Problem 34 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{6 a}{5} \cdot \frac{5}{12 a^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{2a}\).

Step 1: Write the Multiplication of Fractions

The expression given is a multiplication of two fractions: \( \frac{6a}{5} \) and \( \frac{5}{12a^2} \). To multiply fractions, multiply the numerators together and the denominators together. This gives us:\[ \frac{6a \cdot 5}{5 \cdot 12a^2}. \]

Step 2: Multiply the Numerators

Multiply the numerators: \(6a \) and \(5\). This results in:\[6a \cdot 5 = 30a.\]

Step 3: Multiply the Denominators

Multiply the denominators: \(5\) and \(12a^2\). This results in:\[5 \cdot 12a^2 = 60a^2.\]

Step 4: Form the Fraction

Now, place the products obtained in Steps 2 and 3 into a fraction:\[\frac{30a}{60a^2}.\]

Step 5: Simplify the Fraction by Cancelling Common Factors

To simplify the fraction, divide both the numerator and the denominator by common factors. Notice that both the numerator and the denominator have a factor of \(a\) and of \(30\).First, cancel the common factor of \(a\):\[\frac{30 \cdot a}{60 \cdot a^2} = \frac{30}{60a}.\]Next, simplify \(\frac{30}{60}\) by dividing both by 30:\[\frac{30}{60} = \frac{1}{2}.\]Thus, our expression simplifies to:\[\frac{1}{2a}.\]

Step 6: Write the Final Simplified Expression

The simplified form of the expression is \(\frac{1}{2a}\).

Key Concepts

Multiplication of Fractions

Multiplying fractions is a fundamental skill in algebra that involves combining two fractions into one by multiplying their numerators and their denominators, respectively. When you have a problem like \( \frac{6a}{5} \cdot \frac{5}{12a^2} \), the first thing to do is multiply the numerators together and then the denominators. It simplifies the process and can help in further steps of simplification.

To multiply the numerators together in the expression \( \frac{6a}{5} \cdot \frac{5}{12a^2} \), find:\[ 6a \cdot 5 = 30a. \] For the denominators, calculate:\[ 5 \cdot 12a^2 = 60a^2. \]

After performing these multiplications, you form a new fraction with these products:\[ \frac{30a}{60a^2}. \]
This sets the stage for the next steps, which involve simplifying the resulting fraction.

Common Factors

Common factors are numbers or algebraic expressions that divide two or more numbers or expressions evenly. Recognizing and canceling these common factors is crucial in algebra for simplifying expressions.

In our example, we have the fraction \( \frac{30a}{60a^2} \). Notice that both the numerator \(30a\) and the denominator \(60a^2\) share a common factor. First, focus on the variable: both parts contain the variable \(a\). By dividing \(a\) from both, we reduce the fraction to \( \frac{30}{60a} \).

Next, inspect for common numerical factors. Here, the numbers 30 and 60 have a common factor of 30. Dividing both by 30 gives:

• Numerator: \(30 \div 30 = 1\).
• Denominator: \(60 \div 30 = 2\).

Thus, you're left with the simplified fraction \( \frac{1}{2a} \). Recognizing and canceling common factors is a powerful tool to simplify expressions drastically.

Fraction Simplification

Fraction simplification refers to reducing a fraction to its simplest form so that it's easier to understand and use in calculations. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD) or by canceling common factors.

Starting with a fraction such as \( \frac{30a}{60a^2} \), the aim is to express it in the simplest terms possible.

After identifying the factors they share, as we did:

• First cancel the variable part which both share, \(a\), resulting in \( \frac{30}{60a} \).
• Then, divide both parts by 30, their greatest numerical factor, to end up at \( \frac{1}{2a} \).

This final fraction, \( \frac{1}{2a} \), conveys the same information as the original one, but in a more manageable form. Always try to simplify fractions unless instructed otherwise, as it often leads to clearer solutions and a deeper understanding of the problem at hand.