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Chapter 6: Problem 27

Reduce each fraction to simplest form. $$\frac{5 a}{9 a}$$

Short Answer

Expert verified
\( \frac{5}{9} \)

Step by step solution

01

Identify the Common Factor

The given fraction is \( \frac{5a}{9a} \). Notice that both the numerator and the denominator have a common factor \( a \).
02

Cancel the Common Factor

Divide both the numerator and the denominator by the common factor \( a \). This simplifies the fraction to \( \frac{5}{9} \).
03

Verify that the Fraction is in Simplest Form

Since 5 and 9 have no common factors other than 1, \( \frac{5}{9} \) is already in its simplest form.

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Key Concepts

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

Fractions
Fractions are a fundamental part of algebra and mathematics in general. They represent a part of a whole. A fraction consists of two parts: the **numerator** and the **denominator**. Often, when we think about fractions, we imagine cutting a pizza into equal slices and taking some of those slices.
  • The **numerator**, positioned at the top, tells us how many parts we have.
  • The **denominator**, located at the bottom, tells us into how many equal parts the whole is divided.
Understanding fractions is crucial because they frequently appear in everyday life, from simple tasks like cooking to more complex scenarios such as financial calculations. In algebra, fractions often include variables, which can make them seem tricky, but the core concept remains the same.
Simplification
Simplifying a fraction means rewriting it in its simplest form. This involves ensuring that the numerator and the denominator have no common factors other than one.
Here is why simplification is important:
  • **Clarity and Simplicity:** Simplified fractions are easier to understand and compare.
  • **Efficiency:** Simplified forms are simpler to work with when performing operations like addition, subtraction, and multiplication.
To simplify a fraction, you divide both the numerator and the denominator by their largest common factor. If there are variables involved, like in algebra, you might also factor out these variables to reduce the fraction to its basic form.
Common Factor
A common factor refers to a number or expression that evenly divides two or more numbers or expressions. For simplifying fractions, identifying a common factor is crucial.
  • A **factor** of a number is an integer that can be multiplied by another integer to produce the original number.
  • A **common factor** of two numbers is a factor that both numbers share.
To identify a common factor, you can list the factors of each number or expression. The greatest of these common factors is typically used to simplify a fraction. This helps in reducing the fraction to its simplest form by dividing both the numerator and the denominator by this greatest common factor.
Numerator and Denominator
Every fraction has two critical components: the **numerator** and the **denominator**. Understanding these terms can help you manipulate and simplify fractions effectively.
  • The **numerator** is the top number of a fraction and represents how many parts of the whole you have.
  • The **denominator** is the bottom number and signifies into how many parts the whole is divided.
In algebra, both the numerator and the denominator might include variables as well as numbers. When simplifying, you will need to consider both numerical and variable factors. It's essential to treat each of these components carefully to ensure the proper reduction and simplification of algebraic fractions.

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Most popular questions from this chapter

Give the required explanations. Factor \(n^{3}-n,\) and then explain why it represents a multiple of 6 if \(n\) is an integer greater than 1

Solve the given equations and check the results. $$\frac{1}{4 x}+\frac{3}{2 x}=\frac{2}{x+1}$$

Reduce each fraction to simplest form. Each is from the indicated area of application. \(\frac{E^{2} R^{2}-E^{2} r^{2}}{\left(R^{2}+2 R r+r^{2}\right)^{2}} \quad\) (electric power)

Solve the given equations and check the results. $$\frac{2}{s}=\frac{3}{s-1}$$

Solve the given equations and check the results. $$\frac{F-3}{12}-\frac{2}{3}=\frac{1-3 F}{2}$$
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