1. Formulas and Equations

Lesson

In Australia, as in other countries around the world, a significant number of people are admitted to hospital each year for taking an incorrect dosage of medication.

The **dosage** is a measure of the amount of medication to be given or administered (e.g. $30$30 mL every $4$4 hours).

Medicines can be administered in a variety of forms. They can be taken orally in the form of tablets, capsules, caplets, liquids, or through an inhaler. In some cases, they can be administered through the skin by applying a cream or ointment. They can also be given internally by injection or intravenous infusion.

Most medication dosages, prescribed for a particular condition, will depend on a person's **age** and **weight**. The dosage requirements need to be correct in order for the medicine to work appropriately. Correct dosage is particularly important with medicines in liquid form, as these require exact measurements to be made.

The clearest information about dosage directions can be found in the **dosage panel**, displayed clearly on the medication's packaging. The dosage panel contains facts about **active ingredients**, warnings about side effects, and detailed instructions for administering the medication.

Due to the small amounts of active ingredients contained in most medications, we usually use **milligrams** (mg) rather than grams (g). For liquids, we use **millilitres** (mL) rather than litres (L).

A milligram is one thousandth of a gram. So to convert from grams to milligrams we multiply by $1000$1000. The same applies for converting millilitres into litres.

For example,

$0.756$0.756 g | $=$= | $0.756\times1000$0.756×1000 mg |

$=$= | $756$756 mg |

To convert from milligrams to grams, we divide by $1000$1000. The same applies for converting litres into millilitres.

For example,

$9748$9748 mg | $=$= | $\frac{9748}{1000}$97481000 g |

$=$= | $9.748$9.748 g |

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

Most medication comes packaged in a limited number of strengths and concentrations. This is called **stock** medication. The 'active' ingredient or drug is combined with various 'neutral' ingredients to form a tablet or capsule, or it is diluted in a volume of liquid.

The **stock strength** of a medication, is the amount of drug it contains. This amount is displayed on the medication packaging, usually in milligrams (mg).

The **concentration** describes how many milligrams of the drug there are in each tablet or, in the case of liquids, how many milligrams there are in a certain number of millilitres.

In clinics and hospitals, doctors or nurses administering prescribed medications to patients, will use the following formula:

$\text{Volume required}=\frac{\text{strength required}}{\text{stock strength}}\times\frac{\text{volume of stock}}{1}$Volume required=strength requiredstock strength×volume of stock1

A certain medication for children (aged $3$3 months to $12$12 years) contains the drug ibuprofen at a concentration of $100$100 mg in $5$5 mL. If a child is prescribed $2000$2000 mg of ibuprofen, what volume of medication should they take?

**Solution**

There are several ways we can do this question. We can use the formula above or we can use ratios. To use the formula, we notice that the 'stock strength' is $100$100 mg, the 'volume of stock' is $5$5 mL, and the 'strength required' is $2000$2000 mg.

Substituting these values into the formula:

$\text{Volume required}$Volume required | $=$= | $\frac{\text{strength required}}{\text{stock strength}}\times\frac{\text{volume of stock}}{1}$strength requiredstock strength×volume of stock1 |

$=$= | $\frac{2000}{100}\times\frac{5}{1}$2000100×51 | |

$=$= | $100$100 mL |

As an alternative, this question can be done using ratios and proportions. For every $5$5 mL of liquid, there is $100$100 mg of ibuprofen in the medication stock. Because the child's medication dosage must be in the same proportion, we can set up a proportional statement, using $x$`x` as the volume required, then solve for $x$`x`.

$5$5 mL$:$:$100$100 mg | $=$= | $x$x$:$:$2000$2000 mg |
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$\frac{5}{100}$5100 | $=$= | $\frac{x}{2000}$x2000 |
(Convert to fractions) | |

$\frac{5}{100}\times2000$5100×2000 | $=$= | $\frac{x}{2000}\times2000$x2000×2000 |
(Multiply both sides by $2000$2000) | |

$100$100 | $=$= | $x$x |
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volume required | $=$= | $100$100 mL |

If the medication is in tablet or capsule form, we can ignore the 'volume of stock' part of the formula:

$\text{Volume required}=\frac{\text{strength required}}{\text{stock strength}}$Volume required=strength requiredstock strength

A patient is prescribed $750$750 mg of a medication. The available tablets each have a strength of $500$500 mg. How many tablets should be given?

**Solution**

Here the 'strength required' is $750$750 mg and the 'stock strength' is $500$500 mg.

Substituting these values into the formula:

$\text{Volume required}$Volume required | $=$= | $\frac{\text{strength required}}{\text{stock strength}}$strength requiredstock strength |

$=$= | $\frac{750}{500}$750500 | |

$=$= | $1.5$1.5 tablets |

In Australia, giving the wrong dose of medication to children is the most common cause of accidental poisoning in children under $12$12 months old.

Children are particularly sensitive to medicine, even those sold over-the-counter. Many medications can be harmful or fatal to children, particularly if the dosage is incorrect. Even a small change in dosage (too much or not enough), can make a big difference to how their bodies cope with the difference.

Medication dosages that may be suitable for adults, will need to be adjusted for children. Here we look at three different formula that were specifically developed for this purpose. Known as Young's, Clark's and Fried's formulas, they are named after the physicians who first developed them. All of the formulas are based on adapting an adult dosage of medication, to suit the needs of a child.

