# 2.04 Applications of subtraction

Lesson

Remember to check back over the  subtraction strategies  we've seen so far.

### Examples

#### Example 1

Find the value of 58 - 29.

Worked Solution
Create a strategy

Use the subtraction algorithm with regrouping.

Apply the idea

Write the subtraction in a vertical algorithm.\begin{array}{c} & &5 &8 \\ &- &2 &9 \\ \hline & \\ \hline \end{array}

Begin with the ones column. Since we don't have enough ones to subtract 9 from 8, we need to trade 1 ten from the tens place. \begin{array}{c} & &4 & \text{}^1 8 \\ &- &2 &9 \\ \hline & & \\ \hline \end{array}

Now we can subtract 9 from 18 in the ones column and 5 tens becomes 4 tens.\begin{array}{c} & &4 & \text{}^1 8 \\ &- &2 &9 \\ \hline & & & 9 \\ \hline \end{array}

Then subtract the numbers in the tens column, 4- 2 = 2.\begin{array}{c} & &4 & \text{}^1 8 \\ &- &2 &9 \\ \hline & &2 & 9 \\ \hline \end{array}

So, 58 - 29=29.

Idea summary

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

## Subtraction of more than two numbers

What happens when we have to perform subtraction more than once? One thing we can do is use a number line to help us, as this video shows.

### Examples

#### Example 2

Find the value of 30 - 18 - 7.

Worked Solution
Create a strategy

Find the difference of the first and second numbers and subtract the third number from it.

Apply the idea

Write the first two numbers in a vertical algorithm.\begin{array}{c} & &3 &0 \\ &- &1 &8 \\ \hline & \\ \hline \end{array}

Subtract the numbers starting with the units column. We can see that 0 is less than 8, so we need to trade 1 ten from the tens column.\begin{array}{c} & &2 &\text{}^1 0 \\ &- &1 &8 \\ \hline & &1 &2 \\ \hline \end{array}

Subtract 7 from the result using a vertical algorithm.\begin{array}{c} & &1 &2 \\ &- & &7 \\ \hline & \\ \hline \end{array}

In the units column, we can see that 2 is less than 7, so we need to trade 1 ten from the tens column. This gives us 12 - 7 = 5 in the units column and 1 ten becomes 0 ten in the first row. \begin{array}{c} & &0 &\text{}^1 2 \\ &- & &7 \\ \hline & & &5 \\ \hline \end{array}

30 - 18 - 7 = 5

Idea summary

We can subtract two or more numbers by subtracting one number at a time. The second number is subtracted from the first number, and then the third number is subtracted from the result.

## Subtraction of larger numbers

If we have larger numbers, we can use a vertical algorithm to solve our subtraction. Let's take a look.

### Examples

#### Example 3

Find the value of 42\,373 - 25\,215.

Worked Solution
Create a strategy

Use the subtraction algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & &4 &2 &3 &7 &3 \\ &- &2 &5 &2 &1 &5 \\ \hline & \\ \hline \end{array}

Begin with the units column. We can see that 3 is less than 5, so we need to trade 1 ten from the tens column. This gives us 13 - 5 = 8 in the units column and 7 tens becomes 6 tens in the first row. \begin{array}{c} & &4 &2 &3 &6 &\text{}^13 \\ &- &2 &5 &2 &1 &5 \\ \hline & & & & & &8 \\ \hline \end{array}

For the tens place: 6 - 1 = 5.\begin{array}{c} & &4 &2 &3 &6 &\text{}^13 \\ &- &2 &5 &2 &1 &5 \\ \hline & & & & &5 &8 \\ \hline \end{array}

For the hundreds place: 3 - 2 = 1.\begin{array}{c} & &4 &2 &3 &6 &\text{}^13 \\ &- &2 &5 &2 &1 &5 \\ \hline & & & &1 &5 &8 \\ \hline \end{array}

In the thousands column, we can see that 2 is less than 5, so we need to trade 1 ten thousand from the ten thousands column. This gives us 12 - 5 = 7 in the thousands column and 4 ten thousands becomes 3 ten thousands in the first row. \begin{array}{c} & &3 &\text{}^1 2 &3 &6 &\text{}^13 \\ &- &2 &5 &2 &1 &5 \\ \hline & & &7 &1 &5 &8 \\ \hline \end{array}

For the ten thousands place: 3 - 2 = 1.\begin{array}{c} & &3 &\text{}^1 2 &3 &6 &\text{}^13 \\ &- &2 &5 &2 &1 &5 \\ \hline & &1 &7 &1 &5 &8 \\ \hline \end{array}

42\,373 - 25\,215 = 17\,158

Idea summary

We can use a vertical algorithm to subtract large numbers. This will require more steps to subtract each place value.

## Subtraction with story problems

What happens if we have a story problem? We need to look for the keywords or clues to see what we need to do, as we see in this video.

### Examples

#### Example 4

A farmer has 83 sheep in a paddock. 9 sheep are taken away to the wool shed, and while the farmer is away 2 sheep from a neighboring farm get into the paddock through a hole in the fence.

How many sheeps are in the paddock now?

Worked Solution
Create a strategy

The words "taken away" mean subtraction while the words "get into" mean addition.

Apply the idea

We have 83 sheeps in total, 9 of them were taken away and 2 got in. So the number sentence for this is:\text{Number of sheep} = 83 - 9 + 2

From the number sentence above, write the first number and the second number in a vertical algorithm to find their difference. \begin{array}{c} & &8 &3 \\ &- & &9 \\ \hline \\ \hline \end{array}

Begin with the units column. We can see that 3 is less than 9, so we need to trade 1 ten from the tens column. This gives us 13 - 9 = 4 in the units column and 8 tens becomes 7 tens in the first row. \begin{array}{c} & &7 &\text{}^1 3 \\ &- & &9 \\ \hline & & &4 \\ \hline \end{array}

For the tens place: 7 - 0 = 7.\begin{array}{c} & &7 &\text{}^1 3 \\ &- & &9 \\ \hline & &7 &4 \\ \hline \end{array}

Add 2 to the result. Write it in a vertical algorithm. \begin{array}{c} & &7 &4 \\ &+ & &2\\ \hline \\ \hline \end{array}

For the units place: 4 + 2 = 6. \begin{array}{c} & &7 &4 \\ &+ & &2 \\ \hline & & &6 \\ \hline \end{array}

For the tens place: 7 + 0 = 7. \begin{array}{c} & &7 &4 \\ &+ & &2 \\ \hline & &7 &6 \\ \hline \end{array}

So there are 76 sheep in the paddock now.

Idea summary

When we solve subtraction problems, it's useful to have some different methods to use. As our numbers get larger, vertical algorithms may be necessary. Checking the answer on a calculator, or working out an estimate helps to check our answer is reasonable.

### Outcomes

#### MA3-5NA

selects and applies appropriate strategies for addition and subtraction with counting numbers of any size

#### MA3-8NA

analyses and creates geometric and number patterns, constructs and completes number sentences, and locates points on the Cartesian plane