Applications of LCM and HCF - Real Numbers

Introduction:
The Least Common Multiple (LCM) and Highest Common Factor (HCF) are important concepts in mathematics that are extensively used in solving various real-world problems involving real numbers. The LCM is the smallest positive integer that is divisible by two or more numbers, while the HCF is the largest positive integer that divides two or more numbers without leaving a remainder.

Applications of LCM:

1. Calculating Fractions: LCM is used to find a common denominator when adding or subtracting fractions with different denominators. By finding the LCM of the denominators, we can convert the fractions to a common denominator and perform the required operations.

2. Time and Work Problems: LCM is used to determine the time taken by multiple individuals working together to complete a task. If the time taken by each individual to complete the task is given, the LCM of their time durations will give the total time taken when they work together.

3. Repeating or Periodic Decimals: LCM is used to convert repeating or periodic decimals into fractions. By finding the LCM of the number of decimal places, we can express the repeating decimal as a fraction.

4. Calculating Interest: LCM is used in calculating compound interest. When interest is compounded annually, semi-annually, or quarterly, the LCM of the compounding periods is used to determine the total interest and the final amount.

Applications of HCF:

1. Simplifying Fractions: HCF is used to simplify fractions to their lowest terms. By dividing both the numerator and denominator of a fraction by their HCF, we can obtain an equivalent fraction in its simplest form.

2. Ratio and Proportion: HCF is used to simplify ratios and proportions. By dividing both the numerator and denominator of a ratio or a proportion by their HCF, we can simplify it to its simplest form.

3. Divisibility Tests: HCF is used in divisibility tests to check if a number is divisible by another number. If the HCF of the two numbers is greater than 1, then the numbers are not relatively prime, and the smaller number is a factor of the larger number.

4. Sharing and Distribution: HCF is used in sharing and distribution problems. When a certain quantity is to be divided equally among a group of people, the HCF of the quantity and the number of people will determine the maximum number of equal shares that can be distributed without any remainder.

In conclusion, the LCM and HCF have several applications in real-world situations involving real numbers. They are used in calculations involving fractions, time and work problems, interest calculations, simplifying ratios, divisibility tests, and sharing and distribution problems. Understanding these concepts is essential for solving various mathematical problems and practical situations.

LCM*HCF=product of 2 nos.