Vertical Subtraction: Borrowing across zeros

To solve this problem, we'll follow these steps:

• Step 1: Write the numbers vertically aligning by their respective place values:
  $\begin{array}{c} 1000 \\ - \phantom{0}346~~~ \\ \hline \\ \end{array}$

• Step 2: Start subtracting from the units column: - Units column: $0 - 6$, since 0 < 6, we need to borrow. Notice that all digits to the left are zeros, so multiple steps of borrowing are required.

• Step 3: Borrowing Process: - First, borrow from the hundreds place: Borrow 1, making the hundreds column $9$ (since $10 - 1 = 9$), leaving the ten column as $9$ (from borrowing). - Borrow again to the tens column, reducing hundreds column further, ultimately making tens column $9$ and units column $10$.

• Step 4: Perform the subtractions: - Units Column: $10 - 6 = 4$ - Tens Column: $9 - 4 = 5$ - Hundreds Column: $9 - 3 = 6$ - Thousands Column: Remains $0$.

Therefore, putting it all together, we get: $\begin{array}{c} 1000 \\ - \phantom{0}346 ~~~ \\ \hline \phantom{0}654 \\ \end{array}$

Thus, the solution to the problem is $654$.

To solve this problem, we'll follow these steps:

• Step 1: Set up the vertical subtraction with 1000 above 951.
• Step 2: Start subtracting from the rightmost digit.
• Step 3: Borrow as needed across zeros.
• Step 4: Perform the subtraction of each digit, ensuring proper arithmetic adjustment from borrowing.

Now, let's work through each step:
Step 1: We write $1000$ and $951$ in vertical alignment for subtraction.
Step 2: Begin from the units column: We need to subtract $1$ from $0$, so borrowing is required.
Step 3: As there are zeros in both the units and the tens place, we must borrow from the hundreds place and subsequently from the thousands place.
Step 4: Change $1000$ as follows for borrowing: The thousands digit gives 1 to the hundreds, transforming so it looks like $10$ at hundreds, zero at the thousands, from which the hundreds gives 1 to the tens, and similarly tens gives 1 to the units.
The number $1000$ effectively becomes $0990$ temporarily while processing subtraction:
- Now units column: $10 - 1 = 9$.
- In the tens column: $9 - 5 = 4$.
- In the hundreds column: $9 - 9 = 0$.
- Thousands column: $0 - 0 = 0$.

Therefore, the solution to the subtraction problem of $1000 - 951$ is $49$.

To solve this problem, we will subtract 659 from 3008 using borrowing in the subtraction process:

1. Line up the numbers with the larger number on top: $\begin{array}{c} \phantom{2}3008 \\ -659 \\ \end{array}$

2. Begin subtraction from the rightmost column (units place), where we have $8 - 9$. Since $8$ is smaller than $9$, we need to borrow.

3. Move to the tens place, which is also $0$. Since we cannot borrow from a $0$, move left to the hundreds place.

4. The hundreds place is also $0$, so continue to the thousands column, where we have $3$:

5. Decrease the $3$ to $2$ and make the hundreds column $10$. Now borrow from the hundreds column, turning it into $9$ and the tens column into $10$.

6. Finally, borrow $1$ from the tens column to assist the units column: $\begin{array}{c} {2939} \\ -~~~659 \\ \end{array}$ We can now do $18 - 9 = 9$.

7. Proceed to the tens column. $9 - 5 = 4$.

8. The hundreds column: $9 - 6 = 3$.

9. The thousands column: $2 - 0 = 2$.

Writing down the final subtraction gives us:

$\begin{array}{c} \phantom{2939}3008 \\ -~~~~~~~659 \\ \overline{\phantom{2939}2349} \\ \end{array}$

Therefore, the solution to the problem is 2349.

To solve the subtraction problem $300 - 152$, we'll follow these steps:

• Step 1: Align the numbers vertically, positioning 300 over 152.
• Step 2: Start from the rightmost column (units place):
• The units column has $0 - 2$, which requires borrowing since 0 is less than 2.
• Go to the tens column. It contains 0, so borrow from the hundreds column.
• Reduce the hundreds place from 3 to 2, changing the tens 0 into 10 (due to borrowing).
• Step 3: Return to the units place:
• Now, the units place consists of $10 - 2$, which equals $8$.
• Step 4: Move to the tens column:
• With the tens now holding $10$, perform $10 - 5$, resulting in $5$.
• Step 5: Complete the hundreds column:
• Perform $2 - 1$, resulting in $1$.
• Step 6: Combine the results from each column to get the final answer: 148.

