Problem 5 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(6 \frac{8}{9}-3 \frac{4}{9}\)

Step-by-Step Solution

Answer

The result of the subtraction is \(3 \frac{4}{9}\).

Step 1: Convert Mixed Numbers to Improper Fractions

In order to make subtraction easier, we first convert both mixed numbers into improper fractions. A mixed number like \(6 \frac{8}{9}\) can be converted by multiplying the whole number by the denominator (\(6 \times 9 = 54\), then adding the numerator (\(8\)) to get \( \frac{62}{9} \). Similarly, convert \(3 \frac{4}{9}\) into \( \frac{31}{9} \).

Step 2: Subtract Improper Fractions

Since both fractions have the same denominator, simply subtract the numerators: \(62 - 31 = 31\). The result is \( \frac{31}{9} \).

Step 3: Convert Improper Fraction to Mixed Number

Convert \( \frac{31}{9} \) back into a mixed number by dividing \(31\) by \(9\). The quotient is \(3\) (the whole number part), and the remainder is \(4\), giving us a mixed number of \(3 \frac{4}{9}\).

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. This means that the value of the fraction is equal to or greater than 1.
Improper fractions are useful in mathematical operations such as addition, subtraction, multiplication, and division because they simplify complex expressions. For example, converting a mixed number like \(6 \frac{8}{9}\) to an improper fraction involves two simple steps:

• Multiply the whole number (6) by the denominator (9): \(6 \times 9 = 54\).
• Add the numerator (8): \(54 + 8 = 62\).

This gives us the improper fraction \(\frac{62}{9}\). Converting mixed numbers to improper fractions allows for straightforward calculations, especially when the fractions share a common denominator.

Subtraction of Fractions

Subtracting fractions, particularly when dealing with similar denominators, is a straightforward process. If the fractions you're working with share a common denominator, subtraction focuses on the numerators alone. This allows for direct subtraction without the need for further adjustment of the fractions involved. For example:
Given \(\frac{62}{9}\) and \(\frac{31}{9}\), both fractions have a denominator of 9. To find the difference:

• Subtract the numerators: \(62 - 31 = 31\).
• Keep the denominator the same: 9.

This results in the fraction \(\frac{31}{9}\). Using fractions with common denominators can significantly streamline the subtraction process, making calculations efficient and error-free.

Converting Mixed Numbers

Converting mixed numbers from improper fractions is a common step in maintaining clarity in mathematical results. After simplifying operations, such as subtraction, an improper fraction like \(\frac{31}{9}\) can be converted to a mixed number. Here’s how:

• Divide the numerator (31) by the denominator (9) to get the whole number: 31 divided by 9 is 3, with a remainder of 4.
• The quotient (3) becomes the whole number part of the mixed number.
• The remainder (4) is the new numerator, over the original denominator (9), forming the fraction part.

Thus, \(\frac{31}{9}\) converts to \(3 \frac{4}{9}\). This conversion helps in better understanding the size and value of the fraction, especially while communicating results or interpreting real-world measurements.