How to Find the HCF of Fractions: A Comprehensive Guide to 4/5 and 3/7
Understanding how to find the Highest Common Factor (HCF) of fractions is an essential skill in algebra. This guide will walk you through the method of finding the HCF for the fractions 4/5 and 3/7, providing a clear breakdown of the steps involved. Whether you're a student or a professional looking to refresh your algebraic skills, this article will provide you with the necessary insights.
Understanding Fractions and HCF
Before diving into the specific example, let's establish a basic understanding of fractions and HCF.
What is a Fraction?
A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). For example, in 4/5, 4 is the numerator, and 5 is the denominator.
What is HCF?
The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides two or more numbers without leaving a remainder.
Step-by-Step Guide to Finding the HCF of Fractions
Step 1: Expressing Fractions in Their Prime Factor Forms
Let's express the fractions 4/5 and 3/7 in their prime factor forms to simplify the process of finding the HCF.
4/5 can be expressed as (2^2 times 5^{-1}).
3/7 can be expressed as (3 times 7^{-1}).
Step 2: Finding the HCF of the Numerators
We need to find the HCF of the numerators 4 and 3. The prime factorization is as follows:
4 (2^2)
3 (3^1)
Since 4 and 3 have no common factors other than 1, the HCF of 4 and 3 is 1.
Step 3: Finding the LCM of the Denominators
The least common multiple (LCM) of the denominators 5 and 7 is 35. This is because 5 and 7 are prime numbers and do not share any common factors.
Step 4: Calculating the HCF of the Fractions
Now, combine the results from steps 2 and 3. The HCF of the fractions 4/5 and 3/7 is given by the product of the HCF of the numerators and the reciprocal of the LCM of the denominators.
HCF HCF of numerators / LCM of denominators
Simplifying, we get:
HCF of 4/5 and 3/7 (1 / 35)
Therefore, the HCF of the fractions 4/5 and 3/7 is 1/35.
Conclusion: Mastering the HCF of Fractions
Understanding how to find the HCF of fractions is crucial in algebraic operations. By breaking down the problem into steps and using prime factorization, you can easily find the HCF of any pair of fractions. This skill is not only useful in mathematical calculations but also in various real-world applications, such as in engineering and financial calculations.
Additional Tips for Finding HCF of Fractions
1. Always express the fractions in their prime factor forms before finding the HCF.
2. Remember to take the HCF of the numerators and the reciprocal of the LCM of the denominators.
3. Practice regularly to become proficient in these calculations.
Frequently Asked Questions (FAQs)
Q: Why is the HCF of 4 and 3 1?
A: The prime factorization of 4 is (2^2) and of 3 is (3^1). Since they have no common factors other than 1, their HCF is 1.
Q: Can you explain the LCM of 5 and 7?
A: The LCM of 5 and 7 is 35. Since 5 and 7 are prime numbers and do not share any common factors, their LCM is their product, which is 35.
Q: How does the HCF of 4/5 and 3/7 relate to the LCM of the denominators?
A: The HCF of 4/5 and 3/7 is the HCF of the numerators (1) divided by the LCM of the denominators (35), resulting in 1/35.