Lesson Video: Dividing Decimals by Powers of Ten | Nagwa Lesson Video: Dividing Decimals by Powers of Ten | Nagwa

Lesson Video: Dividing Decimals by Powers of Ten Mathematics • 5th Grade

In this video, we will learn how to use patterns to investigate the effects of dividing numbers by increasing and decreasing powers of 10.

16:24

Video Transcript

In this video, we’re going to learn how to use patterns to investigate the effects of dividing numbers by increasing and decreasing powers of 10, such as 10, 100, and 1000. Let’s remind ourselves how we multiply by various powers of 10. Let’s take the number 2.7. We know of course that 2.7 times one is equal to 2.7. This is sometimes called the multiplicative identity property. Then, we also know that 2.7 multiplied by 10 is equal to 27. What happens when we multiply by 10 is that we move every single digit in the number one place to the left. So the two that was in the ones column moves to the tens column. And the seven that was in the tens column moves to the ones column. That leaves us with two tens and seven ones, which is of course 27.

Similarly, if we were to multiply 2.7 by 100, we’d move every single digit in the number 2.7 two places to the left. And so the two moves from the ones column to the hundreds column. The seven moves from the tenths column to the tens column. And then of course we add a zero here as a placeholder. We have two hundreds, seven tens, and zero ones, giving us 270.

We continue using this pattern. This time, if we’re going to multiply 2.7 by 1000, we need to move all of the digits to the left three places. So the number two moves from the ones column to the thousands column. The seven moves from the tenths to the one hundreds. And then we add zeros as placeholders. So we now have two thousands, seven hundreds, zero tens, and zero ones. And that gives us 2700.

So let’s continue this pattern and see if we can work out how we divide by powers of 10. Now, of course, we know that multiplication and division are inverse operations to one another. That is, they are the opposite. Performing multiplication undoes a division and vice versa. And so we can say that dividing by 10 undoes the operation of multiplying by 10 such that 27 divided by 10 gives us 2.7. Similarly, dividing by 100 is the opposite to multiplying by 100. So 270 divided by 100 takes us back to 2.7, and 2700 divided by 1000 will also take us back to 2.7.

And so that should help us spot a pattern. We saw that when we multiplied by 10, we moved the digits to the left one space. And this means then that if I divide a number by 10, I’m going to move the digits to the right one space. And so our place value chart this time is going to look a little bit different. We’re going to be moving all the digits to the right. And so we need to add a few extra columns. I’ve added the hundredths, thousandths, and ten thousandths column.

Moving every single digit in our number to the right one space, and we see that the seven moves into the hundredths column. Then the two moves into the tenths column. And we’ll need a zero in front of the decimal point as a placeholder. We now have zero ones, two tenths, and seven hundredths, and so 2.7 divided by 10 is 0.27.

We’re going to now use the same sort of idea to divide 2.7 by 100. We saw that to multiply a number by 100, we moved every single digit to the left two spaces. This means given that division is the opposite operation to multiplication, we’re going to move every single digit to the right two places when we divide by 100. And so our seven moves once, twice into the thousandths column. Similarly, the two moves once, twice, and it ends up in the hundredths column. We’ll need a couple of zeros as placeholders, and we find that 2.7 divided by 100 is 0.027.

Let’s do this one more time to find the value of 2.7 divided by 1000. Remember, when we multiplied by 1000, we moved all of the digits to the left three spaces. And so we now need to do the opposite. We need to move all of the digits in the number 2.7 three places to the right. The seven moves once, twice, three times, and it ends up in the ten thousandths column. The two also moves once, twice, three times, and the two ends up in the thousandths column. We’ll add the zeros as placeholders. And so we see 2.7 divided by 1000 is 0.0027.

Now, one thing to be really aware of here is we’ve moved the digits, not the decimal point. If we consider the place value table, it doesn’t make a lot of sense to move the decimal point. If we move the decimal point to here, then we have a decimal point between the tenths column and the hundredths column, which we know doesn’t make sense at all. So we always want to consider multiplying and dividing by powers of 10 in terms of moving the digits.

Let’s recap what we’ve found so far. We’ll think about dividing by powers of 10 in terms of the place value of our digits. When we divide by 10, we move all of the digits one space to the right. When we divide by 100, we move all of the digits two spaces to the right. And when we divide by 1000, we move all of the digits three spaces to the right.

Now, we will a little bit later in this video look at the relationship between dividing by 10, 100, and 1000. But for now, if you’re struggling to remember how many places to move the digits, look at the number of zeros in the number you’re dividing by. So when we’re dividing by 10, we have one zero in that number and we move the digits one space to the right. 100 has two zeros and 1000 has three zeros. And the number of zeros tells us how many spaces we move our digits.

Let’s have a look at the application of these ideas.

Find the result of 234.5 divided by 10.

We have a process that helps us divide by powers of 10. We think about the division in terms of the place value of the numbers in the dividend. The dividend in this number of course is the number that we’re going to be dividing. It’s 234.5. And we know that to divide by 10, we move all of the digits in our dividend one space to the right. So our place value chart looks a little something like this. Our number has two hundreds, three tens, four ones, and five tenths.

