// Numbas version: finer_feedback_settings
{"percentPass": "0", "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Addition and subtraction of fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "type": "question", "statement": "<p>Evaluate the following additions and subtractions, giving each fraction in its simplest form.</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"d_coprime": {"description": "", "name": "d_coprime", "group": "Part a", "templateType": "anything", "definition": "d/gcd_cd"}, "hlcm2_j": {"description": "<p>PART B</p>", "name": "hlcm2_j", "group": "Part b", "templateType": "anything", "definition": "h_coprime*lcm2_j"}, "lcm2": {"description": "<p>PART B</p>", "name": "lcm2", "group": "Part b", "templateType": "anything", "definition": "lcm(g_coprime,j_coprime)"}, "denom": {"description": "<p>PART A answer for the denominator of part a</p>", "name": "denom", "group": "Part a", "templateType": "anything", "definition": "lcm/gcd"}, "term3": {"description": "", "name": "term3", "group": "Part c", "templateType": "anything", "definition": "gcd3/o_coprime"}, "a": {"description": "<p>PART A variable a - random number between 1 and 5.</p>", "name": "a", "group": "Part a", "templateType": "anything", "definition": "random(1..5)"}, "d": {"description": "<p>PART A variable d - random number between 5 and 15.</p>", "name": "d", "group": "Part a", "templateType": "anything", "definition": "random(5..15)"}, "gcd_fg": {"description": "<p>PART B gcd of first fraction num and denom</p>", "name": "gcd_fg", "group": "Part b", "templateType": "anything", "definition": "gcd(f,g)"}, "j": {"description": "<p>PART B</p>", "name": "j", "group": "Part b", "templateType": "anything", "definition": "random(2..10 except h)"}, "gcd_lm": {"description": "", "name": "gcd_lm", "group": "Part c", "templateType": "anything", "definition": "gcd(l,m)"}, "c_coprimeb_coprime": {"description": "<p>PART A variable c times variable b</p>", "name": "c_coprimeb_coprime", "group": "Part a", "templateType": "anything", "definition": "c_coprime*b_coprime"}, "l": {"description": "", "name": "l", "group": "Part c", "templateType": "anything", "definition": "random(1..3)"}, "gcd_ab": {"description": "<p>PART A simplification of fractions in the question.</p>", "name": "gcd_ab", "group": "Part a", "templateType": "anything", "definition": "gcd(a,b)"}, "k_simp": {"description": "", "name": "k_simp", "group": "Part c", "templateType": "anything", "definition": "(100k)/(gcd_k100)"}, "o_coprime": {"description": "", "name": "o_coprime", "group": "Part c", "templateType": "anything", "definition": "o/gcd_no"}, "n_coprime": {"description": "", "name": "n_coprime", "group": "Part c", "templateType": "anything", "definition": "n/gcd_no"}, "gcd_k100": {"description": "", "name": "gcd_k100", "group": "Part c", "templateType": "anything", "definition": "gcd(100k,100)"}, "lcm_b": {"description": "<p>PART A lcm of b and d, divided by b</p>", "name": "lcm_b", "group": "Part a", "templateType": "anything", "definition": "lcm/b_coprime"}, "num": {"description": "<p>PART A answer for the numerator input</p>", "name": "num", "group": "Part a", "templateType": "anything", "definition": "alcmclcm/gcd"}, "b": {"description": "<p>PART A variable b - random number between 5 and 10 and not the same value as d.</p>", "name": "b", "group": "Part a", "templateType": "anything", "definition": "random(5..10 except d)"}, "h": {"description": "<p>PART B</p>", "name": "h", "group": "Part b", "templateType": "anything", "definition": "random(1..10)"}, "gcd_no": {"description": "", "name": "gcd_no", "group": "Part c", "templateType": "anything", "definition": "gcd(n,o)"}, "f": {"description": "<p>PART B</p>", "name": "f", "group": "Part b", "templateType": "anything", "definition": "random(1..10)"}, "gcd_numgcd3": {"description": "", "name": "gcd_numgcd3", "group": "Part c", "templateType": "anything", "definition": "gcd(num1,gcd3)"}, "simp": {"description": "", "name": "simp", "group": "Part c", "templateType": "anything", "definition": "(100)/(gcd_k100)"}, "a_coprimed_coprime": {"description": "<p>PART A variable a times variable d</p>", "name": "a_coprimed_coprime", "group": "Part a", "templateType": "anything", "definition": "a_coprime*d_coprime"}, "o": {"description": "", "name": "o", "group": "Part c", "templateType": "anything", "definition": "random(5..15 except m except n except 7 except 11 except 13)"}, "lcm_d": {"description": "<p>PART A lcm of b and d, divided by d</p>", "name": "lcm_d", "group": "Part a", "templateType": "anything", "definition": "lcm/d_coprime"}, "gcd2": {"description": "<p>PART B</p>", "name": "gcd2", "group": "Part b", "templateType": "anything", "definition": "gcd(num2unsim,lcm2)"}, "c": {"description": "<p>PART A variable c - random number between 1 and 5.</p>", "name": "c", "group": "Part a", "templateType": "anything", "definition": "random(1..5)"}, "m": {"description": "", "name": "m", "group": "Part c", "templateType": "anything", "definition": "random(5..12 except 7 except 11)"}, "clcm_d": {"description": "<p>PART A variable c times the lcm of b and d, divided by d</p>", "name": "clcm_d", "group": "Part a", "templateType": "anything", "definition": "c_coprime*lcm_d"}, "h_coprime": {"description": "<p>PART B</p>", "name": "h_coprime", "group": "Part b", "templateType": "anything", "definition": "h/gcd_hj"}, "twolcm2": {"description": "<p>PART B</p>", "name": "twolcm2", "group": "Part b", "templateType": "anything", "definition": "2*lcm2"}, "lcm2_j": {"description": "<p>PART B</p>", "name": "lcm2_j", "group": "Part b", "templateType": "anything", "definition": "lcm2/j_coprime"}, "k": {"description": "", "name": "k", "group": "Part c", "templateType": "anything", "definition": "random(0.01..0.9#0.01)"}, "gcd": {"description": "<p>PART A greatest common divisor of the variables alcmclcm and lcm</p>", "name": "gcd", "group": "Part a", "templateType": "anything", "definition": "gcd(alcmclcm,lcm)"}, "m_coprime": {"description": "", "name": "m_coprime", "group": "Part c", "templateType": "anything", "definition": "m/gcd(l,m)"}, "j_coprime": {"description": "<p>PART B</p>", "name": "j_coprime", "group": "Part b", "templateType": "anything", "definition": "j/gcd_hj"}, "flcmhlcm": {"description": "<p>PART B</p>", "name": "flcmhlcm", "group": "Part b", "templateType": "anything", "definition": "flcm2_g-hlcm2_j"}, "term1": {"description": "", "name": "term1", "group": "Part c", "templateType": "anything", "definition": "gcd3/simp"}, "alcm_b": {"description": "<p>PART