Problem 72 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ 6 \frac{6}{7} \cdot 3 \frac{1}{2} $$

Step-by-Step Solution

Answer

The result of the operation is 24.

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For the first number, \(6 \frac{6}{7}\), multiply the whole number \(6\) by the denominator \(7\) and add the numerator \(6\). This results in \(\frac{48}{7}\). For the second number, \(3 \frac{1}{2}\), multiply the whole number \(3\) by the denominator \(2\) and add the numerator \(1\). This results in \(\frac{7}{2}\).

Step 2: Multiply the Improper Fractions

Multiply the numerators of the fractions together and the denominators together: \(\frac{48}{7} \cdot \frac{7}{2} = \frac{48 \times 7}{7 \times 2}\). This results in \(\frac{336}{14}\).

Step 3: Simplify the Resulting Fraction

To simplify \(\frac{336}{14}\), find the greatest common divisor of \(336\) and \(14\), which is \(14\). Divide both the numerator and the denominator by \(14\), resulting in \(\frac{24}{1}\), which simplifies to \(24\).

Key Concepts

Improper Fractions

Understanding improper fractions is key to handling mixed numbers. A mixed number, such as \(6 \frac{6}{7}\), combines a whole number and a fraction. When converting it to an improper fraction, we focus on the whole number part and the fraction. Here’s how to do it:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator to the result of that multiplication.
• Place this sum over the same denominator.

For example, given \(6 \frac{6}{7}\), multiply the whole number \(6\) by the denominator \(7\) to get \(42\). Add the numerator \(6\), giving a total of \(48\), so the improper fraction is \(\frac{48}{7}\). This allows for simple arithmetic operations like multiplication.

Fractions Multiplication

Multiplying fractions, whether they are improper or regular, is straightforward. You simply need to:

• Multiply the numerators together.
• Multiply the denominators together.

Take \(\frac{48}{7} \cdot \frac{7}{2}\): Multiply the numerators \(48\) and \(7\) to get \(336\), and multiply the denominators \(7\) and \(2\) to get \(14\). Thus, the product is \(\frac{336}{14}\). Fractions multiplication requires attention to both the numerator and denominator interactions. Be sure to simplify if necessary.

Simplifying Fractions

After performing operations like multiplication, simplifying the fraction can make the number easier to understand and work with. The process involves:

• Finding the greatest common divisor (GCD) of the numerator and denominator.
• Dividing both by the GCD.

For \(\frac{336}{14}\), the GCD is \(14\). By dividing both the numerator and denominator by \(14\), you simplify the fraction to \(\frac{24}{1}\), which is simply \(24\). Simplification not only provides a cleaner result but also aids in grasping and using the value in further calculations.