21 63 simplified

2 min read 10-03-2025

The fraction 21/63 might seem daunting at first, but simplifying it is a straightforward process. This article will guide you through the steps, explaining the concept of simplifying fractions and providing practical examples. We'll cover finding the greatest common divisor (GCD) and show you how to apply this method to reduce 21/63 to its simplest form. By the end, you'll be confident in simplifying fractions of any size.

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. Simplifying fractions doesn't change the value of the fraction, only its representation. Think of it like reducing a recipe – you change the amounts of ingredients but maintain the same proportions.

Finding the Greatest Common Divisor (GCD)

The key to simplifying fractions is finding the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. There are several ways to find the GCD:

Method 1: Listing Factors

List all the factors of both the numerator (21) and the denominator (63). The largest number that appears in both lists is the GCD.

Factors of 21: 1, 3, 7, 21
Factors of 63: 1, 3, 7, 9, 21, 63

The largest common factor is 21.

Method 2: Prime Factorization

This method involves breaking down both numbers into their prime factors (numbers only divisible by 1 and themselves).

Prime factorization of 21: 3 x 7
Prime factorization of 63: 3 x 3 x 7 (or 3² x 7)

The common prime factors are 3 and 7. Multiply them together (3 x 7 = 21) to get the GCD.

Simplifying 21/63

Now that we know the GCD of 21 and 63 is 21, we can simplify the fraction:

Divide the numerator by the GCD: 21 ÷ 21 = 1
Divide the denominator by the GCD: 63 ÷ 21 = 3

Therefore, the simplified fraction is 1/3.

Other Examples of Fraction Simplification

Let's look at a few more examples to solidify your understanding:

12/18: The GCD of 12 and 18 is 6. 12 ÷ 6 = 2 and 18 ÷ 6 = 3. Simplified fraction: 2/3
15/25: The GCD of 15 and 25 is 5. 15 ÷ 5 = 3 and 25 ÷ 5 = 5. Simplified fraction: 3/5
10/20: The GCD of 10 and 20 is 10. 10 ÷ 10 = 1 and 20 ÷ 10 = 2. Simplified fraction: 1/2

Conclusion

Simplifying fractions like 21/63 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 to always divide both the numerator and the denominator by their GCD to obtain the equivalent simplified fraction. Practice makes perfect! Now you're equipped to tackle any fraction simplification challenge with confidence.