Problem 9 - Introductory and Intermediate Algebra for College Students 4th

Question

Convert each improper fraction to a mixed number. $$\frac{76}{9}$$

Step-by-Step Solution

Answer

The mixed number for \(\frac{76}{9}\) is \(8 \frac{4}{9}\)

Step 1: Identify the whole number

To identify the whole number, we need to know how many times the denominator divides into the numerator. Starting with \(\frac{76}{9}\), we perform the division \(76 ÷ 9 = 8\) remainder \(4\). So, \(8\) is the whole number.

Step 2: Identify the new numerator

The remainder from step 1 becomes the numerator of the fractional part of the mixed number. In this case, our remainder is \(4\), so that will be the numerator of our fraction.

Step 3: Write your mixed number

We use the whole number and the remaining fraction to write our mixed number. We use the same denominator with the new numerator, the remainder of step 1. So our mixed number is \(8 \frac{4}{9}\)

Key Concepts

Mixed Numbers

Mixed numbers are a way of expressing improper fractions as a combination of a whole number and a proper fraction. An improper fraction, like \( \frac{76}{9} \), is one where the numerator is larger than the denominator. In simpler terms, it's a fraction greater than a whole number.To convert an improper fraction to a mixed number, follow these steps:

• Perform the division of the numerator by the denominator.
• The quotient (whole number from division) becomes the whole number of the mixed number.
• The remainder from this division is used as the new numerator of the fraction part of your mixed number.

So for \( \frac{76}{9} \), you divide 76 by 9 to get 8 as the whole number, and the remainder 4 becomes the new numerator. Together, they form the mixed number \( 8 \frac{4}{9} \). Understanding mixed numbers is useful for simplifying and performing calculations more easily.

Division in Fractions

Division plays a crucial role in converting improper fractions to mixed numbers. It's important to understand how division affects the numerator and the denominator relationship. In \( \frac{76}{9} \), dividing 76 by 9 is our first task. Here's what happens in this division process:

• Divide the numerator by the denominator to determine how many whole times the denominator fits into the numerator. In this example, 9 goes into 76 eight times completely.
• The whole number is 8, which becomes part of the mixed number.
• The remaining part after fitting 9 into 76 eight times is the remainder, which forms the numerator of the fractional part.

This process helps break down an improper fraction into a more easily usable mixed number. Practicing division in fractions makes handling and converting them into mixed numbers straightforward.

Numerator and Denominator Identification

Identifying the numerator and denominator is fundamental in working with fractions, particularly when converting improper fractions. The numerator is the number above the fraction line, which represents how many parts we have. The denominator is below the line, indicating how many parts make up a whole.In the improper fraction \( \frac{76}{9} \):

• 76 is the numerator, representing the total number of parts or the amount to be distributed.
• 9 is the denominator, defining how many parts are needed to form a whole.

When converting to a mixed number, understanding the roles of the numerator and denominator helps perform the division, as the denominator's role as a divisor defines the size of each whole unit. The division's remainder becomes crucial, as it transforms into the new fraction's numerator to express leftover parts accurately.