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Chapter 4: Problem 12

Reduce each of the following rational expressions to lowest terms. $$\frac{4-6}{4-2}$$

Short Answer

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Step by step solution

01

Simplify the Numerator

First, simplify the numerator by performing the subtraction: \[ 4 - 6 = -2 \]
02

Simplify the Denominator

Next, simplify the denominator by performing the subtraction: \[ 4 - 2 = 2 \]
03

Form the Rational Expression

Place the simplified numerator and denominator into the rational expression: \[ \frac{-2}{2} \]
04

Reduce to Lowest Terms

Divide both the numerator and the denominator by their greatest common divisor (GCD), which is 2: \[ \frac{-2 \div 2}{2 \div 2} = \frac{-1}{1} = -1 \]

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Key Concepts

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

simplifying fractions
Simplifying fractions is about making them easier to understand and use. You start with a complex fraction and reduce it to its simplest form. A fraction is in simplest form when you can't divide the numerator and the denominator by any number other than 1. For example, when you have a fraction like \( \frac{4}{8} \), you simplify it by dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD).
This will leave you with \( \frac{1}{2} \).
Simplifying helps in mathematical operations, as dealing with smaller, more manageable numbers makes calculations more straightforward.
numerator and denominator
When working with fractions, it is crucial to understand the roles of the numerator and the denominator. The numerator is the top number in the fraction. It represents how many parts of the whole you have. The denominator is the bottom number and tells you into how many equal parts the whole is divided. For example, in the fraction \( \frac{3}{4} \),
3 is the numerator which means you have 3 parts out of the 4 equal parts (denominator) in total.
Knowing what each part represents helps in performing operations such as addition, subtraction, and most importantly, simplification of fractions.
greatest common divisor
The greatest common divisor (GCD) is the largest number that can evenly divide two or more numbers. To reduce fractions to their simplest form, you first find the GCD of the numerator and the denominator. For instance, take the fraction \( \frac{18}{24} \). The GCD of 18 and 24 is 6 because 6 is the largest number that can divide both of them without leaving a remainder.
Once you find the GCD, you divide both the numerator and the denominator by this number. So, \( \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \).
Reducing fractions using the GCD ensures that the fraction is in its simplest form, making it easier to understand and compare with other fractions.

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Most popular questions from this chapter

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{6}{35 a^{2}}+\frac{1}{30 a^{3}}$$

Solve each of the following problems algebraically. Be sure to label what the variable represents. The sum of the angles of a triangle is \(180^{\circ} .\) If the largest angle is twice the smallest and the third angle is \(20^{\circ}\) more than the smallest, find the size of each angle.

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Lisa has scores of \(79,83,88,\) and 91 on four class exams. What score must she get on her fifth exam so that her exam average is at least \(85 ?\)

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{3 y}{x}-\frac{x}{4 y}+\frac{2 y}{5 x}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. The length of a rectangle is 1 less than twice the width. If the width is 1 more than one seventh of the perimeter, find the dimensions of the rectangle.
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