A variable a times the lcm of b and d, divided by b</p>", "name": "alcm_b", "group": "Part a", "templateType": "anything", "definition": "a_coprime*lcm_b"}, "a_coprime": {"description": "<p>PART A</p>", "name": "a_coprime", "group": "Part a", "templateType": "anything", "definition": "a/gcd_ab"}, "b_coprime": {"description": "<p>PART A&nbsp;</p>", "name": "b_coprime", "group": "Part a", "templateType": "anything", "definition": "b/gcd_ab"}, "g_coprime": {"description": "<p>PART B g_coprime</p>", "name": "g_coprime", "group": "Part b", "templateType": "anything", "definition": "g/gcd_fg"}, "gcd_hj": {"description": "<p>PART B</p>", "name": "gcd_hj", "group": "Part b", "templateType": "anything", "definition": "gcd(h,j)"}, "gcd1": {"description": "", "name": "gcd1", "group": "Part c", "templateType": "anything", "definition": "lcm(simp,m_coprime)"}, "g": {"description": "<p>PART B</p>", "name": "g", "group": "Part b", "templateType": "anything", "definition": "random(2..10 except f except j)"}, "gcd_cd": {"description": "<p>PART A&nbsp;</p>", "name": "gcd_cd", "group": "Part a", "templateType": "anything", "definition": "gcd(c,d)"}, "lcm": {"description": "<p>PART A lowest common multiple of variable b_coprime and variable d_coprime.</p>", "name": "lcm", "group": "Part a", "templateType": "anything", "definition": "lcm(b_coprime,d_coprime)"}, "flcm2_g": {"description": "<p>PART B</p>", "name": "flcm2_g", "group": "Part b", "templateType": "anything", "definition": "f_coprime*lcm2_g"}, "lcm2_g": {"description": "<p>PART B</p>", "name": "lcm2_g", "group": "Part b", "templateType": "anything", "definition": "lcm2/g_coprime"}, "term2": {"description": "", "name": "term2", "group": "Part c", "templateType": "anything", "definition": "gcd3/m_coprime"}, "n": {"description": "", "name": "n", "group": "Part c", "templateType": "anything", "definition": "random(1..5)"}, "alcmclcm": {"description": "<p>PART A&nbsp;</p>", "name": "alcmclcm", "group": "Part a", "templateType": "anything", "definition": "alcm_b+clcm_d"}, "gcd3": {"description": "", "name": "gcd3", "group": "Part c", "templateType": "anything", "definition": "lcm(gcd1,o_coprime)"}, "f_coprime": {"description": "<p>PART B</p>", "name": "f_coprime", "group": "Part b", "templateType": "anything", "definition": "f/gcd_fg"}, "num1": {"description": "", "name": "num1", "group": "Part c", "templateType": "anything", "definition": "(k_simp*term1)+(l_coprime*term2)-(n_coprime*term3)"}, "c_coprime": {"description": "", "name": "c_coprime", "group": "Part a", "templateType": "anything", "definition": "c/gcd_cd"}, "num2unsim": {"description": "<p>PART B</p>", "name": "num2unsim", "group": "Part b", "templateType": "anything", "definition": "flcmhlcm+twolcm2"}, "l_coprime": {"description": "", "name": "l_coprime", "group": "Part c", "templateType": "anything", "definition": "l/gcd_lm"}}, "functions": {}, "tags": ["adding and subtracting fractions", "adding fractions", "converting between decimals and fractions", "converting integers to fractions", "fractions", "Fractions", "integers", "manipulation of fractions", "subtracting fractions", "taxonomy"], "variable_groups": [{"name": "Part a", "variables": ["a", "a_coprime", "b", "b_coprime", "gcd_ab", "c", "c_coprime", "d", "d_coprime", "gcd_cd", "lcm", "a_coprimed_coprime", "c_coprimeb_coprime", "lcm_b", "lcm_d", "alcm_b", "clcm_d", "alcmclcm", "gcd", "num", "denom"]}, {"name": "Part b", "variables": ["f", "f_coprime", "g", "g_coprime", "gcd_fg", "h", "h_coprime", "j", "j_coprime", "gcd_hj", "lcm2", "lcm2_g", "flcm2_g", "lcm2_j", "hlcm2_j", "flcmhlcm", "twolcm2", "num2unsim", "gcd2"]}, {"name": "Part c", "variables": ["k", "gcd_k100", "k_simp", "simp", "l", "l_coprime", "m", "m_coprime", "gcd_lm", "n", "n_coprime", "o", "o_coprime", "gcd_no", "gcd1", "gcd3", "term1", "term2", "term3", "num1", "gcd_numgcd3"]}], "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "num", "showFeedbackIcon": true, "minValue": "num", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "denom", "showFeedbackIcon": true, "minValue": "denom", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "<p>$\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}+\\frac{\\var{c_coprime}}{\\var{d_coprime}}=$ <fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "num2unsim/gcd2", "showFeedbackIcon": true, "minValue": "num2unsim/gcd2", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": "2", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "lcm2/gcd2", "showFeedbackIcon": true, "minValue": "lcm2/gcd2", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "<p>$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}-\\frac{\\var{h_coprime}}{\\var{j_coprime}}+2=$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "num1/gcd_numgcd3", "showFeedbackIcon": true, "minValue": "num1/gcd_numgcd3", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}, {"correctAnswerFraction": false, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "maxValue": "gcd3/gcd_numgcd3", "showFeedbackIcon": true, "minValue": "gcd3/gcd_numgcd3", "correctAnswerStyle": "plain", "allowFractions": false, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "marks": 1, "showCorrectAnswer": true}], "showFeedbackIcon": true, "prompt": "<p>$\\displaystyle \\var{k}+\\frac{\\var{l}}{\\var{m}}-\\frac{\\var{n}}{\\var{o}}=$ <fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction>.</p>", "marks": 0}], "ungrouped_variables": [], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Manipulate fractions in order to add and subtract them. The difficulty escalates through the inclusion of a whole integer&nbsp;and a decimal, which both need to be converted into a fraction before the addition/subtraction can take place.&nbsp;</p>"}, "preamble": {"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", "js": ""}, "advice": "<h4>a)</h4>\n<p>$\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}+\\frac{\\var{c_coprime}}{\\var{d_coprime}}$</p>\n<p>To add or subtract fractions, we need to have a common denominator on both fractions.</p>\n<p>To get a common denominator, we need to find the lowest common multiple of the two denominators.</p>\n<p>The lowest common multiple of $\\var{b_coprime}$ and $\\var{d_coprime}$ is $\\var{lcm}.$</p>\n<p>This will be the new denominator, and we need to multiply each fraction individually to ensure we get this denominator.&nbsp;</p>\n<p>For&nbsp;$\\displaystyle\\frac{\\var{a_coprime}}{\\var{b_coprime}}$, we need to multiply the fraction by $\\displaystyle\\frac{\\var{lcm_b}}{\\var{lcm_b}}$ to give $\\displaystyle\\frac{\\var{alcm_b}}{\\var{lcm}}.