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Written for the Western Sydney University MESH numeracy preparation workshop for the LANTITE test (Australia). Students are given a height in centimetres and another height in metres, and are asked to write the ratio of the two heights in simplest form. There are 16 versions of this question.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "A class conducted a science experiment to investigate the growth of plants. They planted twenty tomato seedlings, and measured their height once per week, until they were mature tomato plants.
\nThe average height of the seedlings on the day they were planted was $\\var{s_ht}$ cm, and the mature plants reached an average height of $\\var{m_ht}$ m.
", "advice": "Did you estimate what the answer should be and eliminate obviously wrong answers first?
\nWhen writing a ratio, the quantities must be in the same units, then the proportion should be fully simplified.
\nThe seedling height is given in centimetres while the mature height is given in metres.
\nThe larger unit should be converted to the smaller unit (to avoid decimals numbers), so convert the measurement in metres (average mature height) to centimetres.
\nThere are $100$ centimetres in $1$ metre, so we multiply $\\var{m_ht}$ by $100$.
\n$\\var{m_ht} \\text{ m} \\times 100 = \\var{m_ht_cm}$ cm.
\nSo the ratio of average seedling height to average mature height is $\\var{s_ht} : \\var{m_ht_cm}$. Note that we omit the units from the ratio.
\nHowever, this is not in simplest form. The largest number that divides exactly into both of these numbers is $\\var{gcd}$.
\nWe need to divide both of the measurements by $\\var{gcd}$, to simplify the ratio.
\n$\\var{s_ht} \\div \\var{gcd} = \\var{s_ratio}$
\n$\\var{m_ht_cm} \\div \\var{gcd} = \\var{m_ratio}$
\nThe ratio of average seedling height to average mature height is $\\var{s_ratio} : \\var{m_ratio}$.
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