Lesson Problems with consecutive odd even integers

Problems with consecutive odd/even integers

Problem 1

Three consecutive integers have a sum of 240.
Find the integers.

Solution 1
This is a very simple problem.
Let be the second (the middle) of these three consecutive integers.
Then the first one is , while the third one is .
Since the sum of these three consecutive integers , and is equal to 240, you can write the equation
.

Simplify this equation. You get
(after combining like terms at the left side),
   (after dividing both sides by 3).

Thus, the three consecutive integers are 79, 80 and 81.
You can easily check this solution:
.

Solution 2
This time denote as the first (the smallest) of these three consecutive integers.
Then the second one is , while the third one is .
Since the sum of these three consecutive integers , and is equal to 240, you can write the equation
.

Simplify this equation step by step. You get
(after combining like terms at the left side),
    (after moving the constant term 3 to the right side with the opposite sign and combining like terms at the right side),
       (after dividing both sides by 3).

Thus, the solution is the same: three consecutive integers are 79, 80 and 81.

Answer.
The three consecutive integers are 79, 80 and 81.

Problem 2

One third of the sum of five consecutive integers is 15.
Find the integers.

Solution
Again, this is a simple problem.
Since one third of the sum of five consecutive integers is 15, the entire sum of five consecutive integers is 45.
Let be the third (the middle) of these five consecutive integers.
Then the first integer is equal to ,
the second integer is equal to       ,
the fourth integer is equal to       , and
the fifth integer is equal to         .
Since the sum of these five consecutive integers , , , and is equal to 45, you can write the equation
.

Simplify this equation. You get
(after combining like terms at the left side),
   (after dividing both sides by 5).

Thus, the five consecutive integers are 7, 8, 9, 10 and 11.

You can easily check that
,
hence one third of the sum is 9. The solution is correct.

Answer.
The five consecutive integers are 7, 8, 9, 10 and 11.

Problem 3

The sum of three consecutive even integers is 18.
Find the integers.

Solution
Let be the second (the middle) of these three consecutive even integers.
Then the first one is , while the third one is .
Since the sum of these three consecutive even integers , and is equal to 18, you can write the equation
.

Simplify this equation. You get
(after combining like terms at the left side),
   (after dividing both sides by 3).

Thus, the three consecutive even integers are 6-2=4, 6, and 6+2=8.
Easy check shows that the solution is correct:
.

Answer.
The three consecutive even integers are 4, 6 and 8.

Problem 4

The sum of three consecutive odd integers is 27.
Find the integers.

Solution
Let be the second (the middle) of these three consecutive odd integers.
Then the first one is , while the third one is .
Since the sum of these three consecutive odd integers , and is equal to 27, you can write the equation
.

Simplify this equation. You get
(after combining like terms at the left side),
   (after dividing both sides by 3).

Thus, the three consecutive odd integers are 9-2=7, 9, and 9+2=11.
Easy check shows that the solution is correct:
.

Answer.
The three consecutive odd integers are 7, 9 and 11.

Problem 5

The sum of four consecutive odd integers is 40.
Find the integers.

Solution
This time there is no the unique middle term of four consecutive odd integers.
Therefore let us introduce the unknown as the first (minimal) of these four consecutive odd integers.
Then the second number is , the third one is and the fourth one is .
Since the sum of these four consecutive odd integers , , and is equal to 40, you can write the equation
.

Simplify this equation. You get
(after combining like terms at the left side),
      (after moving the constant term 12 to the right side with the opposite sign and combining like terms at the right side),
         (after dividing both sides by 4).

Thus, the three consecutive odd integers are 7, 7+2=9, 9+2=11, and 11+2=13.
Easy check shows that the solution is correct:
.

Answer.
The four consecutive odd integers are 7, 9, 11 and 13.