", "description": "", "definition": "random(1..3)"}, "gcd_lcmc": {"name": "gcd_lcmc", "group": "Part c", "templateType": "anything", "description": "", "definition": "gcd((l_coprime)^2,(m_coprime)^2)"}, "m_coprime2": {"name": "m_coprime2", "group": "Part c", "templateType": "anything", "description": "", "definition": "m_coprime^2/gcd_lcmc"}, "gcd_lm": {"name": "gcd_lm", "group": "Part c", "templateType": "anything", "description": "", "definition": "gcd(l,m)"}, "ab": {"name": "ab", "group": "Part a", "templateType": "anything", "description": "<p>Variable a times variable b</p>", "definition": "a_coprime*b_coprime"}, "gcd_bd": {"name": "gcd_bd", "group": "Part a", "templateType": "anything", "description": "", "definition": "gcd(b,d)"}, "gcd2": {"name": "gcd2", "group": "Part b", "templateType": "anything", "description": "", "definition": "gcd(num,denom)"}}, "tags": ["improper fractions", "mixed numbers", "multiplication of fractions", "multiplying fractions", "squared fraction", "taxonomy"], "ungrouped_variables": [], "functions": {}, "name": "Fraction multiplication", "preamble": {"js": "", "css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n"}, "extensions": [], "type": "question", "variable_groups": [{"variables": ["a", "b", "c", "d", "a_coprime", "b_coprime", "c_coprime", "d_coprime", "gcd_ac", "gcd_bd", "ab", "cd", "gcd"], "name": "Part a"}, {"variables": ["f", "g", "g_coprime", "h", "h_coprime", "gcd_gh", "k", "k_coprime", "j", "j_coprime", "gcd_kj", "fh", "numif", "num", "denom", "gcda", "gcdb", "gcd2"], "name": "Part b"}, {"variables": ["aa", "bb", "cc", "dd", "ddcc", "ee"], "name": "Part d"}, {"variables": ["l", "m", "gcd_lm", "l_coprime", "m_coprime", "gcd_lcmc", "l_coprime2", "m_coprime2"], "name": "Part c"}], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Several problems involving the multiplication of fractions, with increasingly difficult examples, including&nbsp;a mixed fraction and a squared fraction. The final part is a word problem.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{ab}/{gcd}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{ab}/{gcd}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "{cd}/{gcd}", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "{cd}/{gcd}", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "<p>$\\displaystyle\\frac{\\var{a_coprime}}{\\var{c_coprime}}\\times\\frac{\\var{b_coprime}}{\\var{d_coprime}}$ =&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "num/gcd2", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "num/gcd2", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "denom/gcd2", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "denom/gcd2", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "<p>$\\displaystyle\\simplify{{k}/{j}}\\times\\var{f}\\frac{\\var{g}}{\\var{h}}$ =&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "l_coprime2", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "l_coprime2", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "m_coprime2", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "m_coprime2", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "<p>$\\displaystyle\\bigg(\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\bigg)^2= $ <fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "ee", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "scripts": {}, "maxValue": "ee", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "showFeedbackIcon": true}], "type": "gapfill", "prompt": "<p>Helen went on holiday in Europe. She spent $\\displaystyle\\frac{\\var{aa}}{7}$ of her time on holiday in Spain. Whilst in Spain, she spent $\\displaystyle\\frac{\\var{dd}}{4}$ of her time in Barcelona.&nbsp;</p>\n<p>If her holiday lasted for $28$ days, how many days was she in Barcelona?&nbsp;</p>\n<p>Helen was in Barcelona for&nbsp;[[0]] days.</p>", "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}]}