Problem 35 - Elementary Algebra

Question

For the following problems, show that the fractions are equivalent. $$ \frac{-9}{10} \text { and } \frac{9}{-10} $$

Step-by-Step Solution

Answer

Answer: Yes, the fractions \(\frac{-9}{10}\) and \(\frac{9}{-10}\) are equivalent, as they both simplify to \(-1 * \frac{9}{10}\).

Step 1: Write given fractions

The given fractions are \(\frac{-9}{10}\) and \(\frac{9}{-10}\).

Step 2: Simplify both fractions

For both fractions, we will observe that the negative sign can be factored out from the numerator and denominator. For the first fraction, \(\frac{-9}{10} = \frac{-1 * 9}{1 * 10} = -1 * \frac{9}{10}\) So, the simplified fraction is \(-1 * \frac{9}{10}\). For the second fraction, \(\frac{9}{-10} = \frac{1*9}{-1 * 10} = -1 * \frac{9}{10}\) So, the simplified fraction is \(-1 * \frac{9}{10}\).

Step 3: Compare the simplified fractions

We can now compare the simplified fractions from Step 2: \(-1 * \frac{9}{10}\) (from the first fraction) is equal to \(-1 * \frac{9}{10}\) (from the second fraction)

Step 4: Conclude the fractions are equivalent

Since both simplified fractions are equal (\(-1 * \frac{9}{10}\)), the given fractions, \(\frac{-9}{10}\) and \(\frac{9}{-10}\), are equivalent.

Key Concepts

Fraction Simplification

Fraction simplification involves reducing a fraction to its most basic form without changing its value. This process is crucial for making fractions easier to understand and work with. To simplify a fraction:

• Identify the greatest common divisor (GCD) for the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

In the given exercise, simplification is focused on the presence of negative signs. By factoring out the negative sign, you simplify how the fraction is perceived, which can ease comparisons and problem-solving. When both fractions are \(rac{-9}{10}\) and \(rac{9}{-10}\), factoring out the negative sign helps show they are equivalent even if they don't look identical at first glance.

Numerator and Denominator

Every fraction has a numerator and a denominator. The numerator is the top number and the denominator is the bottom number.

• Numerator: Indicates how many parts of a whole are being considered.
• Denominator: Represents the total number of equal parts the whole is divided into.

It's essential to understand the roles these two play when working with fractions to properly simplify or manipulate them. In the fractions from the exercise, \(-9\) and \(9\) are numerators, whereas \(10\) and \(-10\) serve as denominators. Recognizing how they interact, especially with negatives, is key to correctly simplifying or understanding the fractions.

Negative Signs in Fractions

Negative signs in fractions can be tricky, but understanding their placement helps simplify and equate fractions easily. In a fraction, a negative sign can be positioned:

• In front of the fraction (e.g., \(-\frac{9}{10}\))
• In the numerator (e.g., \(\frac{-9}{10}\))
• In the denominator (e.g., \(\frac{9}{-10}\))

Regardless of the sign's location, the fraction's overall value remains negative unless it is multiplied by another negative, resulting in a positive. Hence, both \( \frac{-9}{10} \) and \( \frac{9}{-10} \) represent the same value. Understanding this can help students see these fractions are equivalent, merely different stylings of the same expression.