Problem 20 - Intermediate Algebra

Question

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{3}{4 y}+\frac{8}{4 y}$$

Step-by-Step Solution

Answer

\(\frac{11}{4y}\) is the simplified result.

Step 1: Identify Common Denominator

Both fractions have the same denominator, which is \(4y\). This makes it possible to add them directly.

Step 2: Add the Numerators

Since the denominators are the same, add the numerators: \(3 + 8 = 11\).

Step 3: Write the Resulting Fraction

The resulting fraction is \(\frac{11}{4y}\).

Step 4: Simplify the Fraction

The fraction \(\frac{11}{4y}\) is already in its simplest form since 11 is a prime number and does not share any common factors with 4 or \(y\).

Key Concepts

Common Denominator

When dealing with algebraic fractions, one of the first steps in combining them—either through addition or subtraction—is identifying a common denominator. A common denominator is a shared denominator between two or more fractions. In the exercise above, both fractions have the denominator \(4y\). This is advantageous because it allows for straightforward computation without altering the fractions themselves.

• To find a common denominator, if the denominators were different, you would typically find the least common multiple (LCM) of the denominators.
• Once established, the fractions can be rewritten to share this denominator, preparing them for the next steps of addition or subtraction.

Identifying a common denominator is crucial because it sets the foundation for properly combining fractions.

Numerator Addition

Once the common denominator is identified or agreed upon, you proceed to the next step: adding the numerators. With the fractions sharing the same denominator, focus shifts to their numerators. In the above exercise, adding the numerators is straightforward:

• First, observe that the denominator (here, \(4y\)) remains unchanged in this operation.
• Add only the numerators. For example, \(3\) and \(8\) are the numerators, as shown: \(3 + 8 = 11\).

Cleaning up the numerators is an essential step because it determines the new numerator of your resulting fraction. Always remember: only numerators are added or subtracted; the denominator remains constant.

Fraction Simplification

Now that you've successfully added the fractions, the next step is considering simplification. However, not all fractions can be simplified, so it’s vital to understand when a fraction is already in its simplest form.

• The fraction \(\frac{11}{4y}\) in our example is already simplified. This is because the numerator \(11\) is a prime number, meaning it only has two divisors: 1 and itself.
• Check if there is any common factor between the numerator and the denominator. In this case, there is none between \(11\) and \(4y\).
• If simplification is necessary, you would divide both the numerator and the denominator by their greatest common divisor (GCD).

The simplification step ensures that your solution is presented in the most reduced form, making it clean and easy to work with in further operations or comparisons.