Problem 35 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -\frac{3}{8}+\frac{5}{8} $$

Step-by-Step Solution

Answer

The sum is \(\frac{1}{4}\).

Step 1: Identify the Problem

We need to add two fractions: \(-\frac{3}{8}\) and \(\frac{5}{8}\). Both fractions have the same denominator.

Step 2: Check the Denominators

Both fractions have the same denominator, which is 8. This means we can add the numerators directly.

Step 3: Add the Numerators

The numerators are -3 and 5. The sum of these numerators is \(-3 + 5 = 2\).

Step 4: Form the Resulting Fraction

Since the sum of the numerators is 2 and the common denominator is 8, the resulting fraction is \(\frac{2}{8}\).

Step 5: Simplify the Fraction

The fraction \(\frac{2}{8}\) can be simplified by dividing both the numerator and denominator by 2, resulting in \(\frac{1}{4}\).

Key Concepts

Same Denominator

When adding fractions, it's crucial to check whether they share the same denominator. The denominator is the bottom number in a fraction, and it represents how many equal parts the whole is divided into.

If two fractions have the same denominator, it means they are divided into the same number of parts, which simplifies the addition process. You don't need to worry about converting the fractions to have a common denominator — you can immediately add the numerators.

In our example, both fractions, \(-\frac{3}{8}\) and \(\frac{5}{8}\), have a denominator of 8. Since they have the same denominator, we simply proceed to add the numerators.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This often requires dividing both the numerator and denominator by their greatest common factor (GCF).

After adding fractions, like in our example where we get \(\frac{2}{8}\), we simplify by dividing both 2 and 8 by their GCF, which in this case is 2.

By dividing both the top and bottom by 2, we obtain \(\frac{1}{4}\). Simplifying fractions makes them easier to understand and work with. It's considered good practice in mathematics to always simplify your final answer.

Numerators and Denominators

A fraction is composed of two parts: a numerator and a denominator. The numerator is the top number, indicating how many parts we have. The denominator is the bottom number, showing into how many parts the whole is divided.

In addition, like in our example where we add \(-\frac{3}{8}+\frac{5}{8}\), only the numerators \(-3\) and \(5\) get directly involved by being added together, resulting in \(2\). The denominator \(8\) remains the same throughout the process.

• Numerators: Reflect the counted parts (e.g., -3 and 5).
• Denominators: Indicate the size of each part (e.g., 8).

Understanding this relationship is key when performing operations with fractions.