// Numbas version: finer_feedback_settings
{"name": " Numbers II: using fractions (simple operations)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "j", "m", "p", "r", "t", "v", "x", "z", "bb", "dd", "ff", "hh", "jj", "ll", "nn", "pp", "rr", "tt", "h", "k", "l", "n", "o", "q", "s", "u", "w", "y", "aa", "cc", "ee", "gg", "ii", "kk", "mm", "oo", "qq", "ss"], "name": " Numbers II: using fractions (simple operations)", "tags": [], "advice": "<p>When<strong> adding/subtracting</strong> fractions, you must first find a <em>common denominator</em> between the fractions. If they already have the same denominator then you only need to worry about adding/subtracting the numerators and dividing the result by the common denominator.</p>\n<p>For example:<br/>To find a common denominator of $\\frac{2}{5} + \\frac{7}{15}$, the most obvious would be $15$, because $5\\times3=15$. Therefore, you must multiply both sides of the fraction&nbsp;$\\frac{2}{5}$ by $3$ to obtain a new fraction $\\frac{6}{15}$. This is known as 'scaling up'.<br/>Now you can add the two fractions together (by adding the numerators) because they have the same denominator:<br/>$\\frac{6}{15}+\\frac{7}{15}=\\frac{13}{15}$.</p>\n<p>The same applies with&nbsp;subtraction as well as addition.</p>\n<p></p>\n<p>When <strong>multiplying</strong> fractions, you can simply multiply the two numerators and divide this by the multiplication of the two denominators.</p>\n<p>For example:<br/>$\\frac{a}{b}\\times\\frac{c}{d}$ = $\\frac{a\\times{c}}{b\\times{d}}$</p>\n<p></p>\n<p>When <strong>dividing</strong> fractions, you firstly need to reciprocate (flip) <em>one</em> of the fractions, then multiply the numerators and denominators as you would a normal multiplication question.</p>\n<p>For example:<br/>$\\frac{a}{b} \\div \\frac{c}{d}$ would be flipped to become&nbsp;$\\frac{a}{b} \\div \\frac{d}{c}$ and then treated as a normal multiplication question (as explained above).</p>", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>What is the answer to $\\frac{\\var{a}}{\\var{b}} \\times \\frac{\\var{c}}{\\var{b}}$?</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "(a*c)/(b*b)", "minValue": "(a*c)/(b*b)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "<p>Multiply the numerators</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "a*c", "minValue": "a*c", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>Multiply the denominators</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "b*b", "minValue": "b*b", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>Put into a fraction, with the new numerator over the new denominator</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "(a*c)/(b*b)", "minValue": "(a*c)/(b*b)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "scripts": {}, "marks": "3", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "<p>What is the answer to $\\frac{\\var{d}}{\\var{f}} \\div \\frac{\\var{g}}{\\var{f}}$?</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "d/g", "minValue": "d/g", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "<p>Flip the second fraction e.g. $a/b$ becomes $b/a$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "f/g", "minValue": "f/g", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>Multiply the fractions as you would with a normal multiplication question using the flipped fraction above as the new second fraction</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "(d*f)/(g*f)", "minValue": "(d*f)/(g*f)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "scripts": {}, "marks": "3", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "<p>What is the answer to $\\frac{\\var{h}}{\\var{j}} + \\frac{\\var{k}}{\\var{j}}$?</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "(h+k)/j", "minValue": "(h+k)/j", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "<p>Check to see if the denominators are the same. If they are, you only need to add the numerators together and leave the denominator as it is for the final answer.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}, {"prompt": "<p>Add the numerators</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "h+k", "minValue": "h+k", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>Write as a fraction over the similar denominator; cancel down if you can.</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "(h+k)/j", "minValue": "(h+k)/j", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "scripts": {}, "marks": "3", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "<p>What is the answer to $\\frac{\\var{l}}{\\var{m}} - \\frac{\\var{n}}{\\var{m}}$?</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "(l-n)/m", "minValue": "(l-n)/m", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "<p>Check to see if the denominators are the same. If they are, you only need to subtract the numerators and leave the denominator as it is.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}, {"prompt": "<p>Subtract the numerators</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "abs((l-n))", "minValue": "abs((l-n))", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"prompt": "<p>Create a fraction with the original denominator; cancel down if you can.</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "(abs((l-n)))/m", "minValue": "(abs((l-n)))/m", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "scripts": {}, "marks": "3", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": "1", "prompt": "<p>What is the answer to $\\frac{\\var{o}}{\\var{p}} \\times \\frac{\\var{q}}{\\var{r}}$?</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "(o*q)/(p*r)", "minValue": "(o*q)/(p*r)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "<p>Multiply the numerators to make the top of the fraction</p>\n<p>Multiply the denominators to make the bottom of the fraction</p>\n<p>Cancel down if you can</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": "3", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": "1", "prompt": "<p>What is the answer to $\\frac{\\var{s}}{\\var{t}} \\div \\frac{\\var{u}}{\\var{v}}$?