simplify 42/45

2 min read 27-02-2025

Simplifying fractions, like 42/45, is a fundamental math skill. It involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. This process reduces the fraction to its simplest form, while maintaining its value. Let's break down how to simplify 42/45.

Understanding Fraction Simplification

Before we dive into simplifying 42/45, let's quickly review the concept. A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). Simplifying a fraction means expressing it in its lowest terms – a form where the numerator and denominator have no common factors other than 1.

Finding the Greatest Common Divisor (GCD)

The key to simplifying fractions lies in finding the GCD of the numerator and denominator. The GCD is the largest number that divides both numbers without leaving a remainder. There are several ways to find the GCD:

1. Listing Factors

List all the factors of both 42 and 45:

Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Factors of 45: 1, 3, 5, 9, 15, 45

The largest number that appears in both lists is 3. Therefore, the GCD of 42 and 45 is 3.

2. Prime Factorization

Another method involves prime factorization. This breaks down each number into its prime factors (numbers divisible only by 1 and themselves):

Prime factorization of 42: 2 x 3 x 7
Prime factorization of 45: 3 x 3 x 5

The common prime factor is 3. Therefore, the GCD is 3.

3. Euclidean Algorithm (for larger numbers)

For larger numbers, the Euclidean Algorithm provides a more efficient method. It involves repeatedly applying the division algorithm until the remainder is 0. The last non-zero remainder is the GCD. While useful for larger numbers, it's overkill for 42 and 45.

Simplifying 42/45

Now that we've found the GCD (3), we can simplify the fraction:

Divide both the numerator and the denominator by the GCD:

42 ÷ 3 = 14 45 ÷ 3 = 15

Therefore, the simplified fraction is 14/15.

Checking Your Work

Always double-check your answer. Make sure the numerator and denominator have no common factors other than 1. In this case, 14 and 15 share no common factors besides 1, confirming that 14/15 is the simplest form of 42/45.

Practice Problems

Try simplifying these fractions using the methods described above:

18/24
36/60
28/49

Mastering fraction simplification is crucial for various mathematical applications. Understanding the concept of GCD and applying the appropriate method will enable you to confidently simplify any fraction.