Problem 25 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$ \frac{5}{12}, \frac{10}{24} $$

Step-by-Step Solution

Answer

Yes, the fractions \( \frac{5}{12} \) and \( \frac{10}{24} \) are equivalent.

Step 1: Understand the Problem

We need to determine if the two given fractions \( \frac{5}{12} \) and \( \frac{10}{24} \) are equivalent. This means we need to check if they have the same value when simplified to their lowest terms.

Step 2: Simplify the First Fraction

First, we simplify \( \frac{5}{12} \). The greatest common divisor (GCD) of 5 and 12 is 1, so the fraction is already in its simplest form as \( \frac{5}{12} \).

Step 3: Simplify the Second Fraction

Next, we simplify \( \frac{10}{24} \). The GCD of 10 and 24 is 2. We divide both the numerator and the denominator by 2: $$\frac{10 \div 2}{24 \div 2} = \frac{5}{12}.$$

Step 4: Compare the Simplified Fractions

Now we compare the simplified forms of both fractions. The first fraction is \( \frac{5}{12} \) and we've simplified the second fraction to \( \frac{5}{12} \) as well. Since they are the same, the original fractions are equivalent.

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a key concept when working with fractions. It helps us to simplify fractions, making them as easy to work with as possible. The GCD of two numbers is the largest number that divides both without leaving a remainder.
For example, let's look at the numbers 10 and 24. To find the GCD, we need to list the factors of each number.

• Factors of 10: 1, 2, 5, 10
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

In this case, the greatest number that appears in both lists of factors is 2. This means the GCD of 10 and 24 is 2. This common divisor is used to simplify fractions, as it helps reduce the fraction to its simplest form.

Simplifying Fractions

Simplifying fractions is the process of making fractions as simple as possible by reducing them to their lowest terms. This involves dividing both the numerator and the denominator by their GCD.For instance, take the fraction \( \frac{10}{24} \). The GCD of 10 and 24 is 2.
We simplify the fraction by dividing the numerator and denominator by 2: \[\frac{10 \div 2}{24 \div 2} = \frac{5}{12}\]This gives us \( \frac{5}{12} \), which is in its simplest form as no further reduction is possible. Simplifying fractions often makes it easier to see if two fractions are equivalent, simplifying calculations, and ensuring that the fractions used are in their most efficient form.

Comparing Fractions

Comparing fractions can reveal whether two fractions are equivalent or which one is larger or smaller. The goal is to have a common yardstick to make the comparison meaningful. One straightforward method is simplification.
If simplifying two fractions leads to the same result, they are equivalent. For example, compare \( \frac{5}{12} \) and \( \frac{10}{24} \). After simplifying \( \frac{10}{24} \), we find it also equals \( \frac{5}{12} \). This confirms that both fractions are equivalent since they simplify to the same numerator and denominator.
Another approach to comparing fractions involves finding a common denominator, but simplification is a quick method when the objective is to check for equivalency. Using these methods carefully ensures that comparing fractions is both accurate and efficient.