15/63 simplified

2 min read 27-02-2025

Simplifying fractions is a fundamental skill in mathematics. It involves reducing a fraction to its lowest terms by finding the greatest common divisor (GCD) of the numerator and the denominator. This article will guide you through the process of simplifying the fraction 15/63. We'll explore several methods, ensuring you understand the concept thoroughly.

Understanding Fraction Simplification

Before we tackle 15/63, let's review the basics. A fraction represents a part of a whole. It's composed of a numerator (the top number) and a denominator (the bottom number). Simplifying a fraction means finding an equivalent fraction where the numerator and denominator are smaller but maintain the same ratio.

For example, the fraction 2/4 can be simplified to 1/2 because both the numerator and denominator are divisible by 2. The value remains the same; it just represents the fraction in a more concise way.

Method 1: Finding the Greatest Common Divisor (GCD)

The most efficient way to simplify a fraction is to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both numbers without leaving a remainder.

Let's find the GCD of 15 and 63. We can use the prime factorization method:

Prime factorization of 15: 3 x 5
Prime factorization of 63: 3 x 3 x 7

The common factor is 3. Therefore, the GCD of 15 and 63 is 3.

Now, divide both the numerator and the denominator by the GCD:

15 ÷ 3 = 5 63 ÷ 3 = 21

Therefore, the simplified fraction is 5/21.

Method 2: Stepwise Division

If finding the prime factorization seems daunting, you can simplify the fraction stepwise. Look for any common factors and divide until no further simplification is possible.

Start with a known common factor: Both 15 and 63 are divisible by 3.
Divide both numerator and denominator by 3: 15/3 = 5 and 63/3 = 21. This gives us 5/21.
Check for further simplification: 5 and 21 have no common factors other than 1.

Again, the simplified fraction is 5/21.

Method 3: Using a Calculator

Many calculators have a function to simplify fractions. Input 15/63, and the calculator will automatically give you the simplified form: 5/21. However, understanding the underlying process is crucial for more complex problems.

Why Simplify Fractions?

Simplifying fractions makes them easier to understand and work with. It helps to avoid cumbersome calculations and makes it easier to compare fractions. A simplified fraction provides the most concise and efficient representation of the ratio.

Conclusion

Simplifying 15/63 to its lowest terms results in the fraction 5/21. We've explored three different methods to achieve this, illustrating the importance of understanding the concept of the greatest common divisor. Remember, simplifying fractions is a valuable skill in mathematics, and mastering different techniques empowers you to solve a wider range of problems efficiently.