Problem 4 - Algebra 1

Question

A friend tells you that \(-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b} .\) Is your friend correct? Use examples or counterexamples to support your answer.

Step-by-Step Solution

Answer

Yes, your friend is correct. It doesn't matter whether the negative sign is applied to the whole fraction, to the numerator only, or to the denominator only, the overall value of the fraction remains the same.

Step 1: Analyze the given statement

In the equation \(-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b},\) a negative fraction can either be related to the numerator being negative, the denominator being negative, or the whole fraction itself being negative.

Step 2: Understand the properties of negative fractions

If a negative sign is applied to the numerator, the entire fraction becomes negative. Similarly, if the negative sign is applied to the denominator only, the entire fraction also becomes negative. This leads to the equation \(-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b}\.

Step 3: Provide examples

Let's take \(a = 2\) and \(b = 3\). So, \(-\frac{2}{3} = -0.67, \frac{-2}{3} = -0.67,\) and \(\frac{2}{-3} = -0.67\). Hence, they are all equal.

Step 4: Apply these understanding to the given scenario

The examples show that whether the fraction is negative, the numerator is negative, or the denominator is negative, the overall value of the fraction remains the same. Therefore, the friend's statement in the exercise is indeed correct.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole, and they consist of two main parts: the numerator and the denominator. The numerator, placed at the top, indicates how many parts of the whole we are considering. The denominator, on the bottom, tells us how many equal parts the whole is divided into. For example, in the fraction \( \frac{3}{4} \), the numerator is 3, and the denominator is 4. This fraction signifies that we have 3 out of the 4 equal parts.

Fractions are fundamental in mathematics as they allow us to perform operations and comparisons that involve parts of a whole. They also help us understand more complex mathematical concepts like percentages and ratios. When dealing with fractions, it's essential to recognize whether they are positive or negative, as this impacts their value and direction on the number line.

Numerator and Denominator

Understanding the numerator and denominator is crucial in comprehending fractions. The numerator is the top part of the fraction and shows the number of selected parts, while the denominator is at the bottom and provides the total number of equal parts.

Let's take \( \frac{5}{8} \). Here:

• The numerator is 5, representing 5 parts.
• The denominator is 8, indicating that the whole is divided into 8 equal parts.

It's important to note that the fraction's value changes as the numerator or denominator changes. For instance, in negative fractions, the negative sign can be placed with the numerator, the denominator, or in front of the entire fraction, and the fraction remains equivalent.

Thus, given the example \( -\frac{5}{8} \), \( \frac{-5}{8} \), and \( \frac{5}{-8} \) will all yield the same value as they each indicate a portion that is opposite in direction to their positive counterparts.

Properties of Fractions

Fractions have unique properties that help us understand and manipulate them. One vital property is that fractions can exhibit equivalent values despite having different negative placements. For instance, as the exercise explains, the negative sign can be distributed variously across the fraction without altering its value.

In mathematical terms:

• \(-\frac{a}{b}\) places the negative sign before the whole fraction.
• \(\frac{-a}{b}\) applies the negative to the numerator.
• \(\frac{a}{-b}\) applies the negative to the denominator.

Each expression results in the fraction being negative because of the properties that guide fraction calculations. When comparing these forms, it shows that their value remains identical, illustrated with our example fraction \( -\frac{2}{3} \). All variations share a value of -0.67.

This equivalence arises because a negative fraction fundamentally represents a direction on the number line, showing a value opposite to its positive counterpart. Thus, understanding these properties is key to simplifying, comparing, and computing fractions effectively.