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Chapter 7: Problem 25

\(\frac{x}{4}\) and \(\frac{x}{5}\)

Short Answer

Expert verified
The fractions \(\frac{x}{4}\) and \(\frac{x}{5}\) can be expressed with a common denominator as \(\frac{5x}{20}\) and \(\frac{4x}{20}\) respectively.

Step by step solution

01

Identify the Denominators

Here, the denominators are 4 and 5. These are the numbers by which x is divided in each fraction.
02

Find the Least Common Denominator (LCD)

The Least Common Denominator (LCD) is the smallest number that both denominators can divide into evenly. Since 4 and 5 have no common factors other than 1, their LCD is simply their product, which is 20.
03

Express the fractions in terms of the LCD

To express the fractions in terms of the LCD, divide the LCD by the original denominator to find the factor by which the numerator should be multiplied. Then multiply the numerator by this factor. \nThe first fraction \(\frac{x}{4}\) becomes \(\frac{5x}{20}\) and the second fraction \(\frac{x}{5}\) becomes \(\frac{4x}{20}\).

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Key Concepts

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

Fractions
Fractions are numbers that represent parts of a whole. When you see a fraction, it's actually a division of two numbers: a numerator and a denominator. The numerator is the top number and indicates how many parts we have, while the denominator, the bottom number, shows into how many equal parts the whole is divided.
  • For example, in the fraction \(\frac{x}{4}\), "x" is the numerator and 4 is the denominator.
  • This fraction represents "x" divided by 4.
  • Fractions are used to express values that are not whole numbers, often appearing in algebra, geometry, and everyday situations.
Understanding fractions helps us handle scenarios where we need to divide something into smaller, more manageable pieces.
Denominators
The denominator is a crucial part of fractions as it signifies the total number of equal parts the whole is divided into. In algebraic fractions, the denominator might include numbers or variables.
  • Denominators help us understand the size of each part represented by the fraction.
  • Modifying the denominator changes the fraction's meaning and its relation to other fractions.
  • For example, in \(\frac{x}{4}\) and \(\frac{x}{5}\), the fractions have denominators of 4 and 5, respectively.
When dealing with multiple fractions, especially in algebra, finding a common denominator can simplify processes like addition or subtraction of fractions, which we'll explore with algebra.
Algebra
Algebra involves working with symbols and letters, generally representing numbers in formulas and equations. When you include fractions in algebra, you frequently need to manipulate them to simplify expressions.
  • Algebraic operations often involve finding a least common denominator (LCD) to combine fractions.
  • The LCD is the smallest number that is divisible by all denominators in question, which simplifies further operations.
  • In our example, the expressions \(\frac{x}{4}\) and \(\frac{x}{5}\) need to be rewritten using the LCD, which is 20.
    • This adjustment is made by scaling the numerators accordingly: the first fraction becomes \(\frac{5x}{20}\) and the second becomes \(\frac{4x}{20}\). Matching denominators streamline the process of solving for "x" or evaluating expressions.

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    Most popular questions from this chapter

    \(\frac{5}{2}-\frac{12}{x-4}=6\)

    \(\frac{\left(\frac{4}{9}\right)}{\left(\frac{2}{39}\right)}\)

    \(\frac{10}{y+3}+\frac{10}{3}=6\)

    Error Analysis Describe and correct the error. $$ \begin{aligned} \frac{12}{x+5}+\frac{1}{2} &=2 \\ x+5)\left(\frac{12}{x+5}+\frac{1}{2}\right) &=2 \\ x+2(12)+x+5 &=2 \\ x &=-2 x \end{aligned} $$

    \(\frac{x}{x-4}-\frac{9}{x}=\frac{3}{x-4}\)
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