21/63 simplified

2 min read 10-03-2025

Fractions can sometimes seem daunting, but simplifying them is a straightforward process. This article will guide you through simplifying the fraction 21/63, explaining the steps and the underlying mathematical principles. We'll also explore how to simplify other fractions using the greatest common divisor (GCD). By the end, you'll be confident in tackling fraction simplification.

Understanding Fraction Simplification

Simplifying a fraction means reducing it to its lowest terms. This means finding an equivalent fraction where the numerator (top number) and the denominator (bottom number) have no common factors other than 1. The fraction 21/63 isn't in its simplest form because both 21 and 63 share common factors.

Step-by-Step Simplification of 21/63

Here's how to simplify 21/63:

Find the Greatest Common Divisor (GCD): The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For 21 and 63, the GCD is 21. You can find the GCD using different methods, including listing factors or using the Euclidean algorithm (explained further below).
Divide Both Numerator and Denominator by the GCD: Divide both 21 (the numerator) and 63 (the denominator) by the GCD, which is 21:

21 ÷ 21 = 1 63 ÷ 21 = 3
Write the Simplified Fraction: The simplified fraction is 1/3. This means 21/63 is equivalent to 1/3.

Therefore, the simplified form of 21/63 is 1/3.

Finding the Greatest Common Divisor (GCD)

Finding the GCD is crucial for simplifying fractions. Here are two common methods:

Method 1: Listing Factors

List the factors of the numerator (21): 1, 3, 7, 21
List the factors of the denominator (63): 1, 3, 7, 9, 21, 63
Identify the greatest common factor: The largest number appearing in both lists is 21. Therefore, the GCD of 21 and 63 is 21.

Method 2: Euclidean Algorithm

The Euclidean algorithm is a more efficient method for larger numbers. It involves repeated division with remainder:

Divide the larger number (63) by the smaller number (21): 63 ÷ 21 = 3 with a remainder of 0.
Since the remainder is 0, the GCD is the smaller number, which is 21.

Simplifying Other Fractions

Let's try another example: Simplify 15/45.

Find the GCD: The factors of 15 are 1, 3, 5, 15. The factors of 45 are 1, 3, 5, 9, 15, 45. The GCD is 15.
Divide: 15 ÷ 15 = 1; 45 ÷ 15 = 3
Simplified Fraction: 1/3

Conclusion

Simplifying fractions is a fundamental skill in mathematics. By understanding the concept of the greatest common divisor and applying the steps outlined above, you can easily reduce any fraction to its simplest form. Remember, practicing is key to mastering this skill! Now you can confidently tackle fractions like 21/63 and simplify them to their equivalent, simplest form of 1/3.