Problem 11 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-\frac{6}{7}\left(-\frac{7}{6}\right)$$

Step-by-Step Solution

Answer

The product is 1.

Step 1: Understanding the Problem

We need to find the product of two fractions: \(-\frac{6}{7}\) and \(-\frac{7}{6}\). Multiplying these fractions involves multiplying their numerators together and their denominators together.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \(-6\) and \(-7\). Calculate the product: \(-6 \times -7 = 42\).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \(7\) and \(6\). Calculate the product: \(7 \times 6 = 42\).

Step 4: Simplify the Result

Now that we have the product of the numerators and denominators, we find the fraction: \(\frac{42}{42}\). Simplify this fraction by dividing both the numerator and the denominator by their greatest common factor, which is 42: \(\frac{42}{42} = 1\).

Key Concepts

Numerators and Denominators

In the world of fractions, understanding numerators and denominators is essential. A fraction consists of two parts:

• The **numerator**: This is the number on the top of the fraction. It represents how many parts we are considering.
• The **denominator**: This is the number on the bottom of the fraction. It shows the total number of equal parts into which something is divided.

When multiplying fractions, we perform the multiplication separately for numerators and denominators. This means the numerators are multiplied together, and the denominators are multiplied together. In our exercise, for example, the numerators -6 and -7 are multiplied resulting in 42, and the denominators 7 and 6 are also multiplied to give 42.

Fraction Simplification

Once you've multiplied the fractions, you often end up with a fraction that can be simplified. Simplification means making the fraction as simple as possible, and this can be achieved by dividing both the numerator and the denominator by their common factors.
In our example, after multiplying, we obtain a fraction \(\frac{42}{42}\). A fraction where the numerator and the denominator are the same is equal to \(1\) because any number divided by itself is \(1\).
To simplify, we divide \(42\) and \(42\) by their **greatest common factor**. Luckily, simplifying fractions often involves recognizing special cases like this where the fraction equals whole numbers, such as \(1\).

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is crucial to simplifying fractions. The GCF of two or more numbers is the largest number that divides them without leaving a remainder. Finding the GCF helps us reduce fractions effectively.
Here's how our example works: the GCF of \(42\) and \(42\) is indeed \(42\). That’s because \(42\) is the only factor to divide itself exactly, resulting in \(1\).

• List the factors of the numbers involved.
• Identify the greatest factor common to both.

Using the GCF simplifies your fractions efficiently and results in simpler, cleaner answers such as \(1\). This process ensures you’re not left with large or cumbersome numbers and helps reinforce reliable mathematical habits.