Question

Set up the following problems as a proportion and solve. Include labels in the set up and on the answer. If 150 milligrams (mg) of medication is in 2 capsules (caps), how many mg of medication is in 10 caps?

Short answer

There are 750 mg of medication in 10 capsules.

Step-by-step solution

Understand the Problem

We need to determine how many milligrams of medication are in 10 capsules, given that 150 milligrams are in 2 capsules.

Set Up the Proportion

We set up the proportion by comparing the number of capsules to the amount of medication. Write: \( \frac{150 \text{ mg}}{2 \text{ caps}} = \frac{x \text{ mg}}{10 \text{ caps}} \), where \( x \) is the unknown amount of medication in 10 capsules.

Cross-Multiply to Solve for x

To solve the proportion, cross-multiply the fractions. That gives us \( 150 \times 10 = 2 \times x \).

Solve the Equation for x

Calculate \( 1500 = 2x \) and then divide both sides by 2 to find \( x \). This results in \( x = \frac{1500}{2} = 750 \).

Label the Answer

Now that we have calculated \( x = 750 \), label the answer properly: There are 750 milligrams in 10 capsules.

Key concepts

Cross-Multiplication

Cross-multiplication is a method used to simplify and solve equations that involve proportions. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication allows you to solve for one of the variables by multiplying diagonally across the equal sign. In our example, we set up the proportion \( \frac{150 \text{ mg}}{2 \text{ caps}} = \frac{x \text{ mg}}{10 \text{ caps}} \). To solve this, we multiply 150 mg by 10 caps, and 2 caps by \( x \) mg, resulting in the equation \( 150 \times 10 = 2 \times x \).

• Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other.
• This step is crucial for isolating and solving for the unknown variable.

Using cross-multiplication simplifies the proportion to a simple algebra equation, making it easier to solve.

Proportional Reasoning

Proportional reasoning is the ability to see multiplicative relationships between quantities. It's a foundational skill in mathematics that allows us to solve problems relating to ratios and proportions. In this exercise, we understand that if 150 mg is in 2 capsules, the amount in 10 capsules should maintain the same ratio.

• Think of a ratio as a consistent relationship between two quantities.
• Maintaining this relationship helps ensure that any calculation follows the original conditions set through the initial ratio.

Proportional reasoning lets you intuitively set up and anticipate outcomes in real-life mathematical problems, making it an essential concept to grasp.

Mathematical Problem Solving

Mathematical problem solving involves using various methods and strategies to find a solution to a given problem. In this exercise, the goal is to determine how many milligrams are in 10 capsules when given the amount in 2 capsules.

• Identify what you need to find out or solve.
• Choose an appropriate method, in this case, using proportions and cross-multiplication.

Following these steps methodically helps you work through the problem efficiently, often revealing insights into the nature of mathematical relationships. It’s about applying logical thinking to tackle mathematical challenges.

Unit Labeling in Calculations

Unit labeling in calculations is crucial for clarity and accuracy, especially in mathematical problems involving measurements. Throughout the solution, labeling each term with its proper unit, like milligrams (mg) and capsules (caps), was essential.

• Helps avoid mix-ups and ensures that each part of the equation represents the correct type of quantity.
• Units provide a clear understanding of what numbers mean and how they relate to the problem being solved.

It's easy to misinterpret numbers without unit labels, but keeping them in your calculations helps prevent errors and ensures a precise, meaningful solution.