Question

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. There is 40 milligrams (mg) of a medication in every 5 milliliters (mL) of liquid. How much liquid is required to administer \(120 \mathrm{mg}\) of medication?

Short answer

15 mL of liquid is needed to administer 120 mg of medication.

Step-by-step solution

Understand the Problem

We are given that there are 40 mg of medication in every 5 mL of liquid. We need to find the amount of liquid required to administer 120 mg of medication.

Set Up a Proportion

We will set up a proportion to represent the relationship between the amount of medication and the amount of liquid. If 40 mg correspond to 5 mL, then 120 mg corresponds to an unknown amount, which we'll call "x" mL.\[\frac{40 \text{ mg}}{5 \text{ mL}} = \frac{120 \text{ mg}}{x \text{ mL}}\]

Solve the Proportion

Cross-multiply to solve for "x":\[40 \cdot x = 120 \cdot 5\]Calculate the right side:\[40x = 600\]Divide both sides by 40 to solve for "x":\[x = \frac{600}{40}\]\[x = 15\]

Label the Answer

The answer is that 15 milliliters (mL) of liquid are required to administer 120 milligrams (mg) of medication.

Key concepts

Medication Dosage Calculations

When it comes to medication dosage calculations, the precision is paramount. It's essential to calculate the exact amount of liquid medicine required to deliver a specific dose of the active ingredient. This process often involves setting up and solving proportions, which compare the known concentration of medication to the desired dose. For example:

• If 40 mg of medication is present in 5 mL of liquid, and the desired dose is 120 mg, we need to figure out how much liquid is necessary to achieve this.

Setting up a proportion involves matching the known ratio to an unknown ratio where one of the terms (the amount of liquid, in this case) is the variable to solve for. To achieve an accurate dosage, healthcare professionals rely on such calculations, ensuring that patients receive the correct amount required for treatment. Always remember to label the final answer with appropriate units to avoid any potential confusion.

Unit Conversions

Understanding unit conversions is crucial for accurately calculating medication dosages. We frequently encounter various units of measurement, such as milligrams (mg) and milliliters (mL), which are essential in both the medical and pharmaceutical fields. In the given problem, we deal with two primary units:

• Milligrams (mg): This is a unit of mass, used to quantify the amount of medication.
• Milliliters (mL): This is a unit of volume, used to indicate the amount of liquid containing the medication.

When setting up proportions for solving dosage problems like the example exercise, it is important to keep the units consistent. This means making sure that corresponding terms in a proportion are in the same units, allowing for straightforward problem-solving. Incorrect unit conversion could lead to incorrect dosages, potentially compromising patient safety, so it's a vital skill.

Problem-Solving in Mathematics

Problem-solving forms the backbone of mathematics, aiding in understanding and devising solutions for everyday scenarios. When tackling problems like dosage calculations, clear steps are the key:

• Firstly, understand the problem. Break it down into known and unknown variables and think about the relationships between them.
• Next, in cases of proportional relationships, set up the equation. Proportions allow us to represent and solve problems where two ratios are equal.
• Then, solve the equation. Always cross-multiply before simplifying to find the unknown variable.
• Finally, check the solution by contextualizing it within the problem, ensuring the answer is sensible and properly labeled.

Mathematical problem-solving is invaluable not just in academic contexts but also in real-world applications, including our health and well-being. Understanding and practicing these steps can enhance one's ability to effectively approach and solve various mathematical challenges.