Consecutive Integers: Find 3 That Sum To 39

Let's dive into a fun little math problem that involves finding three consecutive integers that add up to 39. It's a classic problem that's great for brushing up on your algebra skills and logical thinking. So, grab your thinking caps, and let's get started!

Understanding Consecutive Integers

First things first, what exactly are consecutive integers? Consecutive integers are numbers that follow each other in order, each number increasing by 1. For example, 1, 2, and 3 are consecutive integers. Similarly, 10, 11, and 12 are also consecutive integers. The key thing to remember is that there's no gap between them – they're right next to each other on the number line. When we say consecutive, we imply a sequence of whole numbers, but don't forget that consecutive even integers (like 2, 4, 6) or consecutive odd integers (like 1, 3, 5) exist too! But for this problem, we're focusing on the regular, run-of-the-mill consecutive integers.

Representing these integers algebraically is super useful. If we let the first integer be n, then the next consecutive integer is n + 1, and the one after that is n + 2. This way, we can easily set up equations and solve for the unknown. So, whenever you encounter a problem involving consecutive integers, remember to think of them as n, n + 1, n + 2, and so on. This simple trick can make solving these problems a whole lot easier. Basically, we're using algebra to turn a word problem into something we can actually work with. It’s like translating from English to Math!

Setting Up the Equation

Now that we know what consecutive integers are, let's get back to our problem. We need to find three consecutive integers that add up to 39. Using our algebraic representation, we can write this as an equation:

n + (n + 1) + (n + 2) = 39

This equation simply states that the sum of the first integer (n), the second integer (n + 1), and the third integer (n + 2) is equal to 39. The beauty of algebra is how it lets us take a verbal description and turn it into a concise mathematical statement. It is a powerful equation to work with, and, in a way, it mirrors the problem statement directly. Make sure you understand how we arrived at this equation before moving on. If you're unsure, reread the previous sections and make sure you're comfortable with the concept of consecutive integers and their algebraic representation.

Solving the Equation

Alright, guys, let's solve this equation! The first step is to simplify the left side by combining like terms. We have n + n + n, which gives us 3n. And we also have 1 + 2, which gives us 3. So, our equation becomes:

3n + 3 = 39

Now, we want to isolate the term with n on one side of the equation. To do this, we subtract 3 from both sides:

3n + 3 - 3 = 39 - 3

This simplifies to:

3n = 36

Finally, to solve for n, we divide both sides by 3:

3n / 3 = 36 / 3

Which gives us:

n = 12

So, we've found that the first integer, n, is 12. But we're not done yet! We need to find the other two consecutive integers as well.

Finding the Integers

We know that the first integer, n, is 12. The next consecutive integer is n + 1, which is 12 + 1 = 13. And the integer after that is n + 2, which is 12 + 2 = 14. Therefore, the three consecutive integers are 12, 13, and 14. To be absolutely sure we have the correct answer, let's quickly check if they add up to 39:

12 + 13 + 14 = 39

Yep, they do! So, we've successfully found the three consecutive integers that sum to 39.

Verification and Conclusion

To make absolutely sure we didn't make a mistake, we should always verify our answer. We already did this in the previous section by adding the three integers together to confirm that they indeed sum to 39. This step is crucial, especially in exams or when accuracy is paramount. It's easy to make a small arithmetic error, so taking a moment to verify can save you from a lot of headaches.

In conclusion, the three consecutive integers that add up to 39 are 12, 13, and 14. We solved this problem by using algebra to represent the consecutive integers and setting up an equation. Then, we solved the equation to find the value of the first integer and used that to find the other two. This problem is a great example of how algebra can be used to solve real-world problems. And remember, practice makes perfect! The more you practice these types of problems, the easier they will become. Keep up the great work, and happy solving!

Now, let’s change the problem a bit. What if we wanted to find four consecutive integers that add up to a certain number? The process would be very similar. We would represent the integers as n, n + 1, n + 2, and n + 3, and then set up an equation like this:

n + (n + 1) + (n + 2) + (n + 3) = Some Number

Then, we would solve for n just like before. The key is to understand the concept of consecutive integers and how to represent them algebraically. Once you have that down, you can tackle any problem involving consecutive integers! Understanding the underlying principles is more important than memorizing formulas. So, take your time, practice, and don't be afraid to ask for help when you need it. Math can be fun, especially when you understand what you're doing.