Problem 71 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{2}{3}+\frac{5}{18}-\frac{1}{6} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{7}{9} \).

Step 1: Determine the Least Common Denominator (LCD)

To add or subtract fractions, they must have a common denominator. The denominators we have are 3, 18, and 6. The Least Common Denominator (LCD) is the smallest number that each of these denominators can divide into without a remainder. In this case, the LCD is 18.

Step 2: Convert Each Fraction to the LCD

Convert each fraction to have the denominator of 18. - The first fraction is \( \frac{2}{3} \). To convert it, multiply both the numerator and the denominator by 6: \[ \frac{2}{3} \times \frac{6}{6} = \frac{12}{18} \] - The second fraction is already \( \frac{5}{18} \), so it stays the same.- The third fraction is \( \frac{1}{6} \). To convert it, multiply both the numerator and the denominator by 3: \[ \frac{1}{6} \times \frac{3}{3} = \frac{3}{18} \]

Step 3: Perform the Addition and Subtraction of the Fractions

Now that all fractions have a common denominator, simply add and subtract the numerators. \[ \frac{12}{18} + \frac{5}{18} - \frac{3}{18} = \frac{12 + 5 - 3}{18} = \frac{14}{18} \]

Step 4: Simplify the Resulting Fraction

Simplify \( \frac{14}{18} \) by finding the Greatest Common Divisor (GCD) of the numerator and the denominator. The GCD of 14 and 18 is 2. Divide both the numerator and the denominator by 2:\[ \frac{14 \div 2}{18 \div 2} = \frac{7}{9} \]

Step 5: Conclusion: Present the Simplified Result

The simplified result of the operation \( \frac{2}{3} + \frac{5}{18} - \frac{1}{6} \) is \( \frac{7}{9} \).

Key Concepts

Least Common Denominator

When it comes to adding or subtracting fractions, having a shared denominator is essential. This is where the concept of the Least Common Denominator (LCD) comes into play. The LCD is the smallest number that all denominators of the fractions involved can divide into evenly. Think of it as the smallest common *multiple* that the denominators share. For example, if your denominators are 3, 18, and 6, the LCD would be 18 because 18 is the smallest number that 3, 6, and 18 can all divide into without leaving a remainder.

To find the LCD:

• Write down the multiples of each denominator.
• Identify the smallest common multiple among them.

Having the LCD makes the fractions compatible for addition or subtraction as it transforms them into a uniform structure that is easy to work with.

Greatest Common Divisor

Simplifying fractions once you obtain the result is where the Greatest Common Divisor (GCD) shines. The GCD is the largest number that divides both the numerator and the denominator without a remainder. It helps you reduce fractions to their simplest form. In the exercise solution, you found the fraction \( \frac{14}{18} \). To simplify it, you determined the GCD of 14 and 18, which turned out to be 2.

To simplify a fraction using the GCD, follow these steps:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

For instance, dividing both 14 and 18 by their GCD (which is 2) gives you \( \frac{14 \div 2}{18 \div 2} = \frac{7}{9} \). This step ensures that fractions are expressed in their simplest and most understandable form.

Adding and Subtracting Fractions

Adding or subtracting fractions requires a methodical approach once you have a common denominator. Once your fractions share the same denominator, you only need to add or subtract the numerators, while keeping the denominator the same. This is because the denominator indicates how many equal parts the whole is divided into, so it remains constant.

Let's break down the process:

• Start by writing the fractions with their new common denominator.
• Add or subtract the numerators as indicated.
• Keep the denominator unchanged.

In our exercise, the fractions \( \frac{12}{18} \), \( \frac{5}{18} \), and \( \frac{3}{18} \) shared the LCD of 18. Therefore, you calculated \( 12 + 5 - 3 \) for the numerators, yielding \( \frac{14}{18} \). This method ensures precision in the final result, keeping the process straightforward and consistent.