// Numbas version: exam_results_page_options {"name": "Rates", "metadata": {"description": "

A quick practice set of problems for education students to take in preparation for their numeracy test.

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Calculating rates and scaling rates. Drops per mL and drops per minute questions unit rate to equivalent rate.

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Write the following question down on paper and evaluate it without using a calculator.

If you are unsure of how to do a question, click on Show steps to see the full working. Then, once you understand how to do the question, click on Try another question like this one to start again.

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A certain IV drip delivers $\\var{dpml}$ drops per mL. This is equivalent to [[0]] drops per $\\var{volume}$ mL.

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We can approach these questions like equivalent fractions by replacing the word 'per' with the operation of division.

The following are all equivalent ways of writing the same rate:

$\\var{dpml}$ drops per mL = $\\var{dpml}$ drops/mL = $\\var{dpml}$ drops / $1$ mL = $\\dfrac{\\var{dpml} \\text{ drops}}{1 \\text{ mL}}$.

We can multiply the top and bottom by any number other than zero and keep the rate the same. Since we are asked about $\\var{volume}$ mL, we multiply the top and bottom by $\\var{volume}$ so the bottom of the fraction is $\\var{volume}$.

$\\dfrac{\\var{dpml} \\text{ drops}}{1 \\text{ mL}}\\times\\dfrac{\\var{volume}}{\\var{volume}}=\\dfrac{\\var{ans1} \\text{ drops}}{\\var{volume}\\text{ mL}}$

In other words, the rate is equivalent to $\\var{ans1}$ drops per $\\var{volume}$ mL.

We can also approach these questions like ratios.

Think of $\\var{dpml}$ drops per mL as the ratio $\\var{dpml}$ drops : $1$ mL.

We can multiply both sides of this ratio by any non-zero number to get an equivalent ratio. Since we want to know about $\\var{volume}$ mL, we multiply both sides by $\\var{volume}$.

$\\var{dpml} \\text{ drops} \\,:\\, 1 \\text{ mL} = \\var{dpml}\\times \\var{volume}\\text{ drops}\\, :\\, 1 \\times \\var{volume} \\text{ mL} = \\var{ans1} \\text{ drops}\\,:\\, \\var{volume}\\text{ mL}$.

In other words, there are $\\var{ans1}$ drops in $\\var{volume}$ mL.

A patient requires $\\var{dpm}$ drops per minute from an IV. How many drops will they need over $\\var{duration}$ minutes?

[[0]] drops.

We can approach these questions like equivalent fractions by replacing the word 'per' with the operation of division.

The following are all equivalent ways of writing the same rate:

$\\var{dpm}$ drops per minute = $\\var{dpm}$ drops/min = $\\var{dpm}$ drops / $1$ min = $\\dfrac{\\var{dpm} \\text{ drops}}{1 \\text{ min}}$.

We can multiply the top and bottom by any number other than zero and keep the rate the same. Since we are asked about $\\var{duration}$ minutes, we multiply the top and bottom by $\\var{duration}$ so the bottom of the fraction is $\\var{duration}$.

$\\dfrac{\\var{dpm} \\text{ drops}}{1 \\text{ min}}\\times\\dfrac{\\var{duration}}{\\var{duration}}=\\dfrac{\\var{ans2} \\text{ drops}}{\\var{duration}\\text{ min}}$

In other words the rate is equivalent to $\\var{ans2}$ drops per $\\var{duration}$ minutes.

We can also approach these questions like ratios.

Think of $\\var{dpm}$ drops per minute as the ratio $\\var{dpm}$ drops : $1$ minute.

We can multiply both sides of this ratio by any non-zero number to get an equivalent ratio. Since we want to know about $\\var{duration}$ minutes, we multiply both sides by $\\var{duration}$.

$\\var{dpm} \\text{ drops} \\,:\\, 1 \\text{ min} = \\var{dpm}\\times \\var{duration}\\text{ drops}\\, :\\, 1 \\times \\var{duration} \\text{ min} = \\var{ans2} \\text{ drops}\\,:\\, \\var{duration}\\text{ min}$.

In other words there are $\\var{ans2}$ drops in $\\var{duration}$ minutes.

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Write the following question down on paper and evaluate it without using a calculator.

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If a patient received $\\var{drops}$ drops over $\\var{minutes}$ minutes, then the patient received [[0]] drops per minute.

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We can approach these questions like equivalent fractions by replacing the word 'per' with the operation of division.

The following are all equivalent ways of writing the same rate:

$\\var{drops}$ drops per $\\var{minutes}$ minutes $= \\var{drops}$ drops/ $\\var{minutes}$ min $= \\dfrac{\\var{drops} \\text{ drops}}{\\var{minutes} \\text{ min}}=\\dfrac{\\var{drops}}{\\var{minutes}} \\text{ drops/min}$.

So we just need to do the division to determine the rate per minute.

$\\dfrac{\\var{drops}}{\\var{minutes}} \\text{ drops/min}=\\var{ans3} \\text{ drops/min}$

In other words the rate is equivalent to $\\var{ans3}$ drops per minute.

We can also approach these questions like ratios.

Think of $\\var{drops}$ drops per $\\var{minutes}$ minutes as the ratio $\\var{drops}$ drops : $\\var{minutes}$ min.

We can multiply or divide both sides of this ratio by any non-zero number to get an equivalent ratio. Since we want to know about drops per minute, we divide both sides by $\\var{minutes}$.

