Question

\(\frac{10}{22}\)

Short answer

The simplified form of \( \frac{10}{22} \) is \( \frac{5}{11} \).

Step-by-step solution

Identify the Problem

The problem requires us to simplify the fraction \( \frac{10}{22} \). This means we need to reduce the fraction to its simplest form by finding the greatest common divisor (GCD) of the numerator (10) and the denominator (22).

Determine the GCD

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. We do this by listing the factors of each number. The factors of 10 are 1, 2, 5, and 10. The factors of 22 are 1, 2, 11, and 22. The greatest common factor is the largest number that both sets of factors share, which is 2.

Simplify the Fraction

Once we have the GCD, which is 2, we divide both the numerator and the denominator by this number. So, we divide 10 by 2 to get 5, and 22 by 2 to get 11. Thus, the fraction \( \frac{10}{22} \) simplifies to \( \frac{5}{11} \).

Verify the Simplification

To ensure the simplification is correct, we can check that 5 and 11 have no common factors other than 1, confirming that \( \frac{5}{11} \) is in its simplest form.

Key concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept when simplifying fractions. It is the largest number that divides both numbers in question without leaving a remainder. To simplify a fraction properly, knowing how to find the GCD is essential.
To discover the GCD, follow these steps:

• List all the factors of the numerator.
• List all the factors of the denominator.
• Identify the largest factor that appears in both lists.

For example, with the fraction \( \frac{10}{22} \), the factors of 10 are 1, 2, 5, and 10, and the factors of 22 are 1, 2, 11, and 22. Both numbers share the factors 1 and 2. The greatest of these shared factors is 2, hence the GCD is 2. This efficient method helps reduce fractions to their simplest forms easily.

Numerator and Denominator

The numerator and denominator are the two main parts of a fraction. The numerator is the number above the line in a fraction, representing how many parts of a whole are being considered. The denominator, on the other hand, is below the line and indicates the total number of equal parts the whole is divided into.
In the fraction \( \frac{10}{22} \), 10 is the numerator and 22 is the denominator. Understanding these terms is important because simplifying a fraction requires adjusting both the numerator and the denominator by the same factor. This technique ensures that the fraction remains equivalent, just in a simpler form.
When simplifying fractions, think about dividing the numerator and denominator by the greatest common divisor to preserve the relationship between the two numbers. Keeping in mind what each part of a fraction represents helps in achieving accurate simplification.

Fraction Simplification Process

Simplifying fractions involves reducing them to their simplest form so they are easier to use in calculations. The process is straightforward if you follow these easy steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

Take the fraction \( \frac{10}{22} \) as an example:

• The greatest common divisor is 2.
• Divide 10 (numerator) by 2 to get 5.
• Divide 22 (denominator) by 2 to get 11.

The simplified fraction is \( \frac{5}{11} \). To verify, check that the new numerator and denominator have no common divisors other than 1. Simplification reduces complexity and makes fractions easier to understand in mathematical operations.