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Chapter 8: Problem 37

Perform each indicated operation. $$ \frac{1}{2} \cdot \frac{1}{3} $$

Short Answer

Expert verified
The result of \( \frac{1}{2} \cdot \frac{1}{3} \) is \( \frac{1}{6} \).

Step by step solution

01

Understand the Problem

The problem asks us to multiply two fractions: \( \frac{1}{2} \) and \( \frac{1}{3} \).
02

Multiply the Numerators

To multiply fractions, start by multiplying the numerators (the top numbers of the fractions). In this case: \( 1 \times 1 = 1 \).
03

Multiply the Denominators

Next, multiply the denominators (the bottom numbers of the fractions). Here: \( 2 \times 3 = 6 \).
04

Write the Result as a Fraction

Combine the results from Step 2 and Step 3 to form a new fraction: \( \frac{1}{6} \).
05

Simplify the Fraction

The fraction \( \frac{1}{6} \) is already in its simplest form since 1 and 6 do not have any common factors other than 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Numerators
The numerators of a fraction are the numbers on top when you write the fraction. They represent how many parts of a whole you have. For example, in the fraction \( \frac{1}{2} \), the numerator is 1. This tells us that there is one part out of two total parts.
When multiplying fractions, you multiply the numerators together to get the numerator of the answer. In our exercise, we multiply the numerators of \( \frac{1}{2} \) and \( \frac{1}{3} \), giving us \( 1 \times 1 = 1 \).
  • Numerators are important when combining parts of different wholes.
  • In multiplication, think of the numerators as telling you how many pieces you are working with.
  • Keep in mind that multiplying numerators is just one step in the process of fraction multiplication.
The Role of Denominators
Denominators are the numbers at the bottom of a fraction. They tell you into how many equal parts the whole is divided. In \( \frac{1}{3} \), the denominator is 3, meaning the whole is divided into three equal parts.
When multiplying fractions, you need to multiply the denominators too. In our example, this is done by multiplying 2 from \( \frac{1}{2} \) with 3 from \( \frac{1}{3} \), resulting in \( 2 \times 3 = 6 \).
  • Denominators stay consistent when you're considering equivalent fractions.
  • In multiplication, consider the denominators as working together to form the new 'whole'.
  • Remember that the result serves as the new denominator in the product.
Simplifying Fractions
Simplifying fractions is the process of making the fraction as simple as possible, or expressing it in the smallest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our example, \( \frac{1}{6} \) is already simplified because there are no common factors between 1 and 6 other than 1 itself.
  • Simplifying helps fractions look cleaner and is often a requirement in mathematical writing.
  • Always check your fraction after performing operations to see if it can be simplified.
  • Simplification doesn't change the value of the fraction; it just presents it in an easier-to-read form.

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