Problem 2 - Beginning and Intermediate Algebra

Question

Multiply. $$\frac{2}{5} \cdot \frac{3}{11}$$

Step-by-Step Solution

Answer

The short answer to the question is: \(\frac{6}{55}\).

Step 1: Multiply the numerators

To multiply two fractions, first multiply the numerators (the top numbers) together. In this case, multiply 2 and 3 together: \(2 \cdot 3 = 6\).

Step 2: Multiply the denominators

Now, multiply the denominators (the bottom numbers) together. In this case, multiply 5 and 11 together: \(5 \cdot 11 = 55\).

Step 3: Write the final fraction

Combine the results from Step 1 and Step 2 to form the new fraction. The new numerator is 6, and the new denominator is 55. So, the fraction is: \(\frac{6}{55}\).

Step 4: Simplify the fraction, if possible

In this case, there are no common factors between 6 and 55, so the fraction is already in its simplest form. The final answer is: \(\frac{6}{55}\).

Key Concepts

Fraction Simplification

Simplifying a fraction means making it as simple as possible by ensuring the numerator and the denominator have no common factors other than 1. This process doesn't change the value of the fraction, but only reduces it to its simplest form.
For example, if you have a fraction like \( \frac{8}{12} \), simplifying involves finding the greatest common factor (GCF) of 8 and 12, which is 4.
After dividing both the numerator and the denominator by their GCF, you get \( \frac{2}{3} \), which is the simplified form.

• Always check for common factors in the numerator and denominator.
• Divide both parts by the common factor to simplify.
• If no common factors exist, the fraction is already simplified.

In the original exercise, the fraction \( \frac{6}{55} \) had no common factors, meaning it was already simplified.

Numerator and Denominator

Understanding the roles of the numerator and denominator is crucial when working with fractions.
The numerator is the top number, which indicates how many parts we have or are considering. On the other hand, the denominator is the bottom number and represents the total number of equal parts into which a whole is divided.
For instance, in \( \frac{3}{4} \), 3 is the numerator, telling us the count of one-quarter parts, whereas 4 is the denominator, showing how the whole is divided into four equal parts.

• Numerator: Part of the fraction above the line.
• Denominator: Part of the fraction below the line.
• They together form the entire fraction.

Each part plays a distinct role in the value of the fraction, as seen in the multiplication process involving \( \frac{2}{5} \) and \( \frac{3}{11} \).

Step-by-Step Math Solution

A structured approach makes multiplying fractions easy and manageable. Let's explore this process step by step:
1. **Multiply the Numerators**: Begin by multiplying the numerators of both fractions. Using the example \( \frac{2}{5} \cdot \frac{3}{11} \), multiply 2 by 3 to get 6. This result becomes the new numerator.
2. **Multiply the Denominators**: Next, multiply the denominators. Here, multiply 5 by 11, yielding 55. This becomes the new denominator.
3. **Form the New Fraction**: Combine the results to create the new fraction \( \frac{6}{55} \).
4. **Simplify if Possible**: Finally, check if the new fraction can be simplified. Since 6 and 55 have no common factors, \( \frac{6}{55} \) is in its simplest form.
Using a step-by-step method keeps the process clear and helps ensure no step is missed. Try practicing with different fractions to get a feel for the method!