Problem 26 - A Graphical Approach to College Algebra

Question

Find each product or quotient. $$\frac{3 m-15}{4 m-20} \cdot \frac{m^{2}-10 m+25}{12 m-60}$$

Step-by-Step Solution

Answer

The simplified product is \(\frac{1}{16}\).

Step 1: Factor Common Terms

Start by factoring the expressions in each numerator and denominator. The first expression's numerator is \(3(m-5)\) and its denominator is \(4(m-5)\). For the second expression, the numerator \(m^2 - 10m + 25\) is a perfect square, factored as \((m-5)(m-5)\), and the denominator \(12m - 60\) can be simplified to \(12(m-5)\).

Step 2: Simplify the Fractions

Rewrite the expression using the factored forms: \(\frac{3(m-5)}{4(m-5)} \cdot \frac{(m-5)(m-5)}{12(m-5)}\). Cancel out the \((m-5)\) terms present in each numerator and denominator. This simplification reduces the expression to \(\frac{3}{4} \cdot \frac{m-5}{12}\).

Step 3: Multiply Simplified Fractions

Multiply the simplified fractions. Multiply the numerators: \(3 \cdot 1 = 3\), and the denominators: \(4 \cdot 12 = 48\). This gives the result \(\frac{3}{48}\).

Step 4: Simplify the Final Result

Simplify \(\frac{3}{48}\) by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3. So, \(\frac{3}{48} = \frac{1}{16}\).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler expressions, called factors, that can be multiplied together to produce the original expression. In the problem provided, factoring is used to simplify both the numerators and the denominators of the fractions.

• For the first fraction, the expression in the numerator, \(3m-15\), is factored into \(3(m-5)\) by taking out the greatest common factor, which is 3.
• The denominator, \(4m-20\), is similarly factored into \(4(m-5)\).
• The second fraction’s numerator \(m^2-10m+25\) uses the perfect square trinomial identification, resulting in \((m-5)^2\).
• Its denominator \(12m-60\) is factored into \(12(m-5)\).

Factoring exposes common components, like \((m-5)\), making simplification possible. This step is crucial as it allows terms to be cancelled out in the next steps.

Simplification

Simplification involves reducing an expression to its simplest form while maintaining its value. After factoring, the expressions are primed for simplification by canceling out terms that are common between a numerator and its corresponding denominator.

• In the expression \(\frac{3(m-5)}{4(m-5)} \cdot \frac{(m-5)(m-5)}{12(m-5)}\), notice that all fractions have \((m-5)\) in both the numerator and denominator.
• These terms can be cancelled out since they appear in identical formation both above and below the fraction line, effectively reducing the expressed complexity.

Once the common terms are cancelled, the simplified form is \(\frac{3}{4} \cdot \frac{1}{12}\). This step greatly reduces the clutter and focuses on the core values left to be calculated.

Multiplication of Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. After simplification, the fractions \(\frac{3}{4}\) and \(\frac{1}{12}\) are ready to be multiplied.

• The numerators are multiplied: \(3 \times 1 = 3\).
• The denominators are multiplied: \(4 \times 12 = 48\).

Thus, the product is \(\frac{3}{48}\). The multiplication step is straightforward after simplification simplifies the expression into manageable numbers. This process emphasizes the mathematical operation's simplicity once unnecessary complexity is removed.

Greatest Common Divisor

The concept of the greatest common divisor (GCD) is used to simplify fractions to their lowest terms. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

• In \(\frac{3}{48}\), the greatest common factor of 3 and 48 is 3.
• Divide both the numerator and the denominator by this GCD: \(\frac{3}{3} = 1\) and \(\frac{48}{3} = 16\).

The fraction simplifies to \(\frac{1}{16}\). Understanding and appropriately applying the GCD ensures the final result is as simple as possible, offering clear and concise solutions.