2.03 Subtraction

Lesson

When we work on subtraction problems, it helps to remember some of the  addition strategies  we've used so far.

Examples

Example 1

Let's find the value of 705 + 205, by partitioning the numbers.

a

Fill in the box with the missing number.

705 = 700 + ⬚

Worked Solution
Create a strategy

Put the numbers in a place value table.

Apply the idea

The only place where that the two numbers differ is in the units column. The first number has 5 units where as the second number has 0 units.

So we need to add 5 more to the number 700 to equal 705.

705 = 700 + 5

b

Fill in the box with the missing number.

205 = ⬚ + 5

Worked Solution
Create a strategy

Put the numbers in a place value table.

Apply the idea

In the place value table, both 205 and 5 have 5 units but they don't match up in the tens and hundreds columns.

So we need to add 0 tens and 2 hundreds or 200 to 5 to get 205.

205 = 200 + 5

c

Find the value of 705 + 205.

Worked Solution
Create a strategy

Add the partitions of the two numbers.

Apply the idea

In parts (a) and (b), we have the partitions of these two numbers:

705 = 700 + 5 \\ 205 = 200 + 5

We can use these to add the two numbers.

705 + 205 = 910

Idea summary

To add large numbers, we can first partition the numbers and then add their partitions.

Subtraction by place value

Examples

Example 2

Find the value of 93 - 24.

Worked Solution
Create a strategy

Use a number line.

Apply the idea

Locate where 93 is.

Jump 24 units to the left from 94.

Idea summary

We can use a number line to subtract numbers, by plotting the first number and jumping left the number of spaces given by the second number.

Subtraction without regrouping

We can also subtract using an algorithm, as we see in this video.

Examples

Example 3

Find the value of 398 - 152.

Worked Solution
Create a strategy

Use the subtraction algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & &3 &9 &8 \\ &- &1 &5 &2 \\ \hline \\ \hline \end{array}

Begin with the units column: 8 -2 = 6.\begin{array}{c} & &3 &9 &8 \\ &- &1 &5 &2 \\ \hline & & & &6 \\ \hline \end{array}

For the tens place: 9 -5 = 4. \begin{array}{c} & &3 &9 &8 \\ &- &1 &5 &2 \\ \hline & & &4 &6 \\ \hline \end{array}

For the hundreds place: 3 -1 = 2. \begin{array}{c} & &3 &9 &8 \\ &- &1 &5 &2 \\ \hline & &2 &4 &6 \\ \hline \end{array}

So 398 - 152 = 246.

Idea summary

We can subtract two numbers using a vertical algorithm. We start at the units column and move left.

Subtraction of large numbers

Let's take a look at how a vertical algorithm helps us solve subtraction with large numbers.

Examples

Example 4

Find the value of 81\,579 -27\,398.

Worked Solution
Create a strategy

Use the subtraction algorithm method.

Apply the idea

Write it in a vertical algorithm.\begin{array}{c} & &8 &1 &5 &7 &9 \\ &- &2 &7 &3 &9 &8 \\ \hline \\ \hline \end{array}

Begin with the units column: 9 -8 = 1.\begin{array}{c} & &8 &1 &5 &7 &9 \\ &- &2 &7 &3 &9 &8 \\ \hline & & & & & &1 \\ \hline \end{array}

In the tens column we can see that 7 is less than 9, so we need to trade 1 hundred from the hundreds column. This gives us 17 - 9 = 8 in the tens column and 5 hundreds becomes 4 hundreds in the first row. \begin{array}{c} & &8 &1 &4 &\text{}^1 7 &9 \\ &- &2 &7 &3 &9 &8 \\ \hline & & & & &8 &1 \\ \hline \end{array}

For the hundreds place: 4 - 3 = 1. \begin{array}{c} & &8 &1 &4 &\text{}^1 7 &9 \\ &- &2 &7 &3 &9 &8 \\ \hline & & & &1 &8 &1 \\ \hline \end{array}

In the thousands column we can see that 1 is less than 7, so we need to trade 1 ten thousand from the ten thousands column. This gives us 11 - 7 = 4 in the thousands column and 8 ten thousands becomes 7 ten thousands in the first row. \begin{array}{c} & &7 &\text{}^1 1 &4 &\text{}^1 7 &9 \\ &- &2 &7 &3 &9 &8 \\ \hline & & &4 &1 &8 &1 \\ \hline \end{array}

For the ten thousands place: 7 - 2 = 5.\begin{array}{c} & &7 &\text{}^1 1 &4 &\text{}^1 7 &9 \\ &- &2 &7 &3 &9 &8 \\ \hline & &5 &4 &1 &8 &1 \\ \hline \end{array}

So 81\,579 - 27\, 398 = 54\, 181.

Idea summary

When subtracting, especially when using the vertical algorithm, place value is really important. We want to make sure our numbers are lined up correctly before we subtract, and we start from the place to the far right.

Outcomes

VCMNA209

Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers and make estimates for these computations