Equations

Lesson

Another important part of health in which mathematics is very important is the calculation of **medicine dosage**. Most medicines, such as *Panadol* and* Nurofen*, have exact **dosage requirements** that need to be correct in order for the drug to work appropriately. This is most important when administering medicine with **oral suspension**, that is those in liquid form.

The clearest information about dosage directions can be found in the **dosage panel**. This panel contains facts about **active** and **inactive **ingredients, warning about the drug and its side effects, and the dosage chart.

Due to the small amounts of ingredients that these medications contain, it is necessary to use **milligrams** (mg) rather than **grams** (g). In other medication with even smaller dosages, this measurement range can reduce further to **micrograms** (µg).

A milligram is one thousandth of a gram. This means that to convert a gram into a milligram you multiply the gram value by 1000.

- e.g. $\text{0.756 g}\times1000=\text{756 mg}$0.756 g×1000=756 mg

To convert a milligram into a gram, divide the milligram value by 1000.

- e.g. $\frac{\text{9748 mg}}{1000}=\text{9.748 g}$9748 mg1000=9.748 g

Another important aspect of the dosage panel is the information about the **drug’s concentration**. This is found under the title “**Active Ingredients**”. There are three main active ingredients: paracetamol, ibuprofen and codeine. Each targets different symptoms and requires different dosages.

Concentrations are expressed on dosage panels as **mass/volume**.

For example, Nurofen for Children (3 months to 12 years) contains ibuprofen at a concentration of 100 mg / 5 mL. This means that for every 5ml of the fluid there is 100 mg of ibuprofen.

This would mean that if a patient was prescribed 2000 mg of ibuprofen then they would be administered 100 mL of the medication.

If we refer to the formula:

There are several formulae available that give the correct dosage of medicines, depending on the age or weight of the patient and on the normal adult dose. Often, the formula is known by the name of the person who developed it.

When we consider dosages, age plays an important factor in the amount that should be consumed. This is particularly important for children. There are standardised formulae that are used when calculating the dosage for children. These include **Fried's formula**, **Young's formula** and **Clark's formula**.

Used for children aged 1 to 2 years old and is based solely on age as follows:

A 24-month child is prescribed paracetamol. The adult dosage is 600 mg and has a concentration of 100 mg / 5mL. How much must the child take based on Fried’s formula?

$\frac{24\times600}{150}=\text{96 mg}$24×600150=96 mg

*Then substitute into the above 'Volume Required' formula...*

$\frac{96}{100}\times5=\text{4.8 mL}$96100×5=4.8 mL

Used for children aged 1 to 12 years old and relies on the child's age and a factor of their age.

An 11-year-old girl is prescribed codeine. The adult dosage for codeine is 500 mg. How much codeine should she be prescribed based on Young’s formula?

$\frac{11\times500}{11+12}=\text{239 mg}$11×50011+12=239 mg (approx.)

Aimed for use with all children in general and is dependent on the child's weight.

A child is prescribed 386 mg of ibuprofen. The adult dosage is 600 mg. How much does the child weigh approximately based on Clark’s formula?

$\text{386 mg}$386 mg | $=$= | $\frac{Weight\times600}{70}$Weight×60070 |

$Weight$Weight |
$=$= | $\frac{386\times70}{600}$386×70600 |

$Weight$Weight |
$=$= | $\text{45 kg}$45 kg |

Another way in which maths is helpful in medicine is its role in calculating drip rates. Drip rates measure the amount that an intravenous drip (IV) dispenses to a patient. A general formula for this is:

A patient receives 3.4 L at a flow rate of 283 mL / h via an intravenous drip (IV). How many hours were they on the IV?

$\text{283 mL / h}$283 mL / h | $=$= | $\frac{\text{3400 mL}}{\text{X hours}}$3400 mLX hours |

$\text{X hours}$X hours | $=$= | $\frac{3400}{283}$3400283 |

$\text{X hours}$X hours | $=$= | $12$12 (approx.) |

Using the image below, determine the following:

The daily dosage for a $25$25 year old adult.

Select all ages for which these tablets are recommended to be used.

$15$15

A$11$11

B$18$18

C$8$8

D$15$15

A$11$11

B$18$18

C$8$8

D

A twenty three month old child has asthma and needs prednisolone. The normal adult dose is $200$200mg.

Calculate the required dosage for the child using Fried's Formula:

Dosage for children 1-2 years $=$= $\frac{\text{age of child (in months)}\times\text{adult dosage}}{150}$age of child (in months)×adult dosage150

Give your answer correct to the nearest milligram (mg).

A four year old child needs some medication. The packet describes the adult dosage to be $800$800 mg per day. Clark's formula can be used to calculate the appropriate dosage for children over two years of age based on their mass:

Dosage for children over 2 years $=$= $\frac{\text{mass in kg}\times\text{adult dosage}}{70}$mass in kg×adult dosage70

Using Clark's formula, calculate the dosage for a child that has a mass of $40$40kg. Give your answer correct to the nearest milligram.

Using Clark's formula, calculate the dosage for a child that has a mass of $24$24kg. Give your answer correct to the nearest milligram.

Use formulae and equations to solve problems in the workplace