Introduction

We can use place value charts to multiply decimals by 10. For instance, to find the value of 0.4 \times 10, we perform the following steps:

Step 1. Write down the place value chart for 0.4\mathbin{:}

ones tenths
. 4

Step 2. Determine the number of places to move. In this case, notice that 10 has \color{blue}1 zero. So, we move all values to the left by \color{blue}1 place, jumping over the decimal if needed. Then, we put a zero in the empty cell on the right:

ones tenths
4 . 0

Therefore, 0.4 \times 10 = 4.0 = 4 \, .

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Example: Multiplying Decimals by Ten

Complete the place value chart below for 4.31\times 10.


thousands hundreds tens ones tenthshundredths
\fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} . \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
EXPLANATION

Step 1. Write down the place value chart for 4.31\mathbin{:}

thousands hundreds tens ones tenths hundredths
0 0 0 4 . 3 1

Step 2. Determine the number of places to move. Here, 10 has 1 zero. So, we move all values to the left by 1 place, and put a zero in the empty cell on the right:

thousands hundreds tens ones tenths hundredths
\bbox[3pt,Gainsboro]{\color{blue}0} \bbox[3pt,Gainsboro]{\color{blue}0} \bbox[3pt,Gainsboro]{\color{blue}4} \bbox[3pt,Gainsboro]{\color{blue}3} . \bbox[3pt,Gainsboro]{\color{blue}1} \bbox[3pt,Gainsboro]{\color{blue}0}
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Practice: Multiplying Decimals by Ten

Complete the place value chart below for $2.49 \times 10.$

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Practice: Multiplying Decimals by Ten

Complete the place value chart below for $31.2 \times 10.$

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Practice: Multiplying Decimals by Ten

Complete the place value chart below for $8.051 \times 10.$

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Multiplying Decimals by Larger Powers of Ten

We can multiply a decimal by a power of ten by shifting the digits in the place value chart left by the number of zeros in the power of ten.

To illustrate, let's compute 1.68\times 100\, .

Step 1. Write down the place value chart for 1.68\mathbin{:}


hundreds tens ones tenthshundredths
1 . 6 8

Step 2. Determine the number of places to move. Here, 100 has \color{blue}2 zeros. So, we move all values to the left by \color{blue}2 places. Then, we put a zero in the empty cells on the right:


hundreds tens ones tenthshundredths
1 6 8 . 0 0

Therefore, 1.68\times 100 = 168.00 = 168\, .

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Example: Multiplying Decimals by One Hundred

Complete the place value chart below for 8.315\times 100.


thousands hundreds tens ones tenthshundredths thousandths
\fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} . \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]}
EXPLANATION

Step 1. Write down the place value chart for 8.315\mathbin{:}


thousands hundreds tens ones tenthshundredths thousandths
0 0 0 8 . 3 1 5

Step 2. Determine the number of places to move. Here, 100 has 2 zeros, so, we move all values to the left by 2 places. Then, we put zeros in the empty cells on the right:


thousands hundreds tens ones tenthshundredths thousandths
\bbox[3pt,Gainsboro]{\color{blue}0} \bbox[3pt,Gainsboro]{\color{blue}8} \bbox[3pt,Gainsboro]{\color{blue}3} \bbox[3pt,Gainsboro]{\color{blue}1} . \bbox[3pt,Gainsboro]{\color{blue}5} \bbox[3pt,Gainsboro]{\color{blue}0} \bbox[3pt,Gainsboro]{\color{blue}0}
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Practice: Multiplying Decimals by One Hundred

Complete the place value chart below for $34.7 \times 100.$

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Practice: Multiplying Decimals by One Hundred

Complete the place value chart below for $7.4 \times 100.$

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Practice: Multiplying Decimals by One Hundred

Complete the place value chart below for $1.8 \times 100.$

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Example: Multiplying Decimals by One Thousand

Complete the place value chart below for 7.2\times 1000.


ten thousands thousands hundreds tens ones tenths
\fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} \fbox{[math]\phantom{0}[/math]} . \fbox{[math]\phantom{0}[/math]}
EXPLANATION

Step 1. Write down the place value chart for 7.2\mathbin{:}

ten thousands thousands hundreds tens ones tenths
0 0 0 0 7 . 2

Step 2. We determine the number of places to move.

Notice that 1,000 has a block of 3 zeros:

1,\!\!\!\underbrace{\color{blue}000}_{\large\text{[math]3[/math] zeros}}\!\!\!.\:

So, we move all values to the left by 3 places. Then, we place zeros in the empty cells on the right.

ten thousands thousands hundreds tens ones tenths
\bbox[3pt,Gainsboro]{\color{blue}0} \bbox[3pt,Gainsboro]{\color{blue}7} \bbox[3pt,Gainsboro]{\color{blue}2} \bbox[3pt,Gainsboro]{\color{blue}0} \bbox[3pt,Gainsboro]{\color{blue}0} . \bbox[3pt,Gainsboro]{\color{blue}0}
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Practice: Multiplying Decimals by One Thousand

Complete the place value chart below for $1.93\times 1,000.$

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Practice: Multiplying Decimals by One Thousand

Complete the place value chart below for $0.8 \times 1,000.$

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Practice: Multiplying Decimals by One Thousand

Complete the place value chart below for $41.3 \times 1,000.$

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