Question

\(\frac{18}{36}\)

Short answer

The simplified fraction is \( \frac{1}{2} \).

Step-by-step solution

Identify the GCD

The goal is to simplify the fraction \( \frac{18}{36} \). First, determine the greatest common divisor (GCD) of the numerator (18) and the denominator (36). Find the GCD by listing the factors of each number: - Factors of 18 are 1, 2, 3, 6, 9, and 18. - Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.The greatest common divisor is 18.

Divide by the GCD

To simplify, divide both the numerator and the denominator by the GCD (18):\[ \frac{18}{36} = \frac{18 \div 18}{36 \div 18} \]

Simplify the Fraction

Perform the division for both the numerator and the denominator:\[ \frac{18 \div 18}{36 \div 18} = \frac{1}{2} \] The simplified form of the fraction \( \frac{18}{36} \) is \( \frac{1}{2} \).

Key concepts

Greatest Common Divisor

When simplifying fractions, the greatest common divisor (GCD) is an essential concept. The greatest common divisor is the largest integer that can evenly divide two numbers without leaving a remainder. Understanding the GCD helps us in breaking down a fraction to its simplest form.
For example, consider the numbers in the fraction \(\frac{18}{36}\). To find the GCD, identify all the factors of each number:

• Factors of 18 are 1, 2, 3, 6, 9, and 18.
• Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Identifying their common factors, we observe that they are 1, 2, 3, 6, 9, and 18. Among these, 18 is the greatest, hence the GCD is 18.
Using the GCD makes simplifying fractions efficient, ensuring that the fraction is reduced completely and in one step.

Numerator and Denominator

In any fraction, two vital components are the numerator and the denominator. The numerator is the top number, representing how many parts we have. The denominator, on the bottom, shows how many total parts make up a whole. Understanding these two parts is crucial since they guide us in fraction operations like simplification.
Looking at the fraction \(\frac{18}{36}\):

• The numerator 18 indicates the parts of the whole considered.
• The denominator 36 signifies the total parts making up the whole.

When simplifying, always consider both values, especially in processes like finding the GCD, to transform the fraction into its simplest form.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form so they are easier to understand and use in calculations. This often involves using the greatest common divisor (GCD). After finding the GCD, you divide both the numerator and the denominator by this number.
For instance, to simplify \(\frac{18}{36}\), we first determined the GCD as 18. By dividing both the numerator and the denominator by 18, we perform:\[\frac{18 \div 18}{36 \div 18} = \frac{1}{2} \]This shows the fraction in its simplest form, \(\frac{1}{2}\). Simplifying helps us compare fractions more readily and work with them in both practical and problem-solving scenarios. It helps in achieving neatness and consistency, especially in larger calculations or comparing proportions.