Dosage calculation practice as algebra
Dosage calculation practice relies on linear relationships among dose, body mass, concentration, and time. The same algebra supports oral liquids, tablets, IV infusions, and dilutions when units remain consistent.
Core proportional relationships
Weight-based dose (amount): \(D = r \cdot m\), where \(r\) is a dose rate (mg/kg) and \(m\) is mass (kg).
Volume from concentration: \(V = \dfrac{D}{c}\), where \(c\) is concentration (mg/mL) and \(V\) is volume (mL).
Tablet count: \(N = \dfrac{D}{s}\), where \(s\) is strength per tablet (mg/tablet).
Infusion conversion (common): \( \text{mg/hr} = \text{(mg/min)} \cdot 60\), and \( \text{mL/hr} = \dfrac{\text{mg/hr}}{\text{mg/mL}} \).
Dimensional consistency and unit conversions
Units function as algebraic factors. Multiplication and division simplify units exactly as symbols do. Useful equalities include \(1 \text{ g} = 1000 \text{ mg}\), \(1 \text{ mg} = 1000 \text{ mcg}\), and \(1 \text{ L} = 1000 \text{ mL}\).
Worked example
A child has mass \(m = 18 \text{ kg}\). An ordered dose rate is \(r = 15 \text{ mg/kg}\). A suspension label states \(250 \text{ mg} / 5 \text{ mL}\).
The required amount is \(D = r \cdot m = 15 \cdot 18 = 270 \text{ mg}\).
The concentration in mg/mL is \(c = \dfrac{250}{5} = 50 \text{ mg/mL}\), so the volume is \[ V = \frac{D}{c} = \frac{270}{50} = 5.4 \text{ mL}. \]
Practice set
The items below provide dosage calculation practice across common linear formats. Rounding conventions are stated per item when needed.
| Item | Given | Compute |
|---|---|---|
| 1 | Mass \(m = 24 \text{ kg}\). Daily dose rate \(30 \text{ mg/kg/day}\). Three equal doses per day. | Amount per dose (mg). |
| 2 | Ordered amount \(D = 0.25 \text{ g}\). Tablets: \(125 \text{ mg/tablet}\). | Tablet count \(N\). |
| 3 | Mass \(m = 70 \text{ kg}\). Infusion order \(q = 2 \text{ mcg/kg/min}\). Solution concentration \(200 \text{ mg} / 100 \text{ mL}\). | Infusion rate (mL/hr). |
| 4 | Stock \(c_1 = 10 \text{ mg/mL}\). Target \(c_2 = 2 \text{ mg/mL}\). Final volume \(V_2 = 50 \text{ mL}\). | Stock volume \(V_1\) and diluent volume \(V_2 - V_1\) (mL). |
| 5 | Volume \(V = 250 \text{ mL}\) over \(2 \text{ hr}\). Drop factor \(20 \text{ gtt/mL}\). | Drip rate (gtt/min), rounded to the nearest whole drop. |
Answer key and algebra checks
Answers (with compact calculations)
1. Daily amount \(D_{\text{day}} = 30 \cdot 24 = 720 \text{ mg/day}\). Per dose \(D = \dfrac{720}{3} = 240 \text{ mg}\).
2. Convert \(0.25 \text{ g} = 250 \text{ mg}\). Tablets \(N = \dfrac{250}{125} = 2\).
3. Rate in mcg/min: \(2 \cdot 70 = 140 \text{ mcg/min}\). Convert \(140 \text{ mcg/min} = 0.14 \text{ mg/min}\). Per hour: \(0.14 \cdot 60 = 8.4 \text{ mg/hr}\). Concentration: \(\dfrac{200}{100} = 2 \text{ mg/mL}\). Infusion: \(\dfrac{8.4}{2} = 4.2 \text{ mL/hr}\).
4. Conservation of solute: \(c_1 V_1 = c_2 V_2\). \[ V_1 = \frac{c_2 V_2}{c_1} = \frac{2 \cdot 50}{10} = 10 \text{ mL}, \quad \text{diluent} = 50 - 10 = 40 \text{ mL}. \]
5. Time \(2 \text{ hr} = 120 \text{ min}\). Flow \(= \dfrac{250}{120} \approx 2.0833 \text{ mL/min}\). Drops \(= 2.0833 \cdot 20 \approx 41.666\). Rounded drip rate: \(42 \text{ gtt/min}\).
Common pitfalls
Decimal placement errors often arise from mcg–mg conversions and hr–min conversions. A consistent unit path keeps scale factors explicit, for example \( \text{mcg} \to \text{mg}\) via division by \(1000\), and \(\text{per min} \to \text{per hr}\) via multiplication by \(60\).
Concentration labels written as “\(250 \text{ mg} / 5 \text{ mL}\)” behave as a fraction; converting to mg/mL before dividing clarifies the linear equation \(V = D/c\).