Solving for Consecutive Odd Numbers: A Mathematical Exploration

Let's delve into the intriguing world of algebraic equations by solving a problem involving consecutive odd numbers. Specifically, we are tasked with finding two consecutive odd numbers such that the difference between 5 times the larger number and 3 times the smaller number is 120. This problem can be solved through a series of logical steps and straightforward algebraic manipulation.

Understanding the Problem

First, let's define our variables. We will denote the smaller of the two consecutive odd numbers as a, and the larger number as c. Since they are consecutive odd numbers, the larger number c can be expressed as c a 2.

Setting Up the Equation

According to the problem, the difference between 5 times the larger number and 3 times the smaller number is 120. This can be written as the following equation:

5c - 3a 120

Substituting c a 2 into the equation, we get:

5(a 2) - 3a 120

Expanding and simplifying this equation:

5a 10 - 3a 120

2a 10 120

2a 110

a 55

Therefore, the smaller number a is 55. The larger number c, being two more than the smaller number, is:

c a 2 55 2 57

Verification and Proof

To verify our solution, we can substitute the values back into the original equation:

5c - 3a 5(57) - 3(55) 285 - 165 120

This confirms that the solution is correct, as the difference between 5 times the larger number (57) and 3 times the smaller number (55) indeed equals 120.

Conclusion

In conclusion, the two consecutive odd numbers that satisfy the given condition are 55 and 57. This problem not only tests our understanding of basic algebraic concepts but also reinforces the importance of logical reasoning and systematic problem-solving in mathematics.

Related Problems and Further Exploration

Understanding and solving problems involving consecutive odd numbers can lead to further exploration into other forms of number sequences and algebraic problems. Feel free to try other similar problems or explore the broader applications of algebraic equations in mathematics.