Four-Digit Numbers With 3 In Thousands And Tens Place

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Hey guys! Let's dive into a cool math problem today. We're going to figure out how many four-digit numbers have the digit 3 in both the thousands and tens places. This might sound tricky, but we'll break it down step by step, so don't worry!

Understanding the Problem

First off, let's make sure we're all on the same page. A four-digit number looks like this: ABCD, where each letter represents a digit. The digits can be any number from 0 to 9. However, there's a catch: the first digit (A) can't be 0 because then it wouldn't be a four-digit number anymore. It would turn into a three-digit number, a two-digit number, or even a one-digit number. So, for a number to be a valid four-digit number, A must be between 1 and 9.

In our specific problem, we have a couple of extra rules. We need the thousands digit (A) and the tens digit (C) to both be 3. This significantly narrows down our options and makes the problem much more manageable. Think of it like this: we've already filled in two of the four slots! Now we just need to figure out the possibilities for the remaining two digits.

Why is this important? Problems like these aren't just about getting the right answer. They're about building your problem-solving skills. When you encounter a challenging problem, it's essential to break it down into smaller, more manageable parts. This approach is applicable not only in math but also in many real-life situations. Understanding the constraints and conditions of a problem is often the first and most crucial step in finding a solution.

Breaking Down the Possibilities

Okay, let's get down to business. We know our number looks something like this: 3 _ 3 _. The blanks are where we need to figure out the possible digits. Let's call the hundreds digit B and the units digit D. So, our number is 3B3D.

The Hundreds Digit (B)

The hundreds digit (B) can be any digit from 0 to 9. There are no restrictions here. It can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. That's a total of 10 possibilities. Think about it: we can have numbers like 3030, 3130, 3230, and so on. Each of these is a valid four-digit number with 3 in the thousands and tens places.

The Units Digit (D)

Similarly, the units digit (D) can also be any digit from 0 to 9. Again, there are no restrictions. It can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. This gives us another 10 possibilities. We can have numbers like 3030, 3031, 3032, and so on. Each of these fits our criteria.

Combining the Possibilities

Now, here's where the magic happens. For each choice of the hundreds digit (B), we have 10 choices for the units digit (D). This is a fundamental concept in combinatorics: if you have m ways to do one thing and n ways to do another, then you have m * n ways to do both. In our case, we have 10 choices for B and 10 choices for D. So, the total number of possibilities is 10 * 10.

Visualizing the combinations can be helpful. Imagine a table where the rows represent the possible values for the hundreds digit (0-9) and the columns represent the possible values for the units digit (0-9). Each cell in the table represents a unique combination of these digits. For example, one cell might represent the number 3030 (hundreds digit 0, units digit 0), another might represent 3537 (hundreds digit 5, units digit 7), and so on. The total number of cells in the table is 10 rows * 10 columns = 100 cells. This gives us a visual confirmation that there are 100 possible combinations.

Calculating the Total Number

So, we've figured out that there are 10 possibilities for the hundreds digit and 10 possibilities for the units digit. To find the total number of four-digit numbers that meet our criteria, we simply multiply these possibilities together:

10 (possibilities for the hundreds digit) * 10 (possibilities for the units digit) = 100

Therefore, there are 100 four-digit numbers that have the digit 3 in both the thousands and tens places. How cool is that?

Let's think about some examples to make this concrete. The smallest number in this set is 3030, and the largest is 3939. All the numbers in between follow the pattern: 3B3D, where B and D are digits from 0 to 9. We've systematically counted all these possibilities, ensuring that we haven't missed any or counted any twice.

Putting it All Together

To recap, we started with a seemingly complex problem: finding the number of four-digit numbers with specific digits in certain places. We broke the problem down into smaller, more manageable parts. We identified the constraints (the thousands and tens digits must be 3) and then considered the possibilities for the remaining digits. We used the principle of multiplication to combine these possibilities and arrive at our final answer: 100.

The Importance of Problem-Solving Strategies

This exercise isn't just about getting the right answer. It's about developing critical thinking and problem-solving skills. These skills are essential in all areas of life, from academics to your future career. When you're faced with a challenge, remember the strategies we used here:

  • Understand the problem: Make sure you know exactly what you're trying to solve.
  • Identify the constraints: What are the rules and limitations?
  • Break it down: Divide the problem into smaller, more manageable parts.
  • Consider the possibilities: What are the different options?
  • Combine the results: Use mathematical principles (like multiplication) to find the overall solution.
  • Check your work: Make sure your answer makes sense.

By applying these strategies, you can tackle even the most challenging problems with confidence.

Practice Makes Perfect

Now that we've solved this problem together, try applying the same techniques to similar problems. For example, how many four-digit numbers have the thousands digit as 5 and the units digit as 2? Or, how many three-digit numbers have the hundreds digit as 1 and the tens digit as 4? The more you practice, the more comfortable you'll become with these types of problems.

Remember, math isn't just about memorizing formulas and procedures. It's about understanding concepts and developing the ability to think critically. Keep practicing, keep exploring, and most importantly, keep having fun with math!

In conclusion, we've successfully determined that there are 100 four-digit numbers with the thousands and tens digits being 3. We achieved this by systematically analyzing the possibilities for each digit and applying the fundamental principle of multiplication. This problem serves as a great example of how breaking down complex problems into smaller, manageable steps can lead to a clear and concise solution. Keep practicing these techniques, and you'll become a math whiz in no time! And remember, the journey of problem-solving is just as important as the final answer. So, embrace the challenge, enjoy the process, and keep learning! 🚀