Problem 26 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{8}{11}-\frac{9}{11} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{-1}{11}\).

Step 1: Identify the Common Denominator

When adding or subtracting fractions, it's important to determine if they have a common denominator. In this case, we have \(\frac{8}{11}\) and \(\frac{9}{11}\), both of which already have the same denominator: 11.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerator of the second fraction from the numerator of the first. So, calculate \(8 - 9 = -1\).

Step 3: Write the Resulting Fraction

The result of the subtraction is a new fraction with the same denominator. So, the expression becomes \(\frac{-1}{11}\).

Step 4: Simplify the Fraction If Possible

Check to see if the resulting fraction \(\frac{-1}{11}\) can be simplified. The numerator is -1 and the denominator is 11, both of which have no common factors other than 1, meaning the fraction is already in its simplest form.

Key Concepts

Understanding Common Denominator

A common denominator is essential when subtracting fractions. It simplifies the process, enabling you to combine fractions directly. If fractions don't already have the same denominator, you must find a common one to proceed with subtraction or addition. The reason behind this is simple: having different denominators means the fractions are of different sizes or types, and only when they're the same can they be easily compared or combined.

In the exercise \( \frac{8}{11} - \frac{9}{11} \), the common denominator is already present, which is 11. This makes the subtraction process straightforward without needing to manipulate the denominators.

• Look for the smallest common multiple if denominators differ.
• Adjust fractions to have the same denominator.

Having a common denominator simplifies the fraction task significantly.

Process of Numerator Subtraction

Once you have a common denominator, focus on subtracting the numerators. This step is simpler if you have prepared your fractions with the same bottom number. With the example fractions \( \frac{8}{11} \) and \( \frac{9}{11} \), since both denominators are 11, you directly subtract the numerators:

8 from 9 gives \( 8 - 9 = -1 \).

• Align the fractions so their denominators match.
• Subtract the second fraction's numerator from the first's.
• Keep the common denominator the same after subtraction.

The result becomes \( \frac{-1}{11} \). Remember, numerator subtraction defines the difference between two fractions and it can sometimes result in a negative number, as in this example.

Simplifying Fractions

Simplifying fractions is the last step to ensure the result is in its basic form. A fraction is simplified when the numerator and the denominator have no common divisors other than 1. This makes it easier to understand and work with the fraction further.

In this exercise, the resulting fraction \( \frac{-1}{11} \) is already simplified because:

• Both -1 and 11 do not have any common factors except for 1.
• Negative numerators are handled the same as positive ones when simplifying.
• The fraction is in its simplest form as neither number can be reduced further.

Thus, we conclude that \( \frac{-1}{11} \) is the final answer in its simplest form. Simplifying is beneficial as it provides clarity and precision in mathematical expressions.