Problem 19 - Beginning and Intermediate Algebra

Question

Multiply or divide as indicated. Write the answer in lowest terms. $$ \frac{1}{2} \cdot \frac{3}{4} $$

Step-by-Step Solution

Answer

\(\frac{3}{8}\) in simplest form.

Step 1: Multiply the Numerators

Multiply the numerators of both fractions: \(1\) and \(3\). The product is \(1 \times 3 = 3\).

Step 2: Multiply the Denominators

Multiply the denominators of both fractions: \(2\) and \(4\). The product is \(2 \times 4 = 8\).

Step 3: Form the New Fraction

Combine the results from Steps 1 and 2 to form the new fraction: \(\frac{3}{8}\).

Step 4: Simplify the Fraction (if needed)

Check if \(\frac{3}{8}\) can be simplified. Since \(3\) and \(8\) have no common factors other than \(1\), the fraction is already in its simplest form.

Key Concepts

Numerators and Denominators

When you work with fractions, you'll frequently encounter terms such as numerators and denominators. Understanding these components is essential for any fraction operation, including multiplication.

The **numerator** is the top number in a fraction and it represents how many parts we have. For instance, in the fraction \(\frac{1}{2}\), the number \(1\) is the numerator, indicating that we have one part out of the two total parts as indicated by the denominator.

On the other side, the **denominator** is the bottom number in a fraction, and it shows into how many equal parts the whole is divided. In \(\frac{1}{2}\), the denominator is \(2\), informing us that the whole is divided into two equal parts.

When multiplying fractions, you multiply the numerators together and the denominators together, forming a new fraction from those products. For example, multiplying \(\frac{1}{2}\) and \(\frac{3}{4}\) involves multiplying the numerators 1 and 3 for the new numerator, and the denominators 2 and 4 for the new denominator.

Simplest Form

A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. Simplifying fractions is an important skill because it helps make them easier to understand and work with.

To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by this number. Sometimes, like in our solution \(\frac{3}{8}\), the fraction is already in its simplest form because the numerator and the denominator share no common factors apart from 1.

Simplifying fractions is not crucial only when displaying final answers, but it also facilitates further operations, like addition or subtraction. It ensures clear communication and prevents confusion in mathematical expressions.

Fractions

Fractions are an essential part of mathematics, representing parts of a whole in a simple, concise way. They consist of two numbers: a numerator, and a denominator.

Fractions are used for a variety of purposes in everyday life and various fields of study, such as when expressing proportions, mixing ingredients in recipes, or in financial calculations. For example, \(\frac{1}{2}\) signifies half of an item, which is easier to interpret and relate to compared to decimal or percentage representations.

Understanding how to perform operations such as multiplication or division with fractions is key to advancing in math. Just like we saw in the original exercise, multiplying fractions only requires multiplying the numerators and denominators directly. This process is simpler than other operations with fractions, making it a great starting point for learning how to handle fractions effectively.

• **Key Points**: Fractions consist of numerators and denominators.
• Multiplication involves straightforward calculations.
• Simplifying fractions is always the final step.

By practicing these fundamental skills, you'll build a solid foundation for working with more complex mathematical concepts.