Problem 26 - Fundamentals of Mathematics

Question

For the following problems, determine if the pairs of fractions are equivalent. $$ \frac{1}{2}, \frac{3}{6} $$

Step-by-Step Solution

Answer

The fractions \( \frac{1}{2} \) and \( \frac{3}{6} \) are equivalent.

Step 1: Simplify the second fraction

To determine if the fractions are equivalent, we first simplify the second fraction. Simplifying \( \frac{3}{6} \) involves finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 3 and 6 is 3. Divide both the numerator and the denominator by 3: \( \frac{3}{3} = 1 \), \( \frac{6}{3} = 2 \), resulting in \( \frac{1}{2} \).

Step 2: Compare the fractions

Now that we have simplified \( \frac{3}{6} \) to \( \frac{1}{2} \), we can compare it with \( \frac{1}{2} \). Since both fractions are \( \frac{1}{2} \), they are equivalent.

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This is done by dividing the numerator and the denominator by their greatest common factor. Simplified fractions are easier to work with and understand. For example, the fraction \( \frac{3}{6} \) can be simplified. Both 3 and 6 are divisible by 3, their greatest common divisor. When you divide the numerator and the denominator by 3, you get \( \frac{1}{2} \). Simplifying fractions helps not only in recognizing equivalent fractions but also in performing operations like addition and subtraction. Try to always express fractions in their simplest form to make mathematical expressions less complex and more comprehensible.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is the largest whole number that divides two or more integers without leaving a remainder. Finding the GCD is essential when simplifying fractions. It helps identify the factor by which you can reduce the fraction's numerator and denominator. To find the GCD of two numbers, you could list the factors of each number and choose the highest one they have in common. Alternatively, the Euclidean algorithm is a more efficient method for larger numbers. In our example, the GCD of 3 and 6 is 3. Dividing both the numerator and the denominator by this GCD yields the simplest form of the fraction. Understanding and using the GCD effectively makes working with fractions easier and helps in gaining more insight into the mathematical principles behind them.

Comparing Fractions

When comparing fractions, you're essentially checking if two fractions express the same value or if one is larger than the other. One of the easiest ways to compare fractions is to simplify them to their lowest terms and then see if they are identical. Take, for example, our fractions \( \frac{1}{2} \) and \( \frac{3}{6} \). By simplifying \( \frac{3}{6} \) to \( \frac{1}{2} \), we see that both fractions are equivalent. This comparison tells us that they represent the same portion of a whole. If fractions are not equivalent, converting them to have a common denominator is another method. After that, you can compare numerators directly. Learning to compare fractions enhances your ability to evaluate and understand them in everyday situations.