# 2.08 Vertical algorithms

Lesson

Are you able to use a  vertical algorithm to add numbers  with no regrouping?

### Examples

#### Example 1

Find the value of 41+56.

Worked Solution
Create a strategy

Use the vertical algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & &4 &1 \\ &+ &5 &6 \\ \hline \\ \hline \end{array}

Add the numbers down each column starting from the ones column, then the tens column to get:\begin{array}{c} & &4 &1 \\ &+ &5 &6 \\ \hline & &9 &7 \\ \hline \end{array}So 41 + 56 = 97.

Idea summary

You might notice that sometimes the standard algorithm is called the 'vertical algorithm'. Let's think about why. When we use the standard algorithm, we line our numbers up in 'vertical' place value columns.

## Add and subtract with regrouping

When we need to regroup to add or subtract, there are some different ways we can do this. Let's see how to solve an addition problem, then a subtraction problem.

### Examples

#### Example 2

Find the value of 997 - 648.

Worked Solution
Create a strategy

Use the subtraction algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & & &9 &9 &7 \\ &- & &6 &4 &8 \\ \hline & \\ \hline \end{array}

Begin with the units column. We can see that 7 is less than 8, so we need to trade 1 ten from the tens place.

So we get 17 - 8 = 9 in the units column and 9 tens becomes 8 tens in the first row.\begin{array}{c} & & &9 &8 &\text{ }^1 7 \\ &- & &6 &4 &8 \\ \hline & & & & &9 \\ \hline \end{array}

For the tens place: 8 - 4 = 4.\begin{array}{c} & & &9 &8 &\text{ }^1 7 \\ &- & &6 &4 &8 \\ \hline & & & &4 &9 \\ \hline \end{array}

For the hundreds place: 9 - 6 = 3.\begin{array}{c} & & &9 &8 &\text{ }^1 7 \\ &- & &6 &4 &8 \\ \hline & & &3 &4 &9 \\ \hline \end{array}

So 997 - 648 = 349.

Idea summary

We always start from the ones place, when we work down our page. If we don't have enough ones to subtract, we need to trade from the tens place. If we don't have enough tens to subtract, we need to trade from the hundreds place.

## Add and subtract 4 digit numbers

What happens with larger numbers? We can use a standard algorithm to solve addition and subtraction, as we see in this video.

### Examples

#### Example 3

Find the value of 2124 + 3351.

Worked Solution
Create a strategy

Use the vertical algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & & &2 &1 &2 &4 \\ &+ & &3 &3 &5 &1 \\ \hline & \\ \hline \end{array}

Add the ones column first. So 4 + 1 = 5:

\begin{array}{c} & & &2 &1 &2 &4 \\ &+ & &3 &3 &5 &1 \\ \hline & & & & & &5\\ \hline \end{array}

Then add the tens column, 2 + 5 = 7:\begin{array}{c} & & &2 &1 &2 &4 \\ &+ & &3 &3 &5 &1 \\ \hline & & & & &7 &5\\ \hline \end{array}

Then add the hundreds column, 1 + 3 = 4:\begin{array}{c} & & &2 &1 &2 &4 \\ &+ & &3 &3 &5 &1 \\ \hline & & & &4 &7 &5\\ \hline \end{array}

Then add the thousands column, 2 + 3 = 5

\begin{array}{c} & & &2 &1 &2 &4 \\ &+ & &3 &3 &5 &1 \\ \hline & & &5 &4 &7 &5\\ \hline \end{array}

So 2124 + 3351 = 5475.

Idea summary

When we are adding or subtracting down the page, it's really important to start from the right (ones place) and work to the left, no matter how many digits our numbers have.

## Add and subtract 4 digit numbers with regrouping

When we need to do some trading, or regrouping, the standard algorithm helps us line up place value digits. Let's see how to add or subtract, when we need to regroup.

### Examples

#### Example 4

Find the value of 7263 - 2418.

Worked Solution
Create a strategy

Use the vertical algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & & &7 &2 &6 &3 \\ &- & &2 &4 &1 &8 \\ \hline & \\ \hline \end{array}

Subtract the ones column first. In the ones column we can see that 3 is less than 8, so we need to trade 1 ten from the tens place.

So we get 13-8=5 in the ones column and 6 tens becomes 5 tens in the first row.\begin{array}{c} & & &7 &2 &5 &\text{}^1 3 \\ &- & &2 &4 &1 &8 \\ \hline & & & & & &5\\ \hline \end{array}

So for the tens column we have 5 - 1 = 4:\begin{array}{c} & & &7 &2 &5 &\text{}^1 3 \\ &- & &2 &4 &1 &8 \\ \hline & & & & &4 &5\\ \hline \end{array}

In the hundreds column, we can see that 2 is less than 4, so we need to trade 1 thousand from the thousands place.

So we get 12-4=8 in the hundreds column and 7 thousands becomes 6 thousands in the first row.\begin{array}{c} & & &6 &\text{}^1 2 &5 &\text{}^1 3 \\ &- & &2 &4 &1 &8 \\ \hline & & & &8 &4 &5\\ \hline \end{array}

And for the thousands place 6 - 2 = 4:\begin{array}{c} & & &6 &\text{}^1 2 &5 &\text{}^1 3 \\ &- & &2 &4 &1 &8 \\ \hline & & &4 &8 &4 &5\\ \hline \end{array}

So 7263 - 2418 = 4845.

Idea summary

Each time we move one place to the left, the value of the digit is ten times bigger. This means we are able to regroup, or trade, to help us with addition and subtraction.

### Outcomes

#### MA2-5NA

uses mental and written strategies for addition and subtraction involving two-, three-, four and five-digit numbers