210/180 simplified

2 min read 27-02-2025

Fractions can seem daunting, but simplifying them is a straightforward process. This guide will walk you through simplifying the fraction 210/180, explaining the steps so you can apply them to other fractions. We'll also explore the concept of simplifying fractions in general and why it's important.

Understanding Fraction Simplification

Simplifying a fraction, also known as reducing a fraction to its lowest terms, means finding an equivalent fraction where the numerator (top number) and denominator (bottom number) have no common factors other than 1. This makes the fraction easier to understand and work with.

Think of a fraction like a pizza. 210/180 represents 210 slices out of a possible 180 slices – quite a lot! Simplifying helps us represent this in a more manageable way.

How to Simplify 210/180

To simplify 210/180, we need to find the greatest common divisor (GCD) of 210 and 180. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Here's how to find the GCD:

Find the factors of each number:
- Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
- Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
Identify the common factors: Notice that both 210 and 180 share several factors, including 1, 2, 3, 5, 6, 10, 15, and 30.
Find the greatest common factor: The largest number that appears in both lists is 30. This is our GCD.
Divide both the numerator and the denominator by the GCD:
- 210 ÷ 30 = 7
- 180 ÷ 30 = 6

Therefore, the simplified fraction is 7/6.

Alternative Method: Prime Factorization

Another way to find the GCD is through prime factorization. This involves breaking down each number into its prime factors (numbers divisible only by 1 and themselves).

Prime factorization of 210: 2 x 3 x 5 x 7
Prime factorization of 180: 2 x 2 x 3 x 3 x 5

The common prime factors are 2, 3, and 5. Multiplying these together (2 x 3 x 5 = 30) gives us the GCD. Then, we divide as before.

Why Simplify Fractions?

Simplifying fractions is crucial for several reasons:

Clarity: Simplified fractions are easier to understand and compare. 7/6 is much clearer than 210/180.
Accuracy: In calculations, using simplified fractions reduces the risk of errors and simplifies further computations.
Efficiency: Working with smaller numbers is generally faster and more efficient.

Working with Improper Fractions

Notice that 7/6 is an improper fraction (the numerator is larger than the denominator). This can be converted to a mixed number: 1 1/6. This represents one whole and one-sixth. The choice between improper and mixed fractions depends on the context of the problem.

Conclusion

Simplifying the fraction 210/180 to its simplest form, 7/6 (or 1 1/6), is a straightforward process. By understanding the concept of the greatest common divisor and using either the factor method or prime factorization, you can effectively simplify any fraction. This skill is foundational in mathematics and will serve you well in various applications. Remember to always check for common factors to ensure your fraction is in its simplest form.