Topics
7. Decimal operations
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AustraliaVIC
Level 6

7.06 Multiply decimals by single digit numbers

Lesson
Practice
Lesson

Are you ready?

We have looked at  visual models for multiplying whole numbers  in previous lessons. Let's look to an area model as a review.

Examples

Example 1

Let's use the area model to find 38\times7.

a

Fill in the areas of each rectangle.

A rectangle with a length of 38 and a height of 7 divided into 4 rectangles. Ask your teacher for more information.
Worked Solution
Create a strategy

For each rectangle, use the formula \text{Area} = \text{length} \times \text{width}.

Apply the idea
\displaystyle \text{Area of top left rectangle}\displaystyle =\displaystyle 30 \times 2Multiply the sides
\displaystyle =\displaystyle 60
\displaystyle \text{Area of top right rectangle}\displaystyle =\displaystyle 8 \times 2Multiply the sides
\displaystyle =\displaystyle 16
\displaystyle \text{Area of bottom left rectangle}\displaystyle =\displaystyle 30 \times 5Multiply the sides
\displaystyle =\displaystyle 150
\displaystyle \text{Area of bottom right rectangle}\displaystyle =\displaystyle 8 \times 5Multiply the sides
\displaystyle =\displaystyle 40

Filling in the rectangle, we get:

A rectangle with a length of 38 and a height of 7 divided into 4 rectangles. Ask your teacher for more information.
b

Find the sums of each column.

A rectangle with a length of 38 and a height of 7 divided into 4 rectangles. Ask your teacher for more information.
Worked Solution
Create a strategy

Add the areas of each column using a place value table.

Apply the idea

We can add the numbers in the first column using a vertical algorithm:

\begin{array}{c} &&{}^11&5&0 \\ &+&&6&0 \\ \hline &&2&1&0 \\ \hline \end{array}

We can add the numbers in the second column using a vertical algorithm:

\begin{array}{c} &&4&0 \\ &+&1&6 \\ \hline &&5&6 \\ \hline \end{array}

A rectangle with a length of 38 and a height of 7 divided into 4 rectangles. Ask your teacher for more information.
c

What is the total area of the rectangles?

A rectangle with a length of 38 and a height of 7 divided into 2 rectangles. Ask your teacher for more information.
Worked Solution
Create a strategy

Add the areas of the two rectangles in a vertical algorithm.

Apply the idea

\begin{array}{c} &&2&1&0 \\ &+&&5&6 \\ \hline &&2&6&6 \\ \hline \end{array}

The total area is 266. So 38\times 7=266.

Idea summary

We can use area models to multiply two-digit numbers.

Multiply decimals by single digit numbers

This video looks at using visual models for multiplying decimals.

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Examples

Example 2

Use the area model to find 4\times1.209.

3 rectangles joined together. The width is 4 and the lengths are 1, 0.2 and 0.009. Ask your teacher for more information.
a

Fill in the areas of each rectangle.

Worked Solution
Create a strategy

For each rectangle, use the formula \text{Area} = \text{length} \times \text{width}.

Apply the idea
\displaystyle \text{Area of 1st rectangle}\displaystyle =\displaystyle 1\times4Multiply the sides
\displaystyle =\displaystyle 4

For the second rectangle we need to multiply 0.2 by 4. So we can first multiply 2 by 4 then add the decimal point so that the answer has the same decimal places as the question.

\displaystyle 2\times4\displaystyle =\displaystyle 8Multiply the sides ignoring the decimal point
\displaystyle \text{Area of 2nd rectangle}\displaystyle =\displaystyle 0.8Insert the decimal point

We can use the same strategy for the last rectangle:

\displaystyle 9\times4\displaystyle =\displaystyle 36Multiply the sides ignoring the decimal point
\displaystyle \text{Area of 3rd rectangle}\displaystyle =\displaystyle 0.036Insert the decimal point

Filling in the rectangles, we get:

3 rectangles joined together. The width is 4 and the lengths are 1, 0.2 and 0.009. Ask your teacher for more information.
b

What is the total area of all three rectangles, and the answer to 4 \times1.209?

Worked Solution
Create a strategy

Add the areas of each rectangle found in part (a) using a place value table.

Apply the idea

Write each area in a place value table. Use 0s as placeholders.

Units.TenthsHundedthsThousandths
4.000
0.800
+0.036
=4.836

So the total area is 4.836, which is also the answer to 4 \times1.209=4.836.

Idea summary

We can use area models to multiply a decimal by a whole number.

Multiply decimals to thousandths by single digit numbers

This video uses the vertical algorithm and a couple of simple methods to check our answers for reasonableness. While this doesn't always tell us our answer is correct, it can definitely help us check our decimal place is in the correct spot.

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Examples

Example 3

We want to find 6\times 7.003.

a

First, choose the most reasonable approximation for 6\times 7.003.

A
80
B
42
C
4
Worked Solution
Create a strategy

Round the decimal to the nearest whole number, then find the product.

Apply the idea

7.003 to the nearest whole number is 7.

The product of 6\times7 is 42. So the correct answer is option B.

b

Find 6\times 7.003, giving your answer as a decimal.

Worked Solution
Create a strategy

Multiply the numbers together as whole numbers first then insert the decimal point.

Apply the idea

Ignore the decimal point and multiply the numbers together as whole numbers in a vertical algorithm.

\begin{array} {c} &&7&0&{}^10&3 \\ \times &&&&&6\\ \hline &4&2&0&1&8& \\ \hline \end {array}

There are 3 decimal places in 7.003. So we will insert the decimal point so that there is 3 decimal places in our answer to get: 6\times7.003=42.018

Idea summary

Whichever method we use to multiply decimals, the most important thing is to remember that we don't need to learn any new processes at all. The value of the digits change, but the way we solve the problem doesn't.

Outcomes

VCMNA215

Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies

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