Questions: Find the LCD of the set of fractions. (Simplify your answer completely.) 1/3, 5/14, 4/21

Find the LCD of the set of fractions. (Simplify your answer completely.) 1/3, 5/14, 4/21
Transcript text: Find the LCD of the set of fractions. (Simplify your answer completely.) \[ \frac{1}{3}, \frac{5}{14}, \frac{4}{21} \]
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Solution

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Solution Steps

Solution Approach

  1. Finding the LCD of the set of fractions: To find the least common denominator (LCD) of the fractions \(\frac{1}{3}\), \(\frac{5}{14}\), and \(\frac{4}{21}\), we need to determine the least common multiple (LCM) of the denominators 3, 14, and 21. The LCM is the smallest number that is a multiple of each of the denominators.

  2. Performing the indicated operation and simplifying: To add the fractions \(\frac{2}{3}\) and \(\frac{1}{6}\), we first need to find a common denominator. The least common denominator of 3 and 6 is 6. We then convert each fraction to have this common denominator and perform the addition.

Step 1: Find the Least Common Denominator (LCD)

To find the least common denominator of the fractions \(\frac{1}{3}\), \(\frac{5}{14}\), and \(\frac{4}{21}\), we need to determine the least common multiple (LCM) of the denominators 3, 14, and 21. The LCM of these numbers is 42. Therefore, the least common denominator is \(42\).

Step 2: Perform the Addition of Fractions

To add the fractions \(\frac{2}{3}\) and \(\frac{1}{6}\), we first find a common denominator. The least common denominator of 3 and 6 is 6. We convert each fraction to have this common denominator:

\[ \frac{2}{3} = \frac{4}{6} \]

\[ \frac{1}{6} = \frac{1}{6} \]

Now, we can add the fractions:

\[ \frac{4}{6} + \frac{1}{6} = \frac{5}{6} \]

Final Answer

  • The least common denominator of the fractions \(\frac{1}{3}\), \(\frac{5}{14}\), and \(\frac{4}{21}\) is \(\boxed{42}\).
  • The result of the addition \(\frac{2}{3} + \frac{1}{6}\) is \(\boxed{\frac{5}{6}}\).
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