Therefore, the solution to the problem is $148$.

To solve this subtraction problem, we must perform vertical subtraction of 89 from 308 using the borrowing technique:

• Step 1: Write the numbers in vertical format, ensuring that each digit is in the correct place value column:

$\begin{array}{c} & 3 & 0 & 8 \\ - & & 8 & 9 \\ \hline \end{array}$

• Step 2: Start with the rightmost column (units). We need to subtract 9 from 8, but since 8 is smaller than 9, borrow 1 from the tens column.
• With borrowing, the tens digit (0) decreases to 9 after borrowing 1 from the hundreds column.
• This gives us: 2 in the hundreds, 9 in the tens, and 18 in the units column.
• Step 3: Proceed with subtraction:
• Units: Subtract 9 from 18 to get 9.
• Tens: Subtract 8 from 9 to get 1.
• Hundreds: As there is no digit to subtract from 2 in the hundreds place, it remains 2.
• Step 4: Write down the resulting digits in their respective place values.

$\begin{array}{c} & 2 & 9 & 18 \\ - & & 8 & 9 \\ \hline & 2 & 1 & 9 \\ \end{array}$

Therefore, the difference between 308 and 89 is $219$.

The correct choice from the provided answer choices is choice 1: 219.

To solve the subtraction problem $4003 - 855$, we will perform the operation step-by-step, ensuring we manage the borrowing process correctly:

• Step 1: Align the numbers for subtraction:
  $4003$
  $- 855$
  It's important to start from the rightmost digit.
• Step 2: Subtract the digits in the units place:
  $3 - 5$. Since 3 is less than 5, we need to borrow. There is nothing in the tens place to borrow, so we move to the hundreds.
  Borrow 1 from the hundreds place (turning the 0 into a 9 in the tens place), and reborrow 1 from there. So, the new setup is:
  $3$ becomes $13$ (in units place), $0$ becomes $9$ (in tens place), and reduce $0$ to $9$ in hundreds place.
• Step 3: Perform the subtraction in the units place now:
  $13 - 5 = 8$, so write down 8.
• Step 4: Move to the tens place:
  $9 - 5 = 4$, write down 4.
• Step 5: Move to the hundreds place:
  $9 - 8 = 1$, write down 1.
• Step 6: Finally, address the thousands place:
  $3 - 0 = 3$, write down 3.

The result of $4003 - 855$ is $3148$.

The correct choice that matches this result is option 3: $3148$.

Let's solve the subtraction problem $404 - 106$ step-by-step:

• Align the numbers vertically by their place values:
• $\begin{array}{c} \phantom{0}404 \\ -106 \\ \end{array}$
• Subtract starting from the rightmost column (ones):

The ones column: $4 - 6$ requires borrowing.

• We need to borrow from the tens column. However, the tens column in 404 has a zero, so we proceed to the hundreds column.
• Borrow 1 from the hundreds column, turning 4 into 3, and the tens zero into 10.
• Then, borrow again from the new tens column (10), turning it into 9 and adding 10 to the ones, making it 14.
• Subtract: $14 - 6 = 8$.
• Subtract the tens column:

Now, subtract the tens column: $9 - 0 = 9$.

• Subtract the hundreds column:

Finally, for the hundreds column: $3 - 1 = 2$.

So, putting it all together, the result of subtracting $106$ from $404$ is $298$.

Therefore, the solution to the subtraction problem is $298$.

To solve this subtraction problem $406 - 78$, we use the vertical subtraction method, aligning the numbers by their place values:

Step-by-step process:

• Align the numbers columnwise:
      406
  -    78
  - - - - -

• Subtract each column, starting from the right:
  - Units place: $6 - 8$ cannot be done directly, so we need to borrow.
  - Look to the tens column (0), but it is not possible to borrow directly from zero. Instead, we move to the hundreds column.
  - From the hundreds, the 4 becomes a 3 (after borrowing), and the tens place (0) becomes 10. Borrow from 10 in tens, so it becomes 9 while units becomes 16.

• Perform the subtraction with borrowing adjusted:
  - Units place: $16 - 8 = 8$
  - Tens place: $9 - 7 = 2$
  - Hundreds place: $3 - 0 = 3$

The result of subtracting 78 from 406 is $\mathbf{328}$.

