Six-Digit Number Difference: A Math Exercise
Let's break down this math problem step by step, guys! We're going to calculate some differences and sums, and then compare the results. It sounds like a fun numerical adventure, right?
Finding the Largest Six-Digit Number with Distinct Digits
Okay, so first things first: we need to figure out what the largest six-digit number with all different digits is. Think about it – to make it as big as possible, we want the biggest digit in the leftmost place, then the next biggest, and so on. This is a crucial step.
So, what's the biggest digit? It's 9! That goes in the hundred thousands place. Then we want the next biggest, which is 8, in the ten thousands place. Then 7 in the thousands place, 6 in the hundreds place, 5 in the tens place, and finally, 4 in the ones place. Therefore, the largest six-digit number with distinct digits is 987654. This is our starting point for the whole exercise. The beauty of math is in this logical deduction, don't you think?
Understanding place value is super important here. Each digit's position determines its contribution to the overall value of the number. Remember, the further left a digit is, the more it's worth. The number 9 in the hundred thousands place is worth way more than the number 4 in the ones place. Got it?
To make sure we really understand, let's consider why we couldn't use the same digit twice. If we repeated a digit, we'd have to use a smaller digit in another place, which would make the whole number smaller. For example, if we used 9 twice, say 99xxxxx, the remaining digits would have to be smaller than 8, 7, 6, 5, and 4. This would clearly result in a smaller overall number. So, distinct digits are key to maximizing the value!
Another way to think about it is this: we're trying to fill six slots, each with a digit, to create the biggest possible number. We have ten digits to choose from (0-9). We want to use the biggest ones first, and we can't repeat any. It's like arranging the digits in descending order to make the largest possible number. This kind of logical thinking is what makes math so fascinating. I hope you agree, guys!
Calculating the Sum of 342431 and 105123
Next up, we need to find the sum of 342431 and 105123. This is a straightforward addition problem. We need to carefully add each place value column, starting from the right (the ones place) and moving left. If the sum in any column is greater than 9, we need to carry over the tens digit to the next column. Let's do it together.
Starting with the ones place: 1 + 3 = 4. Then the tens place: 3 + 2 = 5. Next, the hundreds place: 4 + 1 = 5. Now, the thousands place: 2 + 5 = 7. The ten thousands place: 4 + 0 = 4. And finally, the hundred thousands place: 3 + 1 = 4. So, the sum of 342431 and 105123 is 447554. See? Easy peasy.
Let's write it out to make it super clear:
342431
- 105123
447554
Make sure you line up the numbers correctly according to their place values. Lining them up ensures that you're adding the correct digits together. A common mistake is to misalign the numbers, which can lead to incorrect sums. Pay close attention to this, and you'll be golden.
Addition is a fundamental operation in mathematics, and it's used everywhere. From calculating your grocery bill to designing bridges, addition is an essential tool. Mastering addition, including carrying over when necessary, is crucial for success in math. And don't worry, it gets easier with practice! Just keep adding, guys!
Now, just to double-check our work, let's think about estimation. 342431 is roughly 340000 and 105123 is roughly 100000. So, their sum should be around 440000. Our calculated sum of 447554 is pretty close to this estimate, which gives us confidence in our answer. Always estimate to check your work!
Finding the Difference Between 987654 and 447554
Now we need to find the difference between the largest six-digit number (987654) and the sum we just calculated (447554). This means we'll be doing subtraction. Again, we need to be careful to subtract each place value column correctly. Let's start with the ones place.
In the ones place, we have 4 - 4 = 0. In the tens place, we have 5 - 5 = 0. In the hundreds place, we have 6 - 5 = 1. In the thousands place, we have 7 - 7 = 0. In the ten thousands place, we have 8 - 4 = 4. And finally, in the hundred thousands place, we have 9 - 4 = 5. So, the difference between 987654 and 447554 is 540100.
Writing it out:
987654
- 447554
540100
Subtraction is the inverse operation of addition. It tells us how much is left when we take away one number from another. It's essential for solving problems involving comparisons and reductions. Make sure you're comfortable with borrowing when the top digit is smaller than the bottom digit in a column.
To verify, we can add the difference (540100) to the smaller number (447554) and see if we get the larger number (987654). 540100 + 447554 = 987654. Yep, it checks out! So, we're confident that our subtraction is correct.
Subtraction, like addition, is a fundamental skill in math. Practice subtraction problems regularly to improve your speed and accuracy. And remember to always check your work by adding the difference back to the smaller number. This is a great way to catch any mistakes!
Calculating the Difference Between 67346 and 321213
Now, let's calculate the difference between 67346 and 321213. Since 321213 is larger, we'll subtract 67346 from it.
321213
- 067346
Starting from the right: 3 - 6. We need to borrow. So, the 1 becomes 0, and the 3 becomes 13. 13 - 6 = 7. Next, 0 - 4. Again, we need to borrow. The 2 becomes 1, and the 0 becomes 10. 10 - 4 = 6. Then, 1 - 3. Borrow again. The 1 becomes 0, and the 1 becomes 11. 11 - 3 = 8. Now, 0 - 7. Borrow again. The 2 becomes 1, and the 0 becomes 10. 10 - 7 = 3. Then, 1 - 6. Borrow again. The 3 becomes 2, and the 1 becomes 11. 11 - 6 = 5. Finally, 2 - 0 = 2.
So, the difference between 321213 and 67346 is 253867.
Calculating the Sum of 67346 and 321213
Now, let's find the sum of 67346 and 321213.
67346 +321213
388559
Starting from the right: 6 + 3 = 9. Next, 4 + 1 = 5. Then, 3 + 2 = 5. Now, 7 + 1 = 8. Then, 6 + 2 = 8. Finally, 0 + 3 = 3.
So, the sum of 67346 and 321213 is 388559.
Finding How Much Smaller the Difference Is Compared to the Sum
Finally, we need to find out how much smaller the difference (253867) is compared to the sum (388559). We do this by subtracting the difference from the sum.
388559
- 253867
134692
Starting from the right: 9 - 7 = 2. Next, 5 - 6. We need to borrow. The 5 becomes 4, and the 5 becomes 15. 15 - 6 = 9. Then, 4 - 8. Borrow again. The 8 becomes 7, and the 4 becomes 14. 14 - 8 = 6. Now, 7 - 3 = 4. Then, 8 - 5 = 3. Finally, 3 - 2 = 1.
So, the difference between the numbers is 134692 smaller than their sum.
Conclusion
And that's it! We've solved the entire problem step by step. We found the largest six-digit number with distinct digits, calculated the sum of two numbers, found the difference between the largest number and the sum, and then compared the difference and sum of two other numbers. Phew! That was quite the mathematical workout! I hope you guys found this explanation helpful and easy to understand. Keep practicing, and you'll become math wizards in no time! Remember, math is all about breaking down problems into smaller, manageable steps. You got this!
