Setting up the math for a two-step quantitative problem
The safe dosage of the IV antibiotic ceftriaxone for children weighing more than 2.0 is 60 per kilogram of body weight. Suppose a pediatric nurse has available 1.5 of a ceftriaxone solution with a concentration of 0.040. How can she calculate the volume of this solution that she should administer to a child weighing 30?
Set the math up. But don’t do any of it. Just leave your answer as a math expression.
Also, be sure your answer includes all the correct unit symbols.
The Correct Answer and Explanation is:
Correct Math Expression:
Solution needed=(60 mgkg)×(30 kg)×(1 g1000 mg)÷(0.040 gmL)\text{Solution needed} = \left(60 \, \frac{\text{mg}}{\text{kg}}\right) \times \left(30 \, \text{kg}\right) \times \left(\frac{1 \, \text{g}}{1000 \, \text{mg}}\right) \div \left(0.040 \, \frac{\text{g}}{\text{mL}}\right)
Explanation
In this problem, we are calculating how much of a ceftriaxone IV solution (in mL) should be administered to a child weighing 30 kg, given a dosage recommendation and the solution’s concentration.
The dosage guideline is given as:
- 60 mg per kg of body weight.
This means for every kilogram the child weighs, they should receive 60 mg of the drug.
- Determine the total drug required in mg:
60 mgkg×30 kg=1800 mg60 \, \frac{\text{mg}}{\text{kg}} \times 30 \, \text{kg} = 1800 \, \text{mg}
But instead of calculating this step, we preserve it in the expression format.
- Convert mg to g:
Since the concentration of the solution is given in g/mL, we need to convert mg to g to keep the units consistent. There are 1000 mg in 1 gram, so: 1 g1000 mg\frac{1 \, \text{g}}{1000 \, \text{mg}}
Multiplying by this factor converts milligrams to grams.
- Use concentration to find volume:
The solution concentration is:
- 0.040 g/mL
This means each mL contains 0.040 grams of ceftriaxone. To find how many mL are needed to deliver the required grams of drug, divide the drug amount (in grams) by the concentration: ÷(0.040 gmL)\div \left(0.040 \, \frac{\text{g}}{\text{mL}}\right)
Putting it all together as a single mathematical expression allows us to solve for the required volume in mL without prematurely simplifying or calculating, as instructed.
This setup ensures dimensional consistency: kilograms cancel out, milligrams convert to grams, and grams cancel out, leaving mL — the correct unit for volume administered.