$</p>\n<p>For&nbsp;$\\displaystyle\\frac{\\var{c_coprime}}{\\var{d_coprime}}$, we need to multiply the fraction by&nbsp;$\\displaystyle\\frac{\\var{lcm_d}}{\\var{lcm_d}}$ to give $\\displaystyle\\frac{\\var{clcm_d}}{\\var{lcm}}.$</p>\n<p>Now that we have each fraction in terms of a common denominator, we can now add the fractions together.&nbsp;</p>\n<p>$\\displaystyle\\frac{\\var{alcm_b}}{\\var{lcm}}+\\frac{\\var{clcm_d}}{\\var{lcm}}=\\frac{(\\var{alcm_b}+\\var{clcm_d})}{\\var{lcm}}=\\frac{\\var{alcmclcm}}{\\var{lcm}}.$</p>\n<p>From this, we can try to simplify the result down by finding the greatest common divisor of the numerator and denominator and dividing the whole fraction by this amount.&nbsp;</p>\n<p>The greatest common divisor of $\\var{alcmclcm}$ and $\\var{lcm}$ is $\\var{gcd}.$</p>\n<p><span data-jme-visible=\"gcd &gt; 1\">Simplifying using this value gives a final answer of $\\displaystyle\\frac{\\var{num}}{\\var{denom}}.$</span></p>\n<p><span data-jme-visible=\"gcd = 1\">Therefore, the expression cannot be simplified further, and $\\displaystyle\\frac{\\var{num}}{\\var{denom}}$ is the final answer.</span></p>\n<p></p>\n<h4>b)</h4>\n<p>$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}-\\frac{\\var{h_coprime}}{\\var{j_coprime}}+2.$</p>\n<p></p>\n<p>The two fractions can be individually multiplied to achieve a common denominator of the lowest common multiple, $\\var{lcm2}.$</p>\n<p>$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}$ becomes $\\displaystyle\\frac{\\var{flcm2_g}}{\\var{lcm2}}$ and $\\displaystyle\\frac{\\var{h_coprime}}{\\var{j_coprime}}$ becomes&nbsp;$\\displaystyle\\frac{\\var{hlcm2_j}}{\\var{lcm2}}.$</p>\n<p>We can now subtract the second fraction from the first.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{flcm2_g}}{\\var{lcm2}}-\\frac{\\var{hlcm2_j}}{\\var{lcm2}}=\\frac{\\var{flcmhlcm}}{\\var{lcm2}}.$</p>\n<p>From this, the question asks us to add $2$. We need to change the mixed number, $2$, into an improper fraction.&nbsp;</p>\n<p style=\"text-align: center;\">$\\displaystyle2=2\\bigg(\\frac{\\var{lcm2}}{\\var{lcm2}}\\bigg)=\\frac{\\var{twolcm2}}{\\var{lcm2}}.$</p>\n<p style=\"text-align: left;\">We can now continue with the question.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{flcmhlcm}}{\\var{lcm2}}+\\frac{\\var{twolcm2}}{\\var{lcm2}}=\\frac{\\var{num2unsim}}{\\var{lcm2}}.$</p>\n<p>We can look to simplify by dividing by the greatest common divisor of $\\var{num2unsim}$ and $\\var{lcm2}$ which is $\\var{gcd2}.$</p>\n<p><span data-jme-visible=\"gcd2 &gt; 1\">Simplifying by this value gives&nbsp;the final answer $\\displaystyle\\simplify{{num2unsim}/{lcm2}}.$</span></p>\n<p><span data-jme-visible=\"gcd2 = 1\">Therefore, no further simplification is possible, and&nbsp;$\\displaystyle\\simplify{{num2unsim}/{lcm2}}$ is the final answer.</span></p>\n<p></p>\n<h4>c)</h4>\n<p>$\\displaystyle\\var{k}+\\frac{\\var{l_coprime}}{\\var{m_coprime}}-\\frac{\\var{n_coprime}}{\\var{o_coprime}}.$</p>\n<p>We need to convert the decimal into a fraction and to do this, we need to multiply it by $10$ for every decimal place.</p>\n<p style=\"text-align: center;\">$\\displaystyle\\frac{\\var{k}}{1}\\times\\frac{100}{100}=\\frac{\\var{100k}}{100}.$</p>\n<p>We should look to simplify by dividing by the greatest common divisor which is $\\var{gcd_k100}.$</p>\n<p><span data-jme-visible=\"gcd_k100 = 1\">Therefore, it is not possible to simplify any further, and the&nbsp;fraction stays as</span></p>\n<p><span data-jme-visible=\"gcd_k100 &gt; 1\">Simplifying by this value gives&nbsp;the fraction</span></p>\n<p style=\"text-align: left;\">\\[\\simplify{{{100k}}/{100}}\\text{.}\\]</p>\n<p style=\"text-align: left;\">The original&nbsp;expression is now $\\displaystyle\\frac{\\var{k_simp}}{\\var{simp}}+\\frac{\\var{l_coprime}}{\\var{m_coprime}}-\\frac{\\var{n_coprime}}{\\var{o_coprime}}.$</p>\n<p style=\"text-align: left;\">We can multiply each fraction individually to achieve the common denominator $\\var{gcd3}$.</p>\n<p style=\"text-align: left;\">\\[\\frac{\\var{k_simp}}{\\var{simp}}\\text{ becomes }\\frac{\\var{k_simp*term1}}{\\var{gcd3}}\\text{, }\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\text{ becomes }\\frac{\\var{l_coprime*term2}}{\\var{gcd3}}\\text{ and }\\frac{\\var{n_coprime}}{\\var{o_coprime}}\\text{&nbsp;becomes }\\frac{\\var{n_coprime*term3}}{\\var{gcd3}}\\text{.}\\]</p>\n<p style=\"text-align: left;\">We can&nbsp;now complete the addition.&nbsp;</p>\n<p style=\"text-align: left;\">\\[\\frac{\\var{k_simp*term1}}{\\var{gcd3}}+\\frac{\\var{l_coprime*term2}}{\\var{gcd3}}-\\frac{\\var{n_coprime*term3}}{\\var{gcd3}}=\\frac{\\var{(k_simp*term1)+(l_coprime*term2)-(n_coprime*term3)}}{\\var{gcd3}}\\text{.}\\]</p>\n<p style=\"text-align: left;\">We should look to simplify this fraction by dividing by the highest common divisor,&nbsp;$\\var{gcd_numgcd3}.$</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd_numgcd3 &gt; 1\">Simplifying by this value gives the final answer</span>&nbsp;</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd_numgcd3 = 1\">Therefore, it is not possible to simplify the fraction any further and the final answer is</span></p>\n<p style=\"text-align: left;\">\\[\\simplify{{num1}/{gcd3}}\\text{.}\\]</p>"}, {"name": "Fraction multiplication", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "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": {}, "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"}, "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"}]}, {"name": "Division of fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "metadata": {"description": "<p>Several problems involving dividing fractions, with increasingly difficult examples, including mixed numbers and complex fractions.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "type": "question", "ungrouped_variables": [], "advice": "<h4 style=\"text-align: left;\">a)</h4>\n<p>When faced with dividing fractions, it much easier to switch one of the fractions around and multiply them together instead of divide them.</p>\n<p style=\"text-align: center;\">\\[ \\left( \\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}} \\right) \\equiv \\left( \\frac{\\var{f_coprime}}{\\var{g_coprime}}\\times\\frac{\\var{j_coprime}}{\\var{h_coprime}} \\right) = \\frac{\\var{fj}}{\\var{gh}} \\]</p>\n<p>Then, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\\var{gcd1}$.