Problem 69 - Algebra 1: Concepts and Skills

Question

Multiply the fractions. $$\frac{2}{3} \cdot \frac{2}{3}$$

Step-by-Step Solution

Answer

The product of \( \frac{2}{3} \) and \( \frac{2}{3} \) is \( \frac{4}{9} \).

Step 1: Determine the numerators and denominators

The numerator of the first fraction is 2, and its denominator is 3. The same is true for the second fraction.

Step 2: Multiply the numerators

Multiply the numerators of the first fraction and the second fraction together, which gives \(2 \cdot 2 = 4\). This will be the numerator of the resulting fraction.

Step 3: Multiply the denominators

Multiply the denominators of the first fraction and the second fraction together, which gives \(3 \cdot 3 = 9\). This will be the denominator of the resulting fraction.

Step 4: Write the new fraction

Combine the new numerator and the new denominator to form a new fraction. The new fraction, which is the product of the original fractions, is \( \frac{4}{9} \)

Key Concepts

Numerators

A numerator is the top number in a fraction. It shows how many parts of a whole are being considered.
In the fraction \( \frac{2}{3} \), the numerator is 2, indicating 2 parts.
When multiplying fractions, you'll multiply the numerators together.

• In our example, both numerators are 2.
• So, \( 2 \times 2 = 4 \) becomes the numerator of the new fraction.

Understanding numerators is crucial because they determine the total amount present after multiplying the fractions.

Denominators

The denominator is the bottom number in a fraction. It shows the total number of equal parts the whole is divided into.
For \( \frac{2}{3} \), the denominator 3 means each "whole" is divided into 3 parts.

• In our problem, both fractions have the same denominator, which is 3.
• When we multiply the denominators, we calculate \( 3 \times 3 = 9 \).

This multiplication gives us the new denominator in our resulting fraction, which forms the bottom part of the fraction \( \frac{4}{9} \). It's crucial to correctly multiply these to form an accurate fraction.

Simplifying Fractions

Simplifying fractions means making them as simple as possible. This involves reducing a fraction to its smallest form with the numerator and denominator having no common factors other than 1.
For \( \frac{4}{9} \), you would check if both 4 and 9 have common factors.

• In this case, they do not share any common factor other than 1.
• Thus, \( \frac{4}{9} \) is already in its simplest form.

This step isn't always required, but it's good to check if a fraction can be simplified to ensure clarity and simplicity.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. Multiplying fractions is a form of arithmetic operation that's fairly straightforward.

• Identify numerators and multiply them.
• Do the same for denominators.
• Combine them to form a new fraction.

This method works because fractions represent division of numbers. Multiplying them involves applying properties of division and multiplication systematically.
Once you multiply, don't forget to simplify the result if possible, although this won't change the original product defined by the operation.