Vertical Subtraction

🏆Practice vertical subtraction

Vertical Subtraction

In order to solve vertical subtraction, we follow these rules:
First rule - write the problem in the correct order!
Ones digits under ones digits, tens digits under tens digits, and so on.
Second rule - when the upper digit is smaller than the lower digit - we borrow 11 from the next digit.
Third rule - a 00 that cannot be borrowed from becomes 99 until we reach a digit that is not 00 from which we can borrow 11.
Note! If there is a third 00 right after, it will become 88, if there is a fourth 00 right after, it will become 77, and so on.

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\( \begin{aligned} &15 \\ -& \\ &~~4 \\ &\underline{\phantom{776}} & \\ \end{aligned} \)

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Vertical Subtraction

What is vertical subtraction?

Vertical subtraction is a way of writing a subtraction problem where the second number is written below the first number vertically and in the correct order - ones under ones, tens under tens, and so on.

Why do we need vertical subtraction?

Sometimes you'll encounter relatively complex subtraction exercises that look like this: 431278=431-278=
By writing them vertically we can easily solve them.

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How do you solve vertical subtraction?

The First Rule - writing the problem in the correct order!

ones digits under ones digits, tens digits under tens digits, hundreds digits under hundreds digits, and thousands digits under thousands digits.
Pay attention! It is extremely important that the first number in the exercise is the first and top number.
For example: 8754=87-54=
We will write it as follows:

A vertical subtraction problem showing the number 87 on top and 54 below it, separated by a minus sign (-). A horizontal line divides the two numbers, awaiting the result to be calculated. The Tutorela logo is displayed at the bottom of the image.

Write the (minus) – sign in order to indicate that this is a subtraction exercise.
Draw a line underneath to separate the exercise from the results line.
We'll start by subtracting the ones digits as follows:

74=37-4=3

A vertical subtraction problem showing the calculation: 87 minus 54. The result, 3, is written below the horizontal line. The Tutorela logo is displayed at the bottom of the image.

Let's continue to subtract the tens digits to obtain the following :
85=38-5=3

A vertical subtraction problem showing the calculation: 87 minus 54. The result, 33, is written below the horizontal line. The Tutorela logo is displayed at the bottom of the image.

We're done! The result is 3333.
Now let's learn the next rule using the following example:

The Second Rule -

When the upper digit is smaller than the lower digit - we borrow 11 from the next digit.


Here's a more advanced exercise!
4529=45-29=

Solution:

A vertical subtraction problem showing the calculation: 45 minus 29. The result is not displayed yet. The Tutorela logo is located at the bottom of the image.

Given that we cannot subtract 55 minus 99 we need to borrow a digit from the next number!
That means:
55 will become 1515 given that we'll place one in front of it and 44 will become 33.

We will write it in the following way:

A vertical subtraction problem showing the calculation: 45 minus 29. Borrowing is indicated, with 4 rewritten as 3 and 5 rewritten as 15 to facilitate subtraction. The Tutorela logo is located at the bottom of the image.

Now we can proceed to solve the problem:
159=615-9=6
32=13-2=1
As seen below:

A vertical subtraction problem showing the calculation: 45 minus 29. Borrowing is indicated, with 4 rewritten as 3 and 5 rewritten as 15. The result of the subtraction is 16. The Tutorela logo is located at the bottom of the image.

The result is 1616!

What do we do when we need to subtract a number from the digit 00? For example in the exercise
4029=40-29=
Here too we'll need to borrow 11 from the digit 44 resulting in the exercise seen below.

A vertical subtraction exercise showing 40 minus 29. Borrowing is demonstrated, with the "4" in the tens place crossed out and replaced with "3," and the "0" in the ones place crossed out and replaced with "10."

We can proceed to solve the problem :
109=110-9=1
32=13-2=1
As seen below:

A vertical subtraction exercise showing 40 minus 29. Borrowing is demonstrated, with the "4" in the tens place crossed out and replaced with "3," and the "0" in the ones place crossed out and replaced with "10." The result of the subtraction, 11, is written at the bottom.


We're done! The result is 1111.
Now let's see what happens when we can't borrow from the next digit given that it's also 00:
For example in the following exercise:
500365=500-365=

The third rule -00 that cannot be borrowed from becomes 99 until we reach a digit that is not 00 from which we can borrow 11.


Note! If there is a third 00 immediately after, it becomes 88, if there is a fourth 00 immediately after, it becomes 77 and so on..

Let's learn the following rule through an example:

A vertical subtraction exercise showing 500 minus 365. The numbers are aligned vertically by place value, with a line separating them, but no calculation steps or result are shown.

We want to borrow 11 for the first 00 making it 1010.
The second 00 will be 99 because we can't borrow from it
and the digit 55 will become 44 given that we borrowed one from it.
As seen below:

A vertical subtraction exercise showing 500 minus 365. The borrowing steps are highlighted: a 1 is borrowed from the hundreds place, resulting in 4, 9, and 10 written above the digits in the top number. Diagonal lines indicate the borrowing process for the calculation.

We can proceed to solve the exercise:
105=510-5=5
96=39-6=3
43=14-3=1
Let's write the solution as follows:

A vertical subtraction exercise showing 500 minus 365 with borrowing. The borrowing steps are indicated above the digits: 4 in the hundreds place, 9 in the tens place, and 10 in the ones place. The result of the subtraction is 135, displayed below the subtraction line.

We're done! The result is 135135
Now let's move on to a very advanced exercise!

Let's solve this problem together –
57003786=5700-3786=

Solution:
Let's write it correctly:

A vertical subtraction exercise showing 5700 minus 3786. Borrowing or calculation steps are not displayed yet.

