Find three consecutive even integers such that the sum of the smallest number and twice the middle number is 20 more than the largest number.

x + 2(x+2) = x+4 + 20

x = 10
So the numbers are 10,12,14

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Why did the even integers start partying? Because they wanted to be "consecutive" dancers! Okay, let's get down to business.

Let's represent the three consecutive even integers as x, x+2, and x+4.

According to the problem, the sum of the smallest number (x) and twice the middle number (2(x+2)) is 20 more than the largest number (x+4).

So we can write the equation:
x + 2(x+2) = (x+4) + 20

Simplifying this equation will lead us to the solution:
x + 2x + 4 = x + 24

Combining like terms, we have:
3x + 4 = x + 24

Subtracting x from both sides, we get:
2x + 4 = 24

Subtracting 4 from both sides, we have:
2x = 20

Dividing both sides by 2, we finally get:
x = 10

Therefore, the three consecutive even integers are 10, 12, and 14.

Let's break down the problem step by step to find the three consecutive even integers.

Step 1: Define the unknowns
Let's assume that the smallest even integer is represented by the variable "x". Since we are looking for three consecutive even integers, the next two even integers can be represented by "x + 2" and "x + 4".

Step 2: Set up the equation
According to the problem statement, the sum of the smallest number (x) and twice the middle number (2 * (x + 2)) is equal to 20 more than the largest number (x + 4) . So, we can set up the equation as follows:

x + 2 * (x + 2) = (x + 4) + 20

Step 3: Solve the equation
Let's solve the equation to find the value of "x":

x + 2x + 4 = x + 24

3x + 4 = x + 24

2x + 4 = 24

2x = 20

x = 10

Step 4: Find the consecutive even integers
Now that we have the value of "x" as 10, we can find the three consecutive even integers:

Smallest: x = 10
Middle: x + 2 = 10 + 2 = 12
Largest: x + 4 = 10 + 4 = 14

Therefore, the three consecutive even integers are 10, 12, and 14.