This question tests the student's ability to identify equivalent fractions through spotting a fraction which is not equivalent amongst a list of otherwise equivalent fractions. It also tests the students ability to convert mixed numbers into their equivalent improper fractions. It then does the reverse and tests their ability to convert an improper fraction into an equivalent mixed number.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "name": "Converting between Mixed Numbers and Improper Fractions", "advice": "A mixed number is a number consisting of an integer and a proper fraction, i.e. a number in the form $ a \\displaystyle \\frac{b}{c}$ where $a$ is an integer and $\\displaystyle\\frac{b}{c}$ is a proper fraction: $b$ is smaller than $c$.
\nAn improper fraction is a fraction where the numerator is larger than the denominator, i.e. a number of the form $\\displaystyle\\frac{d}{e}$ where the numerator, $d$, is greater than the denominator, $e$.
\nTo convert a mixed number into an improper fraction, multiply the integer part of the mixed number, $a$, by the denominator, $c$.
\nThe numerator of the improper fraction will be equal to this added to what was already on the numerator of the proper fraction.
\nThe denominator of the proper fraction will stay the same when it converts to an improper fraction to give a final answer of
\n$\\displaystyle\\frac{({a}\\times{c})+b}{c}$.
\ni)
\n\\[
{\\var{f}\\frac{\\var{g_coprime}}{\\var{h_coprime}}} = \\frac{({\\var{f}}\\times{\\var{h_coprime}})+{\\var{g_coprime}}}{{\\var{h_coprime}}}=\\simplify{{num}/{h_coprime}}\\text{.}
\\]
ii)
\n\\[
{\\var{l}\\frac{\\var{j_coprime}}{\\var{k_coprime}}} = \\frac{({\\var{l}}\\times{\\var{k_coprime}})+{\\var{j_coprime}}}{{\\var{k_coprime}}} =\\simplify{{num2}/{k_coprime}}\\text{.}
\\]
To convert an improper fraction into a mixed number, you have to think about how many times the denominator can go into the numerator, and whatever is left over becomes the new numerator of the proper fraction.
\ni) $\\displaystyle\\frac{\\var{s}}{\\var{t}}$:
\n$\\var{s}$ can be divided by $\\var{t}$ fully $\\var{rounds}$ {if(rounds=1,\"time\",\"times\")}, with $\\var{gap1}$ remaining.
\n$\\var{rounds}$ becomes the integer part of the mixed number and $\\var{gap1}$ becomes the new numerator of the proper fraction.
\n$\\displaystyle\\frac{\\var{s}}{\\var{t}} = \\var{rounds}\\frac{\\var{gap1}}{\\var{t}}$.
\nIt may be possible to simplify the proper fraction further by finding the highest common divisor of the numerator and denominator. In this question, this is $\\var{gcd_gap1t}$.
\nTherefore, it is not possible to simplify this further, and the final answer shall be
\nSimplifying by this value gives the final answer
\n$\\displaystyle\\var{rounds}\\frac{\\var{gap1_coprime}}{\\var{t_coprime}}$.
\nii) $\\displaystyle\\frac{\\var{x}}{\\var{y}}$:
\n$\\var{x}$ can be divided by $\\var{y}$ fully $\\var{roundx}$ {if(roundx=1,\"time\",\"times\")}, with $\\var{gap4}$ left over.
\n$\\var{roundx}$ becomes the integer part of the mixed number and $\\var{gap4}$ becomes the new numerator of the proper fraction.
\n$\\displaystyle\\frac{\\var{x}}{\\var{y}} = \\var{roundx}\\frac{\\var{gap4}}{\\var{y}}$.
\nIt may be possible to simplify the proper fraction further by finding the highest common divisor of the numerator and denominator. In this question, this is $\\var{gcd_gap4y}$.
\nTherefore, it is not possible to simplify this further, and the final answer shall be
\nSimplifying by this value gives the final answer
\n$\\displaystyle\\var{roundx}\\frac{\\var{gap4_coprime}}{\\var{y_coprime}}$.
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\ni)
\n$\\displaystyle{\\var{f}\\frac{\\var{g_coprime}}{\\var{h_coprime}}} =$
ii)
\n$\\displaystyle{\\var{l}\\frac{\\var{j_coprime}}{\\var{k_coprime}}} =$
Write these improper fractions as their mixed number equivalent.
\ni)
\n$\\displaystyle\\frac{\\var{s}}{\\var{t}} =$ [[0]]
ii)
\n$\\displaystyle\\frac{\\var{x}}{\\var{y}} =$ [[3]]