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Chapter 1: Problem 49

Simplify each fraction, if possible. $$ \frac{33}{56} $$

Short Answer

Expert verified
The fraction \( \frac{33}{56} \) is already simplified.

Step by step solution

01

Identify Factors of Numerator and Denominator

To simplify the fraction \( \frac{33}{56} \), first identify the factors of both the numerator (33) and the denominator (56). The factors of 33 are 1, 3, 11, and 33, while the factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.
02

Find the Greatest Common Factor (GCF)

Determine the greatest common factor between 33 and 56. Comparing the factors, 33 and 56 share only 1 as a common factor, making it their greatest common factor.
03

Simplify the Fraction

Since the greatest common factor of 33 and 56 is 1, the fraction \( \frac{33}{56} \) is already in its simplest form and cannot be reduced further.

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Key Concepts

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

Factors
In mathematics, factors are numbers that you can multiply together to get another number. Understanding factors is crucial when it comes to simplifying fractions. A factor of a number divides it exactly without leaving a remainder.
For example:
  • The factors of 33 are 1, 3, 11, and 33. When multiplied in pairs, such as 3 脳 11, they give back the original number, 33.
  • The factors of 56 are a bit more involved: 1, 2, 4, 7, 8, 14, 28, and 56. Each of these divides 56 without any remainder.
Finding the factors helps in determining shared numbers between the numerator and denominator, which is the first step in simplifying a fraction.
By identifying these shared factors, it sets the stage to find the greatest common factor, aiding in the simplification process.
Greatest Common Factor
The greatest common factor, or GCF, is the largest factor that two numbers have in common. It plays a vital role in simplifying fractions, providing the tool needed to reduce a fraction to its simplest form.
When you have two sets of factors, like those for the numbers 33 and 56:
  • 33: 1, 3, 11, 33
  • 56: 1, 2, 4, 7, 8, 14, 28, 56
The only factor they share is 1. This means that 1 is the greatest common factor of 33 and 56.
The GCF is used to divide both the numerator and denominator of a fraction so that the remaining numbers are as small as possible, simplifying the fraction. When the GCF is 1, as in this case, the fraction is already in its simplest form and doesn't need any further reduction.
Numerator and Denominator
In a fraction, the numerator and denominator represent two essential parts:
  • The numerator is the top part of the fraction, which partly represents how many parts we have.
  • The denominator is the bottom part, indicating into how many equal parts the whole is divided.
For the fraction \( \frac{33}{56} \), 33 is the numerator and 56 is the denominator.
Understanding each of their roles is fundamental to working effectively with fractions. To simplify, you typically look for the greatest common factor for these two numbers to reduce both to smaller numbers while maintaining the same value.
In practical terms, if there is no common factor greater than 1 (as in our example), the fraction is already simplified. Knowing the roles of the numerator and denominator helps in performing operations and quick checks like simplification.

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