The correct answer, matching our calculation, is option 3: 328.

We need to subtract 937 from 5004 using vertical subtraction. Let’s break this down step-by-step, especially focusing on borrowing across zeros:

Step 1: Set up the numbers for subtraction.
Write 5004 above 937, aligning them to the right so the units, tens, hundreds, and thousands digits line up:

$\begin{array}{c} & 5 & 0 & 0 & 4 \\ - & & 9 & 3 & 7 \\ \hline & & & & \\ \end{array}$

Step 2: Start subtracting from the rightmost digits.
Begin with the units column:

• Units column: Subtract 7 from 4. Since 4 is less than 7, we need to borrow.
• Borrow 1 from the tens column. However, tens column has a 0, so we must go left of these zeros:
• Hundreds column has 0, so borrow 1 from the hundreds column to make it 10, then the hundreds are reduced to 9. The tens will become 10 (and then get reduced to 9 when passing), and when borrowing:
• Units column: Now, 14 – 7 = 7.

Our subtraction may look as follows:

$\begin{array}{c} & 4 & 9 & 9 & 14 \\ - & & 9 & 3 & 7 \\ \hline & & & & 7 \\ \end{array}$

Step 3: Move to the tens column.
Subtract 3 from 9. Result is 6.

$\begin{array}{c} & 4 & 9 & 9 & 14 \\ - & & 9 & 3 & 7 \\ \hline & & & 6 & 7 \\ \end{array}$

Step 4: Move to the hundreds column.
Subtract 9 from 9. Result is 0.

$\begin{array}{c} & 4 & 9 & 9 & 14 \\ - & & 9 & 3 & 7 \\ \hline & & 0 & 6 & 7 \\ \end{array}$

Step 5: Move to the thousands column.
Here we have 4, and there’s no number to subtract, so we carry down 4.

$\begin{array}{c} & 4 & 9 & 9 & 14 \\ - & & 9 & 3 & 7 \\ \hline & 4 & 0 & 6 & 7 \\ \end{array}$

Therefore, the solution to the subtraction problem is $4067$.

To solve this problem of subtracting $367$ from $505$, we'll follow these steps:

• Step 1: Start at the ones place. Subtract $7$ from $5$. Since $5$ is smaller than $7$, we need to borrow from the tens place.
• Step 2: Move to the tens place, before borrowing. There is $0$ in the tens place, so we cannot borrow directly. We must borrow from the hundreds place.
• Step 3: Borrow $1$ from the hundreds place, reducing it to $4$. The tens place becomes $10$.
• Step 4: Now borrow $1$ from the $10$ in the tens place, making it $9$. The $5$ in the ones place becomes $15$.
• Step 5: Now subtract $7$ from $15$, which gives us $8$.
• Step 6: In the tens place, subtract $6$ from $9$, resulting in $3$.
• Step 7: In the hundreds place, subtract $3$ from $4$, resulting in $1$.

Putting it all together, we have:

$505 - 367 = 138$

Therefore, the solution to the problem is $\textbf{138}$, which corresponds to choice 2.

To solve this problem, we'll follow these steps:

• Step 1: Align the two numbers for vertical subtraction
• Step 2: Subtract digit by digit, starting from the right
• Step 3: Borrow when necessary, particularly across zeros

Now, let's work through each step:
Step 1: Write the numbers vertically, aligning by place values:
$\begin{array}{c} 507 \\ - 69 \\ \hline \end{array}$

Step 2: Start subtracting from the units (right-most) digit:
Units column: 7 (from 507) minus 9 (from 69). Since 7 is smaller than 9, borrow 1 from the tens column.

Step 3: Borrowing across zeros:
Borrowing from the tens column requires rolling up to the hundreds column because there is a 0 in the tens place. Change 5 in the hundreds place to 4, making the tens place 10. Now the tens column has 9 after lending 1 to the units column, making it 17.
Subtract 9 from 17 in the units place to get 8.

Step 4: Subtract the tens column:
9 (from borrowing adjustment) minus 6 (from 69) is 3.

Step 5: Subtract the hundreds column:
4 (from borrowing adjustment) minus 0 (from 69) is 4.

Therefore, the subtraction result is:
$\begin{array}{c} 438 \\ \end{array}$

Thus, the solution to the problem is $438$.

