210/360 simplified

2 min read 10-03-2025

Fractions can seem intimidating, but simplifying them is a straightforward process. This guide will walk you through simplifying the fraction 210/360, step-by-step, and teach you the method to apply to other fractions. Understanding how to simplify fractions is crucial in math, from basic arithmetic to more advanced topics.

Understanding Fraction Simplification

Simplifying a fraction means reducing it to its lowest terms. This means finding the greatest common divisor (GCD) of the numerator (the top number) and the denominator (the bottom number) and dividing both by that number. The result is an equivalent fraction that's easier to understand and work with.

Finding the Greatest Common Divisor (GCD)

The first step in simplifying 210/360 is to find the greatest common divisor (GCD) of 210 and 360. There are a couple of ways to do this:

1. Listing Factors

List the factors of both numbers:

Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

The largest number that appears in both lists is 30. Therefore, the GCD of 210 and 360 is 30.

2. Prime Factorization

This method is particularly useful for larger numbers. Break down each number into its prime factors:

210 = 2 x 3 x 5 x 7
360 = 2 x 2 x 2 x 3 x 3 x 5

The GCD is found by identifying the common prime factors and multiplying them together: 2 x 3 x 5 = 30.

Simplifying 210/360

Now that we know the GCD is 30, we can simplify the fraction:

210 ÷ 30 = 7 360 ÷ 30 = 12

Therefore, 210/360 simplified is 7/12.

How to Simplify Other Fractions

The steps for simplifying any fraction are:

Find the GCD: Use either the listing factors or prime factorization method to find the greatest common divisor of the numerator and denominator.
Divide: Divide both the numerator and denominator by the GCD.
Result: The resulting fraction is the simplified form.

Example: Simplifying 18/24

Let's try another example. To simplify 18/24:

GCD: The GCD of 18 and 24 is 6 (you can find this using either method above).
Divide: 18 ÷ 6 = 3 and 24 ÷ 6 = 4
Result: 18/24 simplifies to 3/4.

Mastering fraction simplification is a valuable skill. By following these steps, you can confidently simplify any fraction and improve your understanding of mathematical concepts. Remember to always find the greatest common divisor for the most simplified form.