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Lesson 111: Consecutive Integer Problems
Many algebra problems ask us to find the values of consecutive integers in a pattern, such as 5, 6, 7, or 20, 30, 40, etc. This lesson will show you how.
First, be sure to review all of the previous lessons on algebra. Make sure that you fully understand them. Here is the type of problem that we will learn how to solve in this lesson: The sum of three consecutive integers is 18. What are the three integers?
In algebra, we can't solve an equation that involves more than one variable. We could rearrange the equation so that we have solved for one variable in terms of the other, but we cannot find the value of two variables at the same time. This means that to solve a problem like the one we have, we need to pick a variable for one of the unknown integers, and then define the other two in terms of the first. Here is what we do.
We'll call the first integer x. Now we're told that we have
three consecutive integers. We can represent the second one as
x + 1. We know it is one higher. Then we can represent
the third one as x + 2. It is two higher than the first one.
Our equation is then
x + x + 1 + x + 2 = 18. We're adding the three integers, and
setting them equal to 18.
Combining like terms, we have 3x + 3 = 18. We've learned how to solve problems like this. Subtract 3 from each side, giving us 3x = 15. Then divide each side by 3 to get x = 5. We're not done yet. x was only the first integer. That means that our second one is 6, and our third is 7. Remember to always check your answer. 5 + 6 + 7 = 18, so it works.
In this case, we had to list the integers, so we would write 5, 6, 7. Be careful. A problem may ask for something specific, such as the largest of the integers. In that case, you must answer 7, and only 7. Anything else would be wrong. Read the problem carefully.
Here's another problem. The sum of 3 consecutive even integers is 30. What are the three integers? Be careful, this problem says that the integers must be even. As before, we'll call the first integer x. What is the second one? It's x + 2. We get to the next even integer by adding 2. What is the third? x + 4. Take a moment to make sure that this makes sense.
Our equation is then x + x + 2 + x + 4 = 30. Combine to get 3x + 6 = 30. If we solve the problem like we've learned how, we'll see that x = 8. As before, we can now see that the second integer is 10, and the third is 12. Check to see that 8 + 10 + 12 = 30. It does.
Another example involves multiples of a number. The sum of three consecutive multiples of 3 is 36. This is a bit tricky. Let's call the first integer x. We're dealing with multiples of 3, so that means that the second one must be x + 3. That's how we get to the next multiple of 3. The third one will be x + 6. Our equation is then x + x + 3 + x + 6 = 36. Combine to get 3x + 9 = 36. Solving this problem like we learned how, we get 3x = 27, x = 9. We can now figure out that our next integer is 12, and our next is 15. We have three consecutive multiples of 3. Check to see that 9 + 12 + 15 = 36, and it does.
You'll have more practice with this topic in later lessons. Just make sure that you understand the concepts presented.
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