// Numbas version: exam_results_page_options
{"name": "Decimals to fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "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"}], "name": "Decimals to fractions", "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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their fraction equivalent.</p>\n<p><em>i)</em></p>\n<p>$\\var{a}=$<em>&nbsp;</em><fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>\n<p><em>ii)</em></p>\n<p><em></em>$\\var{b}=$&nbsp;<fraction> <numerator>[[2]]</numerator> <denominator>[[3]]</denominator> </fraction></p>\n<p><em>iii)</em></p>\n<p>$\\var{d}=$&nbsp;<fraction> <numerator>[[6]]</numerator> <denominator>[[7]]</denominator> </fraction></p>\n<p><em>iv)</em></p>\n<p><em></em>$0.\\dot{\\var{c}}=$&nbsp;<fraction> <numerator>[[4]]</numerator> <denominator>[[5]]</denominator></fraction></p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": "0.5", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "a*10", "allowFractions": false, "minValue": "a*10", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 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"allowFractions": false, "minValue": "4", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}, {"variableReplacementStrategy": "originalfirst", "marks": "0.5", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "cround", "allowFractions": false, "minValue": "cround", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}, {"variableReplacementStrategy": "originalfirst", "marks": "0.5", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "3", "allowFractions": false, "minValue": "3", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}, {"variableReplacementStrategy": "originalfirst", "marks": "0.5", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "d_coprime", "allowFractions": false, "minValue": "d_coprime", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}, {"variableReplacementStrategy": "originalfirst", "marks": "0.5", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "5", "allowFractions": false, "minValue": "5", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}], "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "gapfill", "marks": 0, "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "prompt": "<p>Convert this decimal to a fraction, giving your answer in its simplest form.&nbsp;</p>\n<p>$\\displaystyle\\var{f} = $&nbsp;<fraction> <numerator>[[0]]</numerator> <denominator>[[1]]</denominator> </fraction></p>\n<p><em></em></p>", "gaps": [{"variableReplacementStrategy": "originalfirst", "marks": "2", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "f3", "allowFractions": false, "minValue": "f3", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReducedPC": 0}, {"variableReplacementStrategy": "originalfirst", "marks": "2", "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "maxValue": "f4", "allowFractions": false, "minValue": "f4", "mustBeReduced": false, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, 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