Let's solve $6002 - 373$ using the vertical subtraction method:

• Step 1: Write the numbers in column form, ensuring they are aligned by place value: $\begin{array}{c} 6002 \\ -~~373 \\ \hline \end{array}$

• Step 2: Subtract each digit starting from the right (units place). First, inspect the units place:

  • We need to subtract 3 from 2. Since 3 > 2, we must borrow.

  • Borrow 1 from 600 (next higher order, thousands in this case):

    • The result: Change 6002 to 5992.

  • Now, subtract $12 - 3 = 9$, in the units place.

  $\begin{array}{c} 5992 \\ -~~373~~~ \\ \hline ~~~~~~9 \\ \end{array}$

• Step 3: Move to the tens place:

  • Subtract $9 - 7 = 2$.

  $\begin{array}{c} 5992 \\ -~~373~~~ \\ \hline ~~~29 \\ \end{array}$

• Step 4: Move to the hundreds place:

  • Subtract $9 - 3 = 6$.

  $\begin{array}{c} 5992 \\ -~~373~~~~ \\ \hline ~629 \\ \end{array}$

• Step 5: Move to the thousands place:

  • We have 5 remaining and no subtraction needed here.

  • Therefore, the result is $5629$.

  $\begin{array}{c} 5992 \\ -~~373~~~ \\ \hline 5629 \\ \end{array}$

Therefore, the solution to $6002 - 373$ is $5629$.

To solve this subtraction problem, we need to subtract $247$ from $602$. We'll approach it step-by-step:

• Begin with the rightmost digits: Subtract $7$ from $2$. Since $2$ is smaller than $7$, we must borrow.

• Borrow from the tens place: Reduce the next digit left (0 in the tens place) by 1. However, since it's $0$, continue borrowing from the hundreds place.

• The hundreds place $6$ reduces to $5$, making the tens $10$. Borrow $1$ from these $10$, making the tens $9$ and increasing the units place to 12.

• Now, subtract: $12 - 7 = 5$.

• For the tens place, subtract $4$ from $9$ (after adjustment): $9 - 4 = 5$.

• Finally, the hundreds place: Subtract $2$ from $5$, giving $5 - 2 = 3$.

The subtraction gives us the number $355$.

To solve this vertical subtraction problem, we'll follow these steps:

• Step 1: Align the numbers vertically, ensuring that each column corresponds to the correct place value.

• Step 2: Begin subtracting from the rightmost digit (units column) and move left, borrowing where necessary.

• Step 3: If a zero is present in a column and borrowing is needed, borrow from the next higher place value column.

Let's apply these steps using the numbers $700$ and $145$:

Step 1: Align the numbers:
$\phantom{0}700$
-$\phantom{0}145$
- - - - - -

Step 2: Begin with the units column (rightmost):
0 - 5.
We cannot subtract 5 from 0, so we need to borrow from the tens column.
But, the tens column is also 0, so we need to borrow from the hundreds column.

Step 3: Borrow 1 from the hundreds place (making it 6) and convert it to 10 tens in the tens column (makes the tens column 10).
Now borrow 1 ten (making the tens column 9) to turn the units column into 10.

Now solve:
$\begin{array}{c} &amp;\phantom{0}6\phantom{0}&amp;9&amp;10 \\ - &amp; 1 &amp; 4 &amp; 5 \\ \end{array}$
Units column: 10 - 5 = 5
Tens column: 9 - 4 = 5
Hundreds column: 6 - 1 = 5

Therefore:

$\begin{array}{c} \phantom{0}700 \\ -\phantom{0}145~~~ \\ ~~\overline{555} \\ \end{array}$

The correct answer to the problem is $555$.

To solve this subtraction problem of 1000 minus 257, we will use a systematic vertical subtraction method that involves borrowing:

• Step 1: Start with the numbers aligned vertically:

$\begin{aligned} &amp;1000 \\ -&amp; \\ &amp;~~257 \\ \end{aligned}$

• Step 2: Subtract the rightmost digits (in the units place):

• Since 0 is less than 7, we need to borrow. The next column (tens place) also is 0, so we move to the hundreds place to borrow. Reducing the 1 to 0, we turn the 0 into a 10 in the tens column and then subtract 1 to borrow into the units column, turning it into 9. The soon-to-be 10 in the units column becomes 10.

