Problem 53 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{1}{3} \cdot \frac{2}{3}$$

Step-by-Step Solution

Answer

The product is \(\frac{2}{9}\).

Step 1: Multiply the Numerators Together

To find the product of two fractions, you multiply the numerators (the top numbers) together. Here, multiply the numerators: \(1\) and \(2\). So, \(1 \times 2 = 2\).

Step 2: Multiply the Denominators Together

Next, multiply the denominators (the bottom numbers) together. Here, multiply the denominators: \(3\) and \(3\). So, \(3 \times 3 = 9\).

Step 3: Write the Fraction with the New Numerator and Denominator

Combine the results from Steps 1 and 2 to form a new fraction. So, the fraction is \(\frac{2}{9}\).

Step 4: Check for Reduction

Check if the fraction \(\frac{2}{9}\) can be simplified. Since the greatest common factor of 2 and 9 is 1, \(\frac{2}{9}\) is already in its simplest form.

Key Concepts

Numerators and Denominators

When working with fractions, understanding the roles of numerators and denominators is crucial. A fraction consists of two parts: the numerator and the denominator. The numerator is the number above the fraction line, while the denominator is the number below.

• The numerator represents how many parts we have.
• For example, in the fraction \( \frac{1}{3} \), 1 is the numerator, indicating one part of the whole.
• The denominator shows the total number of equal parts the whole is divided into.
• In the same fraction \( \frac{1}{3} \), 3 is the denominator, indicating the whole is divided into three equal parts.

When multiplying fractions, you multiply both the numerators and the denominators. This involves straightforward multiplication of the numbers at the top and bottom. Make sure you multiply them separately and then form a new fraction with these results.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This means making the fraction as small as possible, without changing its value. Simplification involves dividing both the numerator and the denominator by their greatest common factor (GCF), which is the largest number that divides both without a remainder.

• For instance, consider making the fraction \( \frac{8}{12} \) simpler.
• You look for the GCF of 8 and 12, which is 4, and divide both by this number.
• The result is \( \frac{2}{3} \), which is in simplified form.

It's important to note that not every fraction can be reduced further. For the fraction \( \frac{2}{9} \), the GCF is 1, so it's already as simple as it gets. Ensuring that the fraction is in its simplest form makes it easier to work with and understand.

Greatest Common Factor

The greatest common factor, or GCF, is an essential concept when dealing with fraction simplification. It is the largest number that can evenly divide both numbers, which are typically the numerator and denominator of a fraction.

• To find the GCF, list all the factors of each number (factors are numbers that can divide another number without leaving a remainder).
• Look for the highest number that appears in both lists.
• For example, for the numbers 18 and 24, the factors of 18 are 1, 2, 3, 6, 9, and 18; for 24, the factors are 1, 2, 3, 4, 6, 8, 12, and 24.
• The common factors are 1, 2, 3, and 6, with 6 being the greatest.

Using the GCF helps in reducing fractions by dividing both the numerator and the denominator by this number. Thus, it plays a critical role in simplifying fractions and ensuring they are presented in their simplest form.