Solving the Sum of Three Consecutive Even Integers Problem

Solving problems involving consecutive even integers is a common task in arithmetic and algebra. This article will walk you through the process of solving the specific problem: What are three consecutive even integers whose sum is 246?

Method 1: Using Basic Algebra

Let n be the first of these even integers. Since the integers are consecutive and even, the second number will be n 2, and the third will be n 4. The sum of these three integers is given as 246.

Using this information, we can write the equation:

n (n 2) (n 4) 246

Combining like terms, we get:

3n 6 246

Solving for n by first subtracting 6 from both sides:

3n 240

Then, dividing by 3:

n 80

Therefore, the three consecutive even integers are 80, 82, and 84.

Method 2: Logical Deduction

An alternative approach is to use the concept of the middle number. Since the average of the three numbers is 246/3 82, 82 must be the middle number. Thus, the three consecutive even integers are 80, 82, and 84.

Method 3: Another Algebraic Approach

Assume the numbers are represented by x, x 2, x 4. The sum of these numbers is:

x (x 2) (x 4) 246

Combining like terms, we get:

3x 6 246

Solving for x by first subtracting 6 from both sides:

3x 240

Then, dividing by 3:

x 80

Therefore, the three consecutive even integers are 80, 82, and 84.

Method 4: Substitution Method

Assume the three consecutive even integers are 2X, 2X 2, 2X 4. The sum of these integers is:

2X (2X 2) (2X 4) 246

Combining like terms, we get:

6X 6 246

Solving for X by first subtracting 6 from both sides:

6X 240

Then, dividing by 6:

X 40

Therefore, the three consecutive even integers are 80, 82, and 84.

Method 5: Using an Alternative Variable

Assume the consecutive even integers are X - 2, X, X 2. The sum of these integers is:

(X - 2) X (X 2) 246

Combining like terms, we get:

3X 246

Solving for X by dividing by 3:

X 82

Therefore, the three consecutive even integers are 80, 82, and 84.

These methods provide a comprehensive approach to solving the problem and demonstrate the flexibility in solving such arithmetic problems.