Problem 43 - Fundamentals of Mathematics

Question

For problems 43-47, determine if the pairs of fractions are equivalent. $$ \frac{1}{2}, \frac{15}{30} $$

Step-by-Step Solution

Answer

Yes, the fractions \(\frac{1}{2}\) and \(\frac{15}{30}\) are equivalent.

Step 1: Understanding the Problem

We need to determine if the two fractions \(\frac{1}{2}\) and \(\frac{15}{30}\) are equivalent. Fractions are equivalent if they represent the same value.

Step 2: Simplify the Second Fraction

Simplify the fraction \(\frac{15}{30}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 15. \[\frac{15}{30} = \frac{15 \div 15}{30 \div 15} = \frac{1}{2}\]After simplification, \(\frac{15}{30}\) becomes \(\frac{1}{2}\).

Step 3: Compare the Fractions

Now compare the simplified version of the second fraction to the first fraction. Both \(\frac{1}{2}\) and the simplified \(\frac{1}{2}\) are identical.

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a vital concept when dealing with simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. In our exercise, we simplified \(\frac{15}{30}\) by using its GCD. Finding the GCD helps us understand the common factors shared by the numerator and the denominator. To find the GCD, list the factors of each number. For instance, the factors of 15 are 1, 3, 5, 15 while the factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. Their common factors are 1, 3, 5, 15, and 15 is the greatest among them. Hence, the GCD is 15. Knowing the GCD allows us to simplify fractions efficiently, making them easier to compare or understand.

Simplifying Fractions

Simplifying fractions is the process of rewriting a fraction in its simplest form. This makes it easier to deal with complex fractions or compare them. In the example provided, we simplified \(\frac{15}{30}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 15.Here's a friendly step-by-step guide to simplify any fraction:

• Find the GCD of the numerator and the denominator.
• Divide the numerator by the GCD.
• Divide the denominator by the GCD.

By following these steps, \(\frac{15}{30}\) simplifies to \(\frac{1}{2}\). Simplified fractions, such as \(\frac{1}{2}\), are more convenient to work with because they represent the same value, but in a more concise form.

Comparing Fractions

When comparing fractions, we aim to determine if they are equivalent or if one is greater than the other. Fractions are considered equivalent if they simplify to the same value. For example, \(\frac{1}{2}\) and \(\frac{15}{30}\) appear different at first glance. However, by simplifying \(\frac{15}{30}\) to \(\frac{1}{2}\), we see they are indeed equivalent.Here are some tips on comparing fractions:

• Always simplify the fractions first, if possible.
• Convert them to a common denominator for an easier comparison if they are not simple like half, third, etc.
• For equivalent fractions, if the simplified forms match, they represent the same value.

Simplifying and comparing fractions ensure that you can accurately assess the relationships and values of different fractions. It is a fundamental skill in various math-related tasks.