Problem 34 - Algebra A Combined Function

Question

Multiply or divide as indicated. See Examples 11 through 14 and \(21 .\) $$ \frac{7}{11} \cdot \frac{3}{5} $$

Step-by-Step Solution

Answer

The result is \( \frac{21}{55} \).

Step 1: Understand the Problem

We need to multiply two fractions: \( \frac{7}{11} \) and \( \frac{3}{5} \). This operation involves multiplying the numerators together and the denominators together.

Step 2: Multiply the Numerators

To multiply the numerators, we multiply 7 and 3. Thus, the product of the numerators is: \[ 7 \times 3 = 21 \]

Step 3: Multiply the Denominators

To multiply the denominators, we multiply 11 and 5. Thus, the product of the denominators is: \[ 11 \times 5 = 55 \]

Step 4: Write the Resulting Fraction

Combine the products of the numerators and denominators into a new fraction: \[ \frac{21}{55} \]

Step 5: Simplify the Fraction (if necessary)

Check if the fraction \( \frac{21}{55} \) can be simplified. Since 21 and 55 do not have any common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Understanding Numerators

In any fraction, the numerator is the number written above the line. It's an essential part of a fraction because it tells us how many parts we have. For example, in the fraction \( \frac{7}{11} \), the numerator is 7. This indicates that we have 7 parts out of a total represented by the fraction.

When multiplying fractions, like \( \frac{7}{11} \cdot \frac{3}{5} \), we focus on multiplying the numerators together. Here's how you do it:

• First, identify the numerators: 7 and 3.
• Next, multiply them together: \( 7 \times 3 = 21 \).

The product, 21, becomes the numerator in the resulting fraction. By understanding and correctly multiplying numerators, you take the first step toward combining fractions.

Decoding Denominators

The denominator is the number written below the line in a fraction. It gives us an idea of the total number of equal parts a whole is divided into. Using \( \frac{7}{11} \) as an example, the denominator is 11, showing there are 11 parts in a whole cycle.

In fraction multiplication, understanding denominators works similarly to numerators but with a slight twist. Let’s see it applied to the fractions \( \frac{7}{11} \) and \( \frac{3}{5} \):

• Identify the denominators: 11 and 5.
• Multiply the denominators: \( 11 \times 5 = 55 \).

This results in a new fraction with the denominator 55. It's important to correctly multiply denominators to ensure the accuracy of the fraction result. Remember, the denominator helps in determining the fraction's proportion.

The Art of Simplifying Fractions

Simplifying a fraction involves reducing it to its simplest form. This means adjusting the fraction so that the numerator and denominator no longer have any common factors other than 1. It's an important skill to learn, as it can make fractions easier to understand and work with.

To simplify a fraction, such as \( \frac{21}{55} \), follow these steps:

• Identify the greatest common factor (GCF) of the numerator (21) and the denominator (55).
• If the GCF is 1, this means the fraction is already in its simplest form.

For \( \frac{21}{55} \), the GCF is indeed 1, indicating that no further simplification is needed. Simplifying fractions helps in ensuring that the results are as reduced and manageable as possible. Even though not all fractions can be simplified further, checking is always a good mathematical habit.