Video Transcript
In this video, we will learn how to
use patterns to investigate the effects of multiplying decimals by increasing and
decreasing powers of 10. We will focus on what happens to a
decimal when we multiply it by 10, 100, and 1000. Let’s begin by recalling what
happens when we multiply integers or whole numbers by powers of 10. If we multiply the number seven by
10, 100, and 1000, what happens? We know that seven multiplied by 10
is equal to 70, seven multiplied by 100 is 700, and seven multiplied by 1000 is
7000. This pattern suggests that we just
add zeros to our number.
When multiplying by 10, we add one
zero; 100, two zeros; and 1000, three zeros. This pattern suggests that when we
multiply by 10, 100, and 1000, we simply add one, two, and three zeros,
respectively. This would mean that 4.3 multiplied
by 10 is equal to 4.30. We know, however, that this cannot
be correct as 4.3 is the same as 4.30. Our number has no got bigger. Another common mistake is to write
that 4.3 multiplied by 10 is equal to 40.3, adding a zero between the ones column
and the decimal point. This is also incorrect. Whilst we have multiplied the four
by 10, we have not multiplied the 0.3 by 10. Let’s go back to our first example
and see what was happening in terms of place value.
In the place-value table drawn, we
have a thousands, hundreds, tens, and ones or units column. We wish to multiply the seven ones
or seven units by 10. We know that the answer is 70. The seven in the ones column has
moved to the tens column. It has moved one place to the
left. We need to add in a zero as a
placeholder so that seven multiplied by 10 is equal to 70. When multiplying seven by 100, the
seven needs to move two places to the left. It moves from the ones column to
the hundreds column. This time, we need to add in two
placeholders. Seven multiplied by 100 is 700. In the same way, when multiplying
by 1000, the digit seven moves three places to the left. Seven multiplied by 1000 is
7000.
We can mirror this process when
multiplying a decimal by 10, 100, or 1000. This time, our initial number 4.3
has four ones and three tenths. When multiplying by 10, all of the
digits move one place to the left. The four moves from the ones column
to the tens column and the three moves from the tenths column to the ones
column. Whilst we can put a zero as a
placeholder in the tenths column, this is not required if it is after the decimal
point. 4.3 multiplied by 10 is equal to
43.0, which is the same as 43. When multiplying by 100, the digits
moved two places to the left. The four is now in the hundreds
column and the three is in the tens column.
This time, we do need a placeholder
in the ones column, and we have the option of putting one in the tenths column. 4.3 multiplied by 100 is 430.0 or
just 430. When multiplying by 1000, we move
all the digits three places to the left. Adding placeholders, we see that
4.3 multiplied by 1000 is 4300. We will now look at some questions
where we need to multiply decimals by powers of 10.
The following method shows how to
find 0.4 multiplied by 10 using a place-value table. Follow the same method to find 0.1
multiplied by 10.
We recall that when multiplying any
number, we move all of our digits one place to the left. In the example 0.4 multiplied by
10, the four is in the tenths column. We then multiply this by 10. The four then moves from the tenths
column to the ones column. This gives us four in the ones
column and nothing in the tenths column. 0.4 multiplied by 10 is therefore
equal to 4.0 or just four. When there are only zeros after the
decimal point, we don’t need to read or write them.
We want to use the place-value
table to multiply 0.1 by 10. 0.1 has a zero in the ones column
and a one in the tenths column. We need to move this one from the
tenths column to the ones column as we are multiplying by 10. This means that we end up with one
in the ones column and nothing in the tenths column. 0.1 multiplied by 10 is equal to
1.0, which once again we can just write as one.
In our next question, we will
multiply a decimal by 10 where the place-value table isn’t given.
Calculate 52.35 multiplied by
10.
We recall that when multiplying any
number by 10, all of our digits move one place to the left. In this question, we need to
multiply 52.35 by 10. The five is in the tens column, the
two in the ones column, the three is in the tenths column, and the second five is in
the hundredths column. The decimal point stays in the same
place and all of our digits move one place to the left. The five is now in the hundreds
column. The two is in the tens column. The three is in the ones
column. And finally, the five is in the
tenths column. Whilst we could put a placeholder
in the hundredths column, this is not required, as we are after the decimal
point. 52.35 multiplied by 10 is therefore
equal to 523.5.
In our next two questions, we will
complete tables by multiplying by 10, 100, and 1000.
Complete the following table.
