Question

Set up the following problems as a proportion and solve. Include labels in the set up and on the answer. If 60 milligrams (mg) of a medication is in 500 milliliters (mL), how many \(\mathrm{mL}\) of solution contain \(36 \mathrm{mg}\) of medication?

Short answer

300 mL

Step-by-step solution

Set up the Proportion

We need to set up a proportion to find the unknown quantity of the solution. The problem states that 60 mg of a medication is dissolved in 500 mL. Let this be the first ratio: \( \frac{60 \, \text{mg}}{500 \, \text{mL}} \). We are trying to find how many mL (let's call this \( x \)) contains 36 mg. So, the second ratio will be \( \frac{36 \, \text{mg}}{x \, \text{mL}} \). The proportion is therefore: \[ \frac{60 \, \text{mg}}{500 \, \text{mL}} = \frac{36 \, \text{mg}}{x \, \text{mL}} \]

Cross Multiply to Solve for x

In a proportion, the cross-products are equal, so we can set up the equation: \( 60 \times x = 36 \times 500 \). This simplifies to: \[ 60x = 18000 \]

Solve for x

Now, we need to solve for \( x \) by isolating it on one side of the equation, which requires dividing both sides by 60: \[ x = \frac{18000}{60} \] \[ x = 300 \]

Write the Answer with Labels

The proportion tells us that 36 mg of the medication is contained in 300 mL of solution. Thus, the answer is 300 mL.

Key concepts

Cross-Multiplication

Understanding cross-multiplication can greatly simplify solving proportion word problems. Cross-multiplication involves manipulating an equation consisting of two equal ratios to find an unknown value. For this, we multiply the numerator of one ratio by the denominator of the other ratio.
For example, in the proportion \( \frac{60 \, \text{mg}}{500 \, \text{mL}} = \frac{36 \, \text{mg}}{x \, \text{mL}} \), cross-multiplication is applied as follows:

• Multiply 60 by \( x \) to get \( 60x \).
• Multiply 36 by 500 to get 18000.

Thus, \( 60x = 18000 \).
With this equation, it becomes straightforward to isolate \( x \) by dividing both sides by 60, leading to \( x = 300 \). This method is a powerful technique in solving such proportion problems.

Unit Conversion

Unit conversion frequently plays an important role in solving mathematical problems such as those involving dosages. In this problem, the units — milligrams (mg) and milliliters (mL) — are consistent throughout, which simplifies the process.
However, if units were different, say grams and liters, conversion would be necessary to make the units identical. Understanding how to convert units ensures accuracy, because the mathematical operation assumes the ratios are of the same nature.
Remember:

• 1 gram (g) = 1000 milligrams (mg)
• 1 liter (L) = 1000 milliliters (mL)

Having a strong grasp on unit conversion is crucial for maintaining precision in more elaborate scenarios.

Medication Dosage

Calculating medication dosages correctly is vital for ensuring safe and effective treatment. In healthcare, dosing errors can have significant consequences, which is why it is paramount to approach these problems methodically using precise calculations.
This exercise involves determining how many milliliters of solution are required to deliver a specific milligram dosage. By setting up a proportion, we compared the known concentration of medication to the required amount.
Accuracy is key because even minor miscalculations can lead to an incorrect dosage. For healthcare professionals and students alike, mastering dosage calculations with precision can significantly impact patient care.
Always double-check each step in your calculations to ensure it aligns with the expected result. This practice develops confidence and competence in dosage determination.

Mathematical Reasoning

Mathematical reasoning is crucial to solving problems efficiently and accurately. Employing a logical approach entails breaking down a problem into understandable parts, much like in this case of solving the medication dosage proportion problem.
We began by identifying the known values and the unknown variable. Setting up a logical equation reflects not only understanding of the problem but also the foresight to solve it systematically.
Consider:

• Identifying the relationship and setting up the correct proportion.
• Applying cross-multiplication as a tool to simplify the proportion to a straightforward equation.

Such structured methods strengthen mathematical skills and build a foundation for tackling more complex problems. Encouraging a habit of clear, step-by-step reasoning allows students to not only solve what’s at hand but also apply learned skills to diverse situations.