Understanding and Solving Numerical Problems Involving Ratios and Highest Common Factor (H.C.F.)

When dealing with numerical problems that involve the ratio between two numbers and their highest common factor (H.C.F.), it is crucial to break down the problem into manageable steps and use appropriate mathematical principles.

Introduction to Ratios and H.C.F.

A ratio is a way of expressing the relationship between two or more numbers of the same kind. The highest common factor (H.C.F.) of two or more numbers is the greatest number that divides each of them without leaving a remainder.

Problem Statement

Given that two numbers are in the ratio of 15:11 and their H.C.F. is 13, find the numbers.

Step-by-Step Solution

Let the two numbers be represented in terms of the ratio: First number 15x and Second number 11x.

The highest common factor (H.C.F.) of the two numbers is 13.

Since the H.C.F. of these two numbers is 13, we can set x 13.

Now, we calculate the actual numbers:

First number 15x 15 × 13 195

Second number 11x 11 × 13 143

Therefore, the two numbers are 195 and 143.

Exploring the Principles

The principle of expressing numbers in their shortest form (15:11) with the highest common factor (H.C.F.) of 13 is a fundamental concept in number theory. In this case, the larger valued number is 13 × 15 195.

Additional Calculations and Concepts

Additional calculations can include finding the least common multiple (LCM) of the numbers. For the numbers 195 and 143, the LCM can be calculated as 13 × 11 × 15 2145.

Math Principles and Usage

The problem demonstrates the application of mathematical principles such as ratio, proportion, and H.C.F. in real-world scenarios. These principles are essential for solving various numerical problems and form the basis of more complex mathematical concepts.

Conclusion

Mastering the solution of such numerical problems not only enhances mathematical skills but also improves problem-solving abilities. Understanding the relationships between numbers, their ratios, and H.C.F. is crucial for success in various fields, including engineering, finance, and beyond.