How do you add fractions?

To add fractions, they must have the same denominator. Find the least common multiple (LCM) of the denominators, expand both fractions to that denominator, then add the numerators. Example: 1/4 + 2/3 = 3/12 + 8/12 = 11/12.

How do you multiply fractions?

When multiplying fractions, simply multiply numerator by numerator and denominator by denominator. Example: 2/3 × 3/5 = 6/15 = 2/5 (simplified). A common denominator is not needed.

What does simplifying fractions mean?

Simplifying means dividing numerator and denominator by their greatest common divisor (GCD). The fraction keeps its value but becomes simpler. Example: 12/18 ÷ 6 = 2/3. The GCD of 12 and 18 is 6.

How do you convert fractions to decimals?

Divide the numerator by the denominator. Example: 3/4 = 3 ÷ 4 = 0.75. Some fractions result in repeating decimals, e.g., 1/3 = 0.333... (repeating 3).

What is a mixed number?

A mixed number consists of a whole number and a proper fraction, e.g., 2 3/4. It arises when the numerator is larger than the denominator (improper fraction): 11/4 = 2 3/4, because 11 ÷ 4 = 2 remainder 3.

How do you divide fractions?

To divide by a fraction, multiply by its reciprocal (numerator and denominator are swapped). Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.

What is the GCD (greatest common divisor)?

The GCD is the largest number that divides both numerator and denominator without remainder. It's used to simplify fractions. For 24 and 36, the GCD is 12, because 24 = 2×12 and 36 = 3×12.

What is the LCM (least common multiple)?

The LCM of two numbers is the smallest positive number divisible by both. It's used as the common denominator when adding and subtracting fractions. For 4 and 6, the LCM is 12.

How do you compare fractions?

Bring both fractions to the same denominator (LCM) and compare the numerators. The fraction with the larger numerator is greater. Alternatively, convert both to decimals and compare directly.

In which grade do students learn fractions?

In Germany, fraction arithmetic is typically introduced in 5th and 6th grade. Simple fractions and the concept of sharing are covered in elementary school, while formal fraction arithmetic with all operations follows in secondary school.

Simplifying Fractions with GCD — How It Works | Fraction Calculator

Simplifying Fractions with GCD — How It Works

Simplifying makes fractions clearer and easier to work with. In this article, you'll learn how to systematically simplify fractions using the greatest common divisor (GCD).

What Does Simplifying Mean?

When simplifying, you divide both numerator and denominator by the same number without changing the fraction's value. Example: 12/18 → divide both by 6 → 2/3. The fraction 2/3 has the same value as 12/18 but is simpler.

The Greatest Common Divisor (GCD)

The GCD of two numbers is the largest number that divides both without a remainder. For 12 and 18: divisors of 12 are 1, 2, 3, 4, 6, 12. Divisors of 18 are 1, 2, 3, 6, 9, 18. The greatest common divisor is 6.

Method 1: Prime Factorization

Break down both numbers into prime factors: 12 = 2 × 2 × 3, 18 = 2 × 3 × 3. The GCD is the product of common prime factors: 2 × 3 = 6. So: 12/18 ÷ 6 = 2/3.

Method 2: Euclidean Algorithm

Divide the larger number by the smaller and continue with the remainder: 18 ÷ 12 = 1 remainder 6, then 12 ÷ 6 = 2 remainder 0. The last divisor without remainder (6) is the GCD.

Method 3: Step-by-Step Simplification

You can also simplify step by step using small numbers: 12/18 ÷ 2 = 6/9 ÷ 3 = 2/3. The result is the same, but you don't need to know the GCD directly.

When Is a Fraction Fully Simplified?

A fraction is fully simplified when numerator and denominator share no common factor other than 1. We say: numerator and denominator are **coprime**.

Tip

Our calculator's 'Simplify' tab shows you the prime factorization, the GCD, and the complete step-by-step solution — perfect for learning and checking.