Problem 29 - Beginning and Intermediate Algebra

Question

Multiply or divide as indicated. Write the answer in lowest terms. $$2 \frac{7}{9} \cdot \frac{1}{3}$$

Step-by-Step Solution

Answer

The final answer is \(\frac{25}{27}\).

Step 1: Convert Mixed Number to Improper Fraction

The first step is to convert the mixed number into an improper fraction. The mixed number is \(2 \frac{7}{9}\). To convert it, multiply the whole number \(2\) by the denominator \(9\), which equals \(18\), and add the numerator \(7\) to get \(25\). So, \(2 \frac{7}{9}\) becomes \(\frac{25}{9}\).

Step 2: Set Up the Multiplication

Now multiply the improper fraction \(\frac{25}{9}\) by the given fraction \(\frac{1}{3}\). The multiplication of fractions involves multiplying the numerators together and the denominators together.

Step 3: Perform the Multiplication

Multiply the numerators: \(25 \times 1 = 25\). Multiply the denominators: \(9 \times 3 = 27\). Therefore, the product is \(\frac{25}{27}\).

Step 4: Simplify the Fraction

Check if \(\frac{25}{27}\) can be simplified by finding the greatest common divisor of the numerator and denominator. The greatest common divisor of 25 and 27 is 1, so this fraction is already in its simplest form.

Key Concepts

Mixed Numbers

Mixed numbers consist of a whole number and a fraction combined together, such as \(2 \frac{7}{9}\). They represent values that are greater than a whole number but not quite another whole number. For example, \(2 \frac{7}{9}\) means "two whole things and seven ninths of another thing." Mixed numbers are commonly used to easily express values that fall between two whole numbers.

Mixed numbers can be converted into improper fractions, which makes them easier to manipulate in operations like multiplication or division. To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add this result to the numerator of the fraction.
• The resulting sum is the numerator of the new improper fraction, using the original denominator.

This conversion is important because it allows you to work with a consistent single fraction format when performing operations like multiplication, as seen in math problems.

Improper Fractions

Improper fractions have a numerator larger than or equal to the denominator, such as \(\frac{25}{9}\). This type of fraction represents values greater than one whole. Unlike mixed numbers, improper fractions express the whole part and the fractional part together as a single fraction.

Working with improper fractions is quite straightforward since they consist of only numerators and denominators. When multiplying fractions, for instance, you simply multiply the numerators across to get the new numerator, and multiply the denominators to get the new denominator. Using improper fractions keeps multiplication neat and simple:

• Convert mixed numbers to improper fractions before multiplication.
• Multiply numerators to get the new numerator.
• Multiply denominators to get the new denominator.

Being comfortable with improper fractions allows you to handle complex problems involving fractions more effectively.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest (or lowest) terms, where the numerator and denominator have no common factors other than 1. This process ensures that the fraction is as clear and concise as possible, which is important in both mathematics and everyday applications. To simplify a fraction like \(\frac{25}{27}\):

• Identify any common factors of the numerator and the denominator.
• Divide both numbers by the greatest common divisor (GCD) found.
• If the GCD is 1, the fraction is already as simple as possible.

In this problem, since the GCD of 25 and 27 is 1, the fraction cannot be simplified further. Knowing how to simplify fractions is a valuable skill that aids in clarifying and comparing numerical expressions easily.