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 - when you need to borrow from a $0$ , you cannot borrow directly from it. Instead, keep moving left through any consecutive zeros until you find a non-zero digit. Borrow $1$ from that digit, turning all the zeros you passed through into $9$ s, and the original $0$ (where you needed to borrow) becomes $10$ .

Detailed explanation

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Test yourself on vertical subtraction!

$\begin{aligned} &97 \\ -& \\ &63 \\ &\underline{\phantom{776}} & \\ \end{aligned}$ $$

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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 clearly see which digits align by place value and easily track when we need to borrow from the next column.

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Question 1

$\begin{aligned} &49 \\ -& \\ &31 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Question 2

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

Question 3

$\begin{aligned} &85 \\ -& \\ &81 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

How do you solve vertical subtraction?

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

Align the digits by place value: ones digits under ones digits, tens digits under tens digits, hundreds digits under hundreds digits, and thousands digits under thousands digits.

Pay attention! The first number in the problem must be written on top, and the number being subtracted goes below it.

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.
Always start from the rightmost column (ones place) and work left.
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 to the left.

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

Solution:

Given that we cannot subtract $5$ minus $9$ we need to borrow from the tens column!
When we borrow $1$ ten from the $4$ , we're moving $10$ ones to the ones column:
$5$ 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, but the ones place is $0$ . We borrow $1$ from the tens place (the $4$ ). The $0$ becomes $10$ and the the $4$ becomes $3$ .

Like this:

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 $11$ .

But what happens when we can't borrow from the next digit because it's also $0$ ?
For example in the following exercise:
$500-365=$

The third rule - Borrowing through zeros

When you need to borrow from a $0$ , keep moving left until you find a non-zero digit. Borrow $1$ from that digit, and all the zeros in between become $9$ s. The original $0$ you were borrowing for becomes $10$ .

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.

Step-by-step borrowing process:

We need to borrow for the ones place (the first $0$ ), but the tens place is also $0$
Keep moving left to the hundreds place - the $5$ is not zero, so we can borrow from it
The $5$ becomes $4$ (we borrowed $1$ from it)
The tens place $0$ becomes $9$ (we borrowed through it)
The ones place $0$ becomes $10$ (this is what we were borrowing for

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.

First borrowing (for the ones place):

We need to borrow for the ones place. The tens place is $0$ , so we keep moving left to the $7$ in the hundreds place.

The $7$ becomes $6$ (we borrowed $1$ from it)
The tens place $0$ becomes $9$ (we borrowed through it)
The ones place $0$ becomes $10$ (this is what we were borrowing for)

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.

Second borrowing (for the tens place):

Now we have a new problem! In the tens column, we need to subtract $8$ from $9$ - that works. But wait, in the hundreds column we need to subtract $7$ from $6$ , which we can't do. We need to borrow again!

The $5$ (thousands) becomes $4$ (we borrowed $1$ from it)
The $6$ (hundreds) becomes $16$ (we added $10$ to 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

$\begin{aligned} &37 \\ -& \\ &25 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Question 2

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

Question 3

$\begin{aligned} &91 \\ -& \\ &90 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Examples with solutions for Vertical Subtraction

Exercise #1

$\begin{aligned} &97 \\ -& \\ &63 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Video Solution

Step-by-Step Solution

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

Step 1: Align the numbers vertically by their place values.
Step 2: Subtract the digits in the ones place.
Step 3: Subtract the digits in the tens place, borrowing if needed.

Now, let's work through each step together:

Step 1: Align 97 and 63 vertically:

$\begin{array}{c} \phantom{00\ }9\phantom{0}7 \\ -\phantom{0}6\phantom{0}3 \\ \hline \end{array}$

Step 2: Start with the ones place:

The digits are 7 and 3.

Subtract 3 from 7: $7 - 3 = 4$ .

Step 3: Move to the tens place:

The digits are 9 and 6.

Subtract 6 from 9: $9 - 6 = 3$ .

Write the results in their respective place values:

$\begin{array}{c} \phantom{00\ }9\phantom{0}7 \\ -\phantom{0}6\phantom{0}3 \\ \hline \phantom{00\ }3\phantom{0}4 \\ \end{array}$

Therefore, the solution to the problem is 34.

Answer

Exercise #2

$\begin{aligned} &49 \\ -& \\ &31 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Video Solution

Step-by-Step Solution

To solve the subtraction problem by vertically aligning the numbers, we will work as follows:

Step 1: Write the numbers vertically aligning by place value.
$\begin{array}{c} 49 \\ -31~~~\\ \underline{\phantom{7600}} \\ \end{array}$
Step 2: Start subtracting from the units column (rightmost).
Subtract the units digit: $9 - 1 = 8$ .
Step 3: Move to the tens column.
Subtract the tens digit: $4 - 3 = 1$ .

The subtraction gives us a result.

Therefore, the solution to the problem is $18$ .

Answer

Exercise #3

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

Video Solution

Step-by-Step Solution

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

Step 1: Align the numbers vertically. Place the number 25 over the number 24, ensuring the digits are in the correct columns (tens and units).
Step 2: Subtract the units (rightmost) digits first. In this case, subtract 4 from 5.
$5 - 4 = 1$
Step 3: Subtract the tens digits. In this case, subtract 2 from 2.
$2 - 2 = 0$

Now write the result of the subtraction in each column below the line. The units column has 1, and the tens column has 0, resulting in the final answer.

Therefore, the solution to the subtraction $25 - 24$ is $1$ .

Answer

Exercise #4

$\begin{aligned} &85 \\ -& \\ &81 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Video Solution

Step-by-Step Solution

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

Step 1: Align the numbers for vertical subtraction.
Step 2: Subtract the digits in the units column.
Step 3: Subtract the digits in the tens column if necessary.

Now, let's work through each step:
Step 1: We'll arrange the numbers as follows:
$\begin{array}{c} {85} \\ {-81} ~~\ \\ \hline ? \\ \end{array}$ Step 2: Start by subtracting the digits in the units column: $5 - 1 = 4$ .

Step 3: Move to the tens column: $8 - 8 = 0$ .

The result is 04, which simplifies to just 4.

Therefore, the difference when we subtract 81 from 85 is $\boxed{4}$ .

Answer

Exercise #5

$\begin{aligned} &37 \\ -& \\ &25 \\ &\underline{\phantom{776}} & \\ \end{aligned}$

Video Solution

Step-by-Step Solution

Let's solve the subtraction problem $37 - 25$ using vertical subtraction:

Step 1: Align the numbers vertically:
$\begin{array}{c} \ \ \ \ \ \ \ 37 \\ - \ \ \ \ 25 \\ \hline \end{array}$
Step 2: Start with the rightmost column (units place):
$7 - 5 = 2$
Write $2$ under the line in the units place.
Step 3: Move to the left column (tens place):
$3 - 2 = 1$
Write $1$ under the line in the tens place. This gives us $12$ as the result.

Therefore, the solution to the problem is $12$ .

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 to the left.

The third rule - Borrowing through zeros

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 to the left.