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.

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.

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:

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:

Step-by-step borrowing process:

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

As seen below:

We can proceed to solve the exercise:
$10-5=5$
$9-6=3$
$4-3=1$
Let's write the solution as follows:

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:

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:

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:

We're done! The result is $1914$.