Problem 8 - Introductory and Intermediate Algebra for College Students 4th

Question

Convert each improper fraction to a mixed number. $$\frac{47}{8}$$

Step-by-Step Solution

Answer

The mixed number form of the improper fraction \( \frac{47}{8} \) is 5 \( \frac{7}{8} \)

Step 1: Identify the Improper Fraction

The given improper fraction is \( \frac{47}{8} \)

Step 2: Divide the Numerator by the Denominator

Here, divide the numerator 47 by the denominator 8. The result is 5 with a remainder of 7. Therefore, 5 is the whole number and 7 is the new numerator.

Step 3: Write the Result as a Mixed Number

The given improper fraction \( \frac{47}{8} \) can be written as the mixed number 5 \( \frac{7}{8} \)

Key Concepts

Mixed Numbers

Converting an improper fraction to a mixed number is an essential skill in math education. The concept of mixed numbers is crucial for understanding real-world problems involving quantities. A mixed number consists of two components: a whole number and a fraction. The whole number represents the complete groups that can be formed, while the fraction denotes the leftover parts. For students, understanding mixed numbers simplifies various mathematical tasks, such as measurements and everyday calculations.

• Whole Number: Represents fully completed parts.
• Fraction: Represents the parts that are incomplete.

When you convert an improper fraction, like \( \frac{47}{8} \), into a mixed number, you identify how many entire groups of 8 fit into 47. Any remainder becomes the numerator of the fraction in the mixed number.

Numerator and Denominator

In fractions, the terms numerator and denominator play pivotal roles. The numerator is the top number, indicating how many parts are being considered, while the denominator, the bottom number, shows the total parts that make up a whole. These two elements form the foundation of fractions, including improper fractions like \( \frac{47}{8} \).

• Numerator: Indicates the number of selected parts.
• Denominator: Displays the total number of equal parts into which the whole is divided.

Understanding the relationship between the numerator and the denominator allows students to identify if a fraction is proper or improper. In an improper fraction, the numerator is greater than the denominator, implying there is more than one whole.

Division Process

Division is a key operation in converting improper fractions to mixed numbers. Here, the numerator is divided by the denominator to separate whole numbers from the remainder. Using the fraction \( \frac{47}{8} \) as an example:

• Step 1: Divide 47 by 8 to get 5 as the quotient.
• Step 2: The remainder after division is 7.

This division shows that 8 fits 5 whole times into 47, with 7 left over. Thus, the mixed number representation becomes 5 \( \frac{7}{8} \). This process simplifies complex fractions, making them easier to visualize and use in calculations. Always remember:

• Quotient: The whole number part of the mixed number.
• Remainder: The numerator of the fractional part.

Math Education

Grasping fractions and their conversions form a fundamental part of math education. By learning to turn improper fractions into mixed numbers, students develop a deeper understanding of numerical relationships and fractions' flexibility. This skill extends beyond the classroom into practical applications, such as cooking, building, and other daily tasks that involve measuring and dividing.

• Promotes Numerical Literacy: Enhances ability to manipulate and use numbers in various contexts.
• Improves Problem-Solving Skills: Allows easier manipulation of numbers to solve real-world problems.

Teaching students how to convert improper fractions to mixed numbers also aids in developing critical thinking. It encourages identifying patterns, synthesizing mathematical principles, and applying solutions practically. This teaching method fosters an appreciation for math that is both enjoyable and beneficial to their everyday life.