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Chapter 4: Problem 2

For Problems \(1-8\), express each rational number in reduced form. $$ \frac{14}{21} $$

Short Answer

Expert verified
\( \frac{2}{3} \)

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To reduce a fraction to its simplest form, we first need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 14 and 21 is 7.
02

Divide the Numerator by GCD

Take the numerator, 14, and divide it by the GCD, 7. This results in: \( \frac{14}{7} = 2 \).
03

Divide the Denominator by GCD

Take the denominator, 21, and divide it by the GCD, 7. This results in: \( \frac{21}{7} = 3 \).
04

Write the Reduced Fraction

Combine the results from Steps 2 and 3 to write the reduced fraction. So, \( \frac{14}{21} \) simplifies to \( \frac{2}{3} \).

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Key Concepts

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

Greatest Common Divisor
The greatest common divisor, often abbreviated as GCD, is a fundamental concept in mathematics, particularly when working with fractions. The GCD of two or more integers is the largest positive integer that divides each of them without leaving a remainder. Identifying the GCD is essential when simplifying fractions because it allows us to reduce the fraction to its lowest terms.
  • Start by listing the factors of each number.
  • Identify the shared factors between them.
  • The largest factor they have in common is the GCD.
For example, to find the GCD of 14 and 21, list the factors of each:\[\text{Factors of } 14: 1, 2, 7, 14 \\text{Factors of } 21: 1, 3, 7, 21\]The common factors are 1 and 7, making 7 the greatest common divisor.
Rational Numbers
Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. In essence, any number that can be written in the form \( \frac{a}{b} \) is considered rational. Here, 'a' is the numerator and 'b' is the denominator. Rational numbers include
  • Fractions like \( \frac{3}{4} \)
  • Integers (since any integer can be expressed as a fraction with 1 as the denominator, like \( \frac{5}{1} \))
  • Terminating or repeating decimals (for example, 0.5 or 1.333...)
Understanding the nature of rational numbers helps in recognizing how fractions fit into the broader category of numbers and why simplifying them results in another rational number.
Fraction Reduction
Fraction reduction is the process of simplifying a fraction to its simplest form, where the numerator and denominator are as small as possible. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  • Identify the GCD of the numerator and the denominator.
  • Divide both parts of the fraction by this GCD.
  • The result is a reduced fraction that is equivalent to the original.
Using the example \( \frac{14}{21} \), we found the GCD to be 7. By dividing the numerator (14) and the denominator (21) by 7, we simplify the fraction: \[\frac{14 \div 7}{21 \div 7} = \frac{2}{3}\]This process makes the fraction easier to understand and work with, as it represents the same quantity in a simpler form. It's a key skill in algebra that improves clarity in mathematical expressions and calculations.

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Most popular questions from this chapter

For Problems 45-60, set up an algebraic equation and solve each problem. A sum of \(\$ 1750\) is to be divided between two people in the ratio of 3 to 4 . How much does each person receive?

For Problems \(1-44\), solve each equation. $$ 2-\frac{3 x}{x-4}=\frac{14}{x+7} $$

Why is it important to consider more than one way to do a problem?

For Problems \(31-44\), solve each equation for the indicated variable. $$ \frac{R}{S}=\frac{T}{S+T} \text { for } R $$

For Problems \(31-44\), solve each equation for the indicated variable. $$ y=-\frac{a}{b} x+\frac{c}{d} \text { for } x $$
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