Problem 17 - Elementary Algebra

Question

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{11}{22}\)

Step-by-Step Solution

Answer

Answer: The reduced form of the fraction 11/22 is 1/2.

Step 1: Find the GCD of the numerator and the denominator

To find the GCD of 11 and 22, we consider the factors of each number. The factors of 11 are 1 and 11, and the factors of 22 are 1, 2, 11, and 22. The greatest common factor of both numbers is 11.

Step 2: Reduce the fraction by dividing the numerator and the denominator by the GCD

Next, we divide both the numerator (11) and the denominator (22) by the GCD (11): \(\frac{11}{22} = \frac{11\div11}{22\div11} = \frac{1}{2}\) So, the fraction reduced to its lowest terms is \(\frac{1}{2}\).

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is an essential concept when working with fractions. It refers to the largest positive integer that divides two or more numbers without leaving a remainder. Knowing the GCD helps simplify fractions by identifying the common factor between the numerator and denominator. To determine the GCD, you list the factors of each number and choose the greatest one common to both.

• For example, in the fraction \(\frac{11}{22}\), the numerator is 11 and the denominator is 22.
• The factors of 11 are: 1, 11.
• The factors of 22 are: 1, 2, 11, 22.

The common factors are 1 and 11, with 11 being the greatest. Therefore, the GCD of 11 and 22 is 11.

Reducing Fractions

Reducing fractions involves simplifying them to their smallest possible form while keeping the same value. This means you look for a number that can evenly divide both the numerator and the denominator, which is typically done using the GCD. By dividing both parts of the fraction by the GCD, you effectively reduce the fraction.
Take the fraction \(\frac{11}{22}\), identified earlier as sharing a GCD of 11:

• Divide the numerator 11 by the GCD: \(11 \div 11 = 1\).
• Divide the denominator 22 by the GCD: \(22 \div 11 = 2\).

This results in the simplified fraction \(\frac{1}{2}\). The fraction has been reduced, maintaining the same value, but now in its simplest form.

Numerator and Denominator

In a fraction, the numerator is the number placed above the line, representing the parts of a whole being considered. The denominator is the number underneath the line, indicating the total number of equal parts the whole is divided into. Understanding these roles is key to performing operations like addition, subtraction, multiplication, and division with fractions.

• In \(\frac{11}{22}\), 11 is the numerator.
• 22 is the denominator.

These numbers work together to express the concept of division or part of a whole.

Lowest Terms

When a fraction is expressed in its lowest terms, it means it has been simplified to the point where the numerator and the denominator have no common divisors other than 1. Simplifying to lowest terms doesn't change the value of the fraction but makes it easier to work with, whether solving equations or comparing sizes.
For instance, take \(\frac{11}{22}\), which reduces to \(\frac{1}{2}\) using the GCD of 11. Here:

• \(\frac{1}{2}\) is in its lowest terms, as the only common factor between 1 and 2 is 1.

This means the fraction cannot be simplified further. Working with fractions in lowest terms simplifies mathematical operations and ensures accuracy in comparisons.