91Ó°ÊÓ

Fraction simplification is essential for making fractions easier to work with and understand. In our exercise, we started with the fraction: \[\frac{595}{40}\]. To simplify, we first need to find a number that can divide both the numerator (595) and the denominator (40) evenly. This number is called the Greatest Common Divisor (GCD). In this case, the GCD of 595 and 40 is 5.
Simplifying the fraction involves dividing both the numerator and the denominator by this GCD.
Here are the steps laid out:

• Divide 595 by 5 to get 119.
• Divide 40 by 5 to get 8.

Thus, we end up with the simplified fraction \[\frac{119}{8}\]. By simplifying fractions, we make them easier to interpret and use in future calculations.

The Greatest Common Divisor (GCD) is a key concept when working with fractions. The GCD of two numbers is the largest number that evenly divides both of them without leaving a remainder.
To find the GCD of 595 and 40, we can use a few methods, such as:

• Listing the factors: Write down all factors of each number and find the largest common one.
• Using the Euclidean algorithm: This is a more systematic method where we repeatedly divide the numbers and take remainders until we reach a remainder of 0. The divisor at this step is the GCD.

For our example:

• Factors of 595: 1, 5, 7, 85, 119, 595
• Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

The common factors are 1 and 5, and the greatest of these is 5. Hence, the GCD is 5.
Use this GCD to simplify fractions and make them more manageable.

Understanding the roles of the numerator and denominator is fundamental in working with fractions.
In the fraction \[\frac{595}{40}\], 595 is the numerator, and 40 is the denominator.

• The numerator represents the 'part' or 'quantity' we are interested in. In our case, it is the dollar amount (\(595).
• The denominator represents the 'whole' or 'total' number of equal parts. Here, it is the total hours (40 hours).

When writing rates as fractions, it's pivotal to place these correctly: the numerator over the denominator. This helps to clearly illustrate the relationship between the quantities.
Simplification, as we discussed, involves reducing both the numerator and the denominator by their GCD. After this process, the final simplified fraction still represents the same rate but in its simplest form, which is easier to understand and use.
For our example, the simplified form is \[\frac{119}{8}\], maintaining the essence of representing \)595 over 40 hours in a cleaner form.