Problem 28 - Beginning and Intermediate Algebra

Question

Multiply. $$ \frac{2}{7}\left(-\frac{2}{11}\right) $$

Step-by-Step Solution

Answer

The product is \(-\frac{4}{77}\).

Step 1: Understand the Multiplication of Fractions

When multiplying fractions, multiply their numerators together to get the new numerator, and multiply their denominators together to get the new denominator.

Step 2: Multiply the Numerators

Calculate the product of the numerators: \(2 \times -2 = -4\).

Step 3: Multiply the Denominators

Calculate the product of the denominators: \(7 \times 11 = 77\).

Step 4: Form the Resulting Fraction

Combine the results from Step 2 and Step 3 to form the new fraction: \(\frac{-4}{77}\).

Step 5: Check for Simplification

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

Key Concepts

Understanding Numerator and Denominator

To comfortably work with fractions, it is essential to understand what the numerator and denominator represent. In a fraction, such as \( \frac{3}{5} \), the top number is the numerator, and it describes how many parts of a whole we are considering.
The bottom number, called the denominator, shows how many equal parts the whole is divided into. For example, in \( \frac{3}{5} \), we are dealing with 3 parts out of a total of 5 parts of the whole.

• The numerator can be any integer, which makes the fraction represent part of the divisor determined by the denominator.
• The denominator cannot be zero, as dividing by zero is undefined.

When multiplying fractions, the numerator and denominator of each fraction are multiplied separately to form the new fraction.

Simplifying Fractions

Simplifying fractions is all about reducing the fraction to its simplest form so that it is easier to work with or understand. A fraction is simplified when the numerator and denominator are reduced to the smallest numbers possible while still keeping the fraction equivalent to the original.
This involves finding the greatest common divisor (GCD) of both the numerator and the denominator and dividing them by this number.
Consider the fraction \( \frac{8}{12} \):

• First, determine the GCD of 8 and 12. Here, the GCD is 4.
• Divide both the numerator and the denominator by 4.
• This gives us \( \frac{2}{3} \), which is the simplified version of \( \frac{8}{12} \).

When a fraction is fully reduced, the numerator and denominator have no common factors other than 1.

The Product of Fractions

Multiplying fractions is a straightforward process if you remember one essential rule: multiply across the numerators and denominators. This simple rule makes multiplication of fractions easier than addition or subtraction, which requires a common denominator.
Let's use the given example of multiplying \( \frac{2}{7} \) and \(-\frac{2}{11} \):

• First, multiply the numerators: \( 2 \times (-2) = -4 \).
• Then, multiply the denominators: \( 7 \times 11 = 77 \).
• This gives us the product \( \frac{-4}{77} \).

Before declaring this the final answer, examine whether it can be simplified. Since \( -4 \) and \( 77 \) do not have any common factors other than 1, the fraction \( \frac{-4}{77} \) cannot be simplified further. This is the final product.