Problem 62 - Algebra 1

Question

Simplify the fraction. $$\frac{5 x y}{5 x^{2}}$$

Step-by-Step Solution

Answer

The simplified form of the fraction is \(\frac{y}{x}\).

Step 1: Identify common factors in the numerator and the denominator

Notice the common factor of \(5\) and \(x\) in both the numerator \(5xy\) and the denominator \(5x^2\). Identify these common factors for cancelation.

Step 2: Cancel the common factors

Remove the common factors from both the numerator and the denominator. This is equivalent to dividing both the numerator and the denominator by those factors. So you have \(\frac{(5xy)/(5x)}{(5x^2)/(5x)}\).

Step 3: Simplify the fraction

After dividing both parts of the fractions by the common factor, \(y\) remains in the numerator and \(x\) remains in the denominator. Thus, the simplified form of the given fraction is \(\frac{y}{x}\).

Key Concepts

Common Factors

When it comes to simplifying fractions, identifying common factors is a crucial first step. Common factors are numbers or variables that divide both the numerator and the denominator without leaving any remainder. They are elements that both parts of the fraction share. Identifying these shared elements allows us to simplify the fraction by removing them from both the top and bottom parts.
For the given fraction \(\frac{5xy}{5x^2}\), let's break it down. In the numerator, we have \(5\cdot x \cdot y\), and in the denominator, we have \(5 \cdot x \cdot x\). Notice that both terms contain \(5\) and \(x\) as common factors. By recognizing these, we prepare for the next step, where actual simplification happens. Remember, it's important to compare the numerator and denominator carefully to spot these common factors.

Cancelation

Once common factors are identified, the next step is canceling them out. Cancelation involves removing identical factors from both the numerator and the denominator of a fraction. This process is similar to dividing both the top and the bottom by the same value or element, effectively reducing the fraction without changing its value.
Imagine canceling as a way to cut down unnecessary parts of a fraction to make it simpler. In the case of \(\frac{5xy}{5x^2}\), both the numerator and denominator have the common factors \(5\) and \(x\). By 'canceling' or dividing them out, you are left with the essential components: from \(5xy\), the \(5\) and one \(x\) are gone, and from \(5x^2\), a \(5\) and one \(x\) vanish too. This leaves \(y\) in the numerator and a single \(x\) in the denominator. As a result, the simplified fraction emerges as \(\frac{y}{x}\).

Numerator and Denominator

Understanding the terms 'numerator' and 'denominator' is fundamental when working with fractions. The numerator is the top part of a fraction, indicating how many parts of a whole are being considered. The denominator, situated at the bottom, reveals the total number of equal parts the whole is divided into. Recognizing these components in a fraction set the stage for simplification efforts.
In the exercise \(\frac{5xy}{5x^2}\), \(5xy\) is the numerator, and \(5x^2\) is the denominator. Our goal in simplifying is to analyze these two sections, identify common factors, and then reduce the fraction to its simplest form. Once the common factors \(5\) and \(x\) are removed, the resulting fraction \(\frac{y}{x}\) is simpler because it contains only the essential elements. This process of simplification makes the fraction easier to work with and understand, without changing its original value.