We need to multiply the number
12.24 firstly by 10. We then need to multiply it by
100. And finally, we want to multiply
12.24 by 1000. Let’s begin by setting up a
place-value table. Our initial number is 12.24. We have a one in the tens column, a
two in the ones column, a two in the tenths column, and a four in the hundredths
column. When multiplying a number by 10,
all of our digits move one place to the left. This means that the one moves to
the hundreds column, the two to the tens, the second two to the ones, and the four
to the tenths. 12.24 multiplied by 10 is equal to
122.4.
Let’s now consider what happens
when we multiply by 100. Multiplying by 100 moves all of our
digits two places to the left. This time, the one is in the
thousands column, the two in the hundreds and tens columns, and the four in the ones
column. Whilst we could put one or two
zeros as placeholders after the decimal point, these are not required. 12.24 multiplied by 100 is equal to
1224.
Finally, we will multiply 12.24 by
1000. Multiplying by 1000 moves all of
our digits three places to the left. This is because there are three
zeros in 1000. We are multiplying by 10, by 10
again, and then by 10 again. Our one this time has moved to the
ten thousands column, the first two to the thousands column, the second two to the
hundreds column, and the four to the tens column. This time, we do need to include a
placeholder. We need to place a zero in the ones
column, as this is before the decimal point. 12.24 multiplied by 1000 is equal
to 12240. The three missing numbers in the
table are 122.4, 1224, and 12,240.
Complete the following table.
In this question, we need to
multiply the decimal 13.7319 by 10, 100, and 1000. We recall that when multiplying by
10, all of our digits move one place to the left. This means that 13.7319 multiplied
by 10 is equal to 137.319. 100 is the same as 10 multiplied by
10. This can also be written as 10
squared. When multiplying by 10 squared or
100, all of our digits move two places to the left. This means that the one in the tens
column moves to the thousands column, the three moves from the ones column to the
hundreds column, and so on. 13.7319 multiplied by 100 is equal
to 1373.19. 1000 is equal to 10 cubed as it is
equal to 10 multiplied by 10 multiplied by 10.
This time, we need to move all the
digits three places to the left. Another way of thinking about this
is to move all our digits from the previous answer one further place to the
left. 13.7319 multiplied by 1000 is
therefore equal to 13731.9. Our three answers are 137.319,
1373.19, and then 13731.9. We have multiplied 13.7319 by 10,
by 10 again, and then finally, by 10 again. Each time we multiply by 10, we
move all of our digits one place to the left.
In our last question, we will solve
a real-world problem involving multiplying by powers of 10.
The height of Mount Everest, in
meters, can be found by multiplying 8.848 by 1000. Find the height of Mount
Everest.
This is an interesting problem as
it is based on a real-life fact. We are told that if we multiply
8.848 by 1000, we can calculate the height of Mount Everest in meters. We recall that one kilometer is
equal to 1000 meters. This means that in order to convert
from kilometers to meters, we need to multiply by 1000. But this means that an interesting
fact in this question before we start is that 8.848 is the height of Mount Everest
in kilometers. Even though the question doesn’t
say this, what it is getting us to do is to convert the height of Mount Everest from
kilometers to meters. We do this by multiplying 8.848 by
1000.
We know that 1000 is the same as 10
multiplied by 10 multiplied by 10. This can also be written as 10 to
the third power or 10 cubed. We recall that multiplying by 10
moves all of our digits one place to the left. This means that if we’re
multiplying by 10 then 10 again and 10 again or 10 cubed, we move all of our digits
three places to the left. The eight that is in the ones
column moves to the thousands column, the eight in the tenths column to the hundreds
column, the four moves to the tens column, and the final eight moves to the ones
column. 8.848 multiplied by 1000 is
8848. Whilst we are actually moving the
digits to the left, you might have also seen this as moving the decimal point three
places to the right. Either way, we get the answer that
the height of Mount Everest in meters is 8848 meters.
We will now summarize the key
points from this video. We saw in this video that when
multiplying by 10, we move all of our digits one place to the left. When multiplying by 100, we move
all the digits two places to the left. This is because 100 is equal to 10
multiplied by 10. We are multiplying our number by 10
and then by 10 again. This can also be written as 10
squared. The exponent or power — in this
case, two — tells us how many places we need to move the digits to the left.
When multiplying a number by 1000,
we move all of our digits three places to the left. This is because 1000 is equal to 10
multiplied by 10 multiplied by 10, or 10 cubed. As the exponent is three, we move
the digits three places to the left. In this video, we weren’t
multiplying by any number greater than 1000. However, this pattern holds for any
power of 10. When multiplying by 10 to the 𝑛th
power, we move all the digits 𝑛 places to the left. For example, when multiplying by 10
to the power of six or one million, we move all the digits six places to the
left.
