56 64 simplified

2 min read 06-02-2025

Simplifying fractions is a fundamental math skill. It involves reducing a fraction to its lowest terms by finding the greatest common divisor (GCD) of the numerator and denominator. Let's break down how to simplify the fraction 56/64.

Understanding Fraction Simplification

Simplifying a fraction means finding an equivalent fraction where the numerator and denominator are smaller, but the fraction represents the same value. This makes the fraction easier to understand and work with.

Finding the Greatest Common Divisor (GCD)

The key to simplifying fractions lies in finding the greatest common divisor (GCD) of the numerator (56) and the denominator (64). The GCD is the largest number that divides both numbers evenly.

There are several ways to find the GCD:

Method 1: Listing Factors

List all the factors of 56 and 64:

Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Factors of 64: 1, 2, 4, 8, 16, 32, 64

The largest number that appears in both lists is 8. Therefore, the GCD of 56 and 64 is 8.

Method 2: Prime Factorization

Break down 56 and 64 into their prime factors:

56 = 2 x 2 x 2 x 7 = 2³ x 7
64 = 2 x 2 x 2 x 2 x 2 x 2 = 2⁶

The common prime factors are three 2's (2³). Therefore, the GCD is 2 x 2 x 2 = 8.

Method 3: Euclidean Algorithm (for larger numbers)

The Euclidean algorithm is a more efficient method for finding the GCD of larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCD.

Divide 64 by 56: 64 = 1 x 56 + 8
Divide 56 by the remainder 8: 56 = 7 x 8 + 0

Since the remainder is 0, the GCD is the last non-zero remainder, which is 8.

Simplifying 56/64

Now that we know the GCD is 8, we can simplify the fraction:

56/64 = (56 ÷ 8) / (64 ÷ 8) = 7/8

The Simplified Fraction

Therefore, the simplified form of the fraction 56/64 is 7/8. This is the simplest form because 7 and 8 share no common factors other than 1.

Practice Problems

Try simplifying these fractions using the methods described above:

24/36
18/45
35/49

Conclusion

Simplifying fractions like 56/64 to 7/8 is crucial for working with fractions effectively. Mastering the GCD and applying the simplification process will improve your understanding of fractions and make calculations much easier. Remember to always check for common factors to ensure your fraction is in its simplest form.