Introduction

How much bigger is the size of the digit {\color{blue}9} in 3.{\color{blue}9}6 than the size of the digit {\color{red}9} in 4.3{\color{red}9}?

Let's start by writing down the place values for 3.{\color{blue}9}6. We see that the \color{blue}9 is in the tenths place.


ones tenths hundredths
3 . \color{blue}9 6

Now, we write the place values for 4.3{\color{red}9}. Here, the \color{red}9 is in the hundredths place.


ones tenths hundredths
4 . 3 \color{red}9

To compare the two digits, we move one step to the left in the place value chart. Each step to the left is an increase by a factor of 10 :


ones tenths hundredths
. \color{blue}9 \color{red}9

Therefore, the value of the digit \color{blue}9 in the tenths place is 10 times greater than the value of the digit \color{red}9 in the hundredths place.

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Example: Comparing Place Values by Moving Left One Step

How many times greater is the value of the digit 6 in 2.361 than the value of the digit 6 in 2.856?

EXPLANATION

Let's start by writing down the place values for 2.3{\color{blue}6}1.


ones tenths hundredths thousandths
2 . 3 \color{blue}6 1

Now, we write the place values for 2.85{\color{red}6}.


ones tenths hundredths thousandths
2 . 8 5 \color{red}6

To compare the two digits, we move 1 step to the left in the place value chart. Each step to the left is an increase by a factor of 10 :


ones tenths hundredths thousandths
. \color{blue}6 \color{red}6

Therefore, the value of the digit \color{blue}6 in the hundredths place is 10 times greater than the value of the digit \color{red}6 in the thousandths place.

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Practice: Comparing Place Values by Moving Left One Step

How many times greater is the value of the digit $9$ in $9.47$ than the value of the digit $9$ in $0.92?$

a
 $9$
b
 $100$
c
 $10$
d
 $900$
e
 $90$
Practice: Comparing Place Values by Moving Left One Step

How many times greater is the value of the digit $7$ in $9.74$ than the value of the digit $7$ in $6.97?$

a
 $\dfrac 1 7$
b
 $\dfrac{1}{10}$
c
 $10$
d
 $7$
e
 $100$
Comparing Place Values by Moving Left Multiple Steps

How many times greater is the size of the digit \color{blue}2 in {\color{blue}2}3.5 than the size of the digit \color{red}2 in 98.{\color{red}2}?

Let's start by writing down the place values for {\color{blue}{2}}3.5 :


tens ones tenths
\color{blue}2 3 . 5

Then we write the place values for 98.{\color{red}{2}} :


tens ones tenths
9 8 . \color{red}2

Now, let's jump from the tenths place to the tens place. Remember that the decimal does not correspond to a place value, so we jump right over it.


tens ones tenths
\color{blue}2 . \color{red}2

We have that

\underbrace{10\times10}_{\color{black}2\,\text{steps}}=100.

Therefore, the value of the digit \color{blue}2 in the tens place is 100 times larger than the value of the digit \color{red}2 in the tenths place.

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Example: Comparing Place Values by Moving Left Multiple Steps

Find a number in which the digit 9 has a value that is 1,000 times larger than the value of the digit 9 in 313.9.

EXPLANATION

Let's start by writing down the place values for 313.9.


hundreds tens ones tenths
3 1 3 . \color{red}9

We need to find a digit that is 1,000 times greater than this {\color{red}{9}}.

Notice that

1,000 = \underbrace{10\times 10\times 10}_{3\,\text{steps}}.

Therefore, we move this digit \color{red}9 three steps to the left in the place value chart. This gives:


hundreds tens ones tenths
\color{blue}9 .

Thus, the required number is any number that has the digit \color{blue}9 in the hundreds place. Some examples include:

{\color{blue}9}15.74, \qquad 2,{\color{blue}9}80.2, \qquad 32,{\color{blue}9}01.56.

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Practice: Comparing Place Values by Moving Left Multiple Steps

How many times greater is the value of the digit $5$ in $52.172$ than the value of the digit $5$ in $34.158?$

a
 $100$
b
 $500$
c
 $10$
d
 $1,000$
e
 $50$
Practice: Comparing Place Values by Moving Left Multiple Steps

In a place value chart, how many times larger is the value of the digit $9$ in the tens place than the value of the digit $9$ in the tenths place?

a
 $900$
b
 $10$
c
 $1,000$
d
 $90$
e
 $100$
Comparing Place Values With Decimals by Moving Right

The digit \color{red}3 in 9.15{\color{red}3} is smaller than the digit \color{blue}3 in 4.7{\color{blue}3}, but by how much?

