Problem 140 - Introductory Algebra for College Students

Question

Explain how to add fractions with different denominators. Use \(\frac{5}{6}+\frac{1}{2}\) as an example.

Step-by-Step Solution

Answer

The sum of \(\frac{5}{6}\) and \(\frac{1}{2}\) is \(\frac{4}{3}\).

Step 1: Understanding Fraction Addition

When adding fractions, if the fractions have the same denominator, then you can just add the numerators. However, if they have different denominators, like in the case of \(\frac{5}{6}+\frac{1}{2}\), we need to first find a common denominator.

Step 2: Finding a Common Denominator

A common denominator could be any multiple of the individual denominators, but often the least common denominator is preferable. It simplifies the eventual fraction. The least common multiple of 2 and 6 is 6, which is our common denominator.

Step 3: Converting Fractions to Equivalent Forms

The fraction \(\frac{1}{2}\) has to be transformed to its equivalent form with the common denominator. Multiply both the numerator and denominator of this fraction by 3 to get its equivalent fraction with denominator 6. The fraction becomes \(\frac{3}{6}\). The original \(\frac{5}{6}\) already has 6 as its denominator, so it remains unchanged.

Step 4: Adding the Converted Fractions

Now both fractions have the same denominator. Add the numerators: \(5 + 3 = 8\). Thus, \(\frac{5}{6}+\frac{1}{2} =\frac{8}{6}\).

Step 5: Simplifying the Result

The resulting fraction can be simplified further by dividing the numerator and denominator by their greatest common factor, which is 2. \(\frac{8}{6} = \frac{4}{3}\).