Problem 16 - 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 A tablet has a dosage strength of 35 mg. What was the prescribed dosage if the patient was told to take 1.5 tablets?

Step-by-Step Solution

Answer

The prescribed dosage is 52.5 mg.

Step 1: Understand the Problem

The problem states that one tablet has a dosage of 35 mg. The patient is instructed to take 1.5 tablets. We are asked to find out the total dosage prescribed.

Step 2: Define the Relationship

The relationship involves dosage per tablet and the number of tablets taken. We need to express this relationship as a proportion between the dosage for 1 tablet and the dosage for 1.5 tablets.

Step 3: Set Up the Proportion

The dosage of 1 tablet is 35 mg. Therefore, we can write the proportion as: \( \frac{35}{1} = \frac{x}{1.5} \), where \(x\) is the total dosage for 1.5 tablets.

Step 4: Solve the Proportion

To find \(x\), cross-multiply to get \(35 \times 1.5 = x \times 1\). Simplify the equation: \(x = 35 \times 1.5\). Calculating gives \(x = 52.5\).

Step 5: Conclusion

The prescribed dosage when the patient takes 1.5 tablets is 52.5 mg.

Key Concepts

Understanding Word Problems

Word problems can often be tricky. They require you to translate a real-life scenario into mathematical expressions. This is what we call setting up a proportion. Think of word problems like puzzles. You need to identify the pieces and see how they fit together. In our exercise, the word problem describes a situation where a specific dosage is prescribed. The key is to understand the relationships between different numbers.

In this particular problem, the two numbers you’re dealing with are the dosage per tablet and the number of tablets taken. The challenge is to set up a proportion that properly represents this relationship. This involves associating the known dose of one tablet with the unknown total dosage for 1.5 tablets. This method is useful because proportions allow you to compare two ratios to solve for an unknown quantity.

Dosage Calculation Basics

Dosage calculation involves determining the amount of medication a patient needs based on several factors, such as the concentration of a drug and the prescribed quantity. It is crucial in the healthcare field because incorrect dosages can lead to serious side effects. To perform dosage calculations, use proportions to relate different parts of the situation in a straightforward way.

In our example, the problem tells us that one tablet contains 35 mg of the medication. To find out the total dosage for 1.5 tablets, you multiply the dosage of one tablet by the number of tablets taken. Basically, you're scaling up the known dosage to find the desired total. This technique ensures that the medication is administered safely and effectively.

• Understand known values: What is already provided (the dosage for one tablet).
• Identify what needs to be found: The total dosage for the specified number of tablets.
• Set up a proportion that represents the relationship.

Cross-Multiplication Technique

Cross-multiplication is a method used to solve proportions. It is especially useful when dealing with dosage calculations or any problem involving ratios. In this technique, you multiply across the proportion to eliminate the denominator and solve for the unknown quantity.

When you set up the proportion \( \rac{35}{1} = \rac{x}{1.5} \), you're saying that the ratio of the dosage to one tablet should be the same as the dosage to 1.5 tablets. To find \(x\), cross-multiply:

\(35 \times 1.5 = x \times 1\). This effectively means you're computing the same operation on both sides to find the value for \(x\).

As in our example, solving this gives you the desired result of 52.5 mg. Cross-multiplication simplifies the process and helps ensure that the calculations adhere to proper mathematical principles, yielding accurate results.