Problem 26 - Algebra 1

Question

\(\frac{2}{3} \cdot x\) when \(x=\frac{1}{3}\)

Step-by-Step Solution

Answer

The result of the calculation \(\frac{2}{3} \cdot x\) when \(x=\frac{1}{3}\) is \(\frac{2}{9}\)

Step 1: Substitution

Replace \(x\) with the value \(\frac{1}{3}\) in the equation \(\frac{2}{3} \cdot x\). This results in \(\frac{2}{3} \cdot \frac{1}{3}\)

Step 2: Multiply Fractions

Multiplication of fractions involves multiplying the numerators together and the denominators together. In our case, the multiplication results in \(\frac{2*1}{3*3}\)

Step 3: Simplify the Result

On multiplying the numerators and denominators, the result is \(\frac{2}{9}\)

Key Concepts

Fractions

The concept of fractions is fundamental in mathematics and is often used to represent parts of a whole. A fraction consists of two main components: the numerator and the denominator. The numerator is the number on the top. It represents how many parts we have. The denominator, located at the bottom, shows how many equal parts the whole is divided into.

For instance, in the fraction \(\frac{2}{3}\), 2 is the numerator and indicates 2 parts out of the whole, while 3 is the denominator, revealing that the whole consists of 3 equal parts. Fractions are incredibly useful for conveying amounts smaller than one and are widely used in various fields, including measurements, probabilities, and proportions.

Mathematical Operations

Performing operations with fractions can seem a bit tricky at first, but it follows straightforward rules. Among the basic operations—addition, subtraction, multiplication, and division—multiplication of fractions stands out. When multiplying fractions, unlike addition and subtraction, there is no need for a common denominator.

Instead, you multiply directly:

• Numerator multiplied by numerator
• Denominator multiplied by denominator

Returning to our example, multiplying \(\frac{2}{3}\) by \(\frac{1}{3}\) involves multiplying numerators, 2 and 1, giving us 2. Simultaneously, multiplying the denominators, 3 and 3, results in 9. This process gives the fraction \(\frac{2}{9}\). The simplicity of the operation is quite evident, especially when compared to the additional steps often required for adding or subtracting fractions.

Simplification

Simplification is an essential concept when working with fractions, ensuring that they are presented in the simplest form possible. This means reducing a fraction to its smallest equivalent form where the numerator and denominator have no common factors other than 1.

In our example, the fraction \(\frac{2}{9}\) is already in its simplest form because the only common factor of 2 and 9 is 1. To simplify a fraction, follow these steps:

• Identify any common factors of the numerator and denominator.
• Divide both by their greatest common factor (GCF).

This process may not always be necessary, as demonstrated here, but knowing how to simplify fractions is valuable for presenting cleaner and more understandable answers.

Numerator and Denominator

Understanding the numerator and denominator is crucial when working with fractions. These two components define a fraction’s value and meaning. The numerator indicates how many parts of a fraction are being considered, while the denominator tells us into how many parts the whole is divided.

Thus, in \(\frac{2}{3}\), the numerator is 2, signifying two parts, and the denominator is 3, meaning the whole is divided into three parts. This structure allows fractions to convey precise portions of quantities.

When performing operations like multiplication, each part plays a vital role:

• The numerators are multiplied together to get the new numerator.
• The denominators are multiplied to form the new denominator.

This structured approach ensures clarity and consistency in calculations involving fractions, making them a powerful tool in various mathematical applications.