Problem 27 - Fundamentals of Mathematics

Question

(Section 5.4) Find the difference: \(3 \frac{5}{9}-2 \frac{2}{3}\).

Step-by-Step Solution

Answer

The difference is \(\frac{8}{9}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions. For \(3 \frac{5}{9}\), multiply 3 by 9 and add 5, giving \(\frac{32}{9}\). For \(2 \frac{2}{3}\), multiply 2 by 3 and add 2, giving \(\frac{8}{3}\).

Step 2: Find a Common Denominator

To subtract these fractions, they need a common denominator. The denominators are 9 and 3; the least common denominator is 9.

Step 3: Adjust Fractions to Common Denominator

Convert \(\frac{8}{3}\) to a fraction with a denominator of 9 by multiplying both the numerator and denominator by 3, giving \(\frac{24}{9}\).

Step 4: Subtract the Fractions

Now, subtract \(\frac{24}{9}\) from \(\frac{32}{9}\): \(\frac{32}{9} - \frac{24}{9} = \frac{8}{9}\).

Step 5: Simplify the Result (if necessary)

The fraction \(\frac{8}{9}\) is already in its simplest form, so no further simplification is necessary.

Key Concepts

Improper Fractions

When dealing with mixed numbers, it can be easier to use improper fractions to perform operations like addition or subtraction. Mixed numbers are numbers that have both a whole number part and a fractional part, like \(3 \frac{5}{9}\) or \(2 \frac{2}{3}\). To convert a mixed number into an improper fraction, you follow a simple process:

• Multiply the whole number by the denominator of the fractional part.
• Add this result to the numerator of the fractional part.
• This sum becomes the new numerator, and the denominator stays the same.

For example, the mixed number \(3 \frac{5}{9}\) converts to an improper fraction by multiplying 3 by 9 to get 27, then adding 5 to get 32, resulting in \(\frac{32}{9}\). Similarly, \(2 \frac{2}{3}\) becomes \(\frac{8}{3}\). This way, it becomes easier to perform arithmetic calculations.

Least Common Denominator

Finding a common denominator is crucial when performing operations like adding or subtracting fractions. The least common denominator (LCD) is the smallest number that can be used as a common multiple for the denominators in the fractions involved. This ensures that the fractions are compatible for addition or subtraction.

In the exercise, we need to find the LCD for the fractions \(\frac{32}{9}\) and \(\frac{8}{3}\). The denominators are 9 and 3. The smallest number that is a multiple of both is 9, making it the least common denominator.

• Identify the denominators in both fractions.
• Find the smallest number that both denominators can divide into without leaving a remainder (their least common multiple).
• Use this number to adjust the fractions to equivalent fractions that can be easily subtracted.

Mastering the skill of finding the least common denominator helps streamline the process of working with fractions.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form where the numerator and the denominator have no common factors other than 1. This is important because it makes the fraction easier to understand and use in further calculations.

In our exercise, after performing the subtraction, we end up with \(\frac{8}{9}\). To check if the fraction is in its simplest form:

• Identify any common factors between the numerator and the denominator.
• If there are none other than 1, the fraction is already simplified.
• If there are, divide both by the greatest common factor.

Since 8 and 9 have no common factors other than 1, \(\frac{8}{9}\) is already simplified. Simplifying fractions helps in maintaining clarity and precision in mathematical operations.