Problem 97 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{30}{75}$$

Step-by-Step Solution

Answer

The fraction \(\frac{30}{75}\) reduces to \(\frac{2}{5}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction, start by identifying the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 30 and 75 is the largest number that can divide both without leaving a remainder.

Step 2: Find the GCD of 30 and 75

List the factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. List the factors of 75: 1, 3, 5, 15, 25, 75. The common factors are 1, 3, 5, and 15. Hence, the GCD is 15.

Step 3: Divide Both Numerator and Denominator by the GCD

Divide the numerator 30 and the denominator 75 by their GCD, which is 15. Calculate: \(\frac{30}{15} = 2\) \(\frac{75}{15} = 5\)

Step 4: Write the Reduced Fraction

The fraction \(\frac{30}{75}\) reduces to \(\frac{2}{5}\) when you divide both the numerator and the denominator by 15.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) plays a crucial role in reducing fractions to their simplest form. It is the largest positive integer that divides both numbers in the fraction without a remainder.
For instance, when dealing with the fraction \( \frac{30}{75} \), we seek the GCD of 30 and 75. This means we must find the largest number that divides both 30 and 75 evenly.
To do this, we first list out all factors of each number:

• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• Factors of 75: 1, 3, 5, 15, 25, 75

The common factors between these two lists are 1, 3, 5, and 15. Among these, 15 is the highest, making it the greatest common divisor. Understanding how to find the GCD is fundamental because it directly helps in simplifying fractions.

Numerator and Denominator

A fraction is composed of two key parts: the numerator and the denominator. The numerator is the top number, representing the number of equal parts considered, while the denominator is the bottom number, representing the total number of equal parts.
In the fraction \( \frac{30}{75} \), 30 is the numerator, and 75 is the denominator.
Understanding the roles of the numerator and denominator is essential when reducing fractions, as each has a specific function in this mathematical expression.

• The numerator indicates how many parts are of interest.
• The denominator indicates the total parts that make up a whole.

Both the numerator and denominator are important in the fraction reduction process, as reducing means achieving the ideal balance between them without altering the value of the fraction. Each number serves a purpose and contributes to simplifying the fraction to its simplest form.

Fraction Reduction Process

The fraction reduction process is a method used to simplify a fraction to its simplest form without changing its value. This involves using the greatest common divisor to divide both the numerator and the denominator.
Let's explore how to perform this process with a practical example. In our exercise, the fraction \( \frac{30}{75} \) can be reduced by:

• Calculating the GCD, which is 15, as explained earlier.
• Dividing both the numerator and the denominator by this GCD.

The process involves the following calculations:

• \( \frac{30}{15} = 2 \)
• \( \frac{75}{15} = 5 \)

Thus, the fraction \( \frac{30}{75} \) simplifies to \( \frac{2}{5} \).
By following the fraction reduction process systematically, you ensure the fraction is expressed in its lowest terms, preserving its original value while making it simpler and easier to work with.