Problem 34 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$\frac{10}{16}, \frac{15}{24}$$

Step-by-Step Solution

Answer

The fractions are equivalent; both simplify to \( \frac{5}{8} \).

Step 1: Simplify the First Fraction

Let's simplify the fraction \( \frac{10}{16} \). The greatest common divisor (GCD) of 10 and 16 is 2. Divide both the numerator and the denominator by 2 to simplify.\[\frac{10}{16} = \frac{10 \div 2}{16 \div 2} = \frac{5}{8}\]

Step 2: Simplify the Second Fraction

Now let's simplify the fraction \( \frac{15}{24} \). The GCD of 15 and 24 is 3. Divide both the numerator and the denominator by 3 to simplify.\[\frac{15}{24} = \frac{15 \div 3}{24 \div 3} = \frac{5}{8}\]

Step 3: Compare the Simplified Fractions

After simplifying both fractions, we have \( \frac{5}{8} \) for both. Since the two fractions are equal, \( \frac{10}{16} \) and \( \frac{15}{24} \) are equivalent.

Key Concepts

Simplifying Fractions

Understanding how to simplify fractions is essential in determining whether two fractions are equivalent. Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This process involves several steps:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

For example, to simplify \( \frac{10}{16} \), find the GCD of 10 and 16, which is 2. Then divide both 10 and 16 by 2, resulting in \( \frac{5}{8} \).
This process ensures that the fraction is in its simplest form, making it easier to compare with other fractions and check for equivalency. It also aids in reducing the fraction to a more manageable size to work with in calculations.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, plays a crucial role in simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. To find the GCD, you can use several methods:

• List out the factors of each number and choose the largest common one.
• Use the Euclidean algorithm, a method of successive division.

In our example with \( \frac{10}{16} \), the factors of 10 are 1, 2, 5, and 10, while the factors of 16 are 1, 2, 4, 8, and 16. The largest common factor is 2, which is the GCD. Similarly, for \( \frac{15}{24} \), the GCD is 3.
Using the GCD is effective because it guarantees that you fully reduce a fraction to its simplest form, allowing for easier comparison with other fractions.

Comparing Fractions

Comparing fractions is an important skill in determining whether two fractions are equivalent. After simplifying both fractions, you simply check their numerators and denominators.
If both are the same after simplification, then the fractions are equivalent. In our example, both simplified fractions, \( \frac{10}{16} \) and \( \frac{15}{24} \), reduce to \( \frac{5}{8} \). This confirms that they are equivalent.
Being familiar with comparing fractions is useful in various mathematical tasks, such as solving equations and understanding ratios. By simplifying and comparing, you can see relationships between different fractions more clearly.