Sum Of Largest & Smallest 4-Digit Even Numbers

by SLV Team 47 views

Let's dive into this math problem, guys! We're tasked with finding the sum of the largest and smallest 4-digit even numbers that can be formed using each digit only once. It sounds like a fun challenge, so let's break it down step by step.

Finding the Largest 4-Digit Even Number

To find the largest 4-digit even number, we need to think about how place value works. We want the largest digits in the highest place values. So, naturally, we'll start by trying to put the biggest digit, 9, in the thousands place. Our number will look like 9 _ _ _. Now, what's the next largest digit? It's 8, so let's put that in the hundreds place: 98 _ _. We continue this pattern, placing the next largest digit, 7, in the tens place: 987 _.

Now, here's the catch: we need an even number. That means the units digit must be even. We've already used 9, 8, and 7, so the next largest even digit available is 6. So, our largest 4-digit even number is 9876. See how we strategically placed the digits to maximize the number while ensuring it ends with an even digit? It's like building a number puzzle!

To solidify this understanding, let's consider why other arrangements wouldn't work. For example, if we tried to put 7 in the thousands place, we'd immediately have a smaller number than 9000-something. And if we didn't put an even number in the units place, it wouldn't be an even number at all! This careful thought process is key to solving these kinds of problems.

Remember, the goal is not just to find a large even number, but the largest possible one. That's why we prioritize placing the biggest digits in the most significant place values. This principle applies to many similar math challenges, so it's a good one to keep in your mental toolkit.

Finding the Smallest 4-Digit Even Number

Alright, now let's switch gears and find the smallest 4-digit even number using each digit only once. This is like the reverse of what we just did, but with a little twist! We might be tempted to start with 0 in the thousands place, but hold on! If we do that, it wouldn't be a 4-digit number anymore; it would be a 3-digit number. So, we need to be careful.

Instead, let's start with the smallest non-zero digit, which is 1. Our number will look like 1 _ _ _. Now, we can use 0 in the hundreds place: 10 _ _. This is looking good! We're building the smallest possible number. Now, let's think about the tens place. The next smallest digit we haven't used is 2: 102 _. But wait! Remember our even number requirement? The units digit needs to be even, so 2 can't go there just yet.

Looking at the remaining digits, we have 3, 4, 5, 6, 7, 8, and 9 left. To keep the number as small as possible, we want to use the smallest even digit we haven't used yet in the units place. That's 2. So, our number looks like 10 _ 2. This means the only digit left for the tens place is 3. Therefore, the smallest 4-digit even number we can form is 1032. Tricky, right? It's all about paying attention to the details and the rules of the problem.

Notice how the constraint of needing an even number significantly impacted our approach. If we were just looking for the smallest 4-digit number, we could have simply put the digits in ascending order (1023). But the even requirement forced us to think more strategically and consider the units place first.

Calculating the Sum

Okay, we've found our players: the largest 4-digit even number (9876) and the smallest 4-digit even number (1032). Now for the grand finale: let's add them together! This is the straightforward part, but accuracy is still key. We don't want to make a silly mistake after all that hard work!

So, let's add 9876 + 1032. You can do this using column addition, a calculator, or even mental math if you're feeling confident. When we add them up, we get:

  9876
+ 1032
------
 10908

Therefore, the sum of the largest and smallest 4-digit even numbers that can be written using each digit once is 10908. Ta-da! We solved it!

Isn't it satisfying when you finally arrive at the answer after working through a problem like this? It's like completing a puzzle or cracking a code. This problem combined logic, place value understanding, and basic arithmetic, showcasing how different math concepts can work together.

Key Takeaways

Let's recap the main ideas we used to solve this problem. This will help you tackle similar challenges in the future:

  • Place Value is King: Understanding place value is crucial for building the largest and smallest numbers. The digit in the thousands place has the biggest impact, followed by the hundreds, tens, and units places.
  • Constraints Matter: The requirement for the number to be even significantly influenced our strategy. Always pay close attention to the specific rules and conditions of the problem.
  • Start with the Extremes: When finding the largest or smallest number, start by placing the largest or smallest digits in the highest place values, respectively.
  • Don't Forget Zero: Zero can be a tricky digit. Remember that it can't go in the thousands place for a 4-digit number.
  • Double-Check Your Work: Always double-check your addition (or any other calculations) to avoid simple errors.

By keeping these principles in mind, you'll be well-equipped to tackle a wide range of number problems. Math is all about building skills and strategies, and practice makes perfect! So, keep challenging yourselves and exploring new problems, guys!

This problem also highlights the importance of clear and methodical thinking. We didn't just jump to the answer; we broke the problem down into smaller, manageable steps. We first identified the largest and smallest even numbers separately, and then we added them together. This approach makes complex problems much less daunting.

So, the next time you encounter a challenging math problem, remember to breathe, break it down, and apply the strategies you've learned. You've got this!