It's sometimes helpful to switch between different place value representations of the same number.
For example, let's consider the following three-digit whole number:
The place value chart for this number is shown below:
| hundreds | tens | ones |
So, represents " hundreds, tens and ones."
However, also represents ones, because
Therefore, " hundreds, tens and ones" is the same as ones.
In math, we use the word equivalent (e-quiv-a-lent) to mean "the same as."
" thousands and tens" is equivalent to how many ones?
Let's write " thousands and tens" in a place value chart:
| thousands | hundreds | tens | ones |
Therefore, " thousands and tens" is equivalent to " ones."
"$3$ tens" is equivalent to
|
a
|
$30$ ones |
|
b
|
$13$ ones |
|
c
|
$3$ ones |
|
d
|
$300$ ones |
|
e
|
$31$ ones |
"$3$ thousands and $5$ ones" is equivalent to
|
a
|
$3,050$ ones |
|
b
|
$350$ ones |
|
c
|
$3,500$ ones |
|
d
|
$3,005$ ones |
|
e
|
$305$ ones |
" ones" is equivalent to how many thousands and hundreds?
Let's write the number in a place value chart:
| thousands | hundreds | tens | ones |
Therefore, " ones" is equivalent to " thousands and hundreds."
"$18$ ones" is equivalent to
|
a
|
$8$ ones |
|
b
|
$18$ hundreds |
|
c
|
$18$ tens |
|
d
|
$1$ hundred and $8$ ones |
|
e
|
$1$ ten and $8$ ones |
"$132$ ones" is equivalent to
|
a
|
$132$ tens |
|
b
|
$1$ hundred and $32$ tens |
|
c
|
$1$ hundred, $3$ tens, and $2$ ones |
|
d
|
$13$ hundreds and $2$ ones |
|
e
|
$132$ hundreds |
Let's think about the different place value representations of the number
The number represents " ones." This has the following place value representation:
tens ones The number also represents " ten and ones." This has the following place value representation:
tens ones
To go from the first representation to the second, we moved the first digit one place to the left (into the tens column).
The equivalence of the two place value charts simply means that a list of individual objects (i.e., ones) can be viewed as a single block of length (i.e., ten):
Let's now think about the different place value representations of
As it's written, its place-value chart is as follows:
| hundreds | tens | ones |
There are two digits in the "tens" column, so we move the first digit to the "hundreds" column:
| hundreds | tens | ones |
Therefore, " tens" is equivalent to " hundred."
The equivalence of the two place value charts simply means that a list of individual blocks of length (i.e., tens) can be viewed as a single block with objects (i.e., hundred):
Proceeding similarly, we're able to deduce the following pattern:
ones is equivalent to ten
tens is equivalent to hundred
hundreds is equivalent to thousand
thousands is equivalent to ten-thousand
ten-thousands is equivalent to hundred-thousand
hundred-thousands is equivalent to million
By combining this pattern with our understanding of expanded form, we can find more equivalent representations of numbers.
Let's see an example.
" thousands" is equivalent to how many ten-thousands and thousands?
Since we can write thousands as
Now, since thousands is the same as ten-thousand, the statement above is equal to
Therefore, " thousands" is equivalent to " ten-thousand and thousands".
"$13$ tens" is equivalent to
|
a
|
$1$ thousand and $3$ ones |
|
b
|
$1$ hundred and $3$ tens |
|
c
|
$3$ tens and $1$ ones |
|
d
|
$1$ hundred and $3$ ones |
|
e
|
$13$ hundreds |
"$12$ hundreds and $5$ tens" is equivalent to
|
a
|
$1$ thousand and $5$ ones |
|
b
|
$12$ thousands and $5$ tens |
|
c
|
$1$ thousand, $2$ tens and $5$ ones |
|
d
|
$1$ thousand, $2$ hundreds, and $5$ tens |
|
e
|
$1$ thousand and $5$ tens |