Subtracting Reverse Numbers: Math Problem Solution

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Subtracting Reverse Numbers: Math Problem Solution

Hey guys! Let's dive into a fun math problem today where we'll be calculating the difference between the sum of numbers within a range and the reverse of another number. It might sound a bit complex at first, but don't worry, we'll break it down step by step. So, grab your pencils and let's get started!

Understanding the Problem

Okay, so the problem asks us to do two main things. First, we need to find the sum of all the numbers between 23,564 and 23,574. Then, we need to subtract the reverse of the number 34,446 from that sum. To make sure we're on the same page, let's clarify a few key terms:

  • Sum: This is the total you get when you add numbers together.
  • Numbers between: This means we need to include all the whole numbers starting from 23,565 up to 23,573.
  • Reverse of a number: This is the number you get when you write its digits in reverse order. For example, the reverse of 123 is 321.

With these definitions in mind, we can see that the problem involves a combination of addition and subtraction, along with a twist of reversing digits. This makes it a great exercise for practicing our basic arithmetic skills and thinking logically. We need to carefully follow each step to arrive at the correct solution. Let's start by tackling the first part: finding the sum of the numbers between 23,564 and 23,574. This will involve listing out the numbers and adding them together. There might be a clever way to do this more efficiently, so let's explore our options and see what works best. Remember, math is all about problem-solving, and there's often more than one way to reach the answer. So, stay curious, and let's get calculating!

Step 1: Finding the Sum of Numbers Between 23,564 and 23,574

The first part of our problem is to figure out the sum of all the numbers nestled between 23,564 and 23,574. This means we need to consider the numbers from 23,565 up to 23,573. Let's list them out to make sure we don't miss any:

23,565, 23,566, 23,567, 23,568, 23,569, 23,570, 23,571, 23,572, 23,573

Now, we could simply add these numbers together one by one, but that might take a little while and increase the chance of making a mistake. Is there a smarter, more efficient way to do this? Absolutely! One technique we can use is to look for pairs of numbers that add up to a nice, round number. This can make the addition process much smoother. For instance, notice that 23,565 and 23,573 have a sort of symmetry to them. Let's see what they add up to:

23, 565 + 23, 573 = 47, 138

That's a pretty big number, but let's keep this in mind. Now, let's try pairing up the next numbers in the list: 23,566 and 23,572.

23, 566 + 23, 572 = 47, 138

Hey, that's the same sum! It seems like we're onto something here. Let's continue this pattern and pair up the numbers:

  • 23, 567 + 23, 571 = 47, 138
  • 23, 568 + 23, 570 = 47, 138

We've successfully paired up eight of our nine numbers, and each pair adds up to 47,138! This is great because it simplifies our calculation significantly. We have four pairs that each sum to 47,138, and we're left with the middle number, 23,569. Now, we can calculate the sum of the pairs:

47, 138 * 4 = 188, 552

And finally, we add the remaining number:

188, 552 + 23, 569 = 212, 121

So, the sum of the numbers between 23,564 and 23,574 is 212,121. Phew! We've tackled the first part of the problem. Now, let's move on to the next step: finding the reverse of 34,446.

Step 2: Finding the Reverse of 34,446

Okay, guys, now we need to find the reverse of the number 34,446. This is actually a pretty straightforward step. To reverse a number, we simply write its digits in the opposite order. So, the last digit becomes the first, the second-to-last becomes the second, and so on.

Let's take a look at our number, 34,446. It has five digits:

  • 3 is in the ten-thousands place
  • 4 is in the thousands place
  • 4 is in the hundreds place
  • 4 is in the tens place
  • 6 is in the ones place

To reverse it, we'll write these digits in reverse order:

  • The ones digit (6) becomes the ten-thousands digit
  • The tens digit (4) becomes the thousands digit
  • The hundreds digit (4) becomes the hundreds digit (it stays in the same place!)
  • The thousands digit (4) becomes the tens digit
  • The ten-thousands digit (3) becomes the ones digit

So, the reverse of 34,446 is 64,443. See? It's like looking at the number in a mirror! Now that we've found the reverse of the number, we're ready for the final step. We know the sum of the numbers between 23,564 and 23,574 (which is 212,121), and we know the reverse of 34,446 (which is 64,443). The last thing we need to do is subtract the reverse number from the sum. This will give us the final answer to our problem. Are you ready to finish this? Let's go!

Step 3: Subtracting the Reverse from the Sum

Alright, we're in the home stretch now! We've done the hard work of finding the sum of the numbers between 23,564 and 23,574, and we've successfully reversed the number 34,446. Now, all that's left to do is subtract the reversed number from the sum. This is where our good old subtraction skills come into play.

We have two numbers: 212,121 (the sum) and 64,443 (the reversed number). We need to subtract 64,443 from 212,121. Let's set up the subtraction problem:

  212,121
-  64,443
---------

Now, let's subtract column by column, starting from the rightmost column (the ones place):

  • 1 - 3: We can't subtract 3 from 1 directly, so we need to borrow 1 from the tens place. This makes the 1 in the ones place an 11, and the 2 in the tens place becomes a 1.
    • 11 - 3 = 8. So, we write 8 in the ones place of our answer.
  212,111
-  64,443
---------
        8
  • 1 - 4: Again, we can't subtract 4 from 1, so we need to borrow 1 from the hundreds place. The 1 in the tens place becomes an 11, and the 1 in the hundreds place becomes a 0.
    • 11 - 4 = 7. So, we write 7 in the tens place of our answer.
  212,011
-  64,443
---------
       78
  • 0 - 4: We can't subtract 4 from 0, so we need to borrow 1 from the thousands place. The 0 in the hundreds place becomes a 10, and the 2 in the thousands place becomes a 1.
    • 10 - 4 = 6. So, we write 6 in the hundreds place of our answer.
  211,101
-  64,443
---------
      678
  • 1 - 4: We can't subtract 4 from 1, so we need to borrow 1 from the ten-thousands place. The 1 in the thousands place becomes an 11, and the 1 in the ten-thousands place becomes a 0.
    • 11 - 4 = 7. So, we write 7 in the thousands place of our answer.
  201,111
-  64,443
---------
     7678
  • 0 - 6: We can't subtract 6 from 0, so we need to borrow 1 from the hundred-thousands place. The 0 in the ten-thousands place becomes a 10, and the 2 in the hundred-thousands place becomes a 1.
    • 10 - 6 = 4. So, we write 4 in the ten-thousands place of our answer.
  1101,111
-  64,443
---------
    47678
  • Finally, we bring down the 1 in the hundred-thousands place.
  1101,111
-  64,443
---------
 147,678

So, 212,121 - 64,443 = 147,678. That's our final answer!

Conclusion

Wow, we did it! We successfully solved a multi-step math problem that involved finding the sum of numbers within a range and subtracting the reverse of another number. We broke down the problem into smaller, manageable steps, used a clever pairing strategy to simplify the addition, and carefully performed the subtraction. This problem demonstrates how important it is to understand the problem, plan your approach, and take your time to avoid mistakes.

Remember, guys, math isn't just about getting the right answer; it's about the process of problem-solving. By practicing these kinds of problems, we're building our critical thinking skills and becoming better mathematicians. So, keep challenging yourselves, stay curious, and never stop learning! Awesome job today, and I'll see you in the next math adventure!