 # 3.08 Applications of multiplication and division

Lesson

## Ideas

Let's practice breaking apart the number in this problem to help us get ready for this lesson.

### Examples

#### Example 1

We want to find 2 \times 45.

Use the area model to complete the following:

Worked Solution
Create a strategy

We can find the area of the two smaller rectangles and add the two areas.

Apply the idea

We can think of 2\times 45 as representing the area of this rectangle:

which is equal to the sum of the two smaller rectangles.

The rectangle on the left has an area of 2 \times 40, and the rectangle on the right has an area of 2 \times 5.

Idea summary

If we have a number sentence such as 5\times 12 it can be rewritten as 5\times 10 + 5\times 2.

## Strategies for multiplication

We can use some of the multiplication strategies we know to solve multiplication of larger numbers.

How can some of these strategies be used on larger numbers, do you think?

### Examples

#### Example 2

Find the product of 65 \times 97.

Worked Solution
Create a strategy

Partition one of the numbers to make the multiplication easier.

Apply the idea

97=90+7 so we can write 65\times 97 as 65\times 90+65\times 7. To find each of these products we will use the vertical algorithm.

For 65\times7, set up the vertical algorithm: \begin{array}{c} &&6&5 \\ &\times &&7 \\ \hline &&& \\ \hline \end{array}

7 \times 5 = 35 so we put the 5 in the ones place and carry the 3 to the tens place.

\begin{array}{c} &&{}^36&5 \\ &\times &&7 \\ \hline &&& 5 \\ \hline \end{array}

7 \times 6=42 then add the carried over 3 to get 45. Put the 5 in the tens place and the 4 in the hundreds place.

\begin{array}{c} &&&6&5 \\ &\times &&&7 \\ \hline &&4&5& 5 \\ \hline \end{array}

For 65\times 90:

Since we are multiplying by 90, we will put 0 at the ones place as a place holder:

\begin{array}{c} &6&5 \\ \times &9&0 \\ \hline &&0 \\\hline \end{array}

Now we will multiply 65 by 9.

9\times 5=45. Put the 5 in the tens place and carry the 4 to the tens place.

\begin{array}{c} &{}^46&5 \\ \times &9&0 \\ \hline &5&0 \\\hline \end{array}

9 \times 6=54 then add the carried 4 to get 58. Put the 8 in the hundreds place and the 5 in the thousands place.

\begin{array}{c} &&&&6&5 \\ &\times &&&9&0 \\ \hline &&5&8&5&0 \\\hline \end{array}

\begin{array}{c} &{}^15&{}^18&5&0 \\ +& &4&5&5 \\ \hline &6&3&0&5 \\\hline \end{array}

65\times 97 = 6305

Idea summary

We can multiply two digit numbers by two digit numbers using the standard algorithm by partitioning one of the numbers.

## Strategies for division

Let's take a look at how we divide a 4 digit number by a single digit number, using a few different strategies we've learnt.

### Examples

#### Example 3

Find the value of 1209 \div 3.

Worked Solution
Create a strategy

Use short division.

Apply the idea

The working and steps are shown below:

• 3 goes into 1 zero times with remainder 1 which we carry to the hundreds column.

• 3 goes into 12 four times.

• 3 goes into 0 zero times.

• 3 goes into 9 three times.

1209 \div 3=403

Idea summary

Division is when we share a total into a number of groups, or find out how many items each group has. It is the reverse of multiplication.

## How many times bigger?

We often hear and use the language "twice as big as", or "ten times bigger than", but how do we work this out?

### Examples

#### Example 4

The image below shows a row of squares.

a

How many times more circles are shown below compared to the squares?

Worked Solution
Create a strategy

Count the number of rows of 5 circles.

Apply the idea

There is 1 row of 5 squares, and there are 2 rows 5 circles.

So there are 2 times more circles than squares.

b

How many times more triangles are shown below compared to the squares?

Worked Solution
Create a strategy

Count the number of rows of 5 triangles.

Apply the idea

There are 3 rows 5 triangles.

So there are 3 times more triangles shown compared to the squares.

Idea summary

The vertical algorithm method is really useful for solving multiplication and division problems, but we also have other strategies if we need them, including:

• using the area or arrays method

• partitioning our number

• using the distributive property of multiplication

• looking for patterns

• finding the factors of our numbers

### Outcomes

#### VCMNA183

Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies

#### VCMNA184

Solve problems involving division by a one digit number, including those that result in a remainder

#### VCMNA185

Use efficient mental and written strategies and apply appropriate digital technologies to solve problems

#### VCMNA193

Use equivalent number sentences involving multiplication and division to find unknown quantities