Understanding Even Numbers and Their Average

Arithmetic involving even numbers often requires a clear understanding of their properties. In this article, we explore how to find the product of two consecutive even numbers when given the average of a sequence of four consecutive even numbers. This is a classic problem that combines basic arithmetic with logical reasoning.

Problem Statement and Solution Approach

Given the average of four consecutive even numbers is 27, we need to find the product of the first and the fourth numbers in the sequence. We will solve this step by step.

Step-by-Step Solution

Let's denote the four consecutive even numbers as:

A x B x 2 C x 4 D x 6

The formula for the average of these numbers is:

[ text{Average} frac{A B C D}{4} 27 ]

Substituting the values of A, B, C, and D, we get:

[ frac{x (x 2) (x 4) (x 6)}{4} 27 ]

Combining like terms, we have:

[ frac{4x 12}{4} 27 ]

Multiplying both sides by 4, we get:

[ 4x 12 108 ]

Solving for (x), we subtract 12 from both sides:

[ 4x 96 ]

Dividing by 4, we find:

[ x 24 ]

Therefore, the four consecutive even numbers are:

A 24 B 26 C 28 D 30

The product of A and D is:

[ A times D 24 times 30 720 ]

Verification and Explanation

Let's verify the solution step by step to ensure its correctness:

Denote the numbers as (A x), (B x 2), (C x 4), and (D x 6). Set up the average equation: [ frac{x (x 2) (x 4) (x 6)}{4} 27 ] Simplify: [ frac{4x 12}{4} 27 ] Multiplty both sides by 4: [ 4x 12 108 ] Subtract 12 from both sides: [ 4x 96 ] Divide by 4: [ x 24 ] Confirm the numbers: A 24, B 26, C 28, D 30. Calculate the product: [ 24 times 30 720 ]

Conclusion and Final Answer

The product of the first and the fourth numbers in a sequence of four consecutive even numbers with an average of 27 is 720.

Additional Considerations

It's important to note that even numbers occur at even intervals and their average will also be an even number. Therefore, if the given average is odd (like 27), it indicates an error in the problem statement.