• Step 3: The positions now look like:

$\begin{aligned} &amp;\phantom{10\ \ }~9~10 \\ &amp;~1~0~0~0 \\ - &amp;~~~~2~5~7 \\ \underline{\phantom{70}} \\ \end{aligned}$

• Step 4: Now, 10 minus 7 is 3. Write 3 down.

• Step 5: Move to the tens column: 9 minus 5 equals 4. Write 4 under the line.

• Step 6: Lastly, subtract the hundreds column: 0 minus 2 requires borrowing again, leaving that position as 7.

• The final calculated result is:

  $\begin{aligned} &amp;1000 \\ -&amp;~~257 \\ &amp;~\overline{~743} \\ \end{aligned}$

  Therefore, the solution to the subtraction problem is $743$.

To solve this problem, we will use vertical subtraction with borrowing:

• Step 1: Understand the given numbers
  We need to perform the subtraction $1000 - 872$. Let's write them vertically:

1000
-872
_____

• Step 2: Begin with the units digit
  The units digit for the minuend $1000$ is 0, and the subtrahend $872$ is 2. Because you cannot subtract $2$ from $0$, we need to borrow from the next digit over.

  • The tens digit is also $0$, so we need to borrow from the hundreds digit, $0$.

  • Continue borrowing to the thousands place:

  • The thousands digit $1$ will become $0$, the hundreds digit becomes $10$, then $9$ (after lending $1$ to adjust tens place), tens digit becomes $10$, then $9$ (after lending $1$ to adjust units place), and finally, units digit becomes $10$.

  • Result during subtraction step:

  • The number adjusted pre-subtraction is $09910$.

• Step 3: Perform the subtraction
  

  • Now perform each column subtraction:

  • Units: $10 - 2 = 8$

  • Tens: $9 - 7 = 2$

  • Hundreds: $9 - 8 = 1$

  • There is no digit left to subtract in the thousands place, following the adjustment.

• Conclusion
  Bringing the results together from the adjustments and borrowing:

  • The result of the subtraction is $128$.

Therefore, the solution to the problem is $128$.

To solve the problem $100 - 18$, we'll perform the subtraction using borrowing:

• Step 1: Arrange the numbers vertically:
  $\begin{aligned} & 100 \\ - & \\ & ~~18 \\ & \underline{\phantom{776}} \end{aligned}$
• Step 2: Start subtracting from the rightmost digit (the units place). Subtract 8 from 0. Since 0 is smaller than 8, we need to borrow.
• Step 3: Borrow from the tens place. The tens place is also 0, so we must continue to the hundreds place. Borrow 1 from the hundreds place, turning it from 1 to 0, and making the tens place 10.
• Step 4: Now we have 10 in the tens place. Borrow 1 from this new tens digit, turning it from 10 to 9, and giving the units place 10.
• Step 5: Subtract 8 from 10 in the units place. This gives us 2.
• Step 6: Move to the tens place. Subtract 1 from 9, which gives us 8.
• Step 7: As there's nothing left to subtract in the hundreds place, we are left with 0.
• Step 8: Combine the results to form the final answer: 82.

Therefore, the solution to the subtraction problem is $82$.

To solve this problem, we'll follow these steps:

• Step 1: Write the numbers vertically.

• Step 2: Start subtraction from the rightmost digit.

• Step 3: Borrow where necessary.

Now, let's work through each step:

Step 1: Set up the subtraction vertically:
      $2007$
   -   469

Step 2: Subtract starting from the rightmost digit:

• In the one's place, we have $7 - 9$. Since $7$ is smaller than $9$, we need to borrow. However, $0$ has no value to lend, so we need to also borrow from the hundreds place.

• In the tens place, $0$ becomes $10$ (after borrowing $1$ from the hundreds), but now we have to borrow $1$ for the ones place operation, leaving us with $9$ in the tens place. Thus, the ones place calculation becomes $17 - 9 = 8$.

• In the tens place, we subtract $6$ from the adjusted $9$ (due to earlier borrowing). Thus, $9 - 6 = 3$.

• In the hundreds place, initially $0$, now we have $9$ (after borrowing from the thousands, so it effectively becomes $9$). Thus, $9 - 4 = 5$.

• Finally, the thousands place remains $1$ as we have already borrowed $1$ away, and we have no number to subtract from it. So, it remains $1$.

The complete subtraction sequence gives us:

    2007
-   469
   $\overline{1538}$

Therefore, the solution to the problem is $1538$.