Problem 16 - Prealgebra

Question

Find the following sums and differences, and reduce to lowest terms. (Add or subtract as indicated.) $$\frac{5}{a}+\frac{4}{a}+\frac{3}{a}$$

Step-by-Step Solution

Answer

\(\frac{12}{a}\) is the sum in its simplest form.

Step 1: Identify the Common Denominator

Since all the fractions have the same denominator, which is \(a\), we can add them directly without any further adjustments.

Step 2: Add the Numerators

Add the numerators of the fractions: \(5 + 4 + 3\). This results in \(12\).

Step 3: Combine the Fractions

Combine the fractions by using the common denominator \(a\): \(\frac{12}{a}\).

Step 4: Simplify (if possible)

Check if the fraction can be simplified. Since there are no common factors between 12 and \(a\), the fraction \(\frac{12}{a}\) is in its simplest form.

Key Concepts

Common Denominator

In fraction addition, the common denominator is key to combining fractions effectively. When fractions share the same denominator, it becomes much simpler to add or subtract them. A denominator is the bottom part of a fraction that tells us how many equal parts the whole is divided into. When you're dealing with multiple fractions that already have a common denominator, such as in our example where all fractions have the denominator \(a\), you don't need to modify anything.

• Identify if the denominators of the fractions are the same.
• If they're different, you would find a common multiple of the denominators.
• In this exercise, the common denominator is already \(a\), so we can proceed to add the numerators directly.

Understanding the concept of common denominators helps streamline the process of fraction addition, letting you focus on adding the numerators.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their most basic form. It involves making both the numerator and the denominator as small as possible while still keeping the value of the fraction the same. In the context of our problem, we arrived at the fraction \(\frac{12}{a}\). To determine if a fraction can be simplified, you need to see if the numerator and denominator share any common factors other than 1.

• If the greatest common factor (GCF) is 1, your fraction is already in its simplest form.
• In our example, since \(12\) and \(a\) do not share common factors, \(\frac{12}{a}\) is the simplest form.

Reaching the simplest form is important for clarity and ease of use in further mathematical operations.

Numerator Addition

Adding numerators is an integral step when dealing with fraction addition, especially when fractions share the same denominator. The numerator is the top part of a fraction that describes how many parts are being considered. To add the numerators:

• Simply add them together if the fractions have a common denominator.
• In our example, the numerators are \(5\), \(4\), and \(3\). When we add these, we get \(12\).

Once the numerators are added, the result \(12\) becomes the numerator of the new fraction \(\frac{12}{a}\). This simplification step demonstrates the ease of fraction addition when you have already determined the common denominator.