Problem 26 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. See Examples 2 or \(4 .\) $$ \frac{\frac{1}{6}-\frac{5}{s}}{\frac{2}{s}} $$

Step-by-Step Solution

Answer

\( \frac{s - 30}{12} \)

Step 1: Identify the Components of the Complex Fraction

The given complex fraction is \( \frac{\frac{1}{6} - \frac{5}{s}}{\frac{2}{s}} \). This fraction has a numerator, \( \frac{1}{6} - \frac{5}{s} \), and a denominator, \( \frac{2}{s} \).

Step 2: Find a Common Denominator

For the expression \( \frac{1}{6} - \frac{5}{s} \) in the numerator, find a common denominator, which is the product of 6 and \( s \), giving us \( 6s \). Rewrite \( \frac{1}{6} \) as \( \frac{s}{6s} \) and \( \frac{5}{s} \) as \( \frac{30}{6s} \).

Step 3: Simplify the Numerator

Combine the expressions in the numerator with the common denominator of \( 6s \). This gives us \( \frac{s - 30}{6s} \).

Step 4: Divide by the Denominator

Divide the simplified numerator, \( \frac{s - 30}{6s} \), by the denominator, \( \frac{2}{s} \). This is equivalent to multiplying \( \frac{s - 30}{6s} \) by the reciprocal of \( \frac{2}{s} \), which is \( \frac{s}{2} \).

Step 5: Simplify Further Using Reciprocal Multiplication

Perform the multiplication: \[ \frac{s - 30}{6s} \times \frac{s}{2} = \frac{(s - 30) \cdot s}{6s \cdot 2} = \frac{s^2 - 30s}{12s} \].

Step 6: Reduce the Fraction

Factor out an \( s \) from the numerator: \( s(s - 30) \), and cancel the common \( s \) in the numerator and denominator: \[ \frac{s(s - 30)}{12s} = \frac{s - 30}{12} \].

Key Concepts

Common Denominator

The concept of a common denominator is essential when combining or subtracting fractions. It is the shared multiple of the denominators of two or more fractions, which allows us to perform arithmetic operations like addition or subtraction on the numerators. In our problem, we're dealing with the complex fraction \( \frac{1}{6} - \frac{5}{s} \). Here, the denominators are 6 and \( s \). To combine these fractions, we need a common denominator.To find this common denominator, multiply the two denominators together. This ensures that both fractions can be expressed with the same denominator, making them comparable:

• Original denominators: 6 and \( s \)
• Common denominator: \( 6s \)

Once we have this common denominator, we can rewrite the fractions so that their denominators are alike. This transforms the expression \( \frac{1}{6} \) into \( \frac{s}{6s} \) and \( \frac{5}{s} \) into \( \frac{30}{6s} \). Now, you can easily subtract them, using the same denominators to make the operation simple and straightforward.

Fraction Multiplication

Multiplying fractions is an important operation to simplify complex fractions effectively. After reorganizing the expression with a common denominator, we often need to divide by another fraction. Fraction division is replaced by multiplication using the reciprocal of the divisor.In our specific problem, after subtracting in the numerator, we have:\[ \frac{s - 30}{6s} \]We need to divide this by the denominator \( \frac{2}{s} \). Dividing by a fraction is equivalent to multiplying by its reciprocal.Here's how we handle this by multiplying:

• Reciprocal of \( \frac{2}{s} \) is \( \frac{s}{2} \)
• Multiply \( \frac{s - 30}{6s} \) by \( \frac{s}{2} \)
• Result is \( \frac{(s - 30) \cdot s}{6s \cdot 2} \)

This multiplication gives us a new fraction, \( \frac{s^2 - 30s}{12s} \), that can be further simplified.

Algebraic Simplification

Algebraic simplification is about reducing expressions to their simplest form. It often involves canceling out common terms in the numerator and the denominator.The expression from our example, \( \frac{s^2 - 30s}{12s} \), can be simplified by factoring:

• The numerator \( s^2 - 30s \) can be rewritten as \( s(s - 30) \)
• In the denominator, we have \( 12s \)

By identifying \( s \) as a common factor in both the numerator and denominator, we can cancel it out:\[ \frac{s(s - 30)}{12s} = \frac{s - 30}{12} \]This step completes the simplification. The expression \( \frac{s - 30}{12} \) is in its simplest form, showing only the most reduced components of the initial complex fraction. Through algebraic simplification, students can easily manage complex expressions by breaking them down into simpler parts and eliminating unnecessary complexities.