Solving a Mathematical Puzzle: The Relationship Between Two Numbers

Mathematics often presents intriguing puzzles that can challenge our understanding of relationships between numbers. In this article, we will explore a particular problem and break down the thought process step-by-step to arrive at a clear solution. The problem at hand is: the sum of two numbers is five times the smaller number. Additionally, we will go through various interpretations and solutions to this problem.

Problem Statement and Initial Assumptions

The central question posed is: if the sum of two numbers is five times the smaller number, what is their sum if the greater number is 120? To start, let's assume that the smaller number is x, and the larger number is y, where y 5x.

Interpretation and Solution One

Let's analyze three different interpretations of this problem:

Interpretation 1: Direct Application

In this interpretation, we are given that the sum of two numbers is 5 times the smaller number, and the larger number is 120. Therefore:

Assume x (smaller number) 20. Hence, y (larger number) 5x 100. The sum of these two numbers is 100 20 120

Interpretation 2: More Rigorous Analysis

Let's carefully re-examine the problem from a more analytical standpoint:

Say x (smaller number) 120 / 7, then y (larger number) 720 / 7. The sum of these two numbers is x y 120 / 7 720 / 7 840 / 7 ≈ 120.

Conclusion

Both interpretations provide consistent solutions, confirming that the sum of the two numbers is indeed 120, where the larger number is five times the smaller one. Therefore, the problem is correctly interpreted and solved in these two cases.

Additional Scenarios and Incorrect Interpretations

Let's also explore some scenarios where the problem presents itself differently or where incorrect assumptions may lead to misleading results.

Scenario 1: Incorrect Application of Terms

If one mistakenly interprets five times higher as simply adding 5 to the smaller number, the calculations would be incorrect:

If x (smaller number) 120 / 6, then x 5 115, which is not divisible by 2, thus invalid.

Scenario 2: Substitution and Simplification

Another possible method involves substituting and simplifying the given information:

Start with x y 120 and y 5x. Substitute y 5x into the equation: x 5x 120. This simplifies to 6x 120, thus x 20. So, y 5 * 20 100. The sum of the numbers is 100 20 120.

Final Summary

In conclusion, the problem of finding two numbers where one is five times the other and their sum totals 120 can be solved using several valid methods. The key is to accurately interpret the relationship between the numbers and apply algebraic principles to find the correct solution. The two numbers that satisfy these conditions are 100 and 20. Understanding and practicing these types of problems enhances our mathematical problem-solving skills.

Key Points:

The problem involves two numbers, one being five times the other. The sum of the two numbers is 120. Both the smaller and larger numbers can be found through algebraic manipulation. The correct interpretation of the problem yields the numbers 100 and 20.

Conclusion

By carefully analyzing the problem and solving it through various methods, we gain a deeper understanding of the mathematical principles involved. This exercise not only provides a practical solution to the given problem but also reinforces our ability to handle similar scenarios in the future.