Problem 74 - Algebra 2

Question

Evaluate each expression. \(\frac{1}{2}+\frac{1}{3}\)

Step-by-Step Solution

Answer

\(\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\) in simplest form.

Step 1: Find a Common Denominator

Identify the denominators of the fractions, which are 2 and 3. The smallest common denominator between them is 6.

Step 2: Convert Each Fraction

Convert \(\frac{1}{2}\) and \(\frac{1}{3}\) to equivalent fractions with a denominator of 6. To do this: For \(\frac{1}{2}\), multiply the numerator and the denominator by 3 to get \(\frac{3}{6}\). For \(\frac{1}{3}\), multiply the numerator and the denominator by 2 to get \(\frac{2}{6}\).

Step 3: Add the Fractions

Now, add the two equivalent fractions \(\frac{3}{6}\) and \(\frac{2}{6}\). Since they have the same denominator, add the numerators directly: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Step 4: Simplify the Result

Check if \(\frac{5}{6}\) can be simplified. Since 5 and 6 have no common factors other than 1, \(\frac{5}{6}\) is already in its simplest form.

Key Concepts

Understanding Common Denominator

When you add fractions, especially those with different denominators, finding a common denominator is crucial. A common denominator is the least common multiple of the denominators of the fractions involved. For the fractions \(\frac{1}{2}\) and \(\frac{1}{3}\), the denominators are 2 and 3. The smallest number that both 2 and 3 can be multiplied into evenly is 6. This makes 6 the common denominator.

• Calculating a common denominator ensures that both fractions share the same base number before you proceed to add them.
• This step is fundamental because it standardizes the fractions, making them compatible for addition.

Once you establish a common denominator, you can proceed to adjust the fractions accordingly. This will set the stage for a smooth addition process.

Equivalent Fractions Explained

Equivalent fractions are fractions that may look different but represent the same value or proportion. You find equivalent fractions by multiplying or dividing the numerator and denominator of a fraction by the same number. In the exercise, converting \(\frac{1}{2}\) to an equivalent fraction with a denominator of 6 involves multiplying both the numerator and denominator by 3, resulting in \(\frac{3}{6}\). Similarly, for \(\frac{1}{3}\), multiply both by 2 to get \(\frac{2}{6}\).

• Equivalent fractions are vital in performing operations such as addition or subtraction when fractions have different denominators.
• They allow us to rewrite fractions without changing their value, aligning the denominators as needed.

Understanding equivalent fractions helps you manipulate fractions into a form that facilitates the math without altering the fractions' inherent value.

Simplifying Fractions Made Easy

After adding or subtracting fractions, the resulting fraction might not always be in its simplest form. Simplifying a fraction involves finding the greatest common factor (GCF) of the numerator and the denominator and dividing both by that number. For example, in the result \(\frac{5}{6}\), the numbers 5 and 6 do not share any common factors besides 1. Hence, \(\frac{5}{6}\) is already as simple as it can get.

• Simplifying fractions helps to ensure your answer is in the cleanest, most understandable form.
• It often involves checking for any common factors and reducing them as much as possible, while maintaining the fractional value.

By simplifying, you achieve a neater answer that's easier to work with in future calculations or interpretations.