&nbsp;</p>\n<p>This gives a final answer of $\\displaystyle\\simplify{{fj}/{gh}}$.</p>\n<p></p>\n<h4 style=\"text-align: left;\">b)</h4>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\div\\frac{\\var{h1_coprime}}{\\var{j1_coprime}} \\equiv \\left( \\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\times\\frac{\\var{j1_coprime}}{\\var{h1_coprime}} \\right)=\\frac{\\var{f1j1}}{\\var{g1h1}} \\]</p>\n<p>Then, simplify by finding the highest common divisor in the numerator and denominator which in this case is $\\var{gcd2}$.</p>\n<p>This gives a final answer of $\\displaystyle\\simplify{{f1j1}/{g1h1}}$.</p>\n<p></p>\n<h4>c)</h4>\n<p style=\"text-align: center;\">\\[ {\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\div{\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}} \\]</p>\n<p>The first thing to do is to change the mixed numbers into improper fractions.</p>\n<p>An improper fraction is a fraction where the numerator is greater than the denominator. To change a mixed fraction to an improper fraction, multiply the integer part of the mixed number by the denominator, and add it to the existing numerator to make the new numerator of the improper fraction. The denominator will stay the same.&nbsp;</p>\n<p style=\"text-align: center;\">\\[ {\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\equiv\\frac{(\\var{f3}\\times\\var{h3_coprime})+\\var{g3_coprime}}{\\var{h3_coprime}}=\\frac{\\var{f3h3}+\\var{g3_coprime}}{\\var{h3_coprime}}=\\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}} \\]</p>\n<p style=\"text-align: center;\">\\[ {\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}}\\equiv\\frac{(\\var{f4}\\times\\var{h4_coprime})+\\var{g4_coprime}}{\\var{h4_coprime}}=\\frac{\\var{f4h4}+\\var{g4_coprime}}{\\var{h4_coprime}}=\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}} \\]</p>\n<p>We now have our mixed numbers as improper fractions.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\div\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}} \\]</p>\n<p>Now, use the same method as in parts <strong>a)</strong> and <strong>b)</strong> to divide by switching around one fraction and changing the division symbol to multiplication.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\div\\frac{\\var{f4h4+g4_coprime}}{\\var{h4_coprime}}\\equiv\\frac{\\var{f3h3+g3_coprime}}{\\var{h3_coprime}}\\times\\frac{\\var{h4_coprime}}{\\var{f4h4+g4_coprime}}=\\frac{\\var{num}}{\\var{denom}} \\]</p>\n<p>Finally, the last thing to do is to simplify your answer down by finding the highest common divisor in the numerator and denominator, which in this case is $\\var{gcd3}$.</p>\n<p>By doing this, you will get a final answer of</p>\n<p style=\"text-align: center;\">\\[ \\simplify{{num}/{denom}} \\]</p>\n<h4>d)</h4>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{a}}{(\\simplify[all,!collectNumbers]{{b}-{c}/{d}})} \\]</p>\n<p style=\"text-align: left;\">Consider the denominator first, as following the rules of BODMAS, you should address brackets first.</p>\n<p style=\"text-align: left;\">You need to get a common denominator for both terms on the denominator, like this:</p>\n<p style=\"text-align: center;\">\\[ \\var{b}\\times\\frac{\\var{d}}{\\var{d}} = \\frac{\\var{bd}}{\\var{d}} \\]</p>\n<p style=\"text-align: left;\">This now allows you to complete the addition or subtraction as both terms have a common denominator.&nbsp;</p>\n<p style=\"text-align: center;\">\\[ {\\simplify[all,!collectNumbers]{{bd}/{d}-{c}/{d}}} = \\frac{\\var{bd_c}}{\\var{d}} \\]</p>\n<p style=\"text-align: left;\">This means that the expression is now:</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{a}}{\\frac{\\var{bd_c}}{\\var{d}}} \\]</p>\n<p style=\"text-align: left;\">Dealing with this requires a bit of manipulation. You have to change&nbsp;the divisor of the denominator to be a mulitplier of the numerator. The denominator, ${\\var{bd_c}}$ was being divided by ${\\var{d}}$ but by flipping it around, the numerator, ${\\var{a}}$ will be mulitplied by&nbsp;${\\var{d}}$. The value of the expression remains the same.</p>\n<p style=\"text-align: center;\">\\[ \\frac{\\var{a}}{\\frac{\\var{bd_c}}{\\var{d}}}\\equiv \\frac{(\\var{a})\\times(\\var{d})}{\\var{bd_c}}= \\frac{\\var{ad}}{\\var{bd_c}} \\]</p>\n<p style=\"text-align: left;\">From this, you can try to cancel the expression down by finding the highest common factor of the numerator and denominator, to give a final answer of</p>\n<p style=\"text-align: center;\">\\[ \\simplify{{ad}/{bd_c}} \\]</p>", "variable_groups": [{"name": "part d", "variables": ["a", "b", "c", "d", "bd", "ad", "gcd", "ad_gcd", "bcd_gcd", "bd_c"]}, {"name": "part a", "variables": ["f", "g", "f_coprime", "g_coprime", "h", "j", "h_coprime", "j_coprime", "fj", "gh", "gcd1"]}, {"name": "part b", "variables": ["f1", "g1", "f1_coprime", "g1_coprime", "h1", "j1", "h1_coprime", "j1_coprime", "f1j1", "g1h1", "gcd2"]}, {"name": "part c", "variables": ["f3", "g3", "h3", "g3_coprime", "h3_coprime", "f4", "g4", "h4", "g4_coprime", "h4_coprime", "f3h3", "f4h4", "num", "denom", "gcd3"]}], "statement": "<p>Evaluate the following sums&nbsp;involving division of fractions. Simplify your answers where possible.&nbsp;</p>", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\displaystyle\\frac{\\var{f_coprime}}{\\var{g_coprime}}\\div\\frac{\\var{h_coprime}}{\\var{j_coprime}}=$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "fj/gcd1", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "fj/gcd1", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "gh/gcd1", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "gh/gcd1", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\displaystyle\\frac{\\var{f1_coprime}}{\\var{g1_coprime}}\\div\\frac{\\var{h1_coprime}}{\\var{j1_coprime}}=$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "f1j1/gcd2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "f1j1/gcd2", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "g1h1/gcd2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "g1h1/gcd2", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\displaystyle{\\var{f3}\\frac{\\var{g3_coprime}}{\\var{h3_coprime}}}\\div{\\var{f4}\\frac{\\var{g4_coprime}}{\\var{h4_coprime}}}=$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "num/gcd3", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "num/gcd3", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "denom/gcd3", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "denom/gcd3", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\displaystyle\\frac{\\var{a}}{(\\simplify[all,!collectNumbers]{{b}-{c}/{d}})} =$&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>", "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "ad_gcd", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "ad_gcd", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "showFeedbackIcon": true, "marks": 1, "minValue": "bcd_gcd", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "bcd_gcd", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}]}], "tags": ["dividing fractions", "division of fractions", "fractions", "Fractions", "mixed numbers", "taxonomy"], "preamble": {"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", "js": ""}, "functions": {}, "variables": {"f4h4": {"description": "<p>variable f4 times h4.