Introduction to Simplified Fractions
A fraction represents part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). Fractions can be simplified to their lowest terms where the numerator and denominator have no common factors other than 1. Simplifying fractions makes them easier to work with and compare. In this article, we will explore how to determine which simplified fraction is equal to a given fraction.
Ways to Simplify Fractions
There are several methods to simplify fractions:
- Using Prime Factorization: Find the prime factors of the numerator and denominator and divide out any common factors.
- Dividing by the Greatest Common Divisor (GCD): Divide both the numerator and denominator by their greatest common divisor to simplify the fraction.
- Using Euclidean Algorithm: Find the greatest common divisor of the numerator and denominator using the Euclidean Algorithm and divide both by this value.
Determining Which Simplified Fraction Is Equal To
When given a fraction, we can determine which simplified fraction is equal to it by following these steps:
- Step 1: Simplify the given fraction by dividing the numerator and denominator by their greatest common divisor.
- Step 2: Verify if the simplified fraction is equal to the given fraction by performing the division.
- Step 3: If the simplified fraction is equal to the given fraction, then you have found the answer. If not, continue simplifying until you reach the equivalent fraction.
Examples of Finding Equivalent Simplified Fractions
Let’s look at some examples to better understand how to determine which simplified fraction is equal to a given fraction:
Example 1:
Given Fraction: 12/18
- Simplify the fraction: Divide both 12 and 18 by their greatest common divisor.
- Greatest Common Divisor of 12 and 18 is 6.
- Divide both numerator and denominator by 6: 12 ÷ 6 = 2, 18 ÷ 6 = 3.
- The simplified fraction is 2/3.
- Verify: 12/18 = 2/3 (12 ÷ 18 = 2 ÷ 3).
Example 2:
Given Fraction: 25/35
- Simplify the fraction: Divide both 25 and 35 by their greatest common divisor.
- Greatest Common Divisor of 25 and 35 is 5.
- Divide both numerator and denominator by 5: 25 ÷ 5 = 5, 35 ÷ 5 = 7.
- The simplified fraction is 5/7.
- Verify: 25/35 = 5/7 (25 ÷ 35 = 5 ÷ 7).
When No Simplified Fraction Is Equal To
Sometimes, a given fraction may not simplify to an exact whole number ratio. In such cases, the fraction is already simplified to its lowest terms, and no further simplification is possible. However, we can still express the decimal equivalent of the fraction.
Example:
Given Fraction: 7/10
- Simplify the fraction: 7 and 10 have no common factors other than 1.
- The fraction is already in its simplest form.
- No further simplification is needed.
- Verify: The decimal equivalent of 7/10 is 0.7.
Conclusion
Understanding how to determine which simplified fraction is equal to a given fraction is essential in mathematics. By simplifying fractions to their lowest terms, we can compare and operate on them more efficiently. Remember to utilize methods like prime factorization, greatest common divisor, and the Euclidean Algorithm to simplify fractions accurately. Practice with a variety of fractions to strengthen your skills in simplifying fractions and finding equivalent simplified fractions.
