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Chapter 9: Problem 4

Reduce each of the following fractions as completely as possible. $$\frac{5 x^{2}}{25 x^{2}}$$

Short Answer

Expert verified
\( \frac{1}{5} \)

Step by step solution

01

Identify the Numerator and Denominator

In the given fraction \(\frac{5 x^{2}}{25 x^{2}}\), identify the numerator as \(5 x^{2}\) and the denominator as \(25 x^{2}\).
02

Factor Common Elements

Factor both the numerator and the denominator to identify common elements. Here, the numerator is already in its simplest form \(5 x^{2}\), while the denominator can be factored as \(25 x^{2} = 5 \times 5 x^{2}\).
03

Cancel Common Factors

Cancel the common factors from the numerator and the denominator. Both the numerator and the denominator contain \(5 x^{2}\) which can be cancelled out: \(\frac{5 x^{2}}{25 x^{2}} = \frac{5 x^{2}}{5 \times 5 x^{2}} \rightarrow \frac{1}{5}\).
04

Simplified Fraction

The fraction is now simplified to \(\frac{1}{5}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerator and Denominator
To understand fractions, it's essential to grasp what the numerator and the denominator are. Imagine a fraction like a slice of a cake. The numerator represents the number of slices you have. The denominator, on the other hand, represents the total number of equal parts the cake is divided into. For example, in the fraction \(\frac{5x^2}{25x^2}\), the numerator is 5x\textsuperscript{2}, and the denominator is 25x\textsuperscript{2}. This separation helps us understand each part of the fraction before diving into more complicated operations.
Factoring
Factoring is the process of breaking down a number or expression into its simplest components, which can be multiplied together to get the original number or expression. In our exercise with \(\frac{5x^2}{25x^2}\), when we factor the denominator 25x\textsuperscript{2}, we see that it can be broken down into \((5 \times 5)x^2\). This makes it easier to identify common factors that appear in both the numerator and the denominator. Understanding factoring is crucial because it simplifies the process of reducing fractions.
Canceling Common Factors
Once we have factored both the numerator and the denominator, we can look for common factors and cancel them out. In the fraction \(\frac{5x^2}{25x^2}\), both the numerator and the denominator have a common factor of 5x\textsuperscript{2}. By canceling out the common factor, we simplify the fraction. This step is essential because it turns the complicated fraction into something much easier to work with. If you canceled correctly, \(\frac{5x^2}{25x^2}\) simplifies to \(\frac{1}{5}\).
Simplified Fraction
The result of canceling out common factors is a simplified fraction. A simplified fraction has the smallest possible numerator and denominator that still express the same value. For our exercise, \(\frac{5x^2}{25x^2}\) simplifies to \(\frac{1}{5}\). Simplification makes fractions easier to understand and work with, giving you a clear picture of the problem in its simplest form. Remember, a simplified fraction is always the goal when you're reducing fractions.

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Most popular questions from this chapter

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{5}{2 x} \cdot \frac{8}{x+2}$$

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$(2 x+a)(3 x-b)=c \quad \text { for } a$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{11}{x^{2}-9}-\frac{7}{2 x+6}=\frac{2}{x+3}$$

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$(x+a)(y+b)=c \quad \text { for } x$$

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$4(x-y)-\frac{x-y}{2}=3-(y-x) \text { for } x$$
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