14 72 simplified

Full AI 2025

2 min read 19-02-2025

Fractions can seem daunting, but simplifying them is a valuable skill. This guide will walk you through simplifying 14/72 step-by-step, explaining the process so you can apply it to other fractions. We'll explore the concept of greatest common divisors (GCD) and provide practical examples. By the end, you'll confidently tackle fraction simplification.

Understanding Fraction Simplification

Simplifying a fraction means reducing it to its lowest terms. This means finding the smallest whole numbers that represent the same ratio. For example, 2/4 simplifies to 1/2 because both the numerator (top number) and denominator (bottom number) are divisible by 2. The goal is to find the greatest common divisor (GCD).

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

The GCD is the largest number that divides evenly into both the numerator and the denominator. There are several ways to find it:

1. Listing Factors:

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 appearing in both lists is 2. Therefore, the GCD of 14 and 72 is 2.

2. Prime Factorization:

This method is particularly useful for larger numbers. Let's break down 14 and 72 into their prime factors:

14 = 2 x 7
72 = 2 x 2 x 2 x 3 x 3 = 2³ x 3²

The only prime factor they share is 2 (to the power of 1). Thus, the GCD is 2.

Simplifying 14/72

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

14 ÷ 2 = 7 72 ÷ 2 = 36

Therefore, 14/72 simplified is 7/36.

Further Simplification?

Is 7/36 already in its simplest form? Let's check. The factors of 7 are 1 and 7. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The only common factor is 1, confirming that 7/36 is the simplest form.

Practical Applications and Examples

Simplifying fractions is crucial in many areas, including:

Math: Solving equations, comparing fractions, performing operations.
Baking: Adjusting recipes.
Construction: Calculating measurements.

Example 1: Simplify 18/24

Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
GCD: 6
Simplified: 18/24 = 3/4

Example 2: Simplify 25/100

Factors of 25: 1, 5, 25
Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
GCD: 25
Simplified: 25/100 = 1/4

Conclusion

Simplifying fractions like 14/72 to 7/36 is a fundamental skill in mathematics. Mastering the process of finding the greatest common divisor, whether through listing factors or prime factorization, enables you to effectively reduce fractions to their lowest terms. Remember, practice makes perfect! Try simplifying various fractions using the techniques outlined above.