How To Find Balanced Three-Digit Numbers A Step-by-Step Guide

by Scholario Team 62 views

Hey everyone! Today, we're diving into a fun math problem that involves finding balanced three-digit numbers. What exactly are these, you ask? Well, a balanced three-digit number is one where the difference between any two consecutive digits is no more than one. Think of it like a gentle numerical slope, nothing too steep! Let's explore how we can figure out just how many of these balanced numbers exist. In this comprehensive guide, we'll walk through the problem step-by-step, ensuring you understand every facet of the solution. Our goal is to not only arrive at the correct answer but also to equip you with the problem-solving skills needed to tackle similar challenges. Let's embark on this mathematical journey together!

Understanding Balanced Numbers

Before we jump into counting, let's make sure we're all on the same page about what makes a number "balanced." Imagine you have a three-digit number, say, ABC. For this number to be balanced, the difference between A and B must be no more than 1, and the difference between B and C must also be no more than 1. Mathematically, this means |A - B| ≤ 1 and |B - C| ≤ 1. For example, 123 is a balanced number because |1 - 2| = 1 and |2 - 3| = 1. Similarly, 323 is balanced because |3 - 2| = 1 and |2 - 3| = 1. However, 135 is not balanced because |1 - 3| = 2, which is greater than 1. Understanding this definition is crucial because it sets the foundation for our counting strategy. We need to systematically consider each digit and its possible neighbors to ensure we only count the numbers that fit our criteria. Without a clear understanding of what "balanced" means in this context, we risk including numbers that don't meet the condition or excluding those that do. This careful attention to detail will help us avoid errors and arrive at the correct solution.

Breaking Down the Problem

To tackle this problem effectively, we'll break it down into smaller, more manageable parts. Our approach will involve considering each possible digit for the hundreds place and then determining the possible digits for the tens and units places based on our "balanced" condition. This method allows us to systematically explore all valid combinations without missing any. We'll start by examining the hundreds digit, which can be any number from 1 to 9 (since a three-digit number cannot start with 0). For each choice of the hundreds digit, we'll then consider the possible digits for the tens place. Remember, the tens digit can only be within one unit of the hundreds digit. So, if the hundreds digit is 5, the tens digit can be 4, 5, or 6. Once we have the tens digit, we'll repeat this process for the units digit, ensuring it's within one unit of the tens digit. By systematically working through each possibility, we can create a comprehensive list of all balanced three-digit numbers. This step-by-step approach not only helps us organize our thoughts but also makes it easier to track our progress and verify that we've considered all possible cases. Let's start with the hundreds digit and see where it takes us!

Counting Possibilities

Now comes the fun part: actually counting the balanced numbers! We'll start by considering each possible digit for the hundreds place, from 1 to 9, and then figure out the valid options for the tens and units places.

  • If the hundreds digit is 1: The tens digit can be 1 or 2.
    • If the tens digit is 1, the units digit can be 0, 1, or 2.
    • If the tens digit is 2, the units digit can be 1, 2, or 3.
  • If the hundreds digit is 2: The tens digit can be 1, 2, or 3.
    • If the tens digit is 1, the units digit can be 0, 1, or 2.
    • If the tens digit is 2, the units digit can be 1, 2, or 3.
    • If the tens digit is 3, the units digit can be 2, 3, or 4.
  • We'll continue this pattern for each hundreds digit, carefully listing out the possibilities for the tens and units digits. It's crucial to be methodical and avoid rushing, as missing even one valid combination can lead to an incorrect answer. As we count, we'll keep a running total of the number of balanced numbers we find. This systematic approach ensures we don't double-count any numbers and that we cover all possible cases. By the end of this process, we'll have a complete count of all balanced three-digit numbers. Let's get counting!

Detailed Case Analysis

To ensure accuracy, let's perform a detailed case analysis for each possible hundreds digit. This meticulous approach will help us avoid overlooking any valid combinations and ensure we arrive at the correct total.

  • Hundreds Digit is 1:
    • Tens digit can be 1 or 2.
      • If Tens is 1, Units can be 0, 1, or 2 (3 numbers: 110, 111, 112).
      • If Tens is 2, Units can be 1, 2, or 3 (3 numbers: 121, 122, 123).
    • Total for Hundreds digit 1: 3 + 3 = 6 numbers.
  • Hundreds Digit is 2:
    • Tens digit can be 1, 2, or 3.
      • If Tens is 1, Units can be 0, 1, or 2 (3 numbers: 210, 211, 212).
      • If Tens is 2, Units can be 1, 2, or 3 (3 numbers: 221, 222, 223).
      • If Tens is 3, Units can be 2, 3, or 4 (3 numbers: 232, 233, 234).
    • Total for Hundreds digit 2: 3 + 3 + 3 = 9 numbers.
  • Hundreds Digit is 3 to 7:
    • These digits are symmetrical, meaning they will have the same number of combinations.
    • For each, the Tens digit can be H-1, H, or H+1 (where H is the Hundreds digit).
      • If Tens is H-1, Units can be H-2, H-1, or H (3 numbers).
      • If Tens is H, Units can be H-1, H, or H+1 (3 numbers).
      • If Tens is H+1, Units can be H, H+1, or H+2 (3 numbers).
    • Total for each Hundreds digit from 3 to 7: 3 + 3 + 3 = 9 numbers.
  • Hundreds Digit is 8:
    • This case is symmetrical to Hundreds digit 2.
    • Total for Hundreds digit 8: 9 numbers.
  • Hundreds Digit is 9:
    • This case is symmetrical to Hundreds digit 1.
    • Total for Hundreds digit 9: 6 numbers.

Final Calculation

Okay, guys, we've done the heavy lifting! Now it's time to add up all the possibilities we've meticulously counted. This is where all our hard work pays off, as we'll finally arrive at the answer to our balanced number puzzle. Remember, we broke down the problem by considering each possible digit in the hundreds place and then carefully counting the valid combinations for the tens and units places. We've kept track of these totals, and now we just need to sum them up. Let's take a look at our results:

  • Hundreds digit 1: 6 numbers
  • Hundreds digit 2: 9 numbers
  • Hundreds digits 3 to 7 (5 digits): 9 numbers each, so 5 * 9 = 45 numbers
  • Hundreds digit 8: 9 numbers
  • Hundreds digit 9: 6 numbers

Now, let's add them all together: 6 + 9 + 45 + 9 + 6 = 75. So, we've found a total of 75 balanced three-digit numbers! This is a testament to our systematic approach and careful attention to detail. By breaking down the problem into smaller parts and meticulously counting each possibility, we've not only arrived at the correct answer but also gained a deeper understanding of the problem-solving process. Way to go, team!

Putting It All Together

So, putting it all together, we have:

6 (for hundreds digit 1) + 9 (for hundreds digit 2) + 9 * 5 (for hundreds digits 3 to 7) + 9 (for hundreds digit 8) + 6 (for hundreds digit 9) = 6 + 9 + 45 + 9 + 6 = 75

Therefore, there are 75 balanced three-digit numbers.

Conclusion

In conclusion, after a thorough analysis and step-by-step calculation, we've determined that there are 75 balanced three-digit numbers. This was a fun mathematical journey, and I hope you enjoyed it as much as I did! Remember, the key to solving complex problems is often breaking them down into smaller, more manageable parts. By systematically considering each case and carefully counting the possibilities, we were able to arrive at the correct answer. This problem not only tests our mathematical skills but also our ability to think logically and methodically. The next time you encounter a challenging problem, remember our approach: break it down, consider each case, and count carefully. You've got this!

The correct answer is (A) 75.