14/72 simplified

2 min read 10-03-2025

Fractions can seem daunting, but simplifying them is a straightforward process. Let's break down how to simplify the fraction 14/72, and learn the principles behind it. This guide will walk you through the steps, making fraction simplification easy to understand.

Understanding Fraction Simplification

Simplifying a fraction means reducing it to its lowest terms. This means finding the greatest common divisor (GCD) of both the numerator (top number) and the denominator (bottom number) and dividing both by that number. The resulting fraction will be equivalent to the original but simpler. Think of it like reducing a recipe – you keep the same proportions but use smaller amounts.

Finding the Greatest Common Divisor (GCD) of 14 and 72

The first step in simplifying 14/72 is to find the greatest common divisor (GCD) of 14 and 72. The GCD is the largest number that divides both 14 and 72 without leaving a remainder. There are several ways to find the GCD:

Method 1: Listing Factors

List all the factors of 14 and 72:

Factors of 14: 1, 2, 7, 14
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

The largest number that appears in both lists is 2. Therefore, the GCD of 14 and 72 is 2.

Method 2: Prime Factorization

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

Prime factorization of 14: 2 x 7
Prime factorization of 72: 2 x 2 x 2 x 3 x 3 = 2³ x 3²

The common prime factor is 2 (only one 2 is common to both). Therefore, the GCD is 2.

Simplifying 14/72

Now that we know the GCD is 2, we can simplify the fraction:

Divide the numerator (14) by the GCD (2): 14 ÷ 2 = 7
Divide the denominator (72) by the GCD (2): 72 ÷ 2 = 36

Therefore, the simplified fraction is 7/36.

Verifying the Simplification

To ensure our simplification is correct, we can check if the GCD of 7 and 36 is 1. Since 7 is a prime number and 36 is not divisible by 7, their GCD is indeed 1, meaning 7/36 is in its simplest form.

Conclusion: Mastering Fraction Simplification

Simplifying fractions like 14/72 is a crucial skill in mathematics. By understanding the concept of the greatest common divisor and applying the steps outlined above, you can confidently simplify any fraction. Remember, practice makes perfect! The more you work with fractions, the easier it will become to identify common factors and simplify them quickly.