Problem 2 - Algebra A Combined Function

Question

Simplify by dividing the numerator by the denominator. See Examples 1 through \(6 .\) $$ \frac{19}{19} $$

Step-by-Step Solution

Answer

The fraction simplifies to 1.

Step 1: Identify the Numerator and Denominator

In the fraction \( \frac{19}{19} \), the numerator is 19 and the denominator is 19.

Step 2: Understand Division of Like Terms

When the numerator and the denominator of a fraction are the same non-zero number, their division results in 1 because any non-zero number divided by itself is 1.

Step 3: Perform the Division

Divide the numerator by the denominator: \( \frac{19}{19} = 1 \) because 19 divided by 19 equals 1.

Step 4: Final Result

After simplifying the fraction, the result is 1. There is no further simplification possible.

Key Concepts

Numerator and Denominator

In every fraction, there are two main parts: the numerator and the denominator. The numerator is the top number, while the denominator is the bottom number of the fraction. For example, in the fraction \( \frac{19}{19} \), the number 19 at the top is the numerator, and the number 19 at the bottom is the denominator. These two numbers together tell us how many parts we have (numerator) out of a total number of equal parts (denominator).

It's important to understand this basic structure, as it forms the foundation for operations like simplifying fractions. Knowing the roles of the numerator and denominator helps make sense of actions like dividing or multiplying fractions. To simplify or manipulate fractions, always start by correctly identifying these two components.

Division of Like Terms

Division of like terms is a key concept when simplifying fractions. When both the numerator and the denominator are the same, like in the fraction \( \frac{19}{19} \), they are termed 'like terms.' In such cases, you divide the numerator by the denominator to simplify the fraction.

This division is straightforward: any non-zero number divided by itself is always 1. Therefore, when you divide 19 by 19, the result is 1. This rule holds true regardless of what the number is, as long as it is non-zero. Understanding the division of like terms allows you to quickly simplify fractions and recognize when two terms cancel each other out.

Practice this concept with various fractions to become comfortable with recognizing like terms,

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factor other than 1. In \( \frac{19}{19} \), the fraction simplifies itself since both the numerator and the denominator are the same number.

Here’s the basic idea: when simplifying, you look for numbers that both the numerator and denominator can be divided by evenly (these are known as their common factors). When the numbers are identical, they share the same value as their greatest common factor, resulting in a simplified form of 1.

• If both numbers are the same, the fraction simplifies to 1.
• If there are larger numbers, break them down into their factors to find common ones.
• Once divided, ensure no further simplification is possible.

By mastering simplification, you can reduce fractions to their simplest form, making them easier to work with in various mathematical contexts.