</p>\n<p>Used in part c)</p>", "group": "part c", "definition": "f4*h4_coprime", "name": "f4h4", "templateType": "anything"}, "g4_coprime": {"description": "<p>PART C</p>", "group": "part c", "definition": "g4/gcd(g4,h4)", "name": "g4_coprime", "templateType": "anything"}, "h4": {"description": "<p>Random number but not the same number as variable g4.</p>\n<p>Used in part c.</p>", "group": "part c", "definition": "random(5..8 except g4)", "name": "h4", "templateType": "anything"}, "g": {"description": "<p>Random number between 2 and 10 and not the same number as variable f.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "random(f..12 except f) ", "name": "g", "templateType": "anything"}, "a": {"description": "<p>Random number between 1 and 20</p>\n<p>Used by part d)</p>", "group": "part d", "definition": "random(1..10#1)", "name": "a", "templateType": "randrange"}, "bd_c": {"description": "<p>Unsimplified denominator for part d).</p>", "group": "part d", "definition": "(bd-c)", "name": "bd_c", "templateType": "anything"}, "h3_coprime": {"description": "<p>PART C</p>", "group": "part c", "definition": "h3/gcd(g3,h3)", "name": "h3_coprime", "templateType": "anything"}, "f_coprime": {"description": "<p>PART A</p>", "group": "part a", "definition": "f/gcd(f,g)", "name": "f_coprime", "templateType": "anything"}, "g_coprime": {"description": "<p>PART A</p>", "group": "part a", "definition": "g/gcd(f,g)", "name": "g_coprime", "templateType": "anything"}, "j1_coprime": {"description": "<p>PART B</p>", "group": "part b", "definition": "j1/gcd(h1,j1)", "name": "j1_coprime", "templateType": "anything"}, "gcd2": {"description": "<p>greatest common divisor of variables f1j1 and g1h1.</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "gcd(f1j1,g1h1)", "name": "gcd2", "templateType": "anything"}, "c": {"description": "<p>Random prime number between -10 and 10.</p>\n<p>Used by part d).</p>", "group": "part d", "definition": "random([-7,-5,-3,-2,-1,1,2,3,5,7] except d)", "name": "c", "templateType": "anything"}, "ad_gcd": {"description": "<p>Correct answer for the numerator in part d)</p>", "group": "part d", "definition": "ad/gcd", "name": "ad_gcd", "templateType": "anything"}, "g1_coprime": {"description": "<p>PART B</p>", "group": "part b", "definition": "g1/gcd(f1,g1)", "name": "g1_coprime", "templateType": "anything"}, "h1_coprime": {"description": "<p>PART B</p>", "group": "part b", "definition": "h1/gcd(h1,j1)", "name": "h1_coprime", "templateType": "anything"}, "gcd3": {"description": "<p>greatest common denominator for part c.&nbsp;</p>", "group": "part c", "definition": "gcd(num,denom)", "name": "gcd3", "templateType": "anything"}, "bd": {"description": "<p>Variable b times variable d.</p>\n<p>Used in part d)</p>", "group": "part d", "definition": "b*d", "name": "bd", "templateType": "anything"}, "j1": {"description": "<p>Random number between 2 and 20 and not the same value as variable h1.</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "random(h1..11 except h1)", "name": "j1", "templateType": "anything"}, "g1h1": {"description": "<p>variable g1 times h1.&nbsp;</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "g1_coprime*h1_coprime", "name": "g1h1", "templateType": "anything"}, "f": {"description": "<p>Random number between 2 and 10.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "random(2..10)", "name": "f", "templateType": "anything"}, "b": {"description": "<p>Random number between 1 and 10.</p>\n<p>Used by part d)</p>", "group": "part d", "definition": "random(1..10#1)", "name": "b", "templateType": "randrange"}, "bcd_gcd": {"description": "<p>Correct answer for the denominator in part d).</p>", "group": "part d", "definition": "{bd_c}/gcd", "name": "bcd_gcd", "templateType": "anything"}, "f4": {"description": "<p>Random number.</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(1..3)", "name": "f4", "templateType": "anything"}, "f1": {"description": "<p>Random number between 2 and 20.</p>\n<p>Used in part b)</p>", "group": "part b", "definition": "random(2..10)", "name": "f1", "templateType": "anything"}, "d": {"description": "<p>Random prime number between 10 and 20.</p>\n<p>Used in part d).</p>", "group": "part d", "definition": "random(7,11,13,17)", "name": "d", "templateType": "anything"}, "g3": {"description": "<p>Random number.</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(1..3)", "name": "g3", "templateType": "anything"}, "f3h3": {"description": "<p>variable f3 times h3.</p>", "group": "part c", "definition": "f3*h3_coprime", "name": "f3h3", "templateType": "anything"}, "h": {"description": "<p>Random number from 2 to 10.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "random(2..10)", "name": "h", "templateType": "anything"}, "gh": {"description": "<p>variable g times variable h.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "g_coprime*h_coprime", "name": "gh", "templateType": "anything"}, "j_coprime": {"description": "<p>PART A</p>", "group": "part a", "definition": "j/gcd(h,j)", "name": "j_coprime", "templateType": "anything"}, "denom": {"description": "<p>Unsimplified denominator of part c.</p>", "group": "part c", "definition": "h3_coprime*(f4h4+g4_coprime)", "name": "denom", "templateType": "anything"}, "j": {"description": "<p>Random number between 2 and 10 and not the same value as h.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "random(h..12 except h)", "name": "j", "templateType": "anything"}, "f1j1": {"description": "<p>variable f1 times j1.</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "f1_coprime*j1_coprime", "name": "f1j1", "templateType": "anything"}, "h4_coprime": {"description": "<p>PART C</p>", "group": "part c", "definition": "h4/gcd(g4,h4)", "name": "h4_coprime", "templateType": "anything"}, "g1": {"description": "<p>Random number between 2 and 30 and not the same value as variable f1.