Problem 30 - Beginning and Intermediate Algebra

Question

Multiply or divide as indicated. Write the answer in lowest terms. $$\frac{1}{4} \cdot 5 \frac{5}{6}$$

Step-by-Step Solution

Answer

\(\frac{35}{24}\) is the final answer.

Step 1: Convert Mixed Number to Improper Fraction

The mixed number \(5 \frac{5}{6}\) needs to be converted to an improper fraction. Multiply the whole number 5 by the denominator 6 and add the numerator 5 to get a new numerator: \(5 \times 6 + 5 = 30 + 5 = 35\). So, \(5 \frac{5}{6} = \frac{35}{6}\).

Step 2: Multiply the Fractions

Now, multiply the fractions: \(\frac{1}{4} \cdot \frac{35}{6}\). Multiply the numerators: \(1 \times 35 = 35\). Multiply the denominators: \(4 \times 6 = 24\). So, the product is \(\frac{35}{24}\).

Step 3: Simplify the Fraction

Check if \(\frac{35}{24}\) can be simplified. The numerator 35 and the denominator 24 have no common factors other than 1. Thus, \(\frac{35}{24}\) is already in its lowest terms.

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator. These are called 'improper' because they are greater than or equal to one whole. For example, \(\frac{35}{6}\) is an improper fraction because 35 (the numerator) is greater than 6 (the denominator).

Here’s how you convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator.
• Add the numerator to the result.
• This sum becomes the new numerator, and the denominator stays the same.

Using these steps, the mixed number \(5 \frac{5}{6}\) becomes \(\frac{35}{6}\). Understanding improper fractions is essential when performing operations such as multiplication or division on mixed numbers.

Mixed Numbers

Mixed numbers consist of a whole number and a fraction together. They can be a bit tricky when you need to multiply or divide them, so converting them to improper fractions often simplifies calculations.

Mixed numbers are typically used in everyday language because they are easier to understand and visualize, like \(5 \frac{5}{6}\), which represents 5 whole items and another \(\frac{5}{6}\) of an item. Yet, when performing mathematical operations, it's often more practical to convert them into improper fractions.

For instance, to convert the mixed number \(5 \frac{5}{6}\) to an improper fraction, you multiply 5 by 6 (since 6 is the denominator), and then add the numerator 5, resulting in 35. Thus, \(5 \frac{5}{6} = \frac{35}{6}\). This step is crucial for simplifying operations and achieving correct results.

Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms. A fraction is considered simplified or in lowest terms when the numerators and the denominators have no common factors other than 1.

To simplify a fraction, determine if the numerator and denominator have any common factors. If they do, divide both the numerator and the denominator by their greatest common divisor (GCD).

In our example, the product was \(\frac{35}{24}\). To check if it can be simplified:

• List the factors of 35: 1, 5, 7, 35.
• List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
• Identify common factors: only 1 is common.

Since 1 is the only common factor, \(\frac{35}{24}\) cannot be simplified further and is already in its lowest terms. This step ensures you express fractions as simply as possible, which makes them easier to work with in equations and real-world applications.