14 72 simplified

2 min read 10-03-2025

Meta Description: Learn how to simplify the fraction 14/72 to its lowest terms. This step-by-step guide provides a clear explanation and helpful tips for reducing fractions. Master fraction simplification with easy-to-understand examples.

Understanding how to simplify fractions is a fundamental skill in mathematics. This guide will walk you through the process of simplifying 14/72 step-by-step, and explain the underlying concepts. We'll also show you how to apply this technique to other fractions.

Finding the Greatest Common Divisor (GCD)

The key to simplifying fractions lies in finding the greatest common divisor (GCD) of the numerator (14) and the denominator (72). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

One method to find the GCD is to list the factors of each number.

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.

Another method, particularly useful for larger numbers, is the Euclidean algorithm. We'll cover that in a later section.

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.

Visualizing Fraction Simplification

Imagine you have 14 slices of pizza out of a total of 72 slices. Simplifying the fraction 14/72 means finding an equivalent fraction that represents the same proportion of pizza but uses smaller numbers. We've grouped the pizza slices into pairs (because the GCD is 2), leaving us with 7 pairs out of 36 pairs—the simplified fraction 7/36.

Using the Euclidean Algorithm (for larger numbers)

The Euclidean algorithm is a more efficient method for finding the GCD of larger numbers. Let's illustrate:

Divide the larger number (72) by the smaller number (14): 72 ÷ 14 = 5 with a remainder of 2.
Replace the larger number with the smaller number (14), and the smaller number with the remainder (2): Now we find the GCD of 14 and 2.
Repeat: 14 ÷ 2 = 7 with a remainder of 0.
The GCD is the last non-zero remainder: In this case, the GCD is 2.

This method is especially helpful when dealing with larger numbers where listing all factors would be tedious.

Simplifying Fractions: General Tips

Prime Factorization: Breaking down the numerator and denominator into their prime factors can also help find the GCD. For example, 14 = 2 x 7 and 72 = 2 x 2 x 2 x 3 x 3. The common factor is 2.
Divide by Common Factors: If you spot a common factor immediately (like 2 in this case), you can divide both the numerator and denominator by that factor. You might need to repeat this process until no more common factors exist.
Check for Simplest Form: Always check if your simplified fraction can be further reduced. In this case, 7 and 36 have no common factors other than 1, so 7/36 is in its simplest form.

Conclusion

Simplifying 14/72 to its lowest terms results in the fraction 7/36. By understanding the concept of the greatest common divisor and employing methods like the Euclidean algorithm or prime factorization, 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 reduce them to their simplest forms.