Understanding and Calculating HCF and LCM for Fractions

When dealing with fractions, it's often necessary to understand and calculate the Highest Common Factor (HCF) and the Least Common Multiple (LCM). This guide will walk you through the process of finding the HCF and LCM for the fractions 8/9, 10/27, and 16/81.

Introduction to HCF and LCM

The highest common factor (HCF) of a set of numbers is the largest number that divides all of them without leaving a remainder. The least common multiple (LCM) is the smallest number that is a multiple of each of the numbers.

Calculating HCF and LCM for Fractions

For the fractions 8/9, 10/27, and 16/81, we can find the HCF and LCM by following a step-by-step process:

Prime Factorization

First, find the prime factorization of the numerators and denominators:

Numerators: 8, 10, 16

Denominators: 9, 27, 81

Step 1: Finding the HCF of the Numerators

The HCF is calculated as the product of the lowest powers of the common prime factors:

HCF of Numerators:

For 2, the lowest power is 21.

Cross-check by factoring out: HCF 21 2

Step 2: Finding the LCM of the Denominators

The LCM is calculated as the product of the highest powers of the common prime factors:

LCM of Denominators:

For 3, the highest power is 34.

Cross-check by factoring out: LCM 34 81

Step 3: Calculating HCF and LCM for the Fractions

Using the formulas for HCF and LCM:

HCF of Fractions:

Formula: HCF(left(frac{a}{b} ,frac{c}{d} ,frac{e}{f}right) frac{text{HCF}(a, c, e)}{text{LCM}(b, d, f)}

HCF of fractions frac{2}{81}

LCM of Fractions:

Formula: LCM(left(frac{a}{b} ,frac{c}{d} ,frac{e}{f}right) frac{text{LCM}(a, c, e)}{text{HCF}(b, d, f)}

LCM of numerators LCM(8, 10, 16) 80

LCM of denominators HCF(9, 27, 81) 9

LCM frac{80}{9}

Alternative Method: Converting to Like Denominators

Another method involves converting the fractions to a common denominator:

Then, LCM of 72, 30, and 16/81 720/81

And, HCF of 72, 30, and 16/81 2/81

Summary

The HCF and LCM for the given fractions are:

HCF: 2/81

LCM: 80/9

These steps and formulas can be applied to any set of fractions to find their HCF and LCM accurately.