Exploring the Ratio and Highest Common Factor (HCF) of Two Numbers

Mathematics often involves solving problems related to ratios and the highest common factor (HCF), which helps in understanding the foundational aspects of numbers and their relationships. This article provides a step-by-step guide to solving the problem: 'Two numbers are in the ratio of 5:11, and if their HCF is 7, what is the sum of the numbers?' We will break down the solution, explain the reasoning, and provide a detailed solution process.

Understanding the Problem and Key Concepts

The given problem involves understanding the ratios and the highest common factor (HCF) of two numbers. Here’s how to approach the problem step-by-step:

Step 1: Expressing the Numbers in Terms of a Common Multiplier

Assume the two numbers are in the ratio of 5:11. This means the numbers can be written as 5x and 11x, where x is a common multiplier. Let’s express this formally:

Let the two numbers be 5x and 11x

Step 2: Relating the Common Multiplier to the HCF

The problem states that the highest common factor (HCF) of these numbers is 7. Therefore, x must be a multiple of 7. This can be mathematically represented as:

x 7k, where k is an integer

Thus, the two numbers can be rewritten as:

5x 5 * 7k 35k

11x 11 * 7k 77k

Step 3: Calculating the Sum of the Numbers

To find the sum of the two numbers, we simply add 35k and 77k:

35k 77k 112k

Since k can be any integer, the simplest case occurs when k 1. Therefore, the sum of the numbers is:

112 * 1 112

The final answer is the sum of the numbers is 112.

Additional Examples

Let’s explore a few more examples to solidify our understanding:

Example 1

Two numbers are in the ratio of 5:11. Their HCF is 7. What are the two numbers?

Solution: The HCF 7. Therefore, the numbers are 35 (5 * 7) and 77 (11 * 7).

Sum of these two numbers: 35 77 112.

Example 2

Let the two numbers be 35 and 385. Verify if they satisfy the given conditions.

Since 35 5 * 7 and 385 11 * 35, the HCF of 35 and 385 is 35, but we are given that the HCF is 7. To satisfy the given HCF, we can scale both numbers by a factor of 1/5 to get 35 5 * 7 / 5 and 385 11 * 7 / 5, which simplifies to 35 and 77.

Thus, the two numbers are 35 and 77.

Sum of these two numbers: 35 77 112.

Conclusion

Understanding the relationship between ratios and the highest common factor (HCF) can greatly simplify many mathematical problems. By breaking down the problem into smaller, more manageable steps, we can systematically solve for the unknowns and find the required sums, multiples, and factors.

Keywords

ratio, highest common factor (HCF), solving mathematical problems