Problem 63 - Algebra A Combined Function

Question

Perform each indicated operation. See Section R .2. $$ \frac{1}{5}+\frac{4}{5} $$

Step-by-Step Solution

Answer

The sum is 1.

Step 1: Identify the Denominators

Recognize that both fractions \( \frac{1}{5} \) and \( \frac{4}{5} \) have the same denominator, which is 5. This means they can be added directly.

Step 2: Add the Numerators

Since the denominators are the same, add the numerators: \( 1 + 4 = 5 \). Maintain the common denominator 5, so the result is \( \frac{5}{5} \).

Step 3: Simplify the Fraction

The fraction \( \frac{5}{5} \) can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is 5. Thus, \( \frac{5}{5} = 1 \).

Key Concepts

Understanding the Common Denominator

When adding fractions, one of the most important steps is to ensure they have a common denominator. The denominator is the number located below the fraction line that tells us into how many parts the whole is divided. A common denominator means that two or more fractions share the same denominator, making it easier to combine them.

In our example with the fractions \( \frac{1}{5} \) and \( \frac{4}{5} \), both fractions already have 5 as their denominator. This simplifies the addition process significantly since we do not need to make any adjustments to the denominators. If the fractions had different denominators, we would first find the least common multiple (LCM) of the denominators to create equivalent fractions with a common denominator.

Steps to find a common denominator:

• Identify the denominators of all fractions involved.
• Find the least common multiple (LCM) of these denominators if they are different.
• Convert each fraction to an equivalent fraction with the LCM as the new denominator.

This approach ensures accurate addition of fractions, preventing any errors that might arise from having different denominators.

The Importance of the Numerator

In fraction addition, once we have ensured a common denominator, we focus on the numerators. The numerator is the top number of a fraction, representing how many parts of the whole you have. When adding fractions with the same denominator, you only add the numerators while keeping the denominator unchanged.

In the expression \( \frac{1}{5} + \frac{4}{5} \), we add the numerators 1 and 4 to get a total of 5. So, the fraction result before simplification becomes \( \frac{5}{5} \).

Remember:

• Add only the numerators after establishing the same denominator.
• Keep the denominator constant in the sum.

Being aware of these steps is crucial, as altering the wrong components of a fraction leads to incorrect results.

Steps to Simplify Fractions

Simplifying fractions is an essential skill that makes your answers more elegant and simpler to understand. Simplification involves reducing the fraction to its simplest form, where the numerator and denominator no longer share any common factors other than 1.

In our example, \( \frac{5}{5} \) can be simplified since both the numerator and the denominator are 5. To simplify, find the greatest common divisor (GCD) of the numerator and denominator, which in this case is 5. Divide both the numerator and denominator by the GCD to simplify the fraction. Hence, \( \frac{5}{5} = 1 \).

Easy steps for simplifying fractions:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide the numerator by the GCD.
• Divide the denominator by the GCD.
• Write the simplified fraction formed by the results.

By following these steps, you transform fractions into their simplest form, making calculations and comparisons more straightforward.