Problem 14 - Prealgebra

Question

Solve each of the following word problems by translating the statement into a proportion. Be sure to show the proportion used in each case. [Examples \(1-4]\) Nursing An atropine sulfate injection has a dosage strength of 0.1 \(\mathrm{mg} / \mathrm{mL}\). If 4.5 \(\mathrm{mL}\) was given to the patient, how many milligrams did she receive?

Step-by-Step Solution

Answer

The patient received 0.45 mg of atropine sulfate.

Step 1: Understand the Problem

We are given a dosage strength of atropine sulfate as \(0.1 \, \mathrm{mg/mL}\). The patient received \(4.5 \, \mathrm{mL}\) of this solution. We need to determine the total milligrams (mg) that the patient received.

Step 2: Set Up the Proportion

A proportion is a statement that two ratios are equal. We know the dosage strength is \(0.1 \, \mathrm{mg/mL}\). Let \(x\) be the number of milligrams received. We set up the proportion based on the known dosage strength: \[\frac{0.1 \, \mathrm{mg}}{1 \, \mathrm{mL}} = \frac{x \, \mathrm{mg}}{4.5 \, \mathrm{mL}}\]

Step 3: Solve the Proportion

Cross-multiply to solve the proportion: \[0.1 \times 4.5 = 1 \times x\]This simplifies to:\[x = 0.45\]Therefore, the patient received \(0.45 \, \mathrm{mg}\) of atropine sulfate.

Step 4: Verify Your Solution

Double-check the calculation by ensuring the units are consistent and that the arithmetic operations are correct. \(0.1 \, \mathrm{mg/mL} \times 4.5 \, \mathrm{mL} = 0.45 \, \mathrm{mg}\), confirming that our solution is consistent with the problem's data.

Key Concepts

Dosage Calculation

Dosage calculation is a critical concept in fields such as medicine and pharmacy. It ensures patients receive the correct amount of medication for their specific needs. In this exercise, the dosage strength of atropine sulfate is given as \(0.1 \, \mathrm{mg/mL}\), which means there is \(0.1\) milligram of the drug in every milliliter of solution. Hence, when a patient receives \(4.5 \, \mathrm{mL}\), you need to calculate how many milligrams that would equate to. This is done by multiplying the dosage strength by the total volume given:

• Dosage strength: \(0.1 \, \mathrm{mg/mL}\)
• Total volume: \(4.5 \, \mathrm{mL}\)

By calculating \(0.1 \, \mathrm{mg/mL} \times 4.5 \, \mathrm{mL} = 0.45 \, \mathrm{mg}\), you find the patient received \(0.45 \, \mathrm{mg}\) of atropine sulfate.

Cross-Multiplication

Cross-multiplication is a quick and useful algebraic method used to solve proportions, which are statements that two ratios are equal. This technique involves multiplying the numerator of each ratio by the denominator of the other ratio. It is crucial in solving the proportion set up in our exercise. When we have:\[\frac{0.1 \, \mathrm{mg}}{1 \, \mathrm{mL}} = \frac{x \, \mathrm{mg}}{4.5 \, \mathrm{mL}}\]The process involves cross-multiplying:

• Multiply \(0.1 \, \mathrm{mg}\) by \(4.5 \, \mathrm{mL}\)
• Equate this product to \(1 \, \mathrm{mL} \times x \, \mathrm{mg}\)

This gives: \(0.1 \times 4.5 = 1 \times x\), simplifying to \(x = 0.45\). Thus, the cross-multiplication technique helps verify that the patient indeed received \(0.45\) milligrams of medication.

Unit Conversion

Unit conversion is essential for ensuring a medication is delivered accurately according to the health provider's prescription. In our scenario, understanding the units \(\mathrm{mg}\) per \(\mathrm{mL}\) is crucial. Without consistent units, your calculations can go awry. Each milliliter of the solution contains \(0.1 \, \mathrm{mg}\), simplifying the determination of how many total milligrams were administered when multiple milliliters are given. By knowing that \(1 \, \mathrm{mL}\) equals \(0.1 \, \mathrm{mg}\), you could easily find the total dosage for any given volume:

• \(4.5 \, \mathrm{mL} \rightarrow 0.45 \, \mathrm{mg}\)

This ensures precise drug administration, critical in clinical settings.

Mathematical Ratios

Mathematical ratios depict the relationship between two quantities. In this exercise, the ratio comes from the dosage strength — the ratio of \(0.1 \, \mathrm{mg}\) per \(1 \, \mathrm{mL}\). Ratios allow you to set up equations or proportions, helping you scale quantities up or down while maintaining their inherent proportionality. For instance, if \(0.1 \, \mathrm{mg}\) corresponds to \(1 \, \mathrm{mL}\), then multiplying both parts of this ratio by 4.5 allows you to determine how much \(4.5 \, \mathrm{mL}\) contains, equating to \(0.45 \, \mathrm{mg}\). This method ensures that the relative quantities remain consistent:

• \(1 \, \mathrm{mL} : 0.1 \, \mathrm{mg}\)
• \(4.5 \, \mathrm{mL} : 0.45 \, \mathrm{mg}\)

Thus, ratios are fundamental in expressing and maintaining the proportional relationships in calculations.