Ten Times as Much and One Tenth as Much | Grade 5 Math | LevelKite

Look at the numbers $6$ and $60$ . Both use the digit $6$ , but the $6$ does not stand for the same amount in each number.

A digit is one of the ten symbols from $0$ to $9$ that you use to write numbers. A place is the position of a digit in a number, such as the ones place or the tens place. The value of a digit is the amount it stands for in its place.

In $6$ , the digit $6$ is in the ones place. It stands for $6$ ones, so its value is $6$ . In $60$ , the digit $6$ is in the tens place. It stands for $6$ tens, so its value is $60$ .

So the same digit can stand for different amounts, and its place makes the difference. In this lesson, you will find out exactly how the value changes when a digit moves one place to the left or one place to the right.

Each row moves the digit 6 one place to the right

Place of the digit 6	Value in words	Value as a number
thousands place	$6$ thousands	$6{,}000$
hundreds place	$6$ hundreds	$600$
tens place	$6$ tens	$60$
ones place	$6$ ones	$6$

Read down the last column of the table: $6{,}000$ , $600$ , $60$ , $6$ . Then read it from the bottom up: $6$ , $60$ , $600$ , $6{,}000$ . Do you see a pattern each way?

Going up one row, the digit $6$ moves one place to the left, and its value becomes $10$ times as much. That means $10$ equal groups of the amount you had. For example, $10$ groups of $6$ make $60$ , so $60$ is $10$ times as much as $6$ .

Going down one row, the digit $6$ moves one place to the right, and its value becomes one tenth as much. One tenth, which you write as $\frac{1}{10}$ , is one of $10$ equal parts. To find $\frac{1}{10}$ of an amount, split it into $10$ equal parts and keep one part. That is the same as dividing by $10$ . For example, $60$ split into $10$ equal parts gives $6$ in each part, so $6$ is $\frac{1}{10}$ of $60$ .

The same is true for $6$ hundreds and $6$ tens. In words, $600$ is $10$ times as much as $60$ , and $60$ is $\frac{1}{10}$ of $600$ . In numbers, it looks like this:

600 = 10 \times 60 \qquad \text{and} \qquad 60 = 600 \div 10

Key concept

Each place is worth 10 times the place to its right

A digit in one place stands for $10$ times what it stands for in the place to its right. It stands for $\frac{1}{10}$ of what it stands for in the place to its left. This is true for any two places that are next to each other.

Worked example

The digit 7 in two places that are side by side

In $7{,}000$ , the digit $7$ is in the thousands place. In $700$ , the digit $7$ is in the hundreds place. How does the value of the $7$ in $7{,}000$ compare with the value of the $7$ in $700$ ?

Write what the digit 7 stands for in each number, so that you can compare the two amounts. In 7,000 it stands for 7 thousands. In 700 it stands for 7 hundreds.
$7 \text{ thousands} = 7{,}000 \qquad 7 \text{ hundreds} = 700$
The thousands place is one place to the left of the hundreds place. One place to the left means 10 times as much, so 7 thousands is 10 times as much as 7 hundreds. You can check it: 10 groups of 700 make 7,000.
$7{,}000 = 10 \times 700$
Now compare in the other direction. The hundreds place is one place to the right of the thousands place. One place to the right means one tenth as much, and you find one tenth by dividing by 10.
$7{,}000 \div 10 = 700$

Answer

The $7$ in $7{,}000$ stands for $10$ times as much as the $7$ in $700$ . The $7$ in $700$ stands for $\frac{1}{10}$ of what the $7$ in $7{,}000$ stands for.

The same pattern tells you how the places themselves are related. Think about what the digit $1$ stands for in each place, starting at the ones place and moving left: $1$ , $10$ , $100$ , $1{,}000$ , $10{,}000$ . Each amount is $10$ times the one before it. So it always takes $10$ of one place to make $1$ of the next place to the left:

$10$ ones make $1$ ten.
$10$ tens make $1$ hundred.
$10$ hundreds make $1$ thousand.
$10$ thousands make $1$ ten thousand.

Going the other way, $1$ hundred is $\frac{1}{10}$ of $1$ thousand, because $1{,}000 \div 10 = 100$ .

You can link two of these facts to relate places that are farther apart. The next example shows how.

Worked example

From hundreds to ten thousand

How many hundreds make ten thousand?

