Solving for Consecutive Integers: Summing to 135
Suppose you're faced with a problem where the sum of three consecutive integers is 135. Let's walk through the steps to solve this puzzle using algebraic equations. This problem is not only a fun exercise but also a great way to understand fundamental algebraic concepts.
Introduction to Consecutive Integers
Consecutive integers are numbers that follow each other in a sequence, with the difference between any two consecutive numbers being 1. For example, in the sequence 44, 45, and 46, each number is exactly one more than the previous one.
Solving the Problem
The problem states that the sum of three consecutive integers is 135. To solve for these integers, we start by setting up an equation based on the definition of consecutive integers.
Step 1: Express the Integers Algebraically
Let's denote the middle integer as x. The three consecutive integers can then be expressed as x-1, x, and x 1.
Step 2: Formulate the Equation
The sum of these integers is given as 135. Therefore, we can write the equation as:
(x-1) x (x 1) 135
Step 3: Simplify the Equation
Combine like terms:
3x 135
Step 4: Solve for x
Divide both sides of the equation by 3:
x 135 / 3 45
Step 5: Find the Consecutive Integers
Now, substitute x back into the expressions for the three integers:
First integer: x-1 45-1 44
Second integer: x 45
Third integer: x 1 45 1 46
Conclusion
The three consecutive integers that sum to 135 are 44, 45, and 46. Among these, the greatest integer is 46.
Application and Practice
This problem is not just a theoretical exercise but can be applied in various real-life scenarios, such as dividing a sum of money among people or calculating population distribution in a sequence. Practicing such problems helps in developing a strong foundation in algebraic reasoning and problem-solving skills.
Resources for Further Learning
To delve deeper into the topic, you can explore online resources and textbooks that provide detailed explanations and a wide range of practice problems. Some popular resources include Khan Academy, Coursera, and various math consultation websites like Mathway or Symbolab.
Frequently Asked Questions (FAQ)
Q: Can the process be applied to other sums?
A: Yes, the method is universal. You can follow the same steps to find three consecutive integers for any given sum.
Q: What if the sum is even?
A: In that case, you would need to adjust your approach slightly, as the sum will not be divisible by 3. You can still use the concept of consecutive numbers, but you might need to include fractions or decimals.
Q: Is this important for overall math proficiency?
A: Absolutely! Understanding consecutive integers and their properties is crucial for advanced algebra and can be applied in various fields such as physics, engineering, and everyday life problems.