</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "(s*v)/(u*t)", "minValue": "(s*v)/(u*t)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "<p>Flip the second fraction as you did previously for division</p>\n<p>Multiply through</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": "3", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": "1", "prompt": "<p>What is the answer to $\\frac{\\var{w}}{\\var{x}} + \\frac{\\var{y}}{\\var{z}}$?</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "((w*z)+(y*x))/(x*z)", "minValue": "((w*z)+(y*x))/(x*z)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "<p>Check to see if the denominators are the same</p>\n<p>If they are not - multiply each fraction up to equivalent fractions with <em>equal denominators</em></p>\n<p>Once they are equal add the numerators and put over the equal denominator</p>\n<p>Cancel down if needed</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": "3", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": "1", "prompt": "<p>What is the answer to $\\frac{\\var{aa}}{\\var{bb}} - \\frac{\\var{cc}}{\\var{dd}}$?</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "((aa*dd)-(cc*bb))/(bb*dd)", "minValue": "((aa*dd)-(cc*bb))/(bb*dd)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "<p>Check to see if the denominators are the same</p>\n<p>If they are not - multiply each fraction up to equivalent fractions with<em> equal denominators</em></p>\n<p>Once they are equal subtract&nbsp;the numerators and put over the equal denominator</p>\n<p>Cancel down if needed</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": "3", "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "<p>These are basic questions to help you practice adding, subtracting, multiplying and dividing fractions.</p>\n<p>You should be able to do these without a calculator.</p>\n<p>You can show steps to help you with the methods. &nbsp;If you decide to show a step with a marking penalty, your mark <strong>on that question only</strong> will be lowered, and will not go below 0.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"aa": {"definition": "random(1..bb-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "aa", "description": ""}, "ff": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "ff", "description": ""}, "cc": {"definition": "random(1..dd-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "cc", "description": ""}, "ii": {"definition": "random(1..jj-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "ii", "description": ""}, "pp": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "pp", "description": ""}, "nn": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "nn", "description": ""}, "p": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "ll": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "ll", "description": ""}, "ee": {"definition": "random(1..ff-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "ee", "description": ""}, "oo": {"definition": "random(1..pp-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "oo", "description": ""}, "bb": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "bb", "description": ""}, "dd": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "dd", "description": ""}, "hh": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "hh", "description": ""}, "a": {"definition": "random(1..b-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "jj": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "jj", "description": ""}, "d": {"definition": "random(1..f-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "qq": {"definition": "random(1..rr-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "qq", "description": ""}, "gg": {"definition": "random(1..hh-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "gg", "description": ""}, "c": {"definition": "random(1..b-a) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "rr": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "rr", "description": ""}, "g": {"definition": "random(1..f-d) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "kk": {"definition": "random(1..ll-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "kk", "description": ""}, "h": {"definition": "random(1..j-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}, "k": {"definition": "random(1..j-h) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "k", "description": ""}, "j": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "j", "description": ""}, "m": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "l": {"definition": "random(1..m-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "l", "description": ""}, "o": {"definition": "random(1..p-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "o", "description": ""}, "n": {"definition": "random(1..m-l) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "q": {"definition": "random(1..r-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "description": ""}, "ss": {"definition": "random(1..tt-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "ss", "description": ""}, "s": {"definition": "random(1..t-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "r": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "u": {"definition": "random(1..v-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "t": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "w": {"definition": "random(1..x-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "w", "description": ""}, "v": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "y": {"definition": "random(1..x-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "y", "description": ""}, "x": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "x", "description": ""}, "z": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "z", "description": ""}, "tt": {"definition": "random(2..20) //random denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "tt", "description": ""}, "mm": {"definition": "random(1..nn-1) //random numerator less than denominator", "templateType": "anything", "group": "Ungrouped variables", "name": "mm", "description": ""}}, "metadata": {"description": "<p>This is a set of questions designed to help you practice adding, subtracting, multiplying and dividing fractions.</p>\n<p>All of these can be done without a calculator.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Nasir Firoz Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/909/"}]}]}], "contributors": [{"name": "Nasir Firoz Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/909/"}]}