Question

Calculate the volume necessary (in milliliters) to provide the dosage ordered, using medication labels where available. Express your answer as a decimal fraction to the nearest tenth where indicated. Order: Thorazine concentrate $75\mathrm{mg}$ p.o. daily. Available: Thorazine concentrate labeled $100\mathrm{mg}$ per $\mathrm{mL}$

Short answer

0.8 mL

Step-by-step solution

Understand the Dosage Ordered

The doctor's order is for the patient to receive 75 mg of Thorazine daily by mouth (p.o., which stands for "per os" in Latin).

Identify the Concentration of Available Medication

The Thorazine available is in a liquid concentrate form, labeled as 100 mg per milliliter (mL). This means 1 mL of the concentrate contains 100 mg of Thorazine.

Set up a Proportion to Find the Necessary Volume

To determine how much volume the patient needs, set up a proportion. We are looking for the volume (in mL) that contains 75 mg of Thorazine. Use the relation: $\frac{100\text{ mg}}{1\text{ mL}}=\frac{75\text{ mg}}{x\text{ mL}}$.

Solve the Proportion

Cross-multiply to solve for $x$: $100\times x=75\times 1$. This simplifies to $100x=75$. Divide both sides by 100 to solve for $x$: $x=\frac{75}{100}=0.75$.

Round the Answer to the Nearest Tenth

Since the problem asks for the answer to the nearest tenth, round 0.75 mL to 0.8 mL.

Key concepts

Medication Concentration

When you're dealing with medications, understanding the concentration is crucial. Concentration denotes the amount of a drug contained in a given volume of a solution. In this case, the concentration of Thorazine is specified as 100 mg per milliliter (mL). This implies that in every 1 mL, there are 100 milligrams of the active drug—Thorazine. Knowing this concentration helps in calculating how much liquid volume a patient should receive to get the correct dosage.

Simply put, if you know how much drug is in each mL, you can figure out how many milliliters to give the patient to meet their specific dosage needs. This involves simple arithmetic when you know a formula, ingredient proportions, or a label reveals this concentration value.

Proportion in Mathematics

A proportion is an equation that shows two ratios are equal. In medicine, proportions are often used to solve dosage calculations. Here, the goal is to find out how much of a liquid medication you need. You do this by setting up an equation where the two ratios—one from the medication concentration and the other from the required dosage—are equal.

Take the example: the medication contains 100 mg/mL, and you need 75 mg. Set up a proportion by creating a fraction with the known values:

• Known: 100 mg per 1 mL
• Needed: 75 mg per x mL

In symbol terms, this becomes: $$\frac{100\text{ mg}}{1\text{ mL}}=\frac{75\text{ mg}}{x\text{ mL}}.$$To solve, cross-multiply: 100 times x equals 75 times 1. Simplify it to get: $$100x=75.$$ Then, divide both sides by 100 to isolate x, resulting in: $$x=\frac{75}{100}=0.75\text{ mL}.$$Proportions help make calculations straightforward and understandable, especially when directly handling medication.

Rounding Numbers

Rounding is a mathematical method used to simplify numbers, making them easier to work with or more relevant in practical use. When the result of a proportion gives a decimal, sometimes medical orders specify how you need to round it. These guidelines ensure precision in medication dosing, which is critical.

For example, when you solve the dosage calculation and find out that the patient needs 0.75 mL, you might be asked to round to the nearest tenth. To do this, examine the decimal part (the digits after the point). If it is 5 or higher, round the number up. If it's less than 5, round down.

• Here, 0.75 becomes 0.8 when rounded to the nearest tenth.

Rounding ensures that the calculated dosage is practical and easy to administer, especially if exact decimal doses are not feasible to measure. Always follow specific rounding instructions to prevent mistakes.