HCF and LCM of Fractions Calculator

Understanding HCF and LCM of Fractions

What are HCF and LCM of Fractions?

The Highest Common Factor (HCF) of fractions is the largest fraction that divides all the given fractions without a remainder. The Least Common Multiple (LCM) of fractions is the smallest fraction that is divisible by all the given fractions without a remainder.

Formulas and Representation

For fractions \(\frac{a_1}{b_1}, \frac{a_2}{b_2}, ...\), the formulas are:

\[HCF(\frac{a_1}{b_1}, \frac{a_2}{b_2}, ...) = \frac{GCD(a_1, a_2, ...)}{LCM(b_1, b_2, ...)}\]

\[LCM(\frac{a_1}{b_1}, \frac{a_2}{b_2}, ...) = \frac{LCM(a_1, a_2, ...)}{GCD(b_1, b_2, ...)}\]

Where:

• GCD is the Greatest Common Divisor
• LCM is the Least Common Multiple
• \(a_i\) are the numerators
• \(b_i\) are the denominators

Calculation Steps

To calculate the HCF and LCM of fractions:

1. Find the GCD of all numerators
2. Find the LCM of all denominators
3. The HCF is GCD(numerators) / LCM(denominators)
4. Find the LCM of all numerators
5. Find the GCD of all denominators
6. The LCM is LCM(numerators) / GCD(denominators)
7. Simplify the resulting fractions if possible

Example and Visual Representation

Let's calculate the HCF and LCM of 1/2, 2/3, and 3/4:

This diagram illustrates the HCF and LCM of the fractions 1/2, 2/3, and 3/4. The HCF (1/12) represents the largest fraction that divides all given fractions, while the LCM (12/1) is the smallest fraction divisible by all given fractions.