Problem 48 - Algebra A Combined Function

Question

Perform the indicated operation. \(\frac{2}{7}\left(-\frac{2}{11}\right)\)

Step-by-Step Solution

Answer

The result of the operation is \(-\frac{4}{77}\).

Step 1: Understand the Problem

We need to multiply two fractions: \(\frac{2}{7}\) and \(-\frac{2}{11}\). Multiplying a fraction by a negative fraction will result in a negative product.

Step 2: Multiply the Numerators

To multiply fractions, multiply the numerators (the top numbers) together. Here, we multiply \(2\) from \(\frac{2}{7}\) and \(-2\) from \(-\frac{2}{11}\): \(2 \times -2 = -4\).

Step 3: Multiply the Denominators

Next, multiply the denominators (the bottom numbers) together. For our fractions, this means multiplying \(7\) and \(11\): \(7 \times 11 = 77\).

Step 4: Combine Results

Combine the results from the previous steps to form the new fraction. The result is \(\frac{-4}{77}\) as the product of \(\frac{2}{7}\) and \(-\frac{2}{11}\).

Step 5: Simplify the Fraction

Check if the resulting fraction \(\frac{-4}{77}\) can be simplified. Since 4 and 77 have no common factors other than 1, \(\frac{-4}{77}\) is already in its simplest form.

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers separated by a slash—these numbers are called the numerator and the denominator.

• The numerator is the top part of the fraction and represents how many parts we have.
• The denominator is the bottom part and shows the total number of equal parts that make up a whole.

For example, in the fraction \(\frac{2}{7}\), 2 is the numerator and indicates that we have 2 parts, while 7 is the denominator, showing that the whole is divided into 7 equal parts. Fractions can represent quantities less than one, equal to one, or greater than one, depending on the relationship between the numerator and the denominator.

Numerators and Denominators

When multiplying fractions, it’s important to know the roles of numerators and denominators. Let’s break down these roles further.Every fraction has a numerator and a denominator:

• The numerator (top number) indicates the number of parts being considered.
• The denominator (bottom number) indicates the total number of equal parts that make up a whole.

In the given multiplication task \(\frac{2}{7} \times -\frac{2}{11}\), the numerators are 2 and -2, and the denominators are 7 and 11. To multiply these fractions:1. Multiply the numerators: \(2 \times -2 = -4\)2. Multiply the denominators: \(7 \times 11 = 77\)This results in the new fraction \(\frac{-4}{77}\).

Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its simplest form. This means making the numerator and denominator as small as possible while keeping the same value. Here's how you can simplify a fraction:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCF.

In our example, the fraction \(\frac{-4}{77}\) needs to be checked for simplification. Both 4 and 77 do not share any common factors apart from 1. Hence, \(\frac{-4}{77}\) is already in its simplest form.Simplifying fractions is an important final step in fraction multiplication, as it ensures your answer is as clear and concise as possible.