Problem 62 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{5}{10}$$

Step-by-Step Solution

Answer

The fraction \( \frac{5}{10} \) reduces to \( \frac{1}{2} \).

Step 1: Identify the Greatest Common Divisor (GCD)

Find the greatest common divisor of 5 and 10. The GCD is the largest number that divides both numbers without leaving a remainder. Here, both 5 and 10 can be divided by 5.

Step 2: Divide the Numerator by the GCD

Divide the numerator, which is 5, by the GCD (5). This results in \( \frac{5}{5} = 1 \).

Step 3: Divide the Denominator by the GCD

Divide the denominator, which is 10, by the GCD (5). This results in \( \frac{10}{5} = 2 \).

Step 4: Write the Reduced Fraction

Combine the results from Steps 2 and 3 to write the reduced fraction. The reduced form of \( \frac{5}{10} \) is \( \frac{1}{2} \).

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is a fundamental concept in simplifying fractions. It refers to the largest number that can evenly divide two or more numbers without leaving a remainder. Understanding GCD is essential for reducing fractions to their simplest form.

For example, in the fraction \( \frac{5}{10} \), we need to identify the GCD of the numerator and denominator. The numbers 5 and 10 share the GCD of 5 because:

• 5 is a factor of itself as well as of 10, making it the largest number to satisfy both.

Finding the GCD allows us to divide both the numerator and denominator by this number to simplify the fraction. This technique not only helps in fraction simplification but also has applications in problem-solving scenarios in mathematics.

Fraction Simplification

Fraction simplification is the process of finding an equivalent fraction that is in the simplest form. An equivalent fraction has the same value but a smaller numerator and denominator. Simplifying fractions makes calculations easier and helps us better understand the relationships between quantities.

To simplify, follow these steps:

• Identify the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

Using our example \( \frac{5}{10} \):

• The GCD of 5 and 10 is 5.
• Divide the numerator by 5: \( \frac{5}{5} = 1 \).
• Divide the denominator by 5: \( \frac{10}{5} = 2 \).

Thus, \( \frac{5}{10} \) simplifies to \( \frac{1}{2} \). Simplification is not just a mechanical process; it deepens our understanding of numbers and their relations.

Numerator and Denominator

The terms numerator and denominator are key to understanding fractions. A fraction represents a part of a whole, and these terms define the components of a fraction.

The numerator is the top number in a fraction. It indicates how many parts of the whole are being considered. In the fraction \( \frac{5}{10} \), 5 is the numerator. It tells us that we have 5 parts out of a total indicated by the denominator.

The denominator is the bottom number in a fraction. It shows the total number of equal parts into which the whole is divided. In our example, 10 is the denominator, signifying that the whole is divided into 10 equal parts.

By dividing both the numerator and denominator by their GCD during simplification, we maintain the fraction's value but express it in simpler terms. Understanding these components ensures clarity when working with fractions and aids in effectively reducing them to their simplest form.