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 $1$ from the next digit.
Third rule - a $0$ that cannot be borrowed from becomes $9$ until we reach a digit that is not $0$ from which we can borrow $1$ .
Note! If there is a third $0$ right after, it will become $8$ , if there is a fourth $0$ right after, it will become $7$ , and so on.

Detailed explanation

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Test yourself on vertical subtraction!

$\begin{aligned} &105 \\ -& \\ &~~~~3 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

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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: $431-278=$
By writing them vertically we can easily solve them.

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Question 1

$\begin{aligned} &157 \\ -& \\ &~~~~4 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Question 2

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

Question 3

$\begin{aligned} &178 \\ -& \\ &125 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

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: $87-54=$
We will write it as follows:

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:

$7-4=3$

Let's continue to subtract the tens digits to obtain the following :
$8-5=3$

We're done! The result is $33$ .
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 $1$ from the next digit.

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

Solution:

Given that we cannot subtract $5$ minus $9$ we need to borrow a digit from the next number!
That means:
$5$ will become $15$ given that we'll place one in front of it and $4$ will become $3$ .

We will write it in the following way:

Now we can proceed to solve the problem:
$15-9=6$
$3-2=1$
As seen below:

The result is $16$ !

What do we do when we need to subtract a number from the digit $0$ ? For example in the exercise
$40-29=$
Here too we'll need to borrow $1$ from the digit $4$ 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 :
$10-9=1$
$3-2=1$
As seen below:

We're done! The result is $11$ .
Now let's see what happens when we can't borrow from the next digit given that it's also $0$ :
For example in the following exercise:
$500-365=$

The third rule - $0$ that cannot be borrowed from becomes $9$ until we reach a digit that is not $0$ from which we can borrow $1$ .

Note! If there is a third $0$ immediately after, it becomes $8$ , if there is a fourth $0$ immediately after, it becomes $7$ 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 $1$ for the first $0$ making it $10$ .
The second $0$ will be $9$ because we can't borrow from it
and the digit $5$ will become $4$ 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:
$10-5=5$
$9-6=3$
$4-3=1$
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 $135$
Now let's move on to a very advanced exercise!

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

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:
$0$ The first will become $10$ since we borrow $1$ .
$0$ The second will become $9$ because we can't borrow from it and the $7$ will become $6$ because we borrowed $1$ 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!
$6$ is less than $7$ thus we need to borrow $1$ .
So $6$ will become $16$ and $5$ will become 4 because already borrowed $1$ from 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 $1914$ .

Do you know what the answer is?

Question 1

$\begin{aligned} &19 \\ -& \\ &17 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Question 2

$\begin{aligned} &248 \\ -& \\ &135 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Question 3

$\begin{aligned} &25 \\ -& \\ &24 \\ &\underline{\phantom{776}} & \\ \end{aligned}$ $$

Examples with solutions for Vertical Subtraction

Exercise #1

$\begin{aligned} &105 \\ -& \\ &~~~~3 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Video Solution

Step-by-Step Solution

Let's solve the subtraction problem $105 - 3$ :

Align the numbers vertically to ensure each digit is in the correct place value position:

Write 105 as:
$\begin{array}{c} 1 & 0 & 5 \\ \end{array}$
Place 3 beneath 105 such that it aligns with the rightmost digit (units place):
$\begin{array}{c} - & & 3 \\ \end{array}$
Subtract each column starting from the rightmost side (units digit) to the left:

Step-by-step subtraction:

Units column: $5 - 3 = 2$
Tens column: There is nothing to subtract with 0, so it remains $0$ .
Hundreds column: Similarly, there is nothing to subtract, so it remains $1$ .

Combine the results of these steps to find the answer:

The result of the subtraction $105 - 3 = 102$ .

Therefore, the correct answer is $102$ .

Answer

102

Exercise #2

$\begin{aligned} &157 \\ -& \\ &~~~~4 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Video Solution

Step-by-Step Solution

Let's solve the basic vertical subtraction problem where we subtract 4 from 157 step by step:

Step 1: Write the numbers in a vertical format, ensuring to align them by place values.
$\begin{array}{c} 157 \\ -\ 4 \\ \underline{\phantom{776}} \\ \end{array}$
Step 2: Perform subtraction starting from the rightmost digit (the ones place).
- Subtract 4 from 7 in the ones column, resulting in 3.
Step 3: Since the number being subtracted, 4, has no digit in the tens and hundreds places, bring down the remaining digits from 157. These are 5 in the tens place and 1 in the hundreds place.

Thus, the result of subtracting 4 from 157 is $153$ .

Among the options provided, choice 2 is the correct answer, which states 153.

Therefore, the solution to the problem is $153$ .

Answer

153

Exercise #3

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

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: $5 - 4 = 1$ .
Step 3: Move to the tens column. There is no subtraction to perform here since it's only $1 - 0$ , 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 $15 - 4$ is $11$ .

Therefore, the solution to the problem is $11$ .

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

Answer

Exercise #4

$\begin{aligned} &178 \\ -& \\ &125 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Video Solution

Step-by-Step Solution

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

Step 1: Align the numbers 178 and 125 vertically by place value:
$\begin{array}{c} 178 \\ -125~~~ \\ \underline{\phantom{777}} \\ \end{array}$
Step 2: Subtract beginning with the units column on the right:
"8" minus "5" results in "3".
Step 3: Proceed to the tens column:
"7" minus "2" results in "5".
Step 4: Move to the hundreds column:
"1" minus "1" results in "0".

Therefore, the solution to the subtraction problem $178 - 125$ is $53$ .

Answer

Exercise #5

$\begin{aligned} &19 \\ -& \\ &17 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Video Solution

Step-by-Step Solution

To solve this subtraction problem, follow these steps:

Step 1: Write the numbers vertically, aligning their digits properly. The problem is set up as follows:
$\begin{aligned} &19 \\ -& \\ &17 \\ \end{aligned}$
Step 2: Start with the units column. Subtract the digit 7 from the digit 9:
$9 - 7 = 2$
Step 3: Move to the tens column. Subtract the digit 1 from the digit 1:
$1 - 1 = 0$
Step 4: Combine the results from the tens and units columns. The complete calculation results in the number 2.

Therefore, the solution to the problem is $2$ .

Answer

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Vertical Subtraction

Vertical Subtraction

Test yourself on vertical subtraction!

Vertical Subtraction

What is vertical subtraction?

Why do we need vertical subtraction?

How do you solve vertical subtraction?

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

The Second Rule -

When the upper digit is smaller than the lower digit - we borrow 111 from the next digit.

The third rule -000 that cannot be borrowed from becomes 999 until we reach a digit that is not 000 from which we can borrow 111.

Examples with solutions for Vertical Subtraction

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Vertical Subtraction

When the upper digit is smaller than the lower digit - we borrow $1$ from the next digit.

The third rule - $0$ that cannot be borrowed from becomes $9$ until we reach a digit that is not $0$ from which we can borrow $1$ .