35 48 simplified

2 min read 28-02-2025

Understanding how to simplify fractions is a fundamental skill in mathematics. This article will walk you through the process, using the fraction 35/48 as a practical example. We'll explore the concept of greatest common divisors (GCD) and show you different methods to reach the simplest form.

What Does it Mean to Simplify a Fraction?

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) share no common factors other than 1. In simpler terms, you're making the fraction as small as possible while retaining its value.

Finding the Greatest Common Divisor (GCD)

The key to simplifying fractions lies in finding the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both numbers without leaving a remainder.

Several methods exist for finding the GCD:

1. Listing Factors

This method involves listing all the factors of both the numerator (35) and the denominator (48). Then, identify the largest factor they share.

Factors of 35: 1, 5, 7, 35
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The largest common factor is 1.

2. Prime Factorization

This method breaks down both numbers into their prime factors (numbers divisible only by 1 and themselves). Then, identify the common prime factors and multiply them together to find the GCD.

Prime factorization of 35: 5 x 7
Prime factorization of 48: 2 x 2 x 2 x 2 x 3 = 2⁴ x 3

There are no common prime factors between 35 and 48. Therefore, the GCD is 1.

3. Euclidean Algorithm

This is a more efficient method for larger numbers. It involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD.

Since the GCD of 35 and 48 is 1, the fraction 35/48 is already in its simplest form.

Simplifying 35/48: The Result

Because the greatest common divisor of 35 and 48 is 1, the fraction 35/48 is already in its simplest form. It cannot be reduced further.

Conclusion

Simplifying fractions is crucial for understanding mathematical concepts and solving problems. While some fractions can be significantly reduced, as demonstrated with 35/48, sometimes a fraction is already in its simplest form. Mastering the techniques for finding the GCD, whether through listing factors, prime factorization, or the Euclidean algorithm, is key to efficiently simplifying fractions. Remember, always check for common factors to ensure you've found the simplest form of any given fraction.