## Consecutive Integer Word Problems Lesson

• Demonstrate an understanding of how to solve a Linear Equation in One Variable
• Learn the six-step process for solving any word problem that involves a Linear Equation in One Variable
• Learn how to check the solution for a word problem
• Learn how to set up and solve consecutive integer word problems

## How to Solve a Consecutive Integer Word Problem

Six-step method for applications of linear equations in one variable.

• Write down the main objective of the problem
• If more than one unknown exists, we express the other unknowns in terms of this variable
• Write out an equation that describes the given situation
• Solve the equation
• State the answer using a nice clear sentence
• We need to make sure the answer is reasonable. In other words, if asked how many miles were driven to the store, the answer shouldn't be (-3) as we can't drive a negative amount of miles.

## Consecutive Integer Word Problems

• We are asked to find three consecutive integers.
• Let x = smallest consecutive integer
• Then x + 1 = the middle consecutive integer
• Then (x + 1) + 1 = the largest consecutive integer
• If we combine the three consecutive integers: x, (x + 1), (x + 1) + 1, our result is (equals) 21.
• x + (x + 1) + (x + 1) + 1 = 21
• x + x + 1 + x + 1 + 1 = 21
• 3x + 3 = 21
• 3x + 3 - 3 = 21 - 3
• 3x = 18
• 3/3 x = 18/3
• x = 6
• Since x represents the smallest consecutive integer, we know the smallest consecutive integer is 6.
• The middle consecutive integer is 7, from (6 + 1).
• The largest consecutive integer is 8, from (7 + 1).
• Our three consecutive integers are 6, 7, and 8.
• We know that the sum of the consecutive integers is 21.
• Check: 6 + 7 + 8 = 21
• 13 + 8 = 21
• 21 = 21
• Since our three consecutive integers sum to 21, our answer is correct.
• We are asked to find three consecutive odd integers.
• Consecutive odd integers differ by 2, (1, 3, 5, 7, 9,...)
• Let x = smallest consecutive odd integer
• Then x + 2 = the middle consecutive odd integer
• Then (x + 2) + 2 = the largest consecutive odd integer
• If we combine the three consecutive integers: x, (x + 2), (x + 2) + 2, our result is (equals) 57.
• x + (x + 2) + (x + 2) + 2 = 57
• x + x + 2 + x + 2 + 2 = 57
• 3x + 6 = 57
• 3x + 6 - 6 = 57 - 6
• 3x = 51
• 3/3 x = 51/3
• x = 17
• Since x represents the smallest consecutive odd integer, we know the smallest consecutive odd integer is 17.
• The middle consecutive odd integer is 19, from (17 + 2).
• The largest consecutive odd integer is 21, from (19 + 2).
• Our three consecutive odd integers are 17, 19, and 21.
• We know that the sum of the consecutive integers is 57.
• Check: 17 + 19 + 21 = 57
• 36 + 21 = 57
• 57 = 57
• Since our three consecutive integers sum to 57, our answer is correct.

## Skills Check:

Solve each word problem.

The sum of two pages that face each other in a book is 177. What are the page numbers?

The sum of three consecutive even integers is 102. Find the integers.

When the smaller of two consecutive integers is added to 3 times the larger, the result is -37. Find the two integers.

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## Algebra: Consecutive Integer Problems

In these lessons, we will learn how to solve

• consecutive integer word problems
• consecutive even integer word problems
• consecutive odd integer word problems

Related Pages Integers Word Problems Consecutive Integer Word Problems Consecutive Integers 2 Consecutive Even Integer Problems Consecutive Odd Integer Problems More Algebra Word Problems

## What Are 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 diagram shows an example of a consecutive integer problem. Scroll down the page for more examples and solutions on consecutive integer problems.

Consecutive Integer Problems

Consecutive integer problems are word problems that involve consecutive integers .

The following are common examples of consecutive integer problems.

Example: 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

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.

Consecutive Odd Integers

Example 2: 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

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

Consecutive Even Integers

Example 3: 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.

