Mathematics, the language of the universe, is filled with fascinating problems that challenge our thinking and deepen our understanding of numerical relationships. One such problem that often appears in introductory algebra courses is finding two consecutive even integers whose sum equals a given number. While seemingly simple, this problem introduces fundamental concepts like variables, equations, and problem-solving strategies. It’s a gateway to exploring more complex mathematical ideas and develops essential skills for tackling real-world applications.
Understanding Consecutive Even Integers
Before diving into the solution, let’s clarify what we mean by “consecutive even integers.” Even integers are whole numbers that are divisible by 2. Consecutive even integers follow each other in order, differing by 2. For example, 2 and 4 are consecutive even integers, as are 10 and 12.
Representing the Integers
To solve the problem, we need to represent the integers mathematically. Let’s use the following variables:
- x: the first even integer
- x + 2: the next consecutive even integer
This representation captures the relationship between the two integers, as they differ by 2.
Setting up the Equation
Now that we have our variables, we can set up an equation based on the given information: the sum of the two consecutive even integers is 126. Mathematically, this can be expressed as:
x + (x + 2) = 126
This equation represents the problem in a concise and solvable form.
Solving for x
To find the value of x, we can follow these steps:
1. (See Also: Definition of Percent Error in Chemistry? Explained Simply)
Combine like terms on the left side of the equation: 2x + 2 = 126
2.
Subtract 2 from both sides: 2x = 124
3.
Divide both sides by 2: x = 62
Therefore, the first even integer is 62.
Finding the Second Integer
We know that the second even integer is 2 more than the first. So, we can calculate it as follows:
x + 2 = 62 + 2 = 64 (See Also: How Much For An Addition? Cost Breakdown)
Therefore, the second even integer is 64.
Verification
To verify our solution, we can add the two integers together: 62 + 64 = 126. This confirms that our answer is correct.
Applications of Consecutive Integers
The concept of consecutive integers has numerous applications in various fields:
Age Problems
Consecutive integers are often used to represent the ages of people in word problems. For example, if John is 3 years older than his sister Mary, we can represent their ages as x and x + 3, respectively.
Number Sequences
Many number sequences follow a pattern of consecutive integers. For instance, the sequence 2, 4, 6, 8… consists of consecutive even integers.
Data Analysis
In data analysis, consecutive integers can be used to represent data points in a series. This can help identify trends and patterns in the data.
Conclusion
The problem of finding two consecutive even integers whose sum is 126 serves as a fundamental introduction to algebraic problem-solving. It highlights the importance of:
- Understanding mathematical concepts like consecutive integers and variables.
- Formulating equations to represent real-world problems.
- Solving equations using algebraic techniques.
- Verifying solutions and interpreting the results.
By mastering these skills, we can confidently tackle more complex mathematical challenges and apply them to various real-world scenarios. (See Also: 20 Is 80 Percent Of What Number? Discover The Answer)
Frequently Asked Questions
What are consecutive integers?
Consecutive integers are integers that follow each other in order, differing by 1. For example, 3, 4, and 5 are consecutive integers.
How do I find the sum of consecutive even integers?
To find the sum of consecutive even integers, you can use the formula: Sum = (n/2) * (first even integer + last even integer), where ‘n’ is the number of even integers.
Can you give another example of a problem involving consecutive even integers?
Yes, here’s an example: Find two consecutive even integers whose difference is 16. Let the first even integer be ‘x’. Then the next consecutive even integer is ‘x + 2’. We know that (x + 2) – x = 16. Solving for x, we get x = 8 and the next consecutive even integer is 10.
What if the problem involves odd integers instead of even integers?
The process is similar. You would represent the odd integers using variables and set up an equation based on the given information. Then, you would solve the equation to find the values of the variables.
Why are consecutive integers useful in mathematics?
Consecutive integers are useful because they represent patterns and relationships between numbers. They are often used in algebra, number theory, and other areas of mathematics to solve problems and explore mathematical concepts.