Rates in Context (Concentrations & Heart Rate)
Learn rates in context — concentrations and heart rate — for NSW Year 12 Mathematics Standard 2. A rate is an amount per one unit of another quantity: a concentration is milligrams per millilitre (or mg/L, the same as ppm), and a heart rate is beats per minute.
You will find a medicine dose from its concentration, calculate the maximum heart rate using \(220-\text{age}\) and the \(65\%\)–\(85\%\) target training zone, and convert flow rates such as mL/min to L/h — core Standard 2 skills for solving practical problems with ratios and rates.
Theory
Rates in context apply the idea of a rate to practical problems in Year 12 Standard 2 (NSW). A concentration is an amount per unit volume (mg/mL, or mg/L which equals ppm) and a heart rate is beats per minute. This guide shows how to find a dose from a concentration, work out the maximum heart rate \((220-\text{age})\) and the target training zone, and convert flow rates.
A rate compares two quantities measured in different units, written as an amount per one unit of the other. A concentration is a rate: \(\text{concentration}=\dfrac{\text{amount of substance}}{\text{volume}}\), measured in mg/mL or mg/L. Parts per million (ppm) is the same as mg/L.
A heart rate is measured in beats per minute (bpm). The maximum heart rate is estimated by \(\text{MHR}=220-\text{age}\), and the target training zone is \(65\%\) to \(85\%\) of that maximum.
In this Year 12 Standard 2 (NSW) topic you also read and convert flow rates (mL/min, mL/h, L/h) and compare rates to decide which is faster or more concentrated. The table below shows the target zone for a few ages.
| Age | MHR (bpm) | Lower 65% (bpm) | Upper 85% (bpm) |
|---|---|---|---|
| 20 | 200 | 130 | 170 |
| 40 | 180 | 117 | 153 |
| 60 | 160 | 104 | 136 |
With an amount \(A\), a volume \(V\) and a concentration \(c\):
Maximum heart rate and the target training zone (a percentage of the MHR):
How to solve a rate problem
- Identify the rate — what amount is measured per one unit of what else (mg per mL, beats per minute, mL per hour)?
- Match the units. Convert grams to milligrams, litres to millilitres or minutes to hours so both quantities use the units the rate needs.
- Divide in the right order — amount \(\div\) volume for a concentration, volume \(\div\) time for a flow rate — or rearrange \(A=c\times V\) for the unknown.
- For heart rate, work out \(\text{MHR}=220-\text{age}\) first, then take \(65\%\) and \(85\%\) for the target zone.
Rearrange \(c=\dfrac{\text{dose}}{\text{volume}}\) to make the volume the subject.
| \(\text{volume}\) | \(=\) | \(\dfrac{\text{dose}}{\text{concentration}}\) |
| \(\) | \(=\) | \(\dfrac{100\text{ mg}}{40\text{ mg/mL}}\) |
| \(\) | \(=\) | \(2.5\text{ mL}\) |
A \(2.5\) mL dose is required.
Find the MHR, then take \(65\%\) and \(85\%\) of it.
| \(\text{MHR}\) | \(=\) | \(220-45 = 175\text{ bpm}\) |
| \(\text{lower}\) | \(=\) | \(0.65\times175 \approx 114\text{ bpm}\) |
| \(\text{upper}\) | \(=\) | \(0.85\times175 \approx 149\text{ bpm}\) |
The target zone is \(114\) to \(149\) bpm.
Convert grams to milligrams, then divide by the volume in litres.
| \(4\text{ g}\) | \(=\) | \(4000\text{ mg}\) |
| \(\text{concentration}\) | \(=\) | \(\dfrac{4000\text{ mg}}{500\text{ L}} = 8\text{ mg/L}\) |
| \(\) | \(=\) | \(8\text{ ppm}\) |
The iron concentration is \(8\) ppm.
Flow rate is volume per hour; divide by \(60\) for per-minute.
| \(\text{(i) flow rate}\) | \(=\) | \(\dfrac{900\text{ mL}}{6\text{ h}} = 150\text{ mL/h}\) |
| \(\text{(ii) rate}\) | \(=\) | \(\dfrac{150}{60} = 2.5\text{ mL/min}\) |
The drip runs at \(150\) mL/h \(=2.5\) mL/min.
Common pitfalls
Frequently asked questions
What is a rate in maths?
A rate compares two quantities measured in different units, given as an amount per one unit of the other. Examples include a concentration in milligrams per millilitre, a heart rate in beats per minute, and a flow rate in millilitres per minute.
How do you work out a concentration?
Divide the amount of substance by the volume it is dissolved in. For example, 4 grams of iron in 500 litres is 4000 milligrams divided by 500 litres, which is 8 milligrams per litre, or 8 ppm. Keep the units consistent before you divide.
What does ppm mean?
Ppm stands for parts per million. For a substance dissolved in water, a concentration in ppm is the same number as the concentration in milligrams per litre, so you convert any grams to milligrams and divide by the volume in litres.
How do you calculate maximum heart rate?
A common estimate is 220 minus your age, giving a maximum heart rate in beats per minute. For example, a 45 year old has an estimated maximum heart rate of 220 minus 45, which is 175 bpm.
What is the target heart-rate zone?
The target training zone is 65 percent to 85 percent of your maximum heart rate. Work out the maximum heart rate first with 220 minus age, then take 65 percent for the lower bound and 85 percent for the upper bound.
How do you convert millilitres per minute to litres per hour?
Multiply by 60 to change per minute into per hour, then divide by 1000 to change millilitres into litres. So 50 mL/min becomes 3000 mL/h, which is 3 L/h.