Topics
7. Decimal operations
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AustraliaNSW
Stage 3

7.07 Multiply decimals by whole numbers

Lesson
Practice
Lesson

Are you ready?

Let's review a strategy for  multiplying whole numbers  .

Examples

Example 1

Calculate 26\times39 by completing an area model.

a

Firstly, complete the missing values in the area model.

\times206
30
9
Worked Solution
Create a strategy

Multiply the number at the top of the column by the number on the left of the row.

Apply the idea

For the top left square:

20\times30=600

For the top right square:

6\times30=180

For the bottom left square:

20\times9=180

For the bottom right square:

6\times9=54

Filling in the squares, we get:

\times206
30600180
918054
b

Now add the values you found in part a to calculate 26\times39.

Worked Solution
Create a strategy

Add all the values found in part (a).

Apply the idea
\displaystyle \text{Total}\displaystyle =\displaystyle 600+180+180+54Add all the values
\displaystyle =\displaystyle 1014

26 \times 39=1014

Idea summary

We can use area models to multiply two digit numbers.

Multiply decimals by two digit numbers

This video looks at how to use the area model to multiply numbers with decimals.

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Examples

Example 2

Use the area model to find 47 \times 5.4.

A rectangle with a length of 5.4 and a width of 47 divided into 4 rectangles. 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 top left rectangle}\displaystyle =\displaystyle 5 \times 40Multiply the sides
\displaystyle =\displaystyle 200

For the top right rectangle we need to multiply 0.4 by 40. So we can first multiply 4 by 40 then add the decimal point so that the answer has the same decimal places as the question.

\displaystyle 4\times40\displaystyle =\displaystyle 160Multiply the sides ignoring the decimal point
\displaystyle \text{Area of top right rectangle}\displaystyle =\displaystyle 16.0Insert the decimal point
\displaystyle =\displaystyle 16Simplify
\displaystyle \text{Area of bottom left rectangle}\displaystyle =\displaystyle 5\times 7Multiply the sides
\displaystyle =\displaystyle 35

For the bottom right rectangle we need to multiply 0.4 by 7. So we can first multiply 4 by 7 then add the decimal point so that the answer has the same decimal places as the question.

\displaystyle 4\times7\displaystyle =\displaystyle 28Multiply the sides ignoring the decimal point
\displaystyle \text{Area of top right rectangle}\displaystyle =\displaystyle 2.8Insert the decimal point

Filling in the rectangles, 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

What is the total area of the four rectangles, and the answer to 47\times5.4?

Worked Solution
Create a strategy

Add the areas of the four rectangles found in part (a).

Apply the idea

Put the numbers in a place value table using 0s as placeholders and add them, regrouping where necessary.

HundredsTensUnits.Tenths
2{}^100.0
35.0
16.0
+2.8
=253.8

So the total area and the product of 47\times5.4 is equal to 253.8.

Idea summary

We can use area models to multiply decimals.

Multiply decimals by numbers with more than one digit

This video shows how to multiply a decimal number in the hundredths by a 2 digit number, using the same processes as for whole numbers, while thinking about the placement of the decimal point.

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Examples

Example 3

We want to find 43\times 1.06.

a

First, choose the most reasonable estimate for 43\times1.06.

A
91
B
23
C
43
Worked Solution
Create a strategy

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

Apply the idea

1.06 rounded to the nearest whole number is 1.

The product of 43 \times 1 is 43. The best estimate is option C.

b

Find 43\times 1.06, giving your answer as a decimal.

Worked Solution
Create a strategy

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

Apply the idea

Ignore the decimal point multiply the numbers together as whole numbers in a vertical algorithm. We first multiply 106 by 3 and then by the 4 in the tens column, and then add the results.

\begin{array} {c} &&1&0&6 \\ \times &&&4&3\\ \hline &&3&1&8& \\ + &4&2&4&0 \\ \hline &4&5&5&8 \end {array}

In 43\times 1.06 there are 2 decimal places. So we need to insert the decimal point so that there are 2 decimal places in our answer.

43\times1.06=45.58

Idea summary

When we work with decimals, we can use all of the strategies we used with whole numbers, including the area model and vertical algorithm. The important difference is the value of the digits. Digits to the right of the decimal point are fractions, so they are worth less.

Outcomes

MA3-6NA

selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation

MA3-7NA

compares, orders and calculates with fractions, decimals and percentages

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