Question

One tablet contains 200 milligrams of pain medication. How many milligrams are in \(31 / 2\) tablets? ______

Short answer

700 milligrams

Step-by-step solution

Understand the Problem

We need to find out the total amount of pain medication in \(31/2\) tablets. Each tablet contains 200 milligrams of medication.

Convert Mixed Number to Improper Fraction

Convert the mixed number \(31/2\) into an improper fraction. \(31/2 = 3 \times 2 + 1 = 7/2\).

Calculate Total Milligrams

Multiply the improper fraction representing the number of tablets by the milligrams per tablet. \[\frac{7}{2} \times 200 = \frac{7 \times 200}{2} = \frac{1400}{2} = 700\]

Verify the Calculation

Ensure the calculation of 700 milligrams is correct by double-checking the multiplication process.

Key concepts

Fraction Conversion

Understanding how to convert mixed numbers into improper fractions is an essential skill in mathematics, especially for solving problems that involve operations with fractions. A mixed number like \(31/2\) consists of both a whole number and a fraction. To convert this into an improper fraction, follow these simple steps:

• Multiply the whole number (3) by the denominator (2) of the fractional part. This gives \(3 \times 2 = 6\).
• Add this result to the numerator of the fraction (1), resulting in \(6 + 1 = 7\).
• Write this sum over the denominator of the fractional part. Thus, \(31/2\) becomes \(7/2\).

This process allows you to easily use the fraction in further calculations, such as multiplication.

Multiplication

In mathematics, multiplication is a basic operation that represents repeated addition. When multiplying fractions by whole numbers or vice versa, it's essential to treat the whole number as a fraction by giving it a denominator of 1. Here’s how you multiply the improper fraction with a whole number:

• First, write the whole number as a fraction. For example, 200 becomes \(\frac{200}{1}\).
• Multiply the numerators together: \(7 \times 200 = 1400\).
• Multiply the denominators together: \(2 \times 1 = 2\).
• Combine the results to get \(\frac{1400}{2}\).

Finally, simplify the fraction if necessary to achieve the final result. In our case, dividing 1400 by 2 gives 700 milligrams.

Measurement Units

Measurement units help us understand the quantities of substances, like the amount of active ingredient in medications. In this problem, milligrams are used, which is a common unit for measuring small masses. Understanding these units and how to manipulate them is crucial:

• A milligram (mg) is one-thousandth of a gram (g).
• Being familiar with these units allows for correct conversions and dose calculations.
• You can multiply or divide milligrams, just like numbers, to find total quantities.

When calculating medication doses, always ensure that you are using the correct unit conversions and double-check your results to avoid any errors.

Problem Solving

Effective problem solving in mathematics requires a structured approach. The steps provided in this exercise guide you through tackling such problems efficiently:

• First, understand what is being asked. Identify known quantities, such as 200 mg per tablet in this question.
• Use the information to set up your calculations after converting any mixed numbers.
• Perform the calculations step-by-step to find the total amount, ensuring the process is clear and free of errors.
• Verify your work. Double-check calculations to ensure accuracy, as errors can significantly impact results.

Following these methods helps in breaking down complex problems into manageable parts, leading to successful outcomes in both academic and real-world situations.