3 Digit Subtraction - Regrouping with Zeros

3 Digit Subtraction – Regrouping with Zeros

When subtracting 3-digit numbers, sometimes the digit in the top number (minuend) is smaller than the digit in the bottom number (subtrahend). In such cases, we need to regroup or borrow from the next higher place value. This process becomes a bit more complex when there are zeros involved.

Using pictures to illustrate subtraction, especially when regrouping is involved, can be very helpful for visual learners. Here’s a step-by-step guide with pictures for a subtraction problem that includes regrouping with zeros.

Example: Subtracting 204 from 500

Let’s break down the subtraction 500 – 204 using visual aids.

Step 1: Represent 500 with Blocks

Imagine 500 as a combination of blocks:

5 hundreds blocks (each representing 100).
0 tens blocks.
0 ones blocks.

Visual representation:

Hundreds: 🟩 🟩 🟩 🟩 🟩
Tens:     
Ones:

Step 2: Start with the Ones Place

We need to subtract 4 (from 204) from 0 (in 500). But since we don’t have any blocks in the ones place, we need to borrow.

Step 3: Borrow from the Hundreds Place

Since there are no tens blocks to borrow from, we borrow from the hundreds place:

Convert one of the hundreds blocks into 10 tens blocks.
Then, borrow 1 ten block and convert it into 10 ones blocks.

Updated representation:

Hundreds: 🟩 🟩 🟩 🟩
Tens:     🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩
Ones:     🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩

Step 4: Subtract the Ones

Now, subtract 4 ones blocks from the 10 ones blocks:

Hundreds: 🟩 🟩 🟩 🟩
Tens:     🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩
Ones:     🟩 🟩 🟩 🟩 🟩 🟩

10 – 4 = 6 ones blocks left.

Step 5: Subtract the Tens

Next, subtract 0 tens blocks from the 9 tens blocks. Since subtracting 0 doesn’t change the number, we still have:

Hundreds: 🟩 🟩 🟩 🟩
Tens:     🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩
Ones:     🟩 🟩 🟩 🟩 🟩 🟩

9 tens blocks remain.

Step 6: Subtract the Hundreds

Finally, subtract 2 hundreds blocks from the 4 hundreds blocks:

Hundreds: 🟩 🟩
Tens:     🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩 🟩
Ones:     🟩 🟩 🟩 🟩 🟩 🟩

2 hundreds blocks remain.

Final Answer: 296

So, the visual subtraction shows that:

    5 0 0
  - 2 0 4
  --------
    2 9 6

Example: Subtracting 207 from 500

Step 1: Initial Representation of 500

Hundreds: 🟩🟩🟩🟩🟩 (5 hundred blocks)
Tens: None (0 tens blocks)
Ones: None (0 ones blocks)

Step 2: Borrowing Process

Borrow from Hundreds: Take 1 hundred block from the 5 and convert it into 10 tens blocks.
- New Representation:
  - Hundreds: 🟩🟩🟩🟩 (4 hundred blocks)
  - Tens: 🟨🟨🟨🟨🟨🟨🟨🟨🟨🟨 (10 tens blocks)
  - Ones: None (0 ones blocks)
Borrow from Tens: Take 1 ten block from the 10 and convert it into 10 ones blocks.
- New Representation:
  - Hundreds: 🟩🟩🟩🟩 (4 hundred blocks)
  - Tens: 🟨🟨🟨🟨🟨🟨🟨🟨🟨 (9 tens blocks)
  - Ones: 🟦🟦🟦🟦🟦🟦🟦🟦🟦🟦 (10 ones blocks)

Step 3: Subtraction Process

Subtract 7 from Ones: Remove 7 ones blocks.
- Result: 🟦🟦🟦 (3 ones blocks)
Subtract 0 from Tens: No change in tens blocks.
- Result: 🟨🟨🟨🟨🟨🟨🟨🟨🟨 (9 tens blocks)
Subtract 2 from Hundreds: Remove 2 hundred blocks.
- Result: 🟩🟩 (2 hundred blocks)

Final Answer: 293

Final Representation:

Hundreds: 🟩🟩
Tens: 🟨🟨🟨🟨🟨🟨🟨🟨🟨
Ones: 🟦🟦🟦

This gives us a final result of 293.

Summary

The blocks help visualize how regrouping works:

Hundreds blocks are reduced and converted into tens blocks when needed.
Tens blocks are reduced and converted into ones blocks when necessary.
The subtraction is performed step-by-step, removing blocks according to the numbers in the subtrahend (207 in this case).

Visual Summary

Hundreds blocks (🟩) represent hundreds place values.
Tens blocks (🟩) represent tens place values.
Ones blocks (🟩) represent ones place values.
Borrowing involves converting one block into 10 of the next lower place value.

