Problem 23 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$\frac{1}{2}, \frac{5}{10}$$

Step-by-Step Solution

Answer

Yes, the fractions \( \frac{1}{2} \) and \( \frac{5}{10} \) are equivalent.

Step 1: Understand the Concept of Equivalent Fractions

Equivalent fractions are fractions that represent the same value or proportion of the whole. Two fractions are equivalent if the result of dividing the numerator by the denominator is the same in both fractions.

Step 2: Simplify the Fractions

Simplifying means reducing the fraction to its smallest possible numerator and denominator while keeping the same proportion. The fraction \( \frac{5}{10} \) can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 5. This yields \( \frac{1}{2} \).

Step 3: Compare the Simplified Fractions

Once you have simplified \( \frac{5}{10} \) to \( \frac{1}{2} \), compare it directly with \( \frac{1}{2} \). Since both fractions are now identical, \( \frac{1}{2} \) and \( \frac{5}{10} \) are equivalent.

Key Concepts

Simplify Fractions

Simplifying fractions involves turning a fraction into its simplest form, where the numerator and the denominator share no common factors other than 1. This process is key for identifying equivalent fractions.

To simplify a fraction:

• Find the greatest number that divides both the numerator and the denominator exactly; this is the greatest common divisor (GCD). For instance, with \( \frac{5}{10} \), the GCD is 5.
• Divide both the numerator and the denominator by the GCD. In our example, this means \( 5 \div 5 = 1 \) and \( 10 \div 5 = 2 \), which gives us \( \frac{1}{2} \).

Through this process, the equivalent fraction becomes easier to understand, and comparing fractions is more intuitive.

Numerator and Denominator

In any fraction, two integral components determine its value—the numerator and the denominator. The fraction \( \frac{a}{b} \) has:

• The numerator \( a \) which is the top part and indicates how many parts of the whole are being considered.
• The denominator \( b \) which is the bottom part and shows into how many equal parts the whole is divided.

Understanding these parts is crucial when working with fractions because:

• Changing the numerator alters the quantity or the size of the fraction without affecting its parts division.
• Modifying the denominator changes how the whole is divided, impacting how much each part represents.

This knowledge is essential when determining whether two fractions are equivalent, as they must have the same relative relationship between the numerator and the denominator.

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder.

Finding the GCD allows for fractions to be reduced to their simplest form effortlessly. To find the GCD, employ one of these methods:

• List out the factors of both the numerator and denominator, then choose the biggest number that appears in both lists.
• Use the Euclideanalgorithm, which involves iterative division, ideal for larger numbers.

Once the GCD is known, both the numerator and denominator can be divided by it to simplify the fraction.

For example, with \( \frac{5}{10} \), the GCD is 5, hence dividing both by 5 results in \( \frac{1}{2} \), simplifying the fraction succinctly and revealing equivalence when comparing with other fractions.