Question

The sum of three consecutive even integers is $228$, how do you find the integers?

Answer 1

Hint: First we will assume the first integer to be any variable. Then, find second and third even integers in terms of assumed variables. Next, add these three consecutive even integers and equate them to $228$. Solve the equation in one variable by adding, subtracting, multiplying or dividing both sides of the equation to get the value of the assumed variable. Then put the calculated value of variable and get the required integers.

Complete step-by-step solution:
It is given that the sum of three consecutive even integers is $228$.
We have found these three consecutive even integers which in addition give $228$.
So, first we will assume the first integer to be any variable.
Let $x$ be the first even integer.
Now, we have to find second and third even integers in terms of $x$.
We know that, even integers are of the form$2n$, i.e., if we add even terms to even integer, we will get an even integer.
So, to get consecutive even integers after $x$, add $2$ to get a second integer and $4$ to get a third integer.
After adding $2$ to $x$ and $4$ to $x$, we get
Second even integer = $x + 2$
Third even integer = $x + 4$
Now to get the value of $x$, we have to add these three consecutive even integers and equate them to $228$.
As it is given that the sum of three consecutive even integers is $228$
So, adding $x$, $x + 2$, $x + 4$ and equating it to $228$.
$ \Rightarrow x + \left( {x + 2} \right) + \left( {x + 4} \right) = 228$
Adding variable terms and constant terms in above equation, we get
$ \Rightarrow 3x + 6 = 228$
Subtracting $6$ from both sides of the equation, we get
$ \Rightarrow 3x = 222$
Dividing both sides of the equation by $3$, we get
$ \Rightarrow x = 74$
Now, substitute the value of $x$ and get the required consecutive even integers.
First even integer = $x = 74$
Second even integer = $x + 2 = 74 + 2 = 76$
Third even integer = $x + 4 = 74 + 4 = 78$

Therefore, required consecutive integers are $74,76,78$.

Note: In above question, it should be noted that we add $2$ to $x$ and $4$ to $x$, not $1$ and $2$.
As it is clearly stated that consecutive integers are even, so there will be an odd integer between the numbers, which we have to keep in mind.
We can also consider the consecutive even integers as x, x-2, x-4, etc