This visual approach helps to clearly see where the numbers are coming from and how regrouping works.

Let’s go through this with an example:

Example 1: Subtracting 302 from 500

We are subtracting 302 from 500.

    5 0 0
  - 3 0 2
  --------

Step 1: Start from the Rightmost Digit (Ones Place)

Ones Place: We have 0 (in 500) and 2 (in 302).
- Since 0 is smaller than 2, we need to borrow 1 from the tens place.

Step 2: Move to the Tens Place

Tens Place: We now look at the tens place in 500. It’s a 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 5. We reduce this by 1 (so it becomes 4), and the tens place becomes 10.
Now, go back to the tens place:
- Tens Place: The 10 we borrowed from the hundreds place makes this a 9, and now we borrow 1 from it to add to the ones place.
- Ones Place: The 0 becomes 10.

Step 4: Perform the Subtraction

Now, the subtraction looks like this:

    4 9 10  (The 10 in the ones place and 9 in the tens place after borrowing)
  - 3 0  2
  --------

Ones Place: 10 – 2 = 8
Tens Place: 9 – 0 = 9
Hundreds Place: 4 – 3 = 1

So, the final result is:

    1 9 8

Example 2: Subtracting 207 from 600

We are subtracting 207 from 600.

    6 0 0
  - 2 0 7
  --------

Step 1: Start from the Rightmost Digit (Ones Place)

Ones Place: We have 0 (in 600) and 7 (in 207).
- Since 0 is smaller than 7, we need to borrow 1 from the tens place.

Step 2: Move to the Tens Place

Tens Place: We now look at the tens place in 600. It’s a 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 6. We reduce this by 1 (so it becomes 5), and the tens place becomes 10.
Now, go back to the tens place:
- Tens Place: The 10 we borrowed from the hundreds place makes this a 9, and now we borrow 1 from it to add to the ones place.
- Ones Place: The 0 becomes 10.

Step 4: Perform the Subtraction

Now, the subtraction looks like this:

    5 9 10  (The 10 in the ones place and 9 in the tens place after borrowing)
  - 2 0 7
  --------

Ones Place: 10 – 7 = 3
Tens Place: 9 – 0 = 9
Hundreds Place: 5 – 2 = 3

So, the final result is:

    3 9 3

Example 3: Subtracting 408 from 700

    7 0 0
  - 4 0 8
  --------

Step 1: Start from the Ones Place

Ones Place: We have 0 (in 700) and 8 (in 408).
- Since 0 is smaller than 8, we need to borrow from the tens place.

Step 2: Move to the Tens Place

Tens Place: The digit here is 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 7. We reduce this by 1 (so it becomes 6), and the tens place becomes 10.
Now, go back to the tens place:
- Tens Place: The 10 we borrowed from the hundreds place makes this a 9, and now we borrow 1 from it to add to the ones place.
- Ones Place: The 0 becomes 10.

Step 4: Perform the Subtraction

Now, the subtraction looks like this:

    6 9 10
  - 4 0 8
  --------

Ones Place: 10 – 8 = 2
Tens Place: 9 – 0 = 9
Hundreds Place: 6 – 4 = 2

So, the final result is:

    2 9 2

Example 4: Subtracting 155 from 300

    3 0 0
  - 1 5 5
  --------

Step 1: Start from the Ones Place

Ones Place: We have 0 (in 300) and 5 (in 155).
- Since 0 is smaller than 5, we need to borrow from the tens place.

Step 2: Move to the Tens Place

Tens Place: The digit here is 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 3. We reduce this by 1 (so it becomes 2), and the tens place becomes 10.
Now, go back to the tens place:
- Tens Place: The 10 we borrowed from the hundreds place makes this a 9, and now we borrow 1 from it to add to the ones place.
- Ones Place: The 0 becomes 10.

Step 4: Perform the Subtraction

Now, the subtraction looks like this:

    2 9 10
  - 1 5 5
  --------

Ones Place: 10 – 5 = 5
Tens Place: 9 – 5 = 4
Hundreds Place: 2 – 1 = 1

So, the final result is:

    1 4 5

Example 5: Subtracting 127 from 500

    5 0 0
  - 1 2 7
  --------

Step 1: Start from the Ones Place

Ones Place: We have 0 (in 500) and 7 (in 127).
- Since 0 is smaller than 7, we need to borrow from the tens place.

Step 2: Move to the Tens Place

Tens Place: The digit here is 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 5. We reduce this by 1 (so it becomes 4), and the tens place becomes 10.
Now, go back to the tens place:
- Tens Place: The 10 we borrowed from the hundreds place makes this a 9, and now we borrow 1 from it to add to the ones place.
- Ones Place: The 0 becomes 10.