Let's start by writing down the place values for 9.15{\color{red}{3}}.


ones tenths hundredths thousandths
9 . 1 5 \color{red}3

Then, we write the place values for 4.7{\color{blue}{3}}.


ones tenths hundredths thousandths
4 . 7 \color{blue}3 0

Now, we move from tenths place to thousands place. Each step to the right is a decrease by a factor of 10.


ones tenths hundredths thousandths
. \color{blue}3 \color{red}3

Therefore, the value of the digit \color{red}3 in the thousandths place is 10 times smaller than the value of the digit \color{blue}3 in the hundredths place.

Alternatively, we can also say that the value of the digit \color{red}3 in the thousandths place is \dfrac{1}{10} the value of the digit \color{blue}3 in the hundredths place.

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Example: Comparing Place Values by Moving Right One Step

How many times smaller is the digit \color{blue}6 in 11.{\color{blue}6}2 than the digit \color{red}6 in {\color{red}6}.751?

EXPLANATION

Let's start by writing down the place values for 11.{\color{red}6}2.


tens ones tenths hundredths thousandths
1 1 . \color{red}6 2

Now we write the place values for {\color{blue}6}.751.


tens ones tenths hundredths thousandths
\color{blue}6 . 7 5 1

Now let's jump from the ones place to the tenths place.


tens ones tenths hundredths thousandths
\color{blue}6 . \color{red}6

Therefore, the value of the digit \color{red}6 in the tenths place is 10 times smaller than the value of the digit \color{blue}6 in the ones place.

Alternatively, we can also say that the value of the digit \color{red}6 in the tenths place is \dfrac{1}{10} the value of the digit \color{blue}6 in the ones place.

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Practice: Comparing Place Values by Moving Right One Step

What is missing in the following sentence?

$\qquad$ The size of the digit $6$ in $13.6$ is $\underline{\phantom{{}^{000000000000000000000000000}}}$ the size of the digit $6$ in $6.2.$

a
 $\dfrac{1}{10}$
b
 $\dfrac{1}{100}$
c
 $\dfrac{1}{60}$
d
 $\dfrac{1}{1,000}$
e
 $\dfrac{1}{6}$
Practice: Comparing Place Values by Moving Right One Step

How many times smaller is the value of the digit $4$ in $1.4$ than the value of the digit $4$ in $4.8?$

a
 $10$
b
 $4$
c
 $400$
d
 $100$
e
 $40$
Comparing Place Values With Decimals by Moving Right Multiple Steps

How many times smaller is the value of the digit 5 in the tenths place than the value of the digit 5 in the tens place?

Let's move from the tens place to the tenths place in the place value chart:


hundreds tens ones tenths hundredths thousandths
\color{blue}5 . \color{red}5

We see that the value of the digit 5 in the tenths place is 100 times smaller than the value of the digit 5 in the tens place, because

\underbrace{10\times10}_{\color{black}2\,\text{steps}}=100.

This is the same as saying that the size of the digit \color{red}5 in the tenths place is \dfrac{1}{100} the size digit \color{blue}5 in the tens place.

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Example: Comparing Place Values by Moving Right Multiple Steps

What is missing in the following sentence?

\qquad The digit 4 in 57.934 is \underline{\phantom{{}^{000000000000000000000000000}}} the value of the digit 4 in 0.416.

EXPLANATION

Let's start by writing down the place values for 57.93{\color{red}{4}}.


tens ones tenths hundredths thousandths
5 7 . 9 3 \color{red}4

Then, we write the place values for 0.{\color{blue}{4}}16.


tens ones tenths hundredths thousandths
0 . \color{blue}4 1 6

Now let's jump from the tenths place to the thousandths place.


tens ones tenths hundredths thousandths
. \color{red}4 \color{blue}4

We see that the value of the digit 4 in the thousandths place is

\underbrace{10\times 10}_{\color{black}2\,\text{steps}}=100

times smaller than the value of the digit 4 in the tenths place.

Therefore, the complete sentence is "The digit 4 in 57.934 is \dfrac{1}{100} the value of the digit 4 in 0.416. "

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Practice: Comparing Place Values by Moving Right Multiple Steps

In which of the following numbers does the digit $6$ have a size that is $\dfrac{1}{100}$ the size of the digit $6$ in $76.381?$

a
 $42.647$
b
 $64.371$
c
 $95.136$
d
 $18.56$
e
 $671.1$
Practice: Comparing Place Values by Moving Right Multiple Steps

How many times smaller is the value of the digit $8$ in $64.038$ than the value of the digit $8$ in $29.8?$

a
 $8$
b
 $100$
c
 $10$
d
 $800$
e
 $1,000$
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