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a)

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To multiply $\\displaystyle\\frac{\\var{a_coprime}}{\\var{c_coprime}}\\times\\frac{\\var{b_coprime}}{\\var{d_coprime}}$, address the numerators and denominators separately.

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Multiply the numerators across both fractions.

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$\\var{a_coprime}\\times\\var{b_coprime}=\\var{ab}$,

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and then multiply the denominators across both fractions.

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$\\var{c_coprime}\\times\\var{d_coprime}=\\var{cd}$.

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The values of the multiplied numerators and denominators will be the numerator and denominator of the new fraction: $\\displaystyle\\frac{\\var{ab}}{\\var{cd}}$.

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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.

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The greatest common divisor of $\\var{ab}$ and $\\var{cd}$ is $\\var{gcd}$.

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By using $\\var{gcd}$ to cancel down the fraction, the final answer is $\\displaystyle\\simplify{{ab}/{cd}}$.

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b)

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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. 

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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, $\\var{g_coprime}$.

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$\\displaystyle\\frac{(\\var{fh}+\\var{g_coprime})}{\\var{h_coprime}}= \\displaystyle\\frac{\\var{numif}}{\\var{h_coprime}}$.

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Next, we multiply the numerators and denominators across both fractions separately, as done in part a)

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$\\var{k_coprime}\\times\\var{numif} = \\var{num}$,

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$\\var{j_coprime}\\times\\var{h_coprime}=\\var{denom}$.

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This gives the unsimplified version of the new fraction $\\displaystyle\\frac{\\var{num}}{\\var{denom}}$.

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To simplify, find the greatest common divisor in both the numerator and denominator and divide by this number. 

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The greatest common divisor of $\\var{num}$ and $\\var{denom}$ is $\\var{gcdb}$.

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By using $\\var{gcdb}$ to cancel down the fraction, the final answer is $\\displaystyle\\simplify{{num}/{denom}}$.

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c)

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To square a fraction means to multiply the fraction by itself. To do this, multiply the numerators and denominators across individually.

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$\\displaystyle\\bigg(\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\bigg)^2=\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\times\\frac{\\var{l_coprime}}{\\var{m_coprime}}=\\frac{\\var{l_coprime^2}}{\\var{m_coprime^2}}.$

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From this, we should look if it is possible to simplify by finding the highest common divisor of $\\var{l_coprime^2}$ and $\\var{m_coprime^2}.$

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The greatest common divisor is $\\var{gcd_lcmc}$.

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Therefore, it is not possible to simplify this further, and the final answer is

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By simplifying with this value, the final answer is

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$\\displaystyle\\frac{\\var{l_coprime2}}{\\var{m_coprime2}}$.

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d)

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Helen was on holiday for $28$ days and spent $\\displaystyle\\frac{\\var{aa}}{7}$ of her time in Spain. 

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$\\displaystyle\\frac{\\var{aa}}{7}\\times\\frac{28}{1}=\\frac{\\var{bb}}{7}=\\var{cc}$ days in Spain. 

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Whilst in Spain, she spends $\\displaystyle\\frac{\\var{dd}}{4}$ of her time in Barcelona.

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$\\displaystyle\\frac{\\var{dd}}{4}\\times\\frac{\\var{cc}}{1}=\\frac{\\var{ddcc}}{4}=\\var{ee}$ days in Barcelona. 

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", "statement": "

Evaluate the following multiplications, giving each fraction in its simplest form.

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Random number between 1 and 20

", "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": "

Random number between 1 and 20.

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Variable c times variable d.

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Random number from 1 to 12.

", "definition": "random(2..12 except c)"}, "d": {"name": "d", "group": "Part a", "templateType": "randrange", "description": "

Random number from 1 to 12.

", "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": "

Numerator of the improper fraction converted from a mixed number.

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Variable f times variable h

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Random number between 1 and 4 - integer part of the mixed number.

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Random number from 1 to 12.

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PART A

", "definition": "gcd(a,c)"}, "denom": {"name": "denom", "group": "Part b", "templateType": "anything", "description": "

Denominator of new fraction.

", "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": "

Random number between 1 and 20.

", "definition": "random(7..10#1)"}, "num": {"name": "num", "group": "Part b", "templateType": "anything", "description": "

Numerator of gap 0

", "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": "

gcd of the numerator of the improper fraction

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Random number from 1 to 12.

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Random number between 1 and 20

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Variable a times variable b

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Several problems involving the multiplication of fractions, with increasingly difficult examples, including a mixed fraction and a squared fraction. The final part is a word problem. 

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$\\displaystyle\\frac{\\var{a_coprime}}{\\var{c_coprime}}\\times\\frac{\\var{b_coprime}}{\\var{d_coprime}}$ =  [[0]] [[1]]

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$\\displaystyle\\simplify{{k}/{j}}\\times\\var{f}\\frac{\\var{g}}{\\var{h}}$ =  [[0]] [[1]]

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$\\displaystyle\\bigg(\\frac{\\var{l_coprime}}{\\var{m_coprime}}\\bigg)^2= $ [[0]] [[1]]

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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. 

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If her holiday lasted for $28$ days, how many days was she in Barcelona? 

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Helen was in Barcelona for [[0]] days.

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