Consecutive integers

Question

If the sum of three consecutive even integers is 174, what is the first of the three even integers? Hint: If x and x+2 represent the first two consecutive even integers, then how would the third consecutive even integers be represented?

Answer 1

x + (x + 2) + (x + 4) = 174; 3x + 6 = 174; 3x = 168; x = 56

Answer 2

We can use the letter x (or any letter we want) to represent the first of the three unknown numbers. The second number is the next "consecutive even" number so it must be two more than the first number. Therefore, we will call it x +2. The third number is the next consecutive number so it must be two more than the second number (which we called x +2). So the third number can be called (x +2) +2. It is simpler to write this as x +4.

Therefore the three numbers can be written algebraically (meaning we use a letter for the unknown number) as follows:

x

x + 2

x + 4

When combined, we are told these three numbers will equal 174 ("sum of three consecutive even integers is 174").

By combining similar terms (also known as "like terms"), we get 3x +6 as the total of x, x +2, and x +4.

If 3x +6 is equal to 174, we can subtract the 6 from both sides of the equation to simplify the problem. This will leave us with:

3x = 168

When we divide both sides by 3, we will find the value of a single x. 168 divided by 3 = 56. Therefore, the first number in the sequence of three numbers (what we called x when we did not know what it was) is shown to be 56.

The next two numbers in the sequence are 58 (x +2) and 60 (x +4). 56+58+60=174 which proves we have solved the problem correctly.

These steps of combining like terms and solving for x are standard operations with all algebra problems. They become more automatic and make more sense once you have done them multiple times and with practice. Don't worry if you don't understand it all right away!