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Chapter 1: Problem 144

In the following exercises, simplify. $$ -\frac{104}{48} $$

Short Answer

Expert verified
-\frac{13}{6}

Step by step solution

01

Identify the given fraction

The given fraction is \(-\frac{104}{48}\).
02

Find the GCD of the numerator and the denominator

Use the Greatest Common Divisor (GCD) to simplify the fraction. The GCD of 104 and 48 is 8.
03

Simplify the numerator and the denominator

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

Rewrite the simplified fraction

The fraction \(-\frac{13}{6}\) is already 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 (GCD)
The Greatest Common Divisor (GCD) is an essential concept in simplifying fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder. Finding the GCD helps reduce fractions to their simplest form.
For example, in the exercise given, the GCD of 104 and 48 is found to be 8. This means both numbers can be divided by 8 evenly.
To find the GCD, you can:
  • List the factors of each number.
  • Find the highest common factor that appears in both lists.
By using the GCD, you can simplify fractions more efficiently.
Numerator and Denominator
When dealing with fractions, understanding the terms 'numerator' and 'denominator' is crucial. The numerator is the top part of the fraction, representing the number of parts being considered. The denominator is the bottom part, representing the total number of equal parts.
In the fraction \(-\frac{104}{48}\), 104 is the numerator, and 48 is the denominator. To simplify this fraction, both parts need to be divided by their GCD (which is 8 in this case).
This results in \(-\frac{13}{6}\), where 13 is the new numerator, and 6 is the new denominator.
  • Numerator: The number above the line in a fraction.
  • Denominator: The number below the line in a fraction.
Knowing these terms helps in easily understanding and simplifying fractions.
Fraction Simplification
Fraction simplification is the process of reducing a fraction to its simplest form. It's done by dividing both the numerator and the denominator by their Greatest Common Divisor (GCD).
For instance, in the step-by-step solution, the fraction given is \-\frac{104}{48}\. By finding the GCD of 104 and 48, which is 8, and dividing both the numerator and the denominator by 8, we get \(-\frac{13}{6}\).
Here鈥檚 a quick guide to simplifying fractions:
  • Find the GCD of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.
  • Write down the simplified fraction.
This method ensures that the fraction is as simple as possible, making it easier to work with in calculations.

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