Problem 62 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. (Skills Review p. 763) $$\frac{71}{355}$$

Step-by-Step Solution

Answer

The fraction \( \frac{71}{355} \) in its simplest form is \( \frac{1}{5} \).

Step 1: Identify the Greatest Common Divisor (GCD)

First, we need to identify the greatest common divisor (GCD) of 71 and 355. Here, it is 71.

Step 2: Divide the Numerator and Denominator by the GCD

Next, divide both the numerator (71) and the denominator (355) by their GCD. So, \( \frac{71}{71} \) becomes 1 and \( \frac{355}{71} \) becomes 5.

Step 3: Write the Simplified Fraction

Finally, write down the simplified (or reduced) form of the given fraction, which should be \( \frac{1}{5} \).

Key Concepts

Greatest Common Divisor (GCD)

To simplify any fraction, it's essential to understand the concept of the Greatest Common Divisor (GCD). The GCD of two numbers is the largest number that can evenly divide both of them. Identifying the GCD helps us reduce fractions to their simplest form by providing a common factor. To find the GCD, you can use several methods:

• Prime Factorization: Break down both numbers into their prime factors and identify the common factors.
• Euclidean Algorithm: Continuously divide the larger number by the smaller number and take the remainder, repeating this process until the remainder is zero. The last non-zero remainder is the GCD.
• Listing Factors: List all factors of both numbers and choose the greatest number that appears in both lists.

In the exercise, the GCD of 71 and 355 was calculated to be 71, showing that 71 was the largest number both could be divided by without leaving a remainder.

Numerator and Denominator

A fraction is made up of two parts: the numerator and the denominator. The numerator is the top number and tells us how many parts of a whole we have. The denominator is the bottom number and indicates how many total equal parts the whole is divided into. For example, in the fraction \( \frac{71}{355} \):

• 71 is the numerator, representing the portions you have.
• 355 is the denominator, showing the total parts which make up a whole.

Understanding these concepts is important when simplifying fractions because knowing how to divide both the numerator and denominator by the GCD is the essence of fraction simplification. By focusing on these two components, you can resize the fraction without changing its value.

Simplified Fraction

A simplified fraction is the most reduced form of a fraction, with the numerator and the denominator having no common divisors other than 1. This means the numbers are as small as possible while still correctly representing the value of the fraction.Reducing a fraction enhances its simplicity and often makes it easier to interpret in real-world scenarios or mathematical calculations. To simplify a fraction, follow these steps:

• Identify the GCD of the numerator and the denominator.
• Divide the numerator and the denominator by the GCD.

Once this process is complete, the result is a simplified fraction. In our exercise, we followed these steps to reduce \( \frac{71}{355} \) to its simplest form of \( \frac{1}{5} \). This smaller form maintains the same value, achieving clarity and efficiency in mathematical expression.