Problem 34 - Prealgebra

Question

Simplify each fraction. $$\frac{5}{100}$$

Step-by-Step Solution

Answer

The simplified form of \( \frac{5}{100} \) is \( \frac{1}{20} \).

Step 1: Identify the Common Factor

To simplify a fraction, identify the greatest common factor (GCF) of the numerator and the denominator. For the fraction \( \frac{5}{100} \), the GCF of 5 and 100 is 5.

Step 2: Divide the Numerator by the GCF

Divide the numerator by the GCF to simplify it. \( 5 \div 5 = 1 \). Now the numerator of the simplified fraction is 1.

Step 3: Divide the Denominator by the GCF

Divide the denominator by the GCF to simplify it. \( 100 \div 5 = 20 \). This gives the denominator of the simplified fraction as 20.

Step 4: Write the Simplified Fraction

Combine the simplified numerator and denominator to write the fraction in its simplest form. Thus, \( \frac{5}{100} \) simplifies to \( \frac{1}{20} \).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key idea in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCF helps you reduce a fraction to its simplest form. In the fraction \( \frac{5}{100} \), the GCF of 5 and 100 is 5.To find the GCF, list the factors of each number:

• Factors of 5: 1, 5
• Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

The largest common factor is 5, making it the GCF. Once you identify the GCF, you can use it as a tool to simplify the fraction.

Numerator and Denominator

A fraction is made up of a numerator and a denominator. The numerator is the top number, indicating how many parts you have, while the denominator is the bottom number, showing the total number of equal parts the whole is divided into. In the fraction \( \frac{5}{100} \), 5 is the numerator and 100 is the denominator.Understanding the roles of the numerator and the denominator is vital in performing operations on fractions, such as simplification. By dividing both the numerator and the denominator by their GCF, you effectively scale down the fraction without changing its value. This step is crucial for simplification, enabling easier computations and a clearer understanding of the fraction's value.

Simplified Fractions

A simplified fraction means there are no common factors between the numerator and denominator other than 1. In other words, the fraction is reduced to its simplest form. By nature, a simplified fraction is easier to work with, and it represents the same value as the original fraction but in a more straightforward manner.To simplify \( \frac{5}{100} \), you perform the following steps:

• Divide the numerator (5) and the denominator (100) by the greatest common factor, which is 5.
• This gives you a new numerator of 1 (because \( 5 \div 5 = 1 \)) and a new denominator of 20 (because \( 100 \div 5 = 20 \)).

Thus, the simplified form of \( \frac{5}{100} \) is \( \frac{1}{20} \). This process does not change the fraction's value, but it makes it easier to interpret and use in further mathematical calculations.