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3.04 Multiplying with decimals

Lesson

Introduction

We have previously looked at what happens when we multiply decimals by  powers of 10  , and how we increase the place value of each digit by one place for every 10 that we multiply by. We will now look at how we can multiply any decimal numbers, using the same methods we use to multiply whole numbers.

Multiplication with decimals

Let's follow through the process for the multiplication 4.83\times 5.7.

Before we even begin to calculate the answer it can be a good idea to have an estimation of the answer, especially when dealing with decimals. This will help us confirm our final answer is of the right magnitude.

In this case, we can round both numbers to the nearest whole, giving us the much simpler calculation 5\times6, which we can evaluate to get 30. With this in mind, we would expect our answer to be close to this value, and we can know for sure that it will have digits in the tens and ones columns.

Now, to start the process, we simply ignore the decimal points. In this case we get 483 and 57. We then multiply these together with the method we are used to using for whole numbers:

\begin{array}{c}             & & & &4&8&3 \\             &\times & & &  &5&7 \\ \hline & & &3&3&8&1 \\ &+ &2&4&1&5&0 \\ \hline & &2&7&5&3&1             \end{array}

Now we need to account for the decimal point. To do so, we add the total number of decimal places in the original numbers together.

In this case the original numbers are 4.83, which has two decimal places, and 5.7, which has one decimal place. So their product will have 2+1=3 decimal places.

Then to find the final answer, we take the product that we calculated before and insert the decimal point such that there are 3 decimal places:

Image of 27.531 that shows the decimal point is moved three times from 1 into the front of 5.

So we have found that 4.83\times 5.7=27.531.

This is very close to our original estimate of 30, and as expected we have a digit in each of the tens column and ones column. If we ended up with a final value of 275.31 we know we have written the decimal point in the wrong place.

Remember that we can represent any finite decimal as a fraction by using a power of 10 in the denominator. In this case, 4.83 is equal to \dfrac{483}{100} and 5.7 is equal to \dfrac{57}{100}.

Let's now multiply these numbers by using their fraction forms instead. To do so, recall that we just multiply their numerators together and their denominators together:

\begin{aligned} \dfrac{483}{100}\times \dfrac{57}{10} &= \dfrac{483\times 57}{100\times 10}\\ &= \dfrac{27531}{1000} \end{aligned}

The numerator contains the product 483\times 57=27\,531, which is what we initially calculated above. The denominator then tells us the place value of the number

In this case, the original denominators were 100 and 10. The final denominator is their product, which is 1000. Dividing by 1000 is the same as decreasing the place value of each digit by three place values, and so we get 27.531 as our final answer.

The method of adding the decimals in the original numbers to find the number of decimal places in the answers will always work, but we sometimes end up with a 0 as our last decimal place, in which case we can remove the trailing zero when writing our final answer.

There are other strategies for multiplying decimal numbers including partitioning, or repeated addition. The video below shows some of these methods.

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The way we solve these multiplications is no different to working with whole numbers, and we can use exactly the same methods. We just need to think carefully about where the decimal point goes.

Examples

Example 1

Find 2.2\times4, giving your answer as a decimal.

Worked Solution
Create a strategy

Multiply the numbers together as whole numbers and place the decimal point accordingly.

Apply the idea

Ignore the decimal point for the meantime and multiply the numbers together as whole numbers.

\begin{aligned} 22& \\ \times \quad 4& \\ \hline 88\end{aligned}

Add the total number of decimal places in the original numbers.

\displaystyle \text{Total number of decimal places}\displaystyle =\displaystyle 1+0Add 1 for 2.2 and 0 for 4
\displaystyle =\displaystyle 1Evaluate

We have a total of 1 decimal place.

Take the product calculated earlier and insert the decimal point such that there is 1 decimal point.

We now have:

2.2\times4=8.8

Example 2

Find 1.2\times3.2, giving your answer as a decimal.

Worked Solution
Create a strategy

Multiply the numbers together as whole numbers and place the decimal point accordingly.

Apply the idea

Ignore the decimal point for the meantime and multiply the numbers together as whole numbers.

\begin{aligned} 12& \\ \times \quad 32& \\ \hline 24 \\+ \quad 360 \\ \hline 384 \end{aligned}

The result is 384.

Add the total number of decimal places in the original numbers.

\displaystyle \text{Total number of decimal places}\displaystyle =\displaystyle 1+1Add 1 for 1.2 and 1 for 3.2
\displaystyle =\displaystyle 2Evaluate

We have a total of 2 decimal places.

Take the product calculated earlier and insert the decimal point such that there is 2 decimal points.

We now have:

1.2\times 3.2=3.84

Idea summary

To multiply two decimals together:

  • Multiply the numbers together as whole numbers first.

  • Count the total number of decimal places in the original decimals.

  • Place the decimal point in your answer so that it has the same number of decimal places.

The area method

We have previously used  area to solve multiplication  of whole numbers, and we can use the same idea for decimal problems involving multiplication. Starting with a rectangle, we can break it into a series of smaller rectangles, allowing us to work out our problem in smaller steps. Then we can add our answers together at the end.

We can solve something like 3.4\times34 by thinking of it as a rectangle whose length and width measure 3.4 units and 34 units respectively. Then we can break it into smaller rectangles and solve it in four parts. We finally add the individual solutions together to find our overall answer to our original problem.

\times30.4
30
4
\text{Total}

The video below shows this process.

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Examples

Example 3

In this question, we will use the area model to find the product of a decimal and a whole number.

a

Find the area of each rectangle below as a decimal.

3 rectangles  joined together. They have a height of 9, and widths of 4, 0.6, and 0.07 respectively.
Worked Solution
Create a strategy

Multiply the sides of each rectangle to get the area.

Apply the idea
\displaystyle \text{Area of 1st rectangle}\displaystyle =\displaystyle 4\times9Multiply the sides
\displaystyle =\displaystyle 36Evaluate
\displaystyle \text{Area of 2nd rectangle}\displaystyle =\displaystyle 6\times9Multiply the sides ignoring the decimal point
\displaystyle =\displaystyle 54Evaluate
\displaystyle =\displaystyle 5.4Insert the decimal point
\displaystyle \text{Area of 3rd rectangle}\displaystyle =\displaystyle 7\times9Multiply the sides ignoring the decimal point
\displaystyle =\displaystyle 63Evaluate
\displaystyle =\displaystyle 0.63Insert the decimal point
3 rectangles  joined together. They have a height of 9, widths of 4, 0.6, and 0.07, and areas of 36, 5.4, 0.63 respectively.
b

Using the answer from part (a), find 4.67\times 9 as a decimal.

Worked Solution
Create a strategy

Get the sum of the answers in part (a).

Apply the idea

4.67\times 9=36+5.4+0.63

We can work out this addition by computing vertically. If the total sum in any column is bigger than 9, carry the tens into the next column.

\begin{array}{c} &&3&6 \\&&& 5&.&4 \\&+& &0&.&6&3 \\ \hline &&4&2&.&0&3\end{array}

We now have: 4.67\times 9 = 42.03.

Idea summary

We can multiply decimals by starting with a rectangle, and breaking it into a series of smaller rectangles. This allows us to work out our problem in smaller steps. Then we can add our answers together at the end.

Outcomes

VCMNA244

Multiply and divide fractions and decimals using efficient written strategies and digital technologies.

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