/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 79 Simplify each fraction. $$\fra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 79

Simplify each fraction. $$\frac{21}{24}$$

Short Answer

Expert verified
Simplified fraction is \(\frac{7}{8}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{21}{24}\), we first need to find the greatest common divisor (GCD) of the numerator (21) and the denominator (24). The prime factors of 21 are 3 and 7; the prime factors of 24 are 2, 2, 2, and 3. The only common factor is 3, so the GCD is 3.
02

Divide Both Numerator and Denominator by the GCD

Next, divide both the numerator and the denominator by the GCD (which is 3). This gives us \(\frac{21}{24} = \frac{21 \div 3}{24 \div 3} = \frac{7}{8}\).
03

Verify the Simplified Fraction

Ensure that \(\frac{7}{8}\) is fully simplified by confirming there are no further common divisors other than 1 between the numerator and the denominator. Since 7 is a prime number and 8 is a power of 2, they do not have any common factors. Thus, \(\frac{7}{8}\) is the simplest form.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Greatest Common Divisor (GCD)
The Greatest Common Divisor, or GCD, is a concept that helps when you want to simplify a fraction. It is the largest positive integer that divides the numerator and denominator without leaving a remainder. Finding the GCD is the first step in making fractions simpler.
  • To find the GCD, list all factors of both the numerator and denominator.
  • Identify their common factors.
  • The largest of these common factors is the GCD.
For example, in the fraction \(\frac{21}{24}\):
- The factors of 21 are 1, 3, 7, and 21.
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Both numbers share the factor 3, which is the largest, so the GCD is 3.
Prime Factors
Understanding prime factors is important when working with the Greatest Common Divisor. Prime factors are the building blocks of all numbers and can help identify the GCD.
  • Prime factors are the numbers that are only divisible by one and themselves, such as 2, 3, 5, 7, etc.
  • To find the prime factors of a number, divide it by the smallest prime that fits and continue until only 1 is left.
For instance, with the numbers 21 and 24:
- The prime factors of 21 are 3 and 7.
- The prime factors of 24 are 2, 2, 2, and 3.
You can see they share the prime factor 3, which is key for determining the GCD.
Simplified Fraction
A simplified fraction is when you have reduced the fraction as much as possible, using the GCD. It means there are no common factors between the numerator and the denominator other than 1. This is the final, compact form of the fraction and is easiest to work with or understand.
  • To simplify, divide both the numerator and denominator by their GCD.
  • Check that the resulting numbers have no more common factors, other than 1.
In the example of \(\frac{21}{24}\), we divided by the GCD of 3 to get \(\frac{7}{8}\). Since 7 and 8 have no common factors other than 1, \(\frac{7}{8}\) is the simplest form. It shows the underlying relationship between the numbers without unnecessary complexity.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the distance between each pair of points. Round to the nearest tenth, if necessary. $$Q\left(5 \frac{1}{4}, 3\right), R\left(2,6 \frac{1}{2}\right)$$

Write each improper fraction as a mixed number in simplest form. $$\frac{9}{4}$$

Solve each equation. Check your solution. $$8=\frac{2}{3} d$$

Write each decimal as a fraction or mixed number in simplest form. $$0 . \overline{7}$$

Find each product. Write in simplest form. $$-2 \frac{4}{5} \cdot \frac{3}{8}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.