Problem 33 - Elementary Algebra

Question

For the following problems, show that the fractions are equivalent. $$ -\frac{1}{4} \text { and } \frac{-1}{4} $$

Step-by-Step Solution

Answer

Answer: Yes, the fractions \(-\frac{1}{4}\) and \(\frac{-1}{4}\) are equivalent, as they represent the same value.

Step 1: Understand the given fractions

We are given two fractions: \(-\frac{1}{4}\) and \(\frac{-1}{4}\).

Step 2: Check if the numerators and denominators are equal in magnitude

For the first fraction \(-\frac{1}{4}\), the numerator is -1 and the denominator is 4. For the second fraction \(\frac{-1}{4}\), the numerator is also -1 and the denominator is 4.

Step 3: Compare the fractions

Since both fractions have the same numerator (-1) and the same denominator (4), they represent the same value. Therefore, the two fractions are equivalent.

Step 4: Conclusion

We have shown that the fractions \(-\frac{1}{4}\) and \(\frac{-1}{4}\) are equivalent, as they represent the same value.

Key Concepts

Understanding Numerators and Denominators

Fractions consist of two parts: the numerator and the denominator. The numerator is the top number, indicating how many parts we have. The denominator is the bottom number, showing the total number of equal parts. For example, in the fraction \( \frac{1}{4} \), 1 is the numerator while 4 is the denominator.

• The numerator tells us the specific parts we are focusing on.
• The denominator gives context by representing the whole.

Understanding these two components is essential in grasping how fractions function and relate to each other. In the case of \( -\frac{1}{4} \) and \( \frac{-1}{4} \), the numerators are both -1, and the denominators are both 4. This means they are analyzing the same part of the whole, despite how they are presented.

Working with Negative Fractions

Negative fractions can sometimes be confusing, but they are simply fractions with a negative sign. This sign can appear in different positions: either in front of the fraction, with the numerator, or with the denominator. For example:

• \( -\frac{1}{4} \): The negative sign is in front of the fraction.
• \( \frac{-1}{4} \): The negative sign is with the numerator.

Both expressions are equivalent because placing the negative sign in different parts does not alter the overall value of the fraction. It represents a quota of -1 part out of 4, illustrating how negativity in fractions affects their value.

Comparing Fractions for Equivalence

When comparing fractions to see if they are equivalent, you're checking if they represent the same value or portion of a whole. To do this, you'll need to look at the numerators and denominators:

• The fractions must have equal numerators when the denominators are the same to be equivalent.
• The same rule applies with a flipped order: if the denominators are equal, the numerators must also match.

For example, the fractions \( -\frac{1}{4} \) and \( \frac{-1}{4} \) both have numerators of -1 and denominators of 4. Thus, they represent the same quantity, proving these fractions are indeed equivalent. Understanding this simple comparison method is key to mastering fraction equivalence.