How to find consecutive integers, consecutive odd integers, or consecutive even integers that add up to a given number

• The sum of three consecutive integers is 657; find the integers.
• The sum of two consecutive integers is 519; find the integers.
• The sum of three consecutive even integers is 528; find the integers.
• The sum of three consecutive odd integers is 597; find the integers.

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 consecutive even integers is 30. Find the integers.

How to solve consecutive integer word problems?

Example: The sum of three consecutive integers is 24. Find the integers.

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## Course: 8th grade   >   Unit 2

Sums of consecutive integers.

• Sum of integers challenge
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## SOLVING WORD PROBLEMS INVOLVING CONSECUTIVE INTEGERS

Consecutive numbers are numbers that follow each other in order. They have a difference of 1 between every two numbers.

To solve word problems regarding consecutive integers, it is important to note :

Consecutive Integers :

If the first integer be x, then three consecutive integers are

x, x + 1, x + 2

Consecutive Even Integers :

If the first even integer be x, then three consecutive integers are

x, x + 2, x + 4

Consecutive Odd Integers :

If the first odd integer be x, then three consecutive integers are

Problem 1 :

If the average of three consecutive integers is 26, find the integers.

Let x , ( x + 1) and (x + 2) be the three consecutive integers.

Average of three consecutive integers = 26

Multiply both sides by 3.

3x + 3 = 78

Subtract 3 from both sides.

Divide both sides by 3.

Therefore, the three consecutive integers are 25, 26 and 27.

Problem 2 :

If the average of three consecutive even integers is 14, find the integers.

Let x , ( x + 2) and (x + 4) be the three consecutive even integers.

Average of three consecutive even integers = 14

3x + 6 = 42

Subtract 6 from both sides.

Therefore, the three consecutive even integers are 12, 14 and 16.

Problem 3 :

Among the three consecutive odd integers,  26 less than thrice the largest integer is equal to  twice the smallest integer. Find the integers.

Let x , ( x + 2) and (x + 4) be the three consecutive odd integers.

3(x + 4) - 25 = 2x

3x + 12 - 25 = 2x

3x - 13 = 2x

Therefore, the three consecutive odd integers are 12, 14 and 16.

Problem 4 :

Find two consecutive natural numbers whose product is 30.

Let x , ( x + 1) be the two consecutive integers.

Product of two consecutive integers = 30

x(x + 1) = 30

x 2  + x = 30

x 2  + x - 30 = 0

x 2  - 5x + 6x - 30 = 0

x(x - 5) + 6(x - 5) = 0

(x - 5)(x - 6) = 0

x - 5 = 0  or  x + 6 = 0

x = 5  or  x = -6

Since the natural numbers are always positive, x can not be -6.

Therefore, the required two consecutive natural numbers are 5 and 6.

Verification :

Product of two positive integer is 30.

5(6) = 30

30 = 30

Problem 5 :

There are three consecutive positive integers such that the sum of the square of first and the product of the other two is 154. Find the integers.

Let x , ( x + 1) and ( x + 2) be the required three consecutive integers

The sum of the squares of first and the product of the other two is 154.

x 2  + (x + 1)(x + 2) = 154

x 2  + x 2  + 2x + x + 2 = 154

2x 2  + 3x + 2 = 154

2x 2  + 3x + 2 - 154 = 0

2x 2  + 3x - 152 = 0

2x 2  - 16x + 19x - 152 = 0

2x(x - 8) + 19(x - 8) = 0

(x - 8)(2x + 19) = 0

x - 8 = 0  or  2x + 19 = 0

x = 8  or  x = -9.5

Since the numbers are positive integers, x can not be -9.5.

Therefore, the required three consecutive positive integers are 8, 9 and 10.

The sum of the square of first and the product of the other two is 154.

8 2  + (9) (10) = 154

64 + 90 = 154

154 = 154

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#### IMAGES

1. Consecutive Integer Word Problems 3

2. PPT

3. Solving Integer Word Problems

4. Solving a word problem involving consecutive integers

5. an info sheet with the words, which are in different colors and font options for students to use

6. Consecutive Integer Word Problems

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