</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "random(f1..11 except f1) ", "name": "g1", "templateType": "anything"}, "fj": {"description": "<p>variable f times variable j.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "f_coprime*j_coprime", "name": "fj", "templateType": "anything"}, "gcd": {"description": "<p>Greatest common divisor of ad and bd_c.&nbsp;</p>\n<p>Used in part d).&nbsp;</p>", "group": "part d", "definition": "gcd(ad,bd_c)", "name": "gcd", "templateType": "anything"}, "f3": {"description": "<p>Random number between 2 and 6.</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(1..3#1)", "name": "f3", "templateType": "randrange"}, "f1_coprime": {"description": "<p>PART B</p>", "group": "part b", "definition": "f1/gcd(f1,g1)", "name": "f1_coprime", "templateType": "anything"}, "h3": {"description": "<p>Random number and not the same value as variable g3.&nbsp;</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(5..8)", "name": "h3", "templateType": "anything"}, "gcd1": {"description": "<p>greatest common divisor of variable fj and gh.</p>\n<p>Used in part a).</p>", "group": "part a", "definition": "gcd(fj,gh)", "name": "gcd1", "templateType": "anything"}, "g3_coprime": {"description": "<p>PART C</p>", "group": "part c", "definition": "g3/gcd(g3,h3)", "name": "g3_coprime", "templateType": "anything"}, "h_coprime": {"description": "<p>PART A</p>", "group": "part a", "definition": "h/gcd(h,j)", "name": "h_coprime", "templateType": "anything"}, "g4": {"description": "<p>Random number.</p>\n<p>Used in part c).</p>", "group": "part c", "definition": "random(1..5)", "name": "g4", "templateType": "anything"}, "h1": {"description": "<p>Random number between 2 and 20.&nbsp;</p>\n<p>Used in part b).</p>", "group": "part b", "definition": "random(2..10)", "name": "h1", "templateType": "anything"}, "num": {"description": "<p>numerator of the improper fraction in part c. Unsimplified.&nbsp;</p>", "group": "part c", "definition": "h4_coprime*(f3h3+g3_coprime)", "name": "num", "templateType": "anything"}, "ad": {"description": "<p>Variable a times variable d.</p>\n<p>Used in part d).</p>", "group": "part d", "definition": "a*d", "name": "ad", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Select the fraction not equivalent to the others - small denominators", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "advice": "<p>To find the odd fraction out, reduce each fraction to lowest terms.</p>\n<p>\\begin{align}<br/>\\frac{\\var{mone}}{\\var{none}} &amp;= \\frac{\\var{m_coprime}}{\\var{n_coprime}}\\times\\frac{\\var{one}}{\\var{one}} \\\\[0.5em]<br/>\\frac{\\var{mtwo}}{\\var{ntwo}} &amp;= \\frac{\\var{m_coprime}}{\\var{n_coprime}}\\times\\frac{\\var{two}}{\\var{two}} \\\\[0.5em]<br/>\\frac{\\var{mthree}}{\\var{nthree}} &amp;= \\frac{\\var{m_coprime}}{\\var{n_coprime}}\\times\\frac{\\var{three}}{\\var{three}} \\\\[0.5em]<br/>\\frac{\\var{mfour}}{\\var{nfour}} &amp;= \\frac{\\var{m_coprime}}{\\var{n_coprime}}\\times\\frac{\\var{four}}{\\var{four}} \\\\[0.5em]<br/>\\frac{\\var{mfive}}{\\var{nfive}} &amp;= \\frac{\\var{mfive/gcd(mfive,nfive)}}{\\var{nfive/gcd(mfive,nfive)}} \\times \\frac{\\var{gcd(mfive,nfive)}}{\\var{gcd(mfive,nfive)}}<br/>\\end{align}</p>\n<p>All but one of these fractions are equivalent to&nbsp;$\\displaystyle\\frac{\\var{m_coprime}}{\\var{n_coprime}}$.</p>\n<p>The odd fraction out is $\\displaystyle\\frac{\\var{mfive}}{\\var{nfive}}$.</p>", "statement": "", "variables": {"five": {"name": "five", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..7 except one except two except three except four)"}, "m_coprime": {"name": "m_coprime", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "m/gcd_mn"}, "mthree": {"name": "mthree", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "m_coprime*three"}, "ntwo": {"name": "ntwo", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "n_coprime*two"}, "mfive": {"name": "mfive", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "m_coprime*five+random(-4..4 except 0)"}, "mtwo": {"name": "mtwo", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "m_coprime*two"}, "four": {"name": "four", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(5..20 except one except two except three)"}, "none": {"name": "none", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "n_coprime*one"}, "nthree": {"name": "nthree", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "n_coprime*three"}, "two": {"name": "two", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..15 except one)"}, "m": {"name": "m", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Random number between 1 and 10</p>", "definition": "random(1..7)"}, "nfive": {"name": "nfive", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "n_coprime*five"}, "n": {"name": "n", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(7..15 except m)"}, "one": {"name": "one", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..10)"}, "gcd_mn": {"name": "gcd_mn", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "gcd(m,n)"}, "n_coprime": {"name": "n_coprime", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "n/gcd_mn"}, "nfour": {"name": "nfour", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "n_coprime*four"}, "mone": {"name": "mone", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "m_coprime*one"}, "three": {"name": "three", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..6 except one except two)"}, "mfour": {"name": "mfour", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "m_coprime*four"}}, "tags": ["equivalent fractions", "small denominators", "taxonomy"], "ungrouped_variables": ["m", "n", "gcd_mn", "m_coprime", "n_coprime", "one", "mone", "none", "two", "mtwo", "ntwo", "three", "mthree", "nthree", "four", "mfour", "nfour", "five", "mfive", "nfive"], "functions": {}, "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"}, "type": "question", "variable_groups": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Given five fractions, identify&nbsp;the one which is not equivalent to the others by reducing to lowest terms.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "matrix": [0, 0, "0", 0, "1"], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", "", ""], "shuffleChoices": true, "marks": 0, "prompt": "<p>From the options below, select the&nbsp;fraction which is not equivalent to the others.