Problem 99 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{1}{5} \cdot \frac{3}{5} $$

Step-by-Step Solution

Answer

\(\frac{3}{25}\) is the simplified form.

Step 1: Multiply Numerators

To multiply fractions, start by multiplying the numerators. Here, multiply the numerators of \(\frac{1}{5}\) and \(\frac{3}{5}\): \[ 1 \times 3 = 3 \]

Step 2: Multiply Denominators

Next, multiply the denominators. For the fractions \(\frac{1}{5}\) and \(\frac{3}{5}\): \[ 5 \times 5 = 25 \]

Step 3: Write the Initial Product

Combine the product of the numerators and the product of the denominators to form a new fraction: \[ \frac{3}{25} \]

Step 4: Simplify the Fraction

Check if the fraction \(\frac{3}{25}\) can be simplified further. Since 3 and 25 have no common factors other than 1, \(\frac{3}{25}\) is already in its simplest form.

Key Concepts

Fraction Simplification

Simplifying fractions means making a fraction as simple as possible. This means representing it with the smallest possible whole numbers. To check if a fraction can be simplified, you need to find the greatest common factor (GCF) of the numerator and the denominator. For instance:

• If you have a fraction like \( \frac{6}{9} \), check the GCF which is 3, so you divide both numerator and denominator by 3, simplifying it to \( \frac{2}{3} \).
• In the case of \( \frac{3}{25} \), the GCF is 1. So, it is already in its simplest form.

Understanding simplification helps tidy up mathematical expressions and is essential for efficient calculations in more complex problems.

Numerator and Denominator

Every fraction is made up of two parts: the numerator and the denominator. These parts are separated by a horizontal bar, often called the fraction bar.

• The numerator is the top number and it tells how many parts we have. In \( \frac{1}{5} \), 1 is the numerator, indicating 1 part.
• The denominator is the bottom number and it signifies the total number of equal parts something is divided into. For \( \frac{1}{5} \), 5 is the denominator, meaning the whole is divided into 5 parts.

Understanding these parts is crucial for performing operations like addition, subtraction, multiplication, and division of fractions.

Basic Fraction Operations

Operations on fractions include addition, subtraction, multiplication, and division. Each operation follows specific rules:For multiplication:

• Multiply the numerators together.
• Multiply the denominators together.

For example, to multiply \( \frac{1}{5} \) and \( \frac{3}{5} \), multiply the numerators (1 and 3) to get 3. Then multiply the denominators (5 and 5) to get 25, resulting in \( \frac{3}{25} \).Addition and subtraction require the denominators to be the same, while division of fractions involves multiplying by the reciprocal of the second fraction. Understanding these rules makes working with fractions more manageable and is a key foundation for further mathematical learning.