Problem 85 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{11}{22}$$

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1: Identify the Greatest Common Divisor (GCD)

Find the greatest common divisor of the numerator and the denominator. For the fraction \( \frac{11}{22} \), identify the GCD of 11 and 22.

Step 2: Divide Both Terms by the GCD

Divide both the numerator and the denominator by their GCD. Since the GCD is 11, divide both 11 and 22 by 11.

Step 3: Simplify the Fraction

Perform the division: \( \frac{11 \div 11}{22 \div 11} = \frac{1}{2} \). The fraction is now simplified because the only common factor between the numerator and the denominator is 1.

Key Concepts

Greatest Common Divisor

When you're simplifying fractions, the Greatest Common Divisor (GCD) is your best friend. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For example, in the fraction \( \frac{11}{22} \), 11 is the GCD since it divides both 11 and 22 evenly.
To find the GCD of any two numbers:

• List the factors of each number.
• Find the common factors between them.
• The largest common factor is the GCD.

In practice, this means breaking the numbers down into their prime factors or using the Euclidean algorithm, which involves a bit of dividing and subtracting. Identifying the GCD is crucial because it sets the stage for reducing fractions to their lowest terms.

numerator and denominator

In any fraction, you have a numerator and a denominator, which are the numbers above and below the fraction bar, respectively. Let's break these terms down:

• The numerator is the top number in a fraction. It tells you how many parts of a whole you have.
• The denominator is the bottom number. It tells you into how many parts the whole is divided.

For instance, in \( \frac{11}{22} \), 11 is the numerator, and 22 is the denominator.
Understanding what the numerator and denominator signify helps in simplifying fractions. To do this, we divide both by their GCD, reducing the size of the numbers while keeping the value of the fraction unchanged.

simplified fraction

A simplified fraction is as simple as it sounds. Once you've identified the GCD of the numerator and the denominator, you use it to reduce the fraction to its lowest terms. In other words, you make the numbers smaller but the fraction remains the same amount.
To simplify a fraction:

• Divide the numerator and the denominator by their GCD.
• Replace the original fraction with the new, smaller terms.

For example, with \( \frac{11}{22} \), you divide both the numerator and the denominator by 11, resulting in \( \frac{1}{2} \). This new fraction cannot be reduced further because its GCD is 1. Hence, it is considered simplified. Simplifying fractions is a valuable skill, especially when comparing or adding fractions, as it makes calculations easier to manage.