Problem 24 - A Graphical Approach to College Algebra

Question

Find each product or quotient. $$\frac{5 m+25}{10} \cdot \frac{12}{6 m+30}$$

Step-by-Step Solution

Answer

1

Step 1: Simplify the First Fraction

The first fraction is \( \frac{5m+25}{10} \). Notice that both the numerator and the denominator can be simplified by factoring. The numerator has a common factor of 5: \( 5m + 25 = 5(m + 5) \). Thus, the first fraction becomes \( \frac{5(m+5)}{10} \). Simplify \( \frac{5}{10} \) to get \( \frac{1}{2} \), resulting in \( \frac{m+5}{2} \).

Step 2: Simplify the Second Fraction

The second fraction we have is \( \frac{12}{6m+30} \). Factor the greatest common factor from the denominator: \( 6m + 30 = 6(m+5) \). Thus, the fraction becomes \( \frac{12}{6(m+5)} \). Simplify \( \frac{12}{6} \) to get 2, resulting in \( \frac{2}{m+5} \).

Step 3: Multiply Simplified Fractions

Now multiply the simplified fractions: \( \frac{m+5}{2} \cdot \frac{2}{m+5} \). When multiplying fractions, multiply the numerators and the denominators: \( \frac{(m+5) imes 2}{2 imes (m+5)} \).

Step 4: Cancel Common Factors

Since \( (m+5) \) appears in both the numerator and the denominator, it cancels out. Similarly, the 2 in the numerator and the 2 in the denominator also cancel out, leaving you with \( \frac{1}{1} \), which simplifies to 1.

Key Concepts

Simplifying Fractions

Understanding how to simplify fractions is an essential skill in algebra. A fraction is simplified when the greatest common factor (GCF) of the numerator and the denominator is 1. This means there are no common factors other than 1 between the numerator and the denominator.
To simplify a fraction:

• Identify the common factors of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCF.

In the exercise, the fraction \( \frac{5m+25}{10} \) was simplified. By factoring 5 from the numerator, we rewrite it as \( \frac{5(m+5)}{10} \). Then, the fraction reduces to \( \frac{m+5}{2} \) when we divide both terms by 5. Simplifying makes divisions and multiplications straightforward and helps in revealing the core components of the problem.

Factoring

Factoring is a powerful technique in algebra that involves breaking down an expression into simpler 'factors' that multiply together to give the original expression. This process often reveals hidden structures that are crucial for simplifying, multiplying, or solving equations.
Consider the fraction \( \frac{5m+25}{10} \):

• The numerator, \( 5m + 25 \), has a common factor of 5. By factoring out the 5, it becomes \( 5(m+5) \).
• Similarly, the expression \( 6m + 30 \) in the denominator of another fraction simplifies to \( 6(m+5) \) by factoring out 6.

Recognizing and using these patterns allows us to modify expressions for easier handling in further operations. Factoring is vital because it helps in simplifying fractions and finding solutions more easily in algebraic contexts.

Multiplying Fractions

When multiplying fractions, the process is direct and can be elegantly simple if the fractions are already simplified. To multiply two fractions:

• Multiply the numerators (top numbers) together to get the new numerator.
• Multiply the denominators (bottom numbers) together to get the new denominator.

In the exercise, we had two simplified fractions: \( \frac{m+5}{2} \) and \( \frac{2}{m+5} \). When multiplying these:

• The numerator becomes \((m+5) \times 2\).
• The denominator becomes \(2 \times (m+5)\).

Crucial in fraction multiplication is the opportunity to cancel common factors both between and within the fractions. After multiplication, the common factors \((m+5)\) and 2 were canceled out, ultimately simplifying the fraction to \( \frac{1}{1} \), which is 1. This shows that, with careful simplification and factoring beforehand, multiplication can often be made quite simple.