Problem 5 - Algebra A Combined Function

Question

Simplify by dividing the numerator by the denominator. See Examples 1 through \(6 .\) $$ \frac{13}{1} $$

Step-by-Step Solution

Answer

The simplified form is 13.

Step 1: Identify the Numerator and Denominator

In the fraction \( \frac{13}{1} \), the number on top (13) is the numerator and the number on the bottom (1) is the denominator.

Step 2: Perform Division

To simplify the fraction, divide the numerator by the denominator: \( 13 \div 1 = 13 \).

Step 3: Write the Simplified Form

Since dividing by 1 yields the original number, the simplified form of \( \frac{13}{1} \) is simply 13.

Key Concepts

Numerator and Denominator

When dealing with fractions, it's crucial to understand the two key components: the numerator and the denominator. The fraction is written as \( \frac{a}{b} \), where:

• The **numerator** is the top number and indicates the number of parts we have.
• The **denominator** is the bottom number and signifies how many equal parts the whole is divided into.

To visualize this, think about slicing a pizza into equal pieces. If you have a pizza cut into 4 pieces and you eat 3, the fraction of the pizza eaten is \( \frac{3}{4} \), where 3 is the numerator and 4 is the denominator.
Recognizing the numerator and denominator is the starting point for any fraction-related computation, including simplification.

Division of Fractions

The process of simplifying a fraction often involves performing division. This is especially simple when the fraction's denominator is 1, as in the example \( \frac{13}{1} \).

• To simplify, divide the numerator by the denominator.
• Division here involves checking how many times the denominator can fit into the numerator.

For our example, dividing 13 by 1 is straightforward, resulting in 13. This is equivalent to saying that 13 can wholly fit into itself once because any number divided by 1 remains unchanged.
This step is always crucial as it reduces fractions to their simplest form and makes computations much more manageable.

Fraction Simplification Process

As you simplify fractions, the goal is to reduce them to their simplest form, where no whole numbers can divide both the numerator and denominator evenly except 1. In our example \( \frac{13}{1} \), the fraction is already in its simplest form since no smaller numbers besides 1 are shared factors of both 13 and 1.

• If the denominator is 1, as seen, the process is done instantly by rewriting the fraction as the numerator itself.
• For different fractions, identify any common factors in the numerator and denominator.
• Divide both parts of the fraction by the greatest common divisor (GCD) if applicable.

The simplification makes fractions easier to understand and more straightforward to use in equations or real-world applications. A simplified fraction represents the same value, just in a cleaner, more concise way.