Problem 63 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{9}{11}-\frac{3}{11} $$

Step-by-Step Solution

Answer

\frac{6}{11}

Step 1 - Understand the problem

The exercise asks for the evaluation of the expression \( \frac{9}{11} - \frac{3}{11} \). This is a subtraction of two fractions with the same denominator.

Step 2 - Subtract the numerators

Since both fractions have the same denominator, we subtract the numerators directly. \( 9 - 3 = 6 \). The denominator remains the same.

Step 3 - Write the result as a fraction

After subtracting the numerators, we get \( \frac{6}{11} \).

Key Concepts

Numerators

The numerator is the top number in a fraction. It represents how many parts of the whole are being considered.

For example, in the fraction \( \frac{9}{11} \), 9 is the numerator. Similarly, in the fraction \( \frac{3}{11} \), 3 is the numerator. In our exercise, to subtract the fractions \( \frac{9}{11} - \frac{3}{11} \), we focus on the numerators. We subtract 3 from 9. This gives us 6.

So, the new numerator is 6.

Common Denominators

A common denominator is a shared multiple of the denominators of two or more fractions. When fractions have the same denominator, they are easier to add or subtract.

In our example, both fractions, \( \frac{9}{11} \) and \( \frac{3}{11} \), have the common denominator of 11. This means we can simply subtract the numerators while keeping the denominator the same. Hence, our operation becomes straightforward: \( \frac{9}{11} - \frac{3}{11} \) = \( \frac{6}{11} \).
This simplifies the subtraction process significantly.

Fraction Simplification

Simplification of fractions is the process of making the fraction as simple as possible. This means we reduce it to its simplest form.

To simplify a fraction, we find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by this number.

For instance, in \( \frac{6}{11} \), 6 and 11 have no common divisors other than 1. Therefore, \( \frac{6}{11} \) is already in its simplest form.
Remember, a fraction is simplest when the numerator and the denominator have no common factors besides 1.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These are fundamental operations in math.

In our exercise, we used subtraction. Subtraction of fractions with common denominators involves subtracting their numerators while retaining the common denominator.

Here’s a quick review of the process:

• Check if the denominators are the same. If they are, proceed to the next step.
• Subtract the numerators. In the exercise, 9 - 3 = 6.
• Keep the denominator the same. In this case, it remains 11.

This fundamental operation gives us our final fraction \( \frac{6}{11} \).