Let's begin to solve the problem:
00 The first will become 1010 since we borrow 11.
00 The second will become 99 because we can't borrow from it and the 77 will become 66 because we borrowed 11 from it.
As seen below:

A vertical subtraction exercise showing 5700 minus 3786. Borrowing steps are annotated, with 6, 9, and 10 marked above the respective digits. The partial result in the units column is 14.

Hi! We've encountered a problem!
66 is less than 77 thus we need to borrow 11 .
So 66 will become 1616 and 55 will become 4 because already borrowed 11from it. We'll obtain the following:

A vertical subtraction problem showing 5700 minus 3786, with borrowing steps annotated as 4, 16, 9, and 10 above the respective digits. The final result is calculated as 1914.

We're done! The result is 19141914.

Do you know what the answer is?

Examples with solutions for Vertical Subtraction

Exercise #1

amp;15amp;amp;  4amp;776amp; \begin{aligned} &15 \\ -& \\ &~~4 \\ &\underline{\phantom{776}} & \\ \end{aligned}

Video Solution

Step-by-Step Solution

To solve this problem, we'll perform simple vertical subtraction for the numbers given, 15 and 4:

Step-by-step solution:

  • Step 1: Write the numbers in a column, aligning the digits according to place value.
  • Step 2: Start subtracting from the rightmost column (the ones column).
    In the ones column, subtract 4 from 5:54=1 5 - 4 = 1 .
  • Step 3: Move to the tens column. There is no subtraction to perform here since it's only 101 - 0, which leaves the digit as is.

Thus, there is no borrowing needed because the digits in the minuend are sufficient to carry out the subtraction.

The result of the subtraction 15415 - 4 is 1111.

Therefore, the solution to the problem is 11 11 .

The correct multiple-choice answer is option 1: 11 11 .

Answer

11

Exercise #2

amp;27amp;amp;  3amp;776amp; \begin{aligned} &27 \\ -& \\ &~~3 \\ &\underline{\phantom{776}} & \\ \end{aligned}

Video Solution

Step-by-Step Solution

To solve this problem, we'll perform vertical subtraction:

  • Step 1: Write down the numbers vertically with the larger number (the minuend) on top:

27  3776\begin{aligned} &27 \\ -& \\ &~~3 \\ &\underline{\phantom{776}} &\\ \end{aligned}

  • Step 2: Subtract the digits in the ones place: 77 (from 2727) minus 33 (from 33) equals 44.
  • Step 3: Subtract the digits in the tens place: 22 (from 2727) minus 00 (no tens in 33) equals 22.

Therefore, the difference is 2424.

The solution to the problem is 2424, which corresponds to choice 2.

Answer

24

Exercise #3

amp;39amp;amp;  6amp;776amp; \begin{aligned} &39 \\ -& \\ &~~6 \\ &\underline{\phantom{776}} & \\ \end{aligned}

Video Solution

Step-by-Step Solution

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

  • Step 1: Write the numbers vertically, with each digit aligned in their respective place value.

  • Step 2: Begin subtracting starting from the rightmost column.

  • Step 3: Move to the left, repeating the process for each subsequent column until finished.

Now, let's work through each step:
Step 1: Arrange the numbers vertically, aligning according to the decimal place.
 39
-   6
-----
Step 2: Start subtracting from the right. Subtract the ones place: 96=39 - 6 = 3.
 39
-   6
-----
    3
Step 3: Since there is no need to borrow, move to the tens place:
The tens place comprises '3' from '39', as there is no corresponding digit above '6' to subtract from: 30=33 - 0 = 3.
  39
-   6
-----
   33

Therefore, the solution to the problem is 33 33 .

Answer

33

Exercise #4

amp;48amp;amp;  7amp;776amp; \begin{aligned} &48 \\ -& \\ &~~7 \\ &\underline{\phantom{776}} & \\ \end{aligned}

Video Solution

Step-by-Step Solution

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

  • Step 1: Align the numbers vertically by place value.

  • Step 2: Subtract the units place.

  • Step 3: Subtract the tens place if necessary.

Now, let's work through each step:
Step 1: Write the numbers 48 and 7 in columns where the digits (ones place) are aligned: 48 7\begin{array}{c} 48 \\ -~7 \\ \end{array} Step 2: Subtract the units (8 - 7 = 1) and write the result in the units position.
Step 3: The tens place after subtraction is unchanged because there is no borrowing needed, so the 4 from 48 stays as 4.
Thus, we have: 48 7  41 \begin{array}{c} 48 \\ -~7 ~~\\ \underline{41} \end{array}

Therefore, the solution to the problem is 41.

Answer

41

Exercise #5

amp;56amp;amp;  5amp;776amp; \begin{aligned} &56 \\ -& \\ &~~5 \\ &\underline{\phantom{776}} & \\ \end{aligned}

Video Solution

Step-by-Step Solution

To solve this problem, let's use a methodical approach as follows:

  • Step 1: Write the numbers in a vertical format, aligning the digits by place value.
  • Step 2: Subtract the ones digit: 66 (from 5656) minus 55 equals 11.
  • Step 3: Bring down the tens digit since we are not subtracting anything from it: 55.

In detail:

  1. Align the numbers vertically, with the larger number on top:
    564005 \begin{array}{r} 56 \\ - \phantom{400}5 \\ \hline \end{array}
  2. Subtract the digits in the ones column. The ones digit in 5656 is 66 and the ones digit in 55 is 55. Subtract 55 from 66 to get 11.
  3. Since there are no numbers to subtract from the tens column of the first number, write down the 55 from 5656.

The result of the vertical subtraction is thus:
56400551 \begin{array}{r} 56 \\ - \phantom{400}5 \\ \hline 51 \\ \end{array}

Therefore, the solution to this problem is 51 51 .

Answer

51

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