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Chapter 4: Problem 84

Simplify each fraction. Do not convert any improper fractions to mixed numbers. \(-\frac{104}{48}\)

Short Answer

Expert verified
-\frac{13}{6}

Step by step solution

01

- Identify the Greatest Common Divisor (GCD)

First, find the greatest common divisor (GCD) of the numerator (-104) and the denominator (48). The GCD is the largest number that divides both numbers without leaving a remainder.
02

- Calculate the GCD

To find the GCD of 104 and 48, list the factors of each number:Factors of 104: 1, 2, 4, 8, 13, 26, 52, 104Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48The common factors are 1, 2, 4, 8, and the greatest common factor is 8.
03

- Divide Both the Numerator and the Denominator by the GCD

Now, divide both the numerator (-104) and the denominator (48) by the GCD (8):\(-\frac{104}{48} = -\frac{104 \div 8}{48 \div 8} = -\frac{13}{6}\)
04

- Simplify the Fraction

The fraction \(-\frac{13}{6}\) is already in its simplest form because 13 and 6 have no common factors other than 1.

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Key Concepts

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

Greatest Common Divisor (GCD)
When simplifying fractions, the Greatest Common Divisor (GCD) plays a crucial role. The GCD is the largest number that can evenly divide both the numerator and the denominator of a fraction. Understanding how to find the GCD is essential. Start by listing all factors of each number involved. For example, to find the GCD of 104 and 48, list:
  • Factors of 104: 1, 2, 4, 8, 13, 26, 52, 104
  • Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Identify the largest common factor. Here, both 104 and 48 share several factors, but the greatest is 8. So, the GCD of 104 and 48 is 8. Using this GCD, you can simplify the fraction easily by dividing both the numerator and the denominator by their GCD.
Numerator
The numerator is the top part of a fraction. It represents how many parts of the whole are taken. In the fraction \(-\frac{104}{48}\), the numerator is -104. When simplifying fractions, you divide the numerator by the GCD. For our example, using the GCD 8, you divide -104 by 8 to get -13. Remember:
  • The numerator can be a positive or negative number.
  • It tells you 'how many' or 'how much' of the fraction you have.
In our solution, we transformed the numerator to -13 by dividing it by the GCD.
Denominator
The denominator is the bottom part of a fraction. It shows the total number of equal parts the whole is divided into. In the fraction \( -\frac{104}{48} \), the denominator is 48. Simplifying involves dividing the denominator by the GCD. For this example, you divide 48 by 8, resulting in 6. Key points about denominators:
  • The denominator cannot be zero, as division by zero is undefined.
  • It indicates the 'total parts' that make up a whole.
After using the GCD in our fraction, the denominator was simplified from 48 to 6.
Improper Fractions
An improper fraction is one where the numerator is larger than the denominator. It means the fraction represents a value greater than 1 or less than -1 if the numerator is negative. In our example, \( -\frac{104}{48} \) is an improper fraction because 104 is larger than 48. After simplification, it becomes \( -\frac{13}{6} \), which is still improper. Important points about improper fractions:
  • They can be simplified to their lowest terms.
  • They do not need to be converted to mixed numbers unless specified.
Focus on simplifying the fraction by finding the GCD and dividing both the numerator and the denominator.

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Most popular questions from this chapter

In the following exercises, perform the indicated operation. $$ 4 \frac{1}{3}+9 \frac{1}{3} $$

Translate and solve. The sum of two-thirds and \(n\) is \(-\frac{3}{5}\)

In the following exercises, perform the indicated operation. $$ 2 \frac{3}{10}-1 \frac{9}{10} $$

In the following exercises, perform the indicated operation. $$ 8 \frac{6}{11}-2 \frac{9}{11} $$

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