Problem 59 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ \frac{17}{21}-\frac{10}{21} $$

Step-by-Step Solution

Answer

The result is \( \frac{1}{3} \).

Step 1: Identify the Common Denominator

In the expression \( \frac{17}{21} - \frac{10}{21} \), both fractions have the same denominator, which is 21. This allows us to directly subtract the numerators.

Step 2: Subtract the Numerators

Since the denominators are the same, we subtract the numerators: \( 17 - 10 = 7 \).

Step 3: Write the Resulting Fraction

After subtracting the numerators, we place the result over the common denominator which was 21, giving us the fraction \( \frac{7}{21} \).

Step 4: Simplify the Fraction

The fraction \( \frac{7}{21} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 7. So, \( \frac{7}{21} = \frac{1}{3} \).

Key Concepts

Common Denominator

When working with fractions, a **common denominator** is essential for many operations such as addition and subtraction. The denominator in a fraction signifies how many parts make up a whole. To perform operations on fractions effectively, it's crucial that they share the same denominator. Then, only their numerators—the parts of the fraction that represent how many parts are considered—are dealt with directly. For instance, in the exercise given, both fractions share the denominator 21, meaning they're broken into the same number of parts so the numerators can be easily compared or combined. It's this shared denominator that makes such tasks a lot simpler and directly approachable.

Subtracting Fractions

**Subtracting fractions** involves a straightforward method, especially when a common denominator is already determined. This process requires subtraction of the numerators while keeping the denominator unchanged. For example: Given the fractions \( \frac{17}{21} \) and \( \frac{10}{21} \), since they share the denominator 21, you simply subtract the numerators 17 and 10. Thus, the subtraction process becomes: \( 17 - 10 = 7 \), resulting in the fraction \( \frac{7}{21} \).

• Ensure both fractions have a common denominator.
• Subtract the second numerator from the first.
• Write the difference over the common denominator.

This keeps the operation consistent and the fractions easy to work with.

Simplifying Fractions

Once you have a fraction from an operation, such as \( \frac{7}{21} \), the next step often involves **simplifying fractions**. Simplifying entails reducing the fraction to its simplest form while maintaining equivalence. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).

• Find the GCD of the numerator and denominator. For 7 and 21, it's 7.
• Divide both numbers by the GCD.
• Write the new, simplified fraction.

In this example, divide 7 by 7 and 21 by 7, giving \( \frac{1}{3} \). This simpler form of the fraction makes it easier to interpret and use in further calculations or comparisons. Always aim for this final step to make numbers more manageable.