Problem 36 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -\frac{5}{12}+\frac{7}{12} $$

Step-by-Step Solution

Answer

The sum is \(\frac{1}{6}\).

Step 1: Identify the Common Denominator

In the given exercise, the fractions are \(-\frac{5}{12}\) and \(\frac{7}{12}\). Both fractions already have the same denominator, which is 12.

Step 2: Add Numerators

Since the denominators are the same, you can directly add the numerators of the two fractions. Perform the addition: \(-5 + 7 = 2\). The resulting fraction after adding is \(\frac{2}{12}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{2}{12}\) by finding the greatest common divisor (GCD) of the numerator and the denominator, which is 2. Divide both the numerator and the denominator by 2 to simplify: \(\frac{2}{12} = \frac{1}{6}\).

Key Concepts

Common Denominator

When adding or subtracting fractions, having a common denominator, or the same bottom number for both fractions, is crucial. This allows us to combine the fractions easily. The denominator tells us how many equal parts the whole is divided into, while the numerator tells us how many parts we have. For example, in the exercise \(-\frac{5}{12}+\frac{7}{12}\), both fractions share a denominador of 12, which means they are already compatible for addition. Here’s why a common denominator is important:

• It allows us to directly compare and combine fractions.
• We can easily perform operations like addition and subtraction.
• We don’t have to convert or manipulate the fractions to align them.

When fractions already have a common denominator, like in this exercise, we save time and effort!

Adding Fractions

Once we have fractions with a common denominator, adding them becomes simple. Instead of dealing with the whole fraction, you only need to focus on the numerators—the top numbers.In our example, \(-\frac{5}{12}\) and \(\frac{7}{12}\), both fractions have the denominator 12. Therefore, we keep this denominator and only add their numerators: -5 and 7. This gives us:\[-5 + 7 = 2\]So, the combined fraction becomes \(\frac{2}{12}\). Remember, the denominator remains unchanged when adding fractions with a common denominator. Here's a recap of the steps you take:

• Check that the denominators are the same.
• Add the numerators together while keeping the denominator the same.

Simplifying Fractions

Simplifying a fraction is the process of making it as straightforward as possible. This often involves reducing the fraction to its lowest terms, which means using the smallest numbers possible for the numerator and denominator.Once you have added the fractions and reached \(\frac{2}{12}\), it’s essential to simplify it. To do this, find the greatest common divisor (GCD) of the numerator and denominator. In this case, the GCD of 2 and 12 is 2. We simplify by dividing both the numerator and denominator by this GCD:\[\frac{2}{12} = \frac{1}{6}\]Simplifying fractions helps in:

• Making problems easier to understand and solve.
• Presenting results in a clear and concise manner.
• Finding equivalent fractions that are easier to work with.

By simplifying \(\frac{2}{12}\) to \(\frac{1}{6}\), you ensure the fraction is in its most efficient form, making further calculations straightforward.