Consecutive Integer Word Problems Lesson

Over the course of the last few lessons, we have learned how to solve a linear equation in one variable. In real life, our problems will not be given to us in such a manner. We will need to create an equation based on the context of the given situation. When we first learn how to set up and solve word problems, it can be a real challenge for many students. It is often helpful to follow a step-by-step method for solving word problems.

Six-step method for Applications of Linear Equations in One Variable

1. Read the problem carefully and determine what you are asked to find
   • Write down the main objective of the problem
2. Assign a variable to represent the unknown
   • If more than one unknown exists, we express the other unknowns in terms of this variable
3. Write out an equation that describes the given situation
4. Solve the equation
5. State the answer using a nice clear sentence
6. Check the result by reading back through the problem
   • 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

A common word problem that involves "sums of quantities" is to find unknown consecutive integers. Two consecutive integers will differ by 1. As an example, 1 and 2 are consecutive integers, as are 3 and 4.
Some Consecutive Integers:
1,2
3,4
5,6
7,8
9,10
Let's take a look at an example.
Example 1: Solve each word problem.
When we add three consecutive integers together, the result is 21. What are the integers?

1. Read the problem carefully and determine what you are asked to find
   • We are asked to find three consecutive integers.
2. Assign a variable to represent the unknown
   • Let x = smallest consecutive integer
   • Then x + 1 = the middle consecutive integer
   • Then (x + 1) + 1 = the largest consecutive integer
3. Write out an equation that describes the given situation
   • 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
4. Solve the equation
   • 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
5. State the answer using a nice clear sentence
   • 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.
6. Check the result by reading back through the problem
   • 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.

Example 2: Solve each word problem.
When we add three consecutive odd integers together, the result is 57. What are the integers?

1. Read the problem carefully and determine what you are asked to find
   • We are asked to find three consecutive odd integers.
   • Consecutive odd integers differ by 2, (1, 3, 5, 7, 9,...)
2. Assign a variable to represent the unknown
   • Let x = smallest consecutive odd integer
   • Then x + 2 = the middle consecutive odd integer
   • Then (x + 2) + 2 = the largest consecutive odd integer
3. Write out an equation that describes the given situation
   • 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
4. Solve the equation
   • 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
5. State the answer using a nice clear sentence
   • 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.
6. Check the result by reading back through the problem
   • 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.

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