Question

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. The prescriber orders 15 milligrams (mg) of a medication for every \(10 \mathrm{lb}\) of a client's weight. How many mg of medication will be given for a person who weighs \(120 \mathrm{lb} ?\)

Short answer

The person should receive 180 mg of medication.

Step-by-step solution

Identify the Known Proportion

The problem states that the prescriber orders 15 mg of medication for every 10 lb of a client’s weight. This is the known proportion, which can be represented as the fraction \( \frac{15\text{ mg}}{10\text{ lb}} \).

Set Up the Unknown Proportion

We want to find out how many mg the prescriber should order for someone who weighs 120 lb. Let's call the unknown amount of medication \( x \) mg. This can be set up as a new proportion: \( \frac{x\text{ mg}}{120\text{ lb}} \).

Write the Proportion Equation

Set the known and unknown proportions equal to each other: \[ \frac{15\text{ mg}}{10\text{ lb}} = \frac{x\text{ mg}}{120\text{ lb}} \] This equation represents the relationship where both sides of the equation are equivalent.

Solve for the Unknown Quantity

To solve for \( x \), use cross-multiplication: \[ 15 \cdot 120 = 10 \cdot x \]. Simplifying gives: \[ 1800 = 10x \]. Divide both sides by 10: \[ x = \frac{1800}{10} = 180 \].

Label and Check the Solution

Therefore, \( x = 180 \text{ mg} \). The person who weighs 120 lb should receive 180 mg of the medication. It's good to double-check that this makes sense by ensuring units are consistently used.

Key concepts

Medication Dosage Calculation

Understanding medication dosage calculation is crucial in both healthcare and pharmacology. In this concept, medication doses are adjusted based on a patient's body weight to ensure efficacy and safety. For the problem at hand, the dosage is prescribed relative to every 10 pounds of a client's weight. Hence, using proportions helps in determining the right amount for different weights.

• The dosage ordered is expressed in known units, such as milligrams (mg) per pound (lb).
• Given a patient's weight, we set up a proportion to find the unknown dosage.

Ensuring accuracy in dosage calculations prevents underdosing or overdosing, both of which can lead to ineffective treatment or adverse effects. Notice how the units of measurement are preserved throughout the calculation, maintaining consistency.

Cross-Multiplication

Cross-multiplication is a method used to solve equations where two fractions are set equal to each other. It transforms a proportion into a solvable equation. Here's why it's effective:

1. Simplifies the calculation process by removing fractions.
2. Ensures that the proportional relationship holds true by equating the cross products of the ratios.

In the original exercise, we have the equation:\[ \frac{15\text{ mg}}{10\text{ lb}} = \frac{x\text{ mg}}{120\text{ lb}} \]By cross-multiplying, we connect the known and unknown variables:\[ 15 \cdot 120 = 10 \cdot x \]This allows for easy isolation and calculation of the unknown variable \( x \). Simplifying both sides provides the result directly.

Word Problems Solving

Solving word problems involves translating a narrative into mathematical expressions, which can then be solved logically. In medication dosage problems, establishing a clear approach is essential:

• Identify key information: what is given and what needs to be found out.
• Relate the information using mathematical concepts such as rates or ratios.
• Set up an equation that represents the relationship between the variables.

For the given problem, we read the context, extract numbers, and recognize the process of dosage calculation. This approach ensures comprehensive problem-solving, ultimately enabling us to correctly interpret and calculate the required dosage.

Unit Conversion

Unit conversion is a necessary process when different units are involved in a problem. For consistent calculation, it's crucial to express all measurements within the same system.
The initial task might not require explicit conversion if units are already consistent, as seen in the exercise which uses pounds and milligrams consistently. However, understanding conversion aids in situations where inputs vary:

• Convert between weight units if needed (e.g., kilograms to pounds).
• Ensure dosage units remain the same throughout (e.g., all computations use mg).

Maintaining uniform units ensures clarity and precision, minimizing errors in the problem-solving process. Having this skill enhances flexibility in tackling real-world applications where unit variations are common.