</p>", "choices": ["<p>$\\displaystyle\\frac{\\var{mone}}{\\var{none}}$</p>", "<p>$\\displaystyle\\frac{\\var{mtwo}}{\\var{ntwo}}$</p>", "<p>$\\displaystyle\\frac{\\var{mthree}}{\\var{nthree}}$</p>", "<p>$\\displaystyle\\frac{\\var{mfour}}{\\var{nfour}}$</p>", "<p>$\\displaystyle\\frac{\\var{mfive}}{\\var{nfive}}$</p>"], "type": "1_n_2", "showFeedbackIcon": true, "minMarks": 0}]}, {"name": "Select the fraction not equivalent to the others - large denominators", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "advice": "<p>To find the odd fraction out, reduce each fraction to lowest terms.</p>\n<p>\\begin{align}<br/>\\frac{\\var{osix}}{\\var{psix}}=\\frac{\\var{o_coprime}}{\\var{p_coprime}}\\times\\frac{\\var{six}}{\\var{six}} \\\\[0.5em]<br/>\\frac{\\var{oseven}}{\\var{pseven}}=\\frac{\\var{o_coprime}}{\\var{p_coprime}}\\times\\frac{\\var{seven}}{\\var{seven}} \\\\[0.5em]<br/>\\frac{\\var{oeight}}{\\var{peight}}=\\displaystyle\\frac{\\var{o_coprime}}{\\var{p_coprime}}\\times\\frac{\\var{eight}}{\\var{eight}} \\\\[0.5em]<br/>\\frac{\\var{onine}}{\\var{pnine}}=\\frac{\\var{o_coprime}}{\\var{p_coprime}}\\times\\frac{\\var{nine}}{\\var{nine}} \\\\[0.5em]<br/>\\frac{\\var{oten}}{\\var{pten}}&nbsp;= \\frac{\\var{oten/gcd(oten,pten)}}{\\var{pten/gcd(oten,pten)}} \\times \\frac{\\var{gcd(oten,pten)}}{\\var{gcd(oten,pten)}}<br/>\\end{align}</p>\n<p>The odd fraction out is $\\displaystyle\\frac{\\var{oten}}{\\var{pten}}$.</p>\n<p></p>", "statement": "", "variables": {"five": {"name": "five", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..7 except one except two except three except four)"}, "o": {"name": "o", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..7 except p except m)"}, "six": {"name": "six", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..10 except one except two except three except four except five)"}, "oten": {"name": "oten", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*ten+random(-6..6 except 0)"}, "pten": {"name": "pten", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*ten+random(-2..2 except 0)"}, "m": {"name": "m", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>Random number between 1 and 10</p>", "definition": "random(1..7)"}, "pseven": {"name": "pseven", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*seven"}, "eight": {"name": "eight", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(6..15 except one except two except three except four except five except six except seven)"}, "nine": {"name": "nine", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(4..12 except one except two except three except four except five except six except seven except eight)"}, "three": {"name": "three", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..6 except one except two)"}, "psix": {"name": "psix", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*six"}, "ten": {"name": "ten", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..10 except one except two except three except four except five except six except seven except eight except nine)"}, "four": {"name": "four", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(5..20 except one except two except three)"}, "two": {"name": "two", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..15 except one)"}, "peight": {"name": "peight", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*eight"}, "o_coprime": {"name": "o_coprime", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o/gcd_op"}, "osix": {"name": "osix", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*six"}, "pnine": {"name": "pnine", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p_coprime*nine"}, "seven": {"name": "seven", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(1..6 except one except two except three except four except five except six)"}, "n": {"name": "n", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(7..15 except m)"}, "onine": {"name": "onine", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*nine"}, "one": {"name": "one", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(2..10)"}, "p_coprime": {"name": "p_coprime", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "p/gcd_op"}, "gcd_op": {"name": "gcd_op", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "gcd(o,p)"}, "oeight": {"name": "oeight", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*eight"}, "oseven": {"name": "oseven", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "o_coprime*seven"}, "p": {"name": "p", "group": "Ungrouped variables", "templateType": "anything", "description": "", "definition": "random(7..15 except n)"}}, "tags": ["equivalent fractions", "fractions", "Fractions", "taxonomy"], "ungrouped_variables": ["m", "n", "one", "two", "three", "four", "five", "o", "p", "gcd_op", "o_coprime", "p_coprime", "six", "osix", "psix", "seven", "oseven", "pseven", "eight", "oeight", "peight", "nine", "onine", "pnine", "ten", "oten", "pten"], "functions": {}, "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"}, "type": "question", "variable_groups": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "<p>Given five fractions, identify the odd fraction out. The denominators are&nbsp;mainly two or three digits long.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "prompt": "<p>From the options below, select the&nbsp;fraction which is not equivalent to the others.&nbsp;</p>", "matrix": [0, 0, 0, 0, "1"], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", "", ""], "shuffleChoices": true, "marks": 0, "type": "1_n_2", "choices": ["<p>$\\displaystyle\\frac{\\var{osix}}{\\var{psix}}$</p>", "<p>$\\displaystyle\\frac{\\var{oseven}}{\\var{pseven}}$</p>", "<p>$\\displaystyle\\frac{\\var{oeight}}{\\var{peight}}$</p>", "<p>$\\displaystyle\\frac{\\var{onine}}{\\var{pnine}}$</p>", "<p>$\\displaystyle\\frac{\\var{oten}}{\\var{pten}}$</p>"], "showFeedbackIcon": true, "minMarks": 0}]}, {"name": "Decimals to fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}], "variable_groups": [{"variables": ["a", "b", "c", "d", "b_", "b_coprime", "d_coprime", "answer", "cround"], "name": "Part a"}, {"variables": ["f", "f2", "mygcd", "h", "j", "k", "f3", "f4", "f1000", "numerator", "g", "gcd1", "numerator_coprime", "g_coprime"], "name": "Part b"}], "functions": {}, "rulesets": {}, "ungrouped_variables": [], "metadata": {"description": "<p>Identify well-known fractional equivalents of decimals.&nbsp;Convert obscure decimals and recurring decimals into fractions.