Problem 4 - Fundamentals of Mathematics

Question

Perform the following multiplications. $$ \left(\frac{2}{3}\right)\left(\frac{2}{3}\right) $$

Step-by-Step Solution

Answer

The result of the multiplication is \( \frac{4}{9} \).

Step 1: Identify the Multiplication

The multiplication involves two fractions: \( \frac{2}{3} \) multiplied by \( \frac{2}{3} \). Your task is to multiply these fractions together.

Step 2: Multiply the Numerators

First, multiply the numerators (the top numbers) of the fractions: \( 2 \times 2 = 4 \).

Step 3: Multiply the Denominators

Next, multiply the denominators (the bottom numbers) of the fractions: \( 3 \times 3 = 9 \).

Step 4: Simplify the Result

Combine the results from Step 2 and Step 3 to form the new fraction: \( \frac{4}{9} \). Check if this fraction can be simplified. Since \( 4 \) and \( 9 \) share no common factors (other than 1), \( \frac{4}{9} \) is already in its simplest form.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of two main components: the numerator and the denominator. Picture a pizza divided into equal parts. If you have two slices of this pizza with three total slices, you express this as a fraction, \( \frac{2}{3} \). The number on top, the "2", tells us how many parts we have, while the number on the bottom, the "3", tells us how many equal parts make up the whole pizza. This makes fractions super useful for describing anything that can be divided into equal sections.

Fractions can be used in daily life whenever you're splitting or sharing something, like a cake or a group of people. They also play a pivotal role in math, where they help solve problems and represent numbers that aren't whole.

Numerator

The numerator is the top part of a fraction. It indicates how many parts of the whole are being considered or counted. In our multiplication exercise with the fraction \( \frac{2}{3} \), "2" is the numerator. This means you have 2 parts out of the 3 parts that make up the whole.

When multiplying fractions like \( \frac{2}{3} \times \frac{2}{3} \), you multiply the numerators together. For this example, \( 2 \times 2 = 4 \) becomes the new numerator. Understanding the role of the numerator is crucial because it always tells a part of a story, highlighting the portion you have or need.

To visualize: if each slice of pie is \( \frac{1}{3} \) of the whole pie, having 2 slices means you actually have \( \frac{2}{3} \). "2" is not just any number; it's the key to understanding what portion you've got.

Denominator

The denominator is found at the bottom of a fraction. It tells you the total number of equal parts the whole is divided into. For example, in the fraction \( \frac{2}{3} \), the "3" signifies that whatever the whole is, it's split into 3 equal parts.

During the multiplication process, like in our multiplication exercise, you need to multiply the denominators. For \( \frac{2}{3} \times \frac{2}{3} \), this results in \( 3 \times 3 = 9 \), which is the denominator for the new fraction. This step helps determine the new size of the whole you are working with in that specific problem.

Denominators function as the foundation of a fraction, so comprehending their role is vital for solving problems involving fractions of any kind. They set the stage for understanding how parts are related to the whole.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. For example, in the fraction \( \frac{4}{9} \) obtained after multiplying \( \frac{2}{3} \times \frac{2}{3} \), you check whether 4 and 9 have any common divisors.

Since neither 4 nor 9 have common factors, \( \frac{4}{9} \) is already simplified. Simplifying fractions ensures answers are as concise as possible, making them easier to understand and work with.

This process helps with recognizing when numbers can be more neatly represented, which can greatly assist in both mathematical equations and real-life applications, such as measuring ingredients while cooking or determining the amount of time left for an activity.