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Chapter 6: Problem 82

Perform each indicated operation. See Section 1.3 $$ \frac{2}{9}-\frac{1}{6}+\frac{2}{3} $$

Short Answer

Expert verified
The result of the operation is \( \frac{13}{18} \).

Step by step solution

01

Find the Least Common Denominator

To solve the operation \( \frac{2}{9}-\frac{1}{6}+\frac{2}{3} \), we first need to find the least common denominator (LCD). The denominators are 9, 6, and 3. The least common multiple of these numbers is 18.
02

Rewrite Each Fraction with the LCD

Convert each fraction to have the LCD as the denominator.\[ \frac{2}{9} = \frac{4}{18} \] (Multiply numerator and denominator by 2)\[ \frac{1}{6} = \frac{3}{18} \] (Multiply numerator and denominator by 3)\[ \frac{2}{3} = \frac{12}{18} \] (Multiply numerator and denominator by 6)
03

Perform the Operation

With common denominators, you can subtract and add the numerators. Perform the operation.\[ \frac{4}{18} - \frac{3}{18} + \frac{12}{18} = \frac{4 - 3 + 12}{18} = \frac{13}{18} \]
04

Simplify the Result

The fraction \( \frac{13}{18} \) is already in its simplest form because 13 and 18 have no common factors other than 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Least Common Denominator
When dealing with fractions, the least common denominator (LCD) is crucial for adding or subtracting them. The LCD is the smallest number that can be evenly divided by each of the denominators involved. In this exercise, the denominators are 9, 6, and 3. To find the LCD, follow these steps:
  • List out the multiples of each denominator.
  • Identify the smallest multiple that appears in all lists.
For our problem, the multiples shared by 9, 6, and 3 will lead us to 18, which is the least common denominator. This step ensures all fractions have the same denominator, making calculations straightforward.
Addition and Subtraction of Fractions
Once the fractions have a common denominator, adding or subtracting them becomes simple. Only the numerators need to be calculated, since the denominator remains the same. For example, in this problem, after converting, the fractions become:
  • \( \frac{2}{9} = \frac{4}{18} \)
  • \( \frac{1}{6} = \frac{3}{18} \)
  • \( \frac{2}{3} = \frac{12}{18} \)
With these conversions, perform the operations on the numerators:
  • Subtract \( \frac{3}{18} \) from \( \frac{4}{18} \), resulting in \( \frac{1}{18} \).
  • Add \( \frac{1}{18} \) to \( \frac{12}{18} \), giving \( \frac{13}{18} \).
By maintaining the LCD throughout, working with multiple fractions becomes a streamlined process.
Simplifying Fractions
The final step in working with fractions is simplifying the result. A fraction is simplified when the numerator and the denominator are as small as possible, meaning they have no common factors other than 1. In our example, the fraction \( \frac{13}{18} \) is in its simplest form because:
  • 13 is a prime number, meaning its only factors are 1 and 13.
  • 18 is composed of the factors 1, 2, 3, 6, 9, and 18.
Since 13 and 18 share no common factors, \( \frac{13}{18} \) cannot be reduced any further. Knowing whether a fraction is in its simplest form helps in ensuring accuracy and clarity in results.

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