Step 4: Perform the Subtraction

Now, the subtraction looks like this:

    4 9 10
  - 1 2 7
  --------

Ones Place: 10 – 7 = 3
Tens Place: 9 – 2 = 7
Hundreds Place: 4 – 1 = 3

So, the final result is:

    3 7 3

Example 6: Subtracting 206 from 406

    4 0 6
  - 2 0 6
  --------

Step 1: Start from the Ones Place

Ones Place: 6 (in 406) – 6 (in 206) = 0

Step 2: Move to the Tens Place

Tens Place: 0 (in 406) – 0 (in 206) = 0

Step 3: Move to the Hundreds Place

Hundreds Place: 4 (in 406) – 2 (in 206) = 2

So, the final result is:

    2 0 0

Example 1: Subtracting 358 from 602

   6 0 2
 - 3 5 8
 --------

Step 1: Start with the Ones Place

Ones Place: 2 (in 602) – 8 (in 358).
- Since 2 is smaller than 8, we need to borrow from the tens place.

Step 2: Move to the Tens Place

Tens Place: The digit here is 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 6. We reduce this by 1 (so it becomes 5), and the tens place becomes 10.
Go back to the tens place:
- Borrow 1 from the 10 in the tens place to add to the ones place.
- Ones Place: The 2 becomes 12.

Step 4: Perform the Subtraction

   5 9 12
 - 3 5 8
 --------

Ones Place: 12 – 8 = 4
Tens Place: 9 – 5 = 4
Hundreds Place: 5 – 3 = 2

So, the final result is:

   2 4 4

Example 2: Subtracting 134 from 600

   6 0 0
 - 1 3 4
 --------

Step 1: Start with the Ones Place

Ones Place: 0 (in 600) – 4 (in 134).
- Since 0 is smaller than 4, we need to borrow from the tens place.

Step 2: Move to the Tens Place

Tens Place: The digit here is 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 6. We reduce this by 1 (so it becomes 5), and the tens place becomes 10.
Go back to the tens place:
- Borrow 1 from the 10 in the tens place to add to the ones place.
- Ones Place: The 0 becomes 10.

Step 4: Perform the Subtraction

   5 9 10
 - 1 3  4
 --------

Ones Place: 10 – 4 = 6
Tens Place: 9 – 3 = 6
Hundreds Place: 5 – 1 = 4

So, the final result is:

   4 6 6

Example 3: Subtracting 459 from 700

   7 0 0
 - 4 5 9
 --------

Step 1: Start with the Ones Place

Ones Place: 0 (in 700) – 9 (in 459).
- Since 0 is smaller than 9, we need to borrow from the tens place.

Step 2: Move to the Tens Place

Tens Place: The digit here is 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 7. We reduce this by 1 (so it becomes 6), and the tens place becomes 10.
Go back to the tens place:
- Borrow 1 from the 10 in the tens place to add to the ones place.
- Ones Place: The 0 becomes 10.

Step 4: Perform the Subtraction

   6 9 10
 - 4 5  9
 --------

Ones Place: 10 – 9 = 1
Tens Place: 9 – 5 = 4
Hundreds Place: 6 – 4 = 2

So, the final result is:

   2 4 1

Example 4: Subtracting 381 from 800

   8 0 0
 - 3 8 1
 --------

Step 1: Start with the Ones Place

Ones Place: 0 (in 800) – 1 (in 381).
- Since 0 is smaller than 1, we need to borrow from the tens place.

Step 2: Move to the Tens Place

Tens Place: The digit here is 0.
- We can’t borrow from a 0, so we move to the hundreds place.

Step 3: Borrow from the Hundreds Place

Hundreds Place: The digit here is 8. We reduce this by 1 (so it becomes 7), and the tens place becomes 10.
Go back to the tens place:
- Borrow 1 from the 10 in the tens place to add to the ones place.
- Ones Place: The 0 becomes 10.

Step 4: Perform the Subtraction

   7 9 10
 - 3 8  1
 --------

Ones Place: 10 – 1 = 9
Tens Place: 9 – 8 = 1
Hundreds Place: 7 – 3 = 4

So, the final result is:

   4 1 9

This visual method is effective for helping learners understand the concept of borrowing and subtraction with zeros.

Summary

In each of these examples, the key steps are:

Start subtracting from the rightmost digit (ones place).
If the digit in the top number is smaller, borrow from the next higher place value.
Adjust the numbers accordingly and subtract.

Practice these steps with different numbers to become more comfortable with subtraction, especially when dealing with zeros.

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