Start at the hundreds place and move one place to the left, to the thousands place. Each place is worth 10 times the place to its right, so 10 hundreds make 1 thousand.
$10 \times 100 = 1{,}000$
Move one more place to the left, to the ten thousands place. For the same reason, 10 thousands make 1 ten thousand.
$10 \times 1{,}000 = 10{,}000$
Link the two facts. Ten thousand is 10 thousands, and each of those thousands is 10 hundreds. That makes 10 groups of 10 hundreds.
$10 \times 10 = 100$
Check the answer. If 100 hundreds make ten thousand, then 100 groups of 100 must equal 10,000. They do.
$100 \times 100 = 10{,}000$

Answer

It takes $100$ hundreds to make ten thousand.

The pattern does not stop at the ones place. Right after the ones place comes the decimal point, the dot that separates the ones place from the places for parts of $1$ . Each place after it is still worth $\frac{1}{10}$ of the place to its left.

Tenths place. Split $1$ into $10$ equal parts. Each part is one tenth, or $\frac{1}{10}$ . You already write it as $0.1$ .
Hundredths place. Split each tenth into $10$ equal parts. Now $1$ has $10 \times 10 = 100$ equal parts. Each part is one hundredth, or $\frac{1}{100}$ . You already write it as $0.01$ .
Thousandths place. Split each hundredth into $10$ equal parts. Now $1$ has $100 \times 10 = 1{,}000$ equal parts. Each part is one thousandth, or $\frac{1}{1{,}000}$ . You write it as $0.001$ , with the $1$ in the third place after the decimal point.

The table below follows the digit $6$ into these places. One tenth of $1$ is $1$ tenth, so one tenth of $6$ ones is $6$ tenths.

The digit 6 keeps moving right, past the decimal point

Place of the digit 6	Value in words	Value as a number	Compared with one place to the left
ones place	$6$ ones	$6$	$6$ is $\frac{1}{10}$ of $60$
tenths place	$6$ tenths	$0.6$	$0.6$ is $\frac{1}{10}$ of $6$
hundredths place	$6$ hundredths	$0.06$	$0.06$ is $\frac{1}{10}$ of $0.6$
thousandths place	$6$ thousandths	$0.006$	$0.006$ is $\frac{1}{10}$ of $0.06$

Now read the table from the bottom up. Going up one row still moves the digit $6$ one place to the left, so each value is $10$ times as much as the value below it. This works because it takes $10$ thousandths to make $1$ hundredth, $10$ hundredths to make $1$ tenth, and $10$ tenths to make $1$ .

So $0.06$ is $10$ times as much as $0.006$ , $0.6$ is $10$ times as much as $0.06$ , and $6$ is $10$ times as much as $0.6$ . The digit $6$ crossed the decimal point, and the pattern stayed the same.

Worked example

One place each way from the hundredths place

Answer both questions. $0.04$ is $10$ times as much as what number? $0.04$ is $\frac{1}{10}$ of what number?

Name the value in words. In 0.04 the digit 4 is in the hundredths place, so 0.04 is 4 hundredths.
$0.04 = 4 \text{ hundredths}$
In the first question, 0.04 is 10 times as much as the other number, so the other number is one tenth of 0.04. One tenth as much means one place to the right. The 4 moves from the hundredths place to the thousandths place and stands for 4 thousandths. Zeros fill the tenths and hundredths places.
$4 \text{ thousandths} = 0.004$
Check the first answer. It takes 10 thousandths to make 1 hundredth, so 10 groups of 4 thousandths make 4 hundredths.
$10 \times 0.004 = 0.04$
In the second question, 0.04 is one tenth of the other number, so the other number is 10 times as much as 0.04. Ten times as much means one place to the left. The 4 moves from the hundredths place to the tenths place and stands for 4 tenths.
$4 \text{ tenths} = 0.4$
Check the second answer. It takes 10 hundredths to make 1 tenth, so 10 groups of 4 hundredths make 4 tenths. That means 4 hundredths is one tenth of 4 tenths.
$10 \times 0.04 = 0.4$

Answer

The number $0.04$ is $10$ times as much as $0.004$ , and it is $\frac{1}{10}$ of $0.4$ .

Here is the idea to keep. A digit in any place stands for $10$ times what it would stand for one place to the right. It stands for $\frac{1}{10}$ of what it would stand for one place to the left. The same pattern links thousands and hundreds, ones and tenths, and hundredths and thousandths. The decimal point does not change it.

Next, you will compare what the same digit stands for in two places of one number, such as the digit $4$ in $4.44$ .