Problem 35 - Prealgebra

Question

Simplify each fraction. $$\frac{25}{100}$$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{1}{4} \).

Step 1: Identify Common Factors

We need to simplify the fraction \( \frac{25}{100} \). Start by identifying the greatest common factor (GCF) of the numerator and denominator. The factors of 25 are 1, 5, and 25. The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The greatest common factor is 25.

Step 2: Divide by GCF

Divide both the numerator and the denominator by their greatest common factor, which is 25. \( \frac{25}{100} \div \frac{25}{25} = \frac{1}{4} \).

Step 3: Verify Simplified Fraction

Check if the fraction \( \frac{1}{4} \) is in its simplest form by ensuring that the GCF of the numerator and denominator is 1. There are no common factors between 1 and 4, other than 1, confirming that the fraction is simplified.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an essential concept in fraction simplification. It is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Understanding how to find the GCF can make working with fractions much easier and helps ensure the fraction is simplified correctly.

To identify the GCF, start by listing all factors of both numbers. For instance, with the fraction \( \frac{25}{100} \):

• The factors of 25 are: 1, 5, 25.
• The factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50, 100.

By comparing the lists of factors, you can see that the greatest number they both include is 25. This number is the GCF, and it is crucial for reducing the fraction to its simplest form.

Simplifying Fractions

Simplifying fractions involves reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This process is sometimes also known as reducing fractions.

Once you have found the GCF, you can simplify the fraction by dividing both the numerator and the denominator by this number. For example, with \( \frac{25}{100} \):

• The GCF is 25.
• Divide the numerator by 25: \( 25 \div 25 = 1 \).
• Divide the denominator by 25: \( 100 \div 25 = 4 \).

This results in \( \frac{1}{4} \), which is the fraction simplified to its simplest form. Simplifying fractions helps in making calculations easier and the results more comprehensible.

Fraction Mathematics

Fraction mathematics encompasses various operations including addition, subtraction, multiplication, division, and simplification of fractions. Working with fractions often requires understanding how to manipulate them to perform these operations efficiently.

Simplification is particularly important because it provides clarity and simplifies further mathematical operations. For example, multiplying or dividing fractions becomes less cumbersome when they are in their simplest forms.

Fractions also rely on number concepts such as the greatest common factor to determine the simplest form or least common multiple for operations like addition and subtraction. This means mastering fraction simplification is fundamental to manipulating and using fractions effectively in all areas of mathematics. Keeping fractions as simple as possible ensures calculations are not only more straightforward, but also more accurate.