// Numbas version: exam_results_page_options {"name": "judith's copy of Fractions/division and multiplication, different ways of presenting the same thing (non-algebraic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"variableReplacements": [], "choices": "{choices}", "minMarks": 0, "matrix": "marks", "marks": 0, "maxMarks": 0, "warningType": "none", "displayColumns": "1", "showFeedbackIcon": true, "shuffleChoices": true, "type": "m_n_2", "displayType": "checkbox", "showCorrectAnswer": true, "scripts": {}, "maxAnswers": 0, "variableReplacementStrategy": "originalfirst", "minAnswers": 0}], "extensions": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

Students seem to not realise that $\\frac{a}{b}\\times c=c\\times\\frac{a}{b}=\\frac{a\\times c}{b}=\\frac{c\\times a}{b}=a\\times c \\div b=a\\div b\\times c=c\\div b \\times a \\ne c \\div (b\\times a)\\ldots$ etc. This question is my attempt to help rectify this.

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Recall the following:

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• the bar in a fraction is really just division. For example, $\\frac{1}{2}=1\\div 2$.
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• the order of operations dictates that if a term contains multiplication and division, then you do the operations from left to right. For example, $4\\div 1 \\times 4 = (4\\div 1 )\\times 4=16$.
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• the order of multiplication is irrelevant. For example, $3\\times 4=4\\times 3$.
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• a whole number can be written as a fraction over 1. For example, $3=\\frac{3}{1}$.
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• division by a number is equivalent to multiplication by that numbers reciprocal. For example, $12\\div 3=12\\times \\frac{1}{3}$, and $3 \\div \\frac{2}{3}=3\\times \\frac{3}{2}$.
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• multiplication by a number is equivalent to division by that numbers reciprocal. For example, $10\\times \\frac{1}{5}=10\\div 5$.
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The above gives us (amoung other things) that

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $\\displaystyle \\var{c}\\times \\frac{\\var{a}}{\\var{b}}$ $=\\displaystyle\\var{c}\\times\\var{a}\\div\\var{b}$ $=\\displaystyle\\frac{\\var{a}}{\\var{b}}\\times\\var{c}$ $=\\displaystyle(\\var{a}\\div\\var{b})\\times\\var{c}$ $=\\displaystyle\\frac{\\var{a}}{\\var{b}}\\times\\frac{\\var{c}}{1}$ $=\\displaystyle\\frac{\\var{a}\\times\\var{c}}{\\var{b}}$ $=\\displaystyle\\frac{\\var{c}\\times\\var{a}}{\\var{b}}$ $=\\displaystyle\\frac{\\var{c}}{\\var{b}}\\times\\var{a}$ $=\\displaystyle\\var{c}\\div\\var{b}\\times\\var{a}$ $=\\displaystyle\\var{a}\\times\\frac{\\var{c}}{\\var{b}}$ $=\\displaystyle\\var{a}\\times\\var{c}\\div\\var{b}$ $=\\displaystyle\\var{a}\\times\\var{c}\\times\\frac{1}{\\var{b}}$
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", "functions": {}, "statement": "

Without the use of a calculator and without actually calculating the values of each answer, which of the following are equal to $\\displaystyle \\var{c}\\times \\frac{\\var{a}}{\\var{b}}$?

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$\\frac{a}{b}\\times c=c\\times\\frac{a}{b}=\\frac{a\\times c}{b}=\\frac{c\\times a}{b}=a\\times c \\div b=a\\div b\\times c=c\\div b \\times a \\ne c \\div (b\\times a)\\ldots$ etc

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