Problem 146 - College Algebra Essentials

Question

$$\frac{1}{2}+\frac{2}{3}$$

Step-by-Step Solution

Answer

The sum of the fractions is \(\frac{7}{6}\).

Step 1: Identify Common Denominator

To add fractions, a common denominator is required. A common denominator of 2 and 3 is 6.

Step 2: Transform Fractions

Transform the fraction \(\frac{1}{2}\) to \(\frac{3}{6}\) and \(\frac{2}{3}\) to \(\frac{4}{6}\) using the common denominator.

Step 3: Add Transformed Fractions

Add \(\frac{3}{6}\) and \(\frac{4}{6}\) to get \(\frac{7}{6}\).

Step 4: Simplify Result

The result \(\frac{7}{6}\) is already in its simplest form.

Key Concepts

Understanding the Common Denominator

When adding fractions, it's essential to have the same denominator, also known as a common denominator. The denominator is the bottom number of a fraction, and it tells us into how many equal parts one whole is divided. If the denominators are different, we cannot add the fractions directly. Instead, we must convert them so that they both have the same denominator.

The common denominator is basically a common multiple of the two denominators. Finding it can sometimes be as easy as multiplying the two denominators together, but often, it's better to find the least common multiple (LCM) to keep calculations simpler. For example, in the exercise provided, the common denominator for 2 and 3 is 6, since 6 is the smallest number that both 2 and 3 can evenly divide.

Once we have identified the common denominator, we can adjust the fractions accordingly to ensure they are compatible for addition.

Transforming Fractions to Have a Common Denominator

To transform fractions to have a common denominator, we need to adjust both fractions without changing their value. This means multiplying their numerators and denominators by the same number so that the denominator matches the common denominator.

Let's look at the example: We transformed

• \(\frac{1}{2}\) to \(\frac{3}{6}\)
• \(\frac{2}{3}\) to \(\frac{4}{6}\)

To do this, we multiply both the numerator and the denominator of \(\frac{1}{2}\) by 3, because 6 divided by 2 equals 3. This gives us \(\frac{3}{6}\). Similarly, for \(\frac{2}{3}\), we multiply both by 2 because 6 divided by 3 equals 2, resulting in \(\frac{4}{6}\).

By doing this, both fractions now have a denominator of 6, making them ready to be added together.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. Once we have added our fractions and reached an initial result, it's time to check if this result can be simplified.

In this exercise, after adding \(\frac{3}{6}\) and \(\frac{4}{6}\), we obtained \(\frac{7}{6}\). Since 7 is a prime number that cannot be divided by 6, this fraction is already in its simplest form. However, if the numerator and denominator had any common factors, we would divide both by that factor to simplify the fraction.

Always remember to check your final answer because simplifying fractions makes them easier to interpret and work with, especially in more complex mathematical problems. Simplification is your way of ensuring clarity and precision in your results.