$\\var{drops} \\text{ drops} \\,:\\, \\var{minutes} \\text{ min} = \\var{drops}\\div \\var{minutes}\\text{ drops}\\, :\\, \\var{minutes} \\div \\var{minutes} \\text{ min} = \\var{ans3} \\text{ drops}\\,:\\, 1\\text{ min}$.

In other words there are $\\var{ans3}$ drops per minute.

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If an IV drip delivers $\\var{capacity}$ mL through $\\var{numdrops}$ drops, then there must be [[0]] drops in each mL.

We can approach these questions like equivalent fractions by replacing the word 'per' with the operation of division.

The following are all equivalent ways of writing the same rate:

$\\var{numdrops}$ drops per $\\var{capacity}$ mL $= \\var{numdrops}$ drops/$\\var{capacity}$ mL $= \\dfrac{\\var{numdrops} \\text{ drops}}{\\var{capacity} \\text{ mL}}=\\dfrac{\\var{numdrops}}{\\var{capacity}} \\text{ drops/mL}$.

So we just need to do the division to determine the number of drops per mL.

$\\dfrac{\\var{numdrops}}{\\var{capacity}} \\text{ drops/mL}=\\var{ans4} \\text{ drops/mL}$

In other words the rate is equivalent to $\\var{ans4}$ drops per mL.

We can also approach these questions like ratios.

Think of $\\var{numdrops}$ drops per $\\var{capacity}$ mL as the ratio $\\var{numdrops}$ drops : $\\var{capacity}$ mL.

We can multiply or divide both sides of this ratio by any non-zero number to get an equivalent ratio. Since we want to know about drops per mL, we divide both sides by $\\var{capacity}$.

$\\var{numdrops} \\text{ drops} \\,:\\, \\var{capacity} \\text{ mL} = \\var{numdrops}\\div \\var{capacity}\\text{ drops}\\, :\\, \\var{capacity} \\div \\var{capacity} \\text{ mL} = \\var{ans4} \\text{ drops}\\,:\\, 1\\text{ mL}$.

In other words there are $\\var{ans4}$ drops per mL.

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Calculating rates and scaling rates. Drops per mL and drops per minute questions equivalent rate to unit rate.

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Unit rates

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plural, singular, object

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{person} {thing[0]} {number} {thing[2]} per {integer} {unit}s. This is equivalent to {thing[1]} [[0]] {thing[2]} per {unit}.

Note: If the answer has many decimal places leave your answer as a fraction (using / as the fraction bar) so that your answer is exact (and not an approximation/rounded-answer)

The word 'per' can be replaced with the operation of division.

These questions are very similar to equivalent fractions.

The following methods are equivalent but might appear different.

We are given $\\var{number}$ {thing[2]} / $\\var{integer}$ {unit}s. Divide both sides of the rate by $\\var{integer}$ (so that we are dealing with 'per {unit}' not 'per $\\var{integer}$ {unit}s'). This gives $\\var{number}/\\var{integer}$ {thing[2]}/{unit}, or in reduced form (by removing the common factor of $\\var{cf_a}$), $\\var[fractionNumbers]{number/integer}$ {thing[2]}/{unit}.
\\[\\var{number}\\,\\var{thing[2]}/\\var{integer}\\,\\var{unit}\\text{s}=\\frac{\\var{number}\\,\\var{thing[2]}}{\\var{integer}\\,\\var{unit}\\text{s}}=\\frac{\\var{number}}{\\var{integer}}\\frac{\\var{thing[2]}}{\\var{unit}\\text{s}}=\\var[fractionNumbers]{number/integer} \\,\\var{thing[2]}/\\var{unit}.\\]

A {vehicle} travels {distance} km per {amount} L of petrol.

How many kilometres can be travelled by using {niceamount} L? [[0]] km

How many litres of petrol are needed to travel {nicedistance} km? [[1]] L

Note: If the answer has many decimal places leave your answer as a fraction (using / as the fraction bar) so that your answer is exact (and not an approximation/rounded-answer)

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Just like a fraction, we can multiply or divide both sides of the rate by any number (except 0).

\"A {vehicle} travels $\\var{distance}$ km per $\\var{amount}$ L of petrol\". Note, this can be written as $\\var{distance}$ km / $\\var{amount}$ L.

To determine kilometres per $\\var{niceamount}$ L, first determine how many kilometres per $1$ L:

$\\var{distance} \\text{ km} / \\var{amount}\\text{ L} = \\frac{\\var{distance}}{\\var{amount}} \\text{ km/L}$ $= \\var[fractionNumbers]{distance/amount} \\text{ km/L}$

and then multiply both sides of the rate by $\\var{niceamount}$ to get per $\\var{niceamount}$ L:

$\\var{niceamount}\\times\\var[fractionNumbers]{distance/amount} \\text{ km}/\\var{niceamount} \\text{ L}=\\var[fractionNumbers]{niceamount*distance/amount} \\text{ km}/\\var{niceamount} \\text{ L}$.

To determine litres per $\\var{nicedistance}$ km, first determine how many litres per $1$ km:

$\\var{amount}\\text{ L} / \\var{distance} \\text{ km} = \\frac{\\var{amount}}{\\var{distance}} \\text{ L/km}$ $= \\var[fractionNumbers]{amount/distance} \\text{ L/km}$

(that fraction should look familiar) and then multiply both sides of the rate by $\\var{nicedistance}$ to get per $\\var{nicedistance}$ km:

$\\var{nicedistance}\\times\\var[fractionNumbers]{amount/distance} \\text{ L}/ \\var{nicedistance}\\text{ km}=\\var[fractionNumbers]{nicedistance*amount/distance} \\text{ L}/\\var{nicedistance} \\text{ km}$.

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