Question

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. 15 grams (g) of a medication is dissolved in 300 milliliters (mL) of solution. If \(45 \mathrm{~g}\) of the medication is needed, how many \(\mathrm{mL}\) of the solution are needed?

Short answer

900 milliliters are needed for 45 grams of medication.

Step-by-step solution

Understanding the Problem

We are given that 15 grams of medication is dissolved in 300 milliliters of solution, and we need to find how many milliliters are required for 45 grams of medication. This requires setting a proportion between grams and milliliters.

Setting Up the Proportion

First, set up a proportion using the given information. Write it as \( \frac{15 \text{ g}}{300 \text{ mL}} = \frac{45 \text{ g}}{x \text{ mL}} \), where \(x\) is the unknown amount of milliliters needed.

Cross Multiplying

In a proportion \( \frac{a}{b} = \frac{c}{d} \), cross multiply to solve for the unknown. Apply it here: \(15 \times x = 45 \times 300\).

Solving for x

Calculate \(45 \times 300 = 13500\). So, the equation becomes \(15x = 13500\). Divide both sides by 15 to find \(x\).

Final Calculation

Divide: \(x = \frac{13500}{15}\). Performing the division gives \(x = 900\). Thus, 900 milliliters of solution are needed for 45 grams of medication.

Key concepts

Mathematics Education

Mathematics education plays a crucial role in helping learners understand fundamental concepts essential for real-world applications. One of the cornerstones of mathematics education is understanding proportions, which are often used to solve problems involving comparisons of quantities. Understanding how to set up and solve proportions allows students to apply mathematical reasoning effectively to various scenarios, such as scaling recipes or distributing resources. Why Is It Important?

• Develops problem-solving skills through logical reasoning and analysis.
• Applies mathematical concepts to everyday situations, making learning relevant and practical.
• Builds a strong foundation for advanced mathematical topics, necessary for academic and career pursuits.

Engaging students with exercises like the medication dosage calculation problem familiarizes them with setting up equations, translating word problems into mathematical representations, and systematically solving them. This approach enhances cognitive skills and builds confidence in handling complex mathematical scenarios.
By mastering the art of language translation into math, students can bridge the gap between abstract numbers and real-world context.

Medication Dosage Calculations

Medication dosage calculations are critical in fields where precise measurements are necessary. Ensuring the correct dosage is essential for effective treatment and patient safety. Using proportions is a common method to calculate the right amount of medicine, especially when adjusting doses depending on the needed concentration. How It Works

• Identify the relationship between the medication amount and the solution's volume to maintain consistency in dosing.
• Set up a proportion where one side represents the known ratio, and the other the unknown, solving for the desired quantity.
• Use cross multiplication to find the missing value, ensuring units are consistent throughout the calculation.

Accurate calculations help to avoid administration errors, which can lead to potential health risks. For example, determining how many milliliters of solution are needed for a given amount of medication ensures patients receive the correct dosage, thus enhancing treatment outcomes and minimizing side effects. It is vital for students, especially those entering healthcare professions, to develop competence in these calculations.

Unit Conversion

Unit conversion is a fundamental skill in mathematics and science, enabling one to translate measurement units from one system to another. It is especially important in contexts where different units are commonly used interchangeably, such as in medication dosage calculations. The Basics of Unit Conversion

• Recognize the need to express quantities in different units to facilitate calculations and communication.
• Use conversion factors to switch between units, ensuring equations remain balanced and quantities accurate.
• Understand both metric (e.g., grams, liters) and imperial (e.g., ounces, gallons) systems, if applicable, for comprehensive problem-solving.

In the given problem, grams measure the medication, while milliliters measure the solution. Being adept at converting and comparing these units helps prevent miscalculations, which is crucial in fields where precision is paramount.
Proficiency in unit conversion thus empowers students to tackle a broader array of mathematical problems and scientific experiments with confidence.

Problem-Solving Steps

Problem-solving in mathematics involves a systematic approach to identify, analyze, and resolve a problem. Utilizing a step-by-step strategy, such as setting up a proportion, is a highly effective method. The Approach

• Understand the Problem: Comprehend what is being asked and identify the knowns and unknowns.
• Set Up the Equation: Represent the problem mathematically, using a proportion or equation appropriately.
• Apply Mathematical Techniques: Use techniques such as cross-multiplication to manipulate the equation and solve for the unknown variable.
• Verify the Solution: Check the calculated answer against the problem to ensure it is logical and accurate.

Following these steps breaks down a complex problem into manageable tasks, making it easier to approach and solve efficiently. By practicing these techniques, students enhance their analytical skills, making them adept at solving a wider range of mathematical problems. This structured methodology is at the heart of successful mathematics education, fostering logical thinking and robust problem-solving abilities in students.