Problem 87 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ 6 \cdot 2 \frac{7}{24} $$

Step-by-Step Solution

Answer

The simplified result is \( 13 \frac{3}{4} \).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number to an improper fraction. The mixed number is \( 2 \frac{7}{24} \). To convert, multiply the whole number 2 by the denominator 24 and add the numerator 7. This results in \( 48 + 7 = 55 \). So, \( 2 \frac{7}{24} = \frac{55}{24} \).

Step 2: Perform the Multiplication

Now multiply the integer 6 by the improper fraction \( \frac{55}{24} \). This can be done by writing 6 as a fraction: \( \frac{6}{1} \). The multiplication is as follows: \( \frac{6}{1} \times \frac{55}{24} = \frac{6 \times 55}{1 \times 24} = \frac{330}{24} \).

Step 3: Simplify the Fraction

Now simplify \( \frac{330}{24} \). First, find the greatest common divisor (GCD) of 330 and 24. The GCD is 6. Divide the numerator and the denominator by 6: \( \frac{330 \div 6}{24 \div 6} = \frac{55}{4} \).

Step 4: Convert Improper Fraction to Mixed Number

Finally, convert the improper fraction \( \frac{55}{4} \) to a mixed number. Divide 55 by 4, which gives 13 with a remainder of 3. Thus, \( \frac{55}{4} = 13 \frac{3}{4} \).

Key Concepts

Understanding Mixed Numbers

Mixed numbers contain both a whole number and a fraction. They're often used to represent amounts greater than one. For instance, the mixed number \(2 \frac{7}{24}\) signifies two wholes and the fraction \(\frac{7}{24}\). To fully grasp mixed numbers, remember:

• The whole number illustrates how many complete units you have.
• The fraction tells you about additional parts of the whole you possess.

Knowing how to handle mixed numbers is crucial in operations, especially when transitioning them into improper fractions for calculations.

Introduction to Improper Fractions

Improper fractions have numerators larger than their denominators. They express a quantity greater than one whole. To work with improper fractions, especially when converting from mixed numbers, follow these steps:

• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator of the fraction part.
• Place this sum over the original denominator.

For example, converting the mixed number \(2 \frac{7}{24}\) gives you the improper fraction \(\frac{55}{24}\). This conversion helps simplify mathematical operations like multiplication and division, making improper fractions a common tool in calculations.

Simplification of Fractions

Simplifying fractions means reducing them to their simplest form. The goal is to find the smallest possible numerator and denominator that represent the same proportion. Here’s a step-by-step guide on fraction simplification:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator of the fraction by this GCD.

For instance, in the fraction \(\frac{330}{24}\), the GCD is 6. Divide both terms by 6: \(\frac{330 \div 6}{24 \div 6} = \frac{55}{4}\). Simplifying fractions is essential for making numbers easier to handle and understand, and it ensures you're working with the most concise form in mathematical problems.