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "advice": "<h4>a)</h4>\n<p>To convert a decimal into a fraction, firstly place the decimal over $1$ as a fraction, and then multiply both the numerator and denominator by $10$ for however many decimal places the decimal has. For example, if the decimal was $0.1$, you would multiply the fraction by $10$ as there is one decimal place. If the decimal was $0.01$, you would multiply it by $100$, as there are two decimal places. &nbsp;</p>\n<p><em>i)</em></p>\n<p>$\\var{a}$</p>\n<p>\\[<br/>\\frac{\\var{a}}{1}\\times\\frac{10}{10}=\\frac{\\var{10a}}{10}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\"><em>ii)</em></p>\n<p style=\"text-align: left;\">$\\var{b}$</p>\n<p style=\"text-align: left;\">\\[<br/>\\frac{\\var{b}}{1}\\times\\frac{100}{100}=\\frac{\\var{100b}}{100}=\\simplify{{100b}/{100}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\"><em>iii)</em></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">$\\var{d}$</p>\n<p style=\"text-align: left;\">\\[<br/>\\frac{\\var{d}}{1}\\times\\frac{10}{10}=\\frac{\\var{10d}}{10}=\\simplify{{10d}/{10}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\"><em>iv)</em></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">$0.\\dot{\\var{c}}$</p>\n<p style=\"text-align: left;\">To convert a recurring decimal to a fraction, the first step is to set up a simple equation where</p>\n<p style=\"text-align: left;\">\\[<br/>x=0.\\dot{\\var{c}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\">By multiplying both sides by $10$, we can gain another simple equation where</p>\n<p style=\"text-align: left;\">\\[<br/>10x=\\var{c}.\\dot{\\var{c}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\">By subtracting one equation from the other, we can find the fraction equivalent of the recurring decimal.&nbsp;</p>\n<p>\\[<br/>\\begin{align}<br/>&amp;&amp;\\var{c}.\\dot{\\var{c}}&amp;={10}x\\\\<br/>-&amp;&amp;{0.\\dot{\\var{c}}}&amp;=x\\\\<br/>&amp;&amp;\\overline{\\qquad} &amp;&nbsp;\\overline{\\qquad}\\\\<br/>&amp;&amp;{\\var{c}}&amp;=9x\\\\<br/>\\\\<br/>&amp;&amp;\\frac{\\var{c}}{9}&amp;=x<br/>\\end{align}<br/>\\]</p>\n<p style=\"text-align: left;\">$\\displaystyle\\frac{\\var{c}}{9}$ simplifies to $\\simplify{{{c}}/{9}}$ by dividing by $3$ and therefore, $0.\\dot{\\var{c}}=\\simplify{{{c}}/{9}}$ in its fractional form.}<br/><br/></p>\n<p style=\"text-align: center;\"></p>\n<h4>b)</h4>\n<p>$\\displaystyle\\var{f}$</p>\n<p style=\"text-align: left;\">\\[<br/>\\var{f}\\times\\frac{\\var{f1000}}{\\var{f1000}}=\\frac{\\var{f2}}{\\var{f1000}}\\text{.}<br/>\\]</p>\n<p style=\"text-align: left;\">From this, we can look to see if we can cancel down the fraction by finding the highest common divisor between the numerator and denominator. This is $\\var{mygcd}$.</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"mygcd = 1\">Therefore, it is not possible to simplify the answer any further and the final answer is</span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"mygcd &gt; 1\">Simplifying by this amount gives the final answer</span></p>\n<p style=\"text-align: left;\">\\[\\frac{\\var{f3}}{\\var{f4}}.\\]</p>\n<h4>c)</h4>\n<p style=\"text-align: left;\">$\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}.$</p>\n<p style=\"text-align: left;\">To convert a recurring decimal to a fraction, the first step is to set up a simple equation where,</p>\n<p style=\"text-align: center;\">$x=\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}.$</p>\n<p><br/>By multiplying both sides by $100$ to isolate the recurring section on the left hand side of the decimal point, we can gain another simple equation</p>\n<p style=\"text-align: center;\">$100x=\\var{h}\\var{j}\\var{k}.\\dot{\\var{j}}\\dot{\\var{k}}.$</p>\n<p></p>\n<p style=\"text-align: left;\">Now that we have two equations in terms of $x$, we can subtract one from the other and solve to get a value of $x$.</p>\n<p style=\"text-align: left;\">\\[<br/>\\begin{align}<br/>&amp;&amp;\\var{h}\\var{j}\\var{k}.\\dot{\\var{j}}\\dot{\\var{k}}&amp;=100x\\\\<br/>-&amp;&amp;\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}&amp;=x\\\\<br/>&amp;&amp;\\overline{\\qquad} &amp;&nbsp;\\overline{\\qquad}&nbsp;<br/>\\\\<br/>&amp;&amp;{{\\var{h}}\\var{j}\\var{k-h}}&amp;=99x\\\\<br/>\\\\<br/>&amp;&amp;\\frac{\\var{numerator}}{\\var{g}}&amp;=x\\text{.}\\\\<br/>\\end{align}<br/>\\]</p>\n<p style=\"text-align: left;\">From this, we should look to see if it is possible to simplify by finding the greatest common divisor of the numerator and the denominator. The greatest common divisor is $\\var{gcd1 }.$</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd1 = 1\">Therefore, it is not possible to simplify and&nbsp;so</span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd1 &gt; 1\">Simplifying by this value gives the fraction $\\displaystyle\\simplify{{{numerator}}/{g}}$ and so &nbsp;</span></p>\n<p style=\"text-align: left;\">\\[<br/>\\begin{align}<br/>\\var{h}.\\dot{\\var{j}}\\dot{\\var{k}}=\\simplify{{{numerator}}/{g}}\\text{ in its fractional form.}\\\\<br/>\\end{align}<br/>\\]</p>", "statement": "<p>Fractions can be equivalently represented as decimals and vice versa. One form may be more useful in a context than another and it is useful to practise how to change between them.</p>\n<p>Have a go at these questions involving fractions and decimals, remembering to write your answer in its simplest form.&nbsp;&nbsp;</p>", "preamble": {"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", 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{"templateType": "anything", "description": "", "name": "h", "group": "Part b", "definition": "random(1..5)"}, "b": {"templateType": "anything", "description": "", "name": "b", "group": "Part a", "definition": "random(0.25,0.75)"}, "f4": {"templateType": "anything", "description": "", "name": "f4", "group": "Part b", "definition": "f1000/gcd(f2,f1000)"}, "d_coprime": {"templateType": "anything", "description": "", "name": "d_coprime", "group": "Part a", "definition": "10d/gcd(10d,10)"}, "f1000": {"templateType": "anything", "description": "", "name": "f1000", "group": "Part b", "definition": "1000"}, "numerator_coprime": {"templateType": "anything", "description": "", "name": "numerator_coprime", "group": "Part b", "definition": "numerator/gcd1"}, "c": {"templateType": "anything", "description": "", "name": "c", "group": "Part a", "definition": "random(3,6)"}, "k": {"templateType": "anything", "description": "", "name": "k", "group": "Part b", "definition": "random(h..9 except j except 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