Problem 59 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. (Skills Review p. 763) $$\frac{25}{100}$$

Step-by-Step Solution

Answer

The fraction \(\frac{25}{100}\) in simplest form is \(\frac{1}{4}\).

Step 1: Identifying the Greatest Common Divisor (GCD)

The GCD of 25 and 100 is 25, because 25 is the biggest number that can divide both 25 and 100 without any remainder.

Step 2: Dividing by the GCD

Divide both the numerator (25) and the denominator (100) of the fraction by their GCD (which is 25). Following this process, \(\frac{25}{100}\) becomes \(\frac{25/25}{100/25} = \frac{1}{4}\).

Key Concepts

Greatest Common Divisor

When simplifying fractions, finding the greatest common divisor (GCD) is a crucial step. The GCD is the largest number that evenly divides two or more numbers. In the fraction \(\frac{25}{100}\), the GCD helps us reduce the fraction to its simplest form.
For the numbers 25 and 100, the GCD is 25. This is because 25 is the largest number that can divide both 25 and 100 without leaving a remainder.

• To determine the GCD, you can list all the factors of each number and find the largest common one.
• Alternatively, use methods such as the prime factorization or Euclidean algorithm.

Finding the GCD is essential for simplifying fractions efficiently, as it ensures the fraction is reduced to its simplest form in one step.

Numerator and Denominator

Fractions consist of two parts: the numerator and the denominator. Understanding these components is key to working with fractions.
The numerator is the top number, which represents how many parts of the whole are being considered. For example, in \(\frac{25}{100}\), 25 is the numerator.

• The denominator, on the other hand, is the bottom number indicating the total number of equal parts in the whole. In this case, 100 is the denominator.
• A larger numerator means we have more of the whole portion, while a larger denominator indicates the parts are smaller.

By recognizing the role of the numerator and the denominator, you can better understand how changes to these numbers affect the fraction and its value.

Basic Fraction Reduction

Simplifying, or reducing, fractions is a fundamental math skill. It involves rewriting the fraction in its simplest form, where the numerator and denominator are as small as possible but still maintain the same ratio.
In the fraction \(\frac{25}{100}\), we simplify by using the greatest common divisor.

• First, divide both the numerator and the denominator by the GCD, which is 25 in this situation.
• The calculation goes as follows: \(\frac{25}{25} = 1\) and \(\frac{100}{25} = 4\).

After this division, the fraction reduces to \(\frac{1}{4}\). This is the simplest form of the fraction since there are no common divisors between the numerator 1 and denominator 4 other than 1 itself. Simplifying fractions is an efficient way to make calculations easier and improve understanding of proportionate relationships.