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Chapter 9: Problem 19

Reduce each of the following fractions as completely as possible. $$\frac{3 x-6}{5 x-10}$$

Short Answer

Expert verified
\( \frac{3}{5} \)

Step by step solution

01

Identify the Numerator and Denominator

The given fraction is \( \frac{3x - 6}{5x - 10} \). Here, \( 3x - 6 \) is the numerator and \( 5x - 10 \) is the denominator.
02

Factor the Numerator

Identify common factors in the numerator. The numerator \( 3x - 6 \) can be factored as \( 3(x - 2) \).
03

Factor the Denominator

Similarly, factor the denominator by identifying common factors. The denominator \( 5x - 10 \) can be factored as \( 5(x - 2) \).
04

Cancel Common Factors

The fraction now looks like \( \frac{3(x - 2)}{5(x - 2)} \). Cancel the common factor \( x - 2 \) from the numerator and the denominator, resulting in \( \frac{3}{5} \).

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Key Concepts

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

Numerator
The numerator is the top part of a fraction. It represents how many parts we have out of the total number represented by the denominator. In the fraction given in the exercise, \(\frac{3x - 6}{5x - 10}\), the numerator is \(3x - 6\). Understanding the numerator is essential because it often contains variables or expressions that can be simplified. By factoring, we can simplify the numerator to make it easier to work with. In our example, \(3x - 6\) can be factored to \(3(x - 2)\), helping to reduce the fraction.
Denominator
The denominator is the bottom part of the fraction. It tells us the total number of equal parts in the whole. In our exercise, the denominator is \(5x - 10\). Like the numerator, the denominator can also be simplified through factoring. Factoring the denominator \(5x - 10\) gives us \(5(x - 2)\), making it possible to identify and cancel common factors between the numerator and denominator. This simplifies the fraction, showing a clearer mathematical relationship.
Factoring
Factoring is the process of breaking down complex expressions into simpler components, called factors. These factors, when multiplied together, produce the original expression. In the fraction \(\frac{3x - 6}{5x - 10}\), both the numerator and the denominator can be factored to reveal common factors.
  • For the numerator \(3x - 6\), we factor out the common factor 3 to get \(3(x - 2)\).
  • For the denominator \(5x - 10\), we factor out the common factor 5 to get \(5(x - 2)\).

Once factored, we notice the common factor \((x - 2)\) in both the numerator and denominator, which can be cancelled. This leaves us with the reduced fraction \(\frac{3}{5}\). Factoring not only simplifies the fraction but also makes it easier to see relationships between different parts of an expression.

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Most popular questions from this chapter

Perform the indicated operations and simplify. $$x+\frac{2}{3 x}-\frac{1}{2 y}$$

A student was asked to reduce two fractions and proceeded as follows: (a) \(\frac{x^{2}+5}{x} \stackrel{?}{x} \frac{x^{2}+5}{x} \stackrel{?}{=} x+5\) (b) \(\frac{x^{2}+y^{2}}{x+y} \stackrel{?}{=} \frac{x^{2}+y^{2}}{x+y} \stackrel{?}{=} x+y\) Discuss what the student did in each case and whether or not it was correct.

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. \(\frac{x-\mu}{s}<2\) for \(x \quad\) (Assume that \(s>0\) )

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

The length of a rectangle is 3 more than twice the width. If the ratio of width to length is 2 to \(5,\) find the dimensions of the rectangle.
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