Problem 65 - Algebra A Combined Function

Question

Perform each indicated operation. See Section R .2. $$ \frac{9}{9}-\frac{19}{9} $$

Step-by-Step Solution

Answer

The result is \( \frac{-10}{9} \).

Step 1: Identify the Operation

We are asked to subtract two fractions with the same denominator: \( \frac{9}{9} \) and \( \frac{19}{9} \). The operation is subtraction.

Step 2: Subtract the Numerators

Since the denominators are the same, we subtract the numerators while keeping the denominator constant. The operation is:\[\frac{9 - 19}{9}\]

Step 3: Simplify the Expression

Calculate the subtraction in the numerator:\[9 - 19 = -10\]Thus, the expression becomes:\[\frac{-10}{9}\]

Step 4: Finalize the Fraction

Since \( \frac{-10}{9} \) is already in its simplest form and is an improper fraction, it can also be left as is. The result of \( \frac{9}{9} - \frac{19}{9} \) is \( \frac{-10}{9} \).

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator, or top number, is greater than or equal to the denominator, or bottom number.
The result of the original subtraction problem is \(-10/9\), which is an improper fraction.
Understanding improper fractions is important because it affects how we interpret the value of the fraction.
Here are some key points:

• An improper fraction represents a value greater than or equal to 1 or less than or equal to -1, depending on whether the fraction is positive or negative.
• In the fraction \(-10/9\), since the absolute value of 10 is larger than 9, this fraction represents more than 1 full unit, specifically, a negative value over 1 full negative unit.

Sometimes, we convert improper fractions to mixed numbers for easier readability, although in this exercise, \(-10/9\) is left as an improper fraction.

Numerator and Denominator

Every fraction consists of two main parts: the numerator and the denominator. Understanding these components is crucial for performing operations like addition and subtraction.
Let's explore these terms in detail:

• Numerator: This is the top part of a fraction which signifies how many parts we are considering. In the context of the subtraction problem, the numerators were 9 and 19.
• Denominator: This is the bottom part of a fraction that tells us into how many equal parts the whole is divided. Both fractions \(\frac{9}{9}\) and \(\frac{19}{9}\) have a denominator of 9, which means any whole unit is divided into 9 equal parts.

Knowing how to manipulate these numbers, particularly the numerators, is essential in carrying out arithmetic operations. Since the denominators were equal in the given problem, we only needed to subtract the numerators.

Simplifying Fractions

Simplifying fractions involves reducing fractions to their simplest form, which means the greatest common divisor (GCD) of the numerator and denominator is 1.
Let's have a closer look at this process:

• Sometimes, after performing subtraction, you might end up with a fraction that can be further simplified.
• To simplify, you need to find the GCD of the numerator and the denominator and divide both by this number.

In the provided exercise, after calculating, we arrived at the fraction \(-10/9\). Here, 10 and 9 do not have any common divisors other than 1.
This means that \(-10/9\) is already in its simplest form. When a fraction is already simplified, it cannot be reduced further whilst keeping its value intact. Remember, simplifying makes it easier to work with and understand fractions.