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Chapter 4: Problem 14

Reduce the given fraction to lowest terms. $$\frac{28}{56}$$

Short Answer

Expert verified
The fraction \(\frac{28}{56}\) reduces to \(\frac{1}{2}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \(\frac{28}{56}\), identify the GCD of 28 and 56. The divisors of 28 are 1, 2, 4, 7, 14, and 28. The divisors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. The largest common divisor is 28.
02

Divide the Numerator and Denominator by the GCD

Using the GCD identified in Step 1, simplify the fraction by dividing both the numerator and the denominator by 28: \[ \frac{28}{56} = \frac{28 \div 28}{56 \div 28} = \frac{1}{2} \]
03

Verify the Result

To ensure the fraction is completely reduced, check if 1 and 2 have any common divisors other than 1. Since they do not, \(\frac{1}{2}\) is indeed the fraction in its simplest form.

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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 (GCD) is key to simplifying fractions. It refers to the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD can help reduce fractions to their simplest form. For example, consider the fraction \(\frac{28}{56}\). To find the GCD of 28 and 56:
  • List the divisors of 28: 1, 2, 4, 7, 14, 28.
  • List the divisors of 56: 1, 2, 4, 7, 8, 14, 28, 56.
  • The shared divisors between 28 and 56 are 1, 2, 4, 7, 14, and 28.
  • The greatest of these shared divisors is 28.
By dividing both the numerator and denominator by 28, you reduce the fraction to its simplest form.
Numerator and Denominator
In any fraction, the numerator and denominator play vital roles. The numerator is the top number, showing how many parts we have. The denominator is the bottom number, indicating how many equal parts make up a whole. Examining the fraction \(\frac{28}{56}\):
  • The numerator is 28, meaning 28 parts of something.
  • The denominator is 56, indicating those parts are out of 56 total parts.
When simplifying a fraction, you focus on these numbers to ensure both are divided by the greatest common divisor. This division retains the same value, simply expressed in a simpler form.
Fraction Reduction
Reducing a fraction means rewriting it in the simplest form where the numerator and denominator share no common factors except 1. Let's break down the process:
  • Identify the greatest common divisor of the numerator and the denominator. For \(\frac{28}{56}\), the GCD is 28.
  • Divide both the numerator and the denominator by this GCD, resulting in \(\frac{28 \div 28}{56 \div 28} = \frac{1}{2}\).
  • Check the resulting fraction. Here, 1 and 2 share no common divisors other than 1, confirming \(\frac{1}{2}\) is in its simplest form.
This process of fraction reduction makes complex fractions easier to understand and work with, without altering their value.

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Most popular questions from this chapter

Solve the equation and simplify your answer. $$x-\frac{4}{7}=\frac{7}{8}$$

Solve the equation and simplify your answer. $$x-\frac{5}{6}=\frac{1}{4}$$

At a local soccer game, announcers informed the crowd that attendance for the game was 4,302 . If this is \(2 / 9\) of the capacity, find the full seating capacity for the soccer stadium.

Is \(3 / 8\) a solution of the equation \(x-\frac{5}{9}=-\frac{13}{72}\) ?

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