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How to Find HCF of Fractions Fast?

Published in Fraction HCF 5 mins read

To find the HCF (Highest Common Factor) or GCD (Greatest Common Divisor) of fractions quickly, use the straightforward formula: HCF of fractions = (HCF of numerators) / (LCM of denominators).

This method allows you to efficiently determine the largest fraction that can divide all the given fractions without leaving a remainder. Understanding the HCF and LCM of integers is fundamental to mastering this concept for fractions.

Understanding HCF of Fractions

The Highest Common Factor (HCF) of fractions is the largest fraction that is a common divisor of all the given fractions. Unlike integers, where the HCF is always an integer, the HCF of fractions will itself be a fraction.

Step-by-Step Method to Calculate HCF of Fractions

Follow these steps for a quick and accurate calculation:

Step 1: Find the HCF of All Numerators

Identify all the numerators from the given fractions. Then, calculate their Highest Common Factor (HCF). The HCF is the largest positive integer that divides each of the numerators without any remainder.

Methods to find HCF of integers:

  • Prime Factorization Method:
    1. Express each numerator as a product of its prime factors.
    2. Identify the common prime factors among all numbers.
    3. Multiply these common prime factors, taking the lowest power of each common factor.
  • Long Division Method (Euclidean Algorithm):
    1. Divide the larger number by the smaller number.
    2. Take the divisor as the new dividend and the remainder as the new divisor.
    3. Repeat the process until the remainder is zero. The last non-zero divisor is the HCF. For more than two numbers, find the HCF of two, then find the HCF of the result and the next number.

Step 2: Find the LCM of All Denominators

Next, identify all the denominators from the given fractions. Calculate their Least Common Multiple (LCM). The LCM is the smallest positive integer that is a multiple of all the denominators.

Methods to find LCM of integers:

  • Prime Factorization Method:
    1. Express each denominator as a product of its prime factors.
    2. List all unique prime factors that appear in any of the factorizations.
    3. For each unique prime factor, take the highest power it appears with.
    4. Multiply these highest powers together to get the LCM.
  • Division Method:
    1. Write the denominators in a row.
    2. Divide them by the smallest prime number that divides at least one of the numbers exactly.
    3. Bring down the numbers that are not divisible.
    4. Repeat the process until no two numbers share a common prime factor.
    5. Multiply all the divisors and the remaining undivided numbers.

Step 3: Form the Resulting Fraction

Once you have calculated the HCF of the numerators and the LCM of the denominators, construct the final fraction:

HCF of Fractions = (HCF of Numerators) / (LCM of Denominators)

Example Calculation

Let's find the HCF of the fractions: 2/3, 4/9, 6/15

  • Step 1: HCF of Numerators
    Numerators are 2, 4, 6.

    • Prime factorization:
      • 2 = 2¹
      • 4 = 2²
      • 6 = 2¹ × 3¹
    • The common prime factor is 2, with the lowest power being 2¹.
    • HCF (2, 4, 6) = 2
  • Step 2: LCM of Denominators
    Denominators are 3, 9, 15.

    • Prime factorization:
      • 3 = 3¹
      • 9 = 3²
      • 15 = 3¹ × 5¹
    • Unique prime factors are 3 and 5.
    • Highest power of 3 is 3² (from 9).
    • Highest power of 5 is 5¹ (from 15).
    • LCM (3, 9, 15) = 3² × 5¹ = 9 × 5 = 45
  • Step 3: Form the Resulting Fraction
    HCF of (2/3, 4/9, 6/15) = (HCF of 2, 4, 6) / (LCM of 3, 9, 15) = 2 / 45

So, the HCF of 2/3, 4/9, and 6/15 is 2/45.

Tips for Finding HCF/LCM Quickly

To speed up the process of finding HCF and LCM of integers, which are the core components of the fraction calculation:

  • For small numbers: Practice mental calculations.
  • Prime Factorization: This method is highly efficient for both HCF and LCM, especially for numbers with distinct prime factors or moderate sizes.
  • Euclidean Algorithm (for HCF): For larger numerators, the Euclidean algorithm is significantly faster than prime factorization.
    • For example, to find HCF(81, 108):
      1. 108 = 81 × 1 + 27
      2. 81 = 27 × 3 + 0
        The HCF is 27.
  • Relationship between HCF and LCM: For any two positive integers 'a' and 'b', HCF(a, b) × LCM(a, b) = |a × b|. This can be used to quickly find LCM if you've already found the HCF of two numbers.
    • LCM(a, b) = (|a × b|) / HCF(a, b)
    • Note: This formula applies only to two numbers at a time.

Summary of HCF and LCM Methods for Integers

Method Description Best For
Prime Factorization Find prime factors of each number. For HCF, multiply common prime factors with lowest powers. For LCM, multiply all unique prime factors with highest powers. Versatile; good for both HCF and LCM.
Long Division / Euclidean Algorithm (HCF) Divide larger by smaller, repeat with divisor and remainder until remainder is 0. Last non-zero divisor is HCF. Finding HCF of two large numbers quickly.
Division Method (LCM) Divide numbers by common prime factors until no common factors remain. Multiply divisors and remaining numbers. Finding LCM of multiple numbers efficiently.
List Multiples/Divisors (HCF) List divisors of each number and find the largest common. (LCM) List multiples of each number and find the smallest common. Small numbers, conceptual understanding.

For further learning and practice on finding HCF and LCM of integers, you can refer to resources like Khan Academy's lessons on HCF and LCM or Byju's explanations.