solving a word problem involving consecutive integers

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?

Please choose the best answer.

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.

Congrats, Your Score is 100 %

Better Luck Next Time, Your Score is %

Ready for more? Watch the Step by Step Video Lesson   |   Take the Practice Test

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.

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Video transcript

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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