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21 63 simplified

21 63 simplified

2 min read 05-02-2025
21 63 simplified

The fraction 21/63 might seem daunting at first, but simplifying it is easier than you think. This comprehensive guide will walk you through the process, explaining the concepts and providing you with a step-by-step solution. We'll also explore the broader topic of fraction simplification and provide you with the tools to tackle similar problems independently.

Understanding Fraction Simplification

Simplifying a fraction, also known as reducing a fraction, means finding an equivalent fraction with a smaller numerator and denominator. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that divides both numbers without leaving a remainder.

Think of it like this: Imagine you have a pizza cut into 63 slices. You have 21 of those slices. Simplifying the fraction 21/63 means finding a way to represent your portion of the pizza with fewer slices, while still maintaining the same proportion.

Finding the Greatest Common Divisor (GCD) of 21 and 63

To simplify 21/63, we first need to find the greatest common divisor (GCD) of 21 and 63. There are several ways to do this:

Method 1: Listing Factors

List all the factors of 21 and 63:

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

The largest number that appears in both lists is 21. Therefore, the GCD of 21 and 63 is 21.

Method 2: Prime Factorization

This method involves breaking down each number into its prime factors.

  • 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. Multiplying these together (3 x 7 = 21) gives us the GCD.

Method 3: Euclidean Algorithm (for larger numbers)

For larger numbers, the Euclidean algorithm is a more efficient method. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCD. We won't detail this here as it's unnecessary for 21 and 63.

Simplifying 21/63

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

21/63 = (21 ÷ 21) / (63 ÷ 21) = 1/3

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

Visual Representation

Imagine a rectangle divided into 63 equal squares. If you shade 21 of them, you'll see that exactly one-third of the rectangle is shaded. This visually confirms our simplified fraction. (Include an image here showing a rectangle divided into 63 squares with 21 shaded).

Practice Problems

Try simplifying these fractions using the methods described above:

  • 15/45
  • 24/36
  • 14/28

Conclusion

Simplifying fractions like 21/63 is a fundamental skill in mathematics. By understanding the concept of the greatest common divisor and applying the appropriate methods, you can easily reduce fractions to their simplest forms. Remember, practice is key to mastering this skill! With a little practice, simplifying fractions will become second nature.

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