Problem 12 - Elementary Algebra

Question

For the following problems, show that the fractions are equivalent. $$ \frac{-3}{8},-\frac{3}{8} $$

Step-by-Step Solution

Answer

Answer: Yes, the fractions $\frac{-3}{8}$ and $-\frac{3}{8}$ are equivalent.

Step 1: Identifying the Fractions

We are given the fractions: $$ \frac{-3}{8} \quad \text{and} \quad -\frac{3}{8} $$ Our goal is to show that these two fractions are equivalent.

Step 2: Comparing the Fractions

To show that the fractions are equivalent, we need to compare their values. In this case, we see that the two fractions have the same numerator and denominator, so the only difference is the negative sign. Let's examine this difference in the context of the two fractions. For the first fraction: $$ \frac{-3}{8} = -\frac{3}{8} $$ Here, we see that multiplying the numerator by -1 gives us the same value as when the negative sign is placed outside the fraction. This is due to the property of multiplying with -1, where: $$ -1 \cdot \frac{a}{b} = \frac{-a}{b} $$

Step 3: Conclusion

By comparing the values of the two fractions, we can see that they are equivalent. We've shown that they differ only in the placement of the negative sign, and this difference has no effect on the value of the fractions: $$ \frac{-3}{8} = -\frac{3}{8} $$ Thus, the fractions are proven to be equivalent.

Key Concepts

Negative Numbers

Negative numbers can be a bit confusing at first, but they are an essential concept in mathematics. A negative number is simply a number that is less than zero. It is written with a minus sign (\(-\)) in front of the number. For example, \(-3\) represents three units below zero on the number line.
When working with fractions, a minus sign can appear in different positions:

• Before the fraction: \(-\frac{3}{8}\) means the entire fraction is negative.
• In the numerator: \(\frac{-3}{8}\) indicates that the numerator is negative.

The placement of the negative sign will not affect the overall value as long as it appears only once. This is because multiplying or dividing by a negative number reverses the sign, not the fraction's magnitude. Understanding how negative numbers interact with fractions is crucial when comparing or proving equivalency.

Numerator and Denominator

The terms 'numerator' and 'denominator' define the two main parts of a fraction. In the fraction \(\frac{3}{8}\), **3** is the numerator, describing how many parts we are considering, and **8** is the denominator, indicating the total number of equal parts in a whole.
Here are some key facts:

• The numerator is divided by the denominator, showing a part of a whole or a ratio.
• The fraction \(\frac{-3}{8}\) has the negative sign in its numerator, affecting the fraction's value.

Grasping this is vital when looking at fractions like \(-\frac{3}{8}\) and \(\frac{-3}{8}\); they may look different, but they represent the same part of a negative whole. Always keep in mind that if both the numerator and denominator were negative, the fraction would be positive.

Fraction Properties

Fractions have several properties that help in making comparisons or simplifying expressions. One crucial property is related to the placement of the negative sign:

• A fraction \(\frac{-a}{b}\) is the same as \(-\frac{a}{b}\) because multiplying the entire fraction by \(-1\) will just flip the sign, not the value.
• Despite having negative numerators or denominators, as long as only one of them is negative, the fraction will retain its negative value.

This is why fractions like \(-\frac{3}{8}\) and \(\frac{-3}{8}\) are equivalent; they both signify the same division of negative parts. Recognizing these properties simplifies complex fractional problems, enabling easier solving and comparison of such expressions in mathematics.