Algebra: Consecutive Integer Problems

Consecutive integer problems are word problems that involve consecutive integers.

Consecutive integers are integers that follow in sequence, each number being 1 more than the previous number, represented by n, n +1, n + 2, n + 3, ..., where n is any integer.

For example: 23, 24, 25, …

If we start with an even number and each number in the sequence is 2 more than the previous number then we will get consecutive even integers.

For example: 16,18, 20, …

If we start with an odd number and each number in the sequence is 2 more than the previous number then we will get consecutive odd integers.

For example: 33, 35, 37, …

The following are common examples of consecutive integer problems.

Example 1: Consecutive Integer Problem

The sum of the least and greatest of 3 consecutive integers is 60. What are the values of the 3 integers?

Solution:

Step 1: Assign variables :

Let x = least integer

x + 1 = middle integer

x + 2 = greatest integer

Translate sentence into an equation.

Sentence: The sum of the least and greatest is 60.

Rewrite sentence:

x + (x + 2) = 60

Step 2: Solve the equation

Combine like terms

2x + 2 = 60

Isolate variable x

2x =58

Step 3: Check your answer

29 + 29 + 2 = 60

The question wants all the 3 consecutive numbers: 29, 30 and 31

Answer: The 3 consecutive numbers are 29, 30 and 31.

Example 2: Consecutive Odd Integer

The lengths of the sides of a triangle are consecutive odd numbers. What is the length of the longest side if the perimeter is 45?

Solution:

Step 1: Being consecutive odd numbers we need to add 2 to the previous number.

Assign variables :

Let x = length of shortest side

x + 2 = length of medium side

x + 4 = length of longest side

Sketch the figure

Step 2: Write out the formula for perimeter of triangle.

P = sum of the three sides

Step 3: Plug in the values from the question and from the sketch.

45 = x + x + 2 + x + 4

Combine like terms

45 = 3x + 6

Isolate variable x

3x = 45 – 6

3x = 39

x =13

Step 3: Check your answer

13 + 13 + 2 + 13 + 4 = 45

Be careful! The question requires the length of the longest side.

The length of longest = 13 + 4 =17

Answer: The length of longest side is 17

Example 3: Consecutive Even Integer

John has a board that is 5 feet long. He plans to use it to make 4 shelves whose lengths are to be a series of consecutive even numbers. How long should each shelf be in inches?

Solution:

Step 1: Being consecutive even numbers we need to add 2 to the previous number.

Assign variables :

Let x = length of first shelf

x + 2 = length of second shelf

x + 4 = length of third shelf

x + 6 = length of fourth shelf

Step 2: Convert 5 feet to inches

5 × 12 = 60

Step 3: Sum of the 4 shelves is 60

x + x + 2 + x + 4 + x + 6 = 60

Combine like terms

4x + 12 = 60

Isolate variable x

4x = 60 – 12

4x = 48

x = 12

Step 3: Check your answer

12 + 12 + 2 + 12 + 4 + 12 + 6 = 60

The lengths of the shelves should be 12, 14, 16 and 18.

Answer: The lengths of the shelves in inches should be 12, 14, 16 and 18.

The following video shows how to solve the integer word problems:
The sum of two consecutive integers is 99. Find the value of the smaller integer.
The sum of two consecutive odd integers is 40. What are the integers?
The sum of three consecuitve even integers is 30. Find the integers.

The following video gives another example of how to solve consecutive integer word problems.

Custom Search