Young's formula is named after a physician from England, Dr Thomas Young (1773-1829). It is used for children aged from $1$1 to $12$12 years and is based on the child's age in years.

$\text{Dosage for children 1-12 years}=\frac{\text{age of child (in years)}\times\text{adult dosage}}{\text{age of child (in years)}+12}$Dosage for children 1-12 years=age of child (in years)×adult dosageage of child (in years)+12

An $11$11 year old girl is prescribed codeine. The adult dosage for codeine is $500$500 mg. How much codeine should the girl be prescribed, based on Young’s formula? Round your answer to the nearest milligram.

**Solution**

Substituting the child's age of $11$11 and the adult dosage of $500$500 mg into the formula:

$\text{Dosage }$Dosage | $=$= | $\frac{\text{age of child (in years) }\times\text{adult dosage }}{\text{age of child (in years) }+12}$age of child (in years) ×adult dosage age of child (in years) +12 |

$=$= | $\frac{11\times500}{11+12}$11×50011+12 | |

$=$= | $\frac{5500}{23}$550023 | |

$=$= | $239.1204\ldots$239.1204… | |

$=$= | $239$239 mg (nearest mg) |

A twelve year old child has bronchitis and needs augmentin. The normal adult dose is $450$450mg.

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

Dosage for children 1-12 years $=$= $\frac{\text{age of child (in years)}\times\text{adult dosage}}{\text{age of child (in years) + 12}}$age of child (in years)×adult dosageage of child (in years) + 12

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

Clark's formula is named after Dr Cecil Belfield Clark (1894-1970), a physician originally from Barbados, who worked in London for fifty years.

Clark's formula is used for children aged from $2$2 to $17$17 years and is based on the child's weight in kilograms.

$\text{Child dosage}=\frac{\text{weight in kg}\times\text{adult dosage}}{70}$Child dosage=weight in kg×adult dosage70

A child is prescribed $386$386 mg of ibuprofen. The adult dosage is $600$600 mg. What is the approximate weight of the child, based on Clark’s formula? Give your answer to the nearest kilogram.

**Solution**

Here we are finding the value of a variable that is not the subject of the original formula. We begin by substituting known values into the formula, then solve the resulting equation for the child's weight.

$\text{Child dosage}$Child dosage | $=$= | $\frac{\text{weight in kg}\times\text{adult dosage}}{70}$weight in kg×adult dosage70 | ||

$386$386 | $=$= | $\frac{\text{weight }\times600}{70}$weight ×60070 | (Substitute known values) | |

$386\times70$386×70 | $=$= | $\frac{\text{weight }\times600}{70}\times70$weight ×60070×70 | (Multiply both sides by $70$70) | |

$27020$27020 | $=$= | $\text{weight }\times600$weight ×600 | ||

$\frac{27020}{600}$27020600 | $=$= | $\frac{\text{weight }\times600}{600}$weight ×600600 | (Divide both sides by $600$600) | |

$45.033\ldots$45.033… | $=$= | $\text{weight }$weight | ||

$\text{weight }$weight | $=$= | $45$45 kg (nearest kg) |

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.

Fried's formula is named after an Israeli, Dr Kalman Fried (1914-1999), who developed the formula while observing children at a hospital in Jerusalem.

Fried's formula is similar to Clark's formula, except that it has been modified for infants (children aged from $1$1 to $2$2 years). For this reason, the formula uses the child's age in months, rather than its weight.

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

A $2$2 year old child is prescribed paracetamol. The adult dosage is $600$600 mg and has a concentration of $100$100 mg in $5$5 mL. How much must the child take based on Fried’s formula?

**Solution**

The child's age must be in months when using Fried's formula and $2$2 years is equivalent to $24$24 months. Substituting into Fried's formula we can calculate the correct child dosage:

$\text{Child dosage}$Child dosage | $=$= | $\frac{\text{age (in months)}\times\text{adult dosage}}{150}$age (in months)×adult dosage150 |

$=$= | $\frac{24\times600}{150}$24×600150 | |

$=$= | $\frac{14400}{150}$14400150 | |

$=$= | $96$96 mg |

We can now work out the volume of medicine required for the child that has a strength of $96$96 mg, given that the stock strength is $100$100 mg and the volume of stock is $5$5 mL:

$\text{Volume required}$Volume required | $=$= | $\frac{\text{strength required}}{\text{stock strength}}\times\frac{\text{volume of stock}}{1}$strength requiredstock strength×volume of stock1 |

$=$= | $\frac{96}{100}\times\frac{5}{1}$96100×51 | |

$=$= | $4.8$4.8 mL |

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).

Did you know?

Each of the formulas used above have been expressed in words rather than letters, but they could easily have been written using letters. For example, here is an alternative version of Young's formula:

$D=\frac{yA}{y+12}$`D`=`y``A``y`+12,

where $y=\text{age of child (in years)}$`y`=age of child (in years), $A=\text{adult dosage}$`A`=adult dosage and $D=\text{dosage for children aged 1-12 years}$`D`=dosage for children aged 1-12 years.

We can see that it is exactly the same as the formula using words. The letters chosen could have been any letters. The important thing is that each letter has been clearly defined.

makes predictions about everyday situations based on simple mathematical models