We’re going to move every single one of these digits one space right. We’ll begin with the five. The five moves from the tenths column to the hundredths column. Then we take the four and we move it from the ones column to the tenths column. The three moves from the tens to the ones. And finally, the two moves from the hundreds to the tens. And now when we divide 234.5 by 10, we end up with two tens, three ones, four tenths, and five hundredths. And so we say that 234.5 divided by 10 is 23.45.

Let’s have a look at another example.

What is 668.7 divided by 1000?

We know that to divide a number by 1000, we take all of the digits in our dividend, that’s the number 668.7 here, and we move them three places to the right. When we pop the number 668.7 into a place value chart, it looks a little something like this. We have six hundreds, six tens, eight ones, and seven tenths. We’re going to move every single digit three places to the right.

So let’s begin with the seven. We move it once, twice, three times, and it goes all the way over into the ten thousandths column. So the result of 668.7 divided by 1000 will have seven ten thousandths. We then do the same with the number eight. One, two, three spaces to the right brings it to the thousandths column. We repeat this process with the remaining two digits. We’ve got the number six, which moves all the way over to the hundredths column, and we have another six. And when we move that three digits to the right, we end up in the tenths column.

Now, of course, it doesn’t make sense to just write .6687. And we know we have zero ones. So we add the zero sort of as a placeholder. And so we see that 668.7 divided by 1000 is 0.6687.

In our next example, we’ll look at the relationship between dividing by various powers of 10.

Complete the following table. And then we have a table with the number 231.72 and headings number divided by 10, number divided by 100, and number divided by 1000.

So let’s remind ourselves how we divide by each of these numbers. When we divide by 10, we move our digits to the right one space. When we divide by 100, we move them in the same direction, but this time two spaces. And when we divide by 1000, we move the digits to the right three spaces. So we’re going to pop the number 231.72 into a place value table. This number has two hundreds, three tens, one one, seven tenths, and two hundredths.

We’ll begin then by dividing it by 10. And when we do, we’re going to move every single digit one space to the right. So the two moves from the hundredths column to the thousandths column. Then we move the seven. It moves from the tenths column to the hundredths column. In a similar way, we move the one from the ones column to the tenths column. And we continue with the final two digits. And so we find that when we divide our number by 10, we end up with 23.172.

So we’re now going to divide it by 100. Now, when we divide by 100, we move every single digit two spaces to the right. The thing is when we divided by 10, we’d already moved the digits one space to the right. So we can, in fact, take every single digit in our new number and just move those one space to the right. This will result in an overall movement of two spaces.

So we take the digit two, and we move that from the thousandths column to the ten thousandths column. Then we move the seven from the hundredths column into the thousandths column. The one moves from the tenths column into the hundredths column. And then, once again, we repeat this process with the remaining two digits. So our number 231.72 divided by 100 is 2.3172.

Our last job is to divide the number by 1000. And we know that to do so, we move every digit in the original number to the right three spaces. But actually, we moved every digit right once when we divided by 10 and then once again when we divided by 100. So actually, we can take the digits in our newest number and just move those to the right ones.

So the two moves from the ten thousandths column into the hundred thousandths column. The seven moves from the thousandths column into the ten thousandths column. And then we just continue with the rest of our digits. We do need to add a placeholder in the ones column. And when we do, we find that our number divided by 1000 is 0.23172.

And so the numbers in our table are 23.172, 2.3172, and 0.23172.

Now, we’re actually able to find a relationship between dividing by 10, 100, and 1000. We saved ourselves a little bit of time when dividing by 100 by taking the digits that we had after dividing by 10 and just moving them to the right one space. Now, this makes a lot of sense because 100 is the same as 10 squared. So when we divide by 100, we can say that that is the same as dividing by 10 and then by dividing by 10 again. Similarly, when we divided by 1000, we took the digits that we’d already moved twice to the right and then just moved them once again. And this also makes a lot of sense because 10 cubed is the same as 1000. So dividing by 1000 is just like dividing by 10, then 10 again, and then 10 a third time.

And so we see that there are, in fact, two different ways that we can perform the same calculation. Dividing by 100 is the same as dividing by 10 once and then twice. And dividing by 1000 is the same as dividing by 10 three times in a row.

Let’s consider one final example.

Complete the figure shown.

Our figure contains three series of numbers. We have 348.9, which becomes 3.489 and then becomes 0.3489. We then have 8.5, which goes along in a similar pattern, and 21.5, and we’re given 0.215. But we’re not told what the final number in this list is. We are, however, told that the first thing that happens to each of our numbers is that they’re divided by 100. And this makes a lot of sense since when we divide by 100, we move all of the digits in our dividend two spaces to the right.

If we look at 348.9, for instance, the three has moved from the hundreds column to the ones column. The four has moved from the tens column to the tenths column, and so on. And so what has happened between column two and column three? Well, let’s take the three in the number 3.489. That has moved one space to the right from the ones column into the tenths column. Then if we look at the four, that’s moved one space to the right too, from the tenths column into the hundredths column.

A similar pattern happens in this number down here. The eight moves exactly one space to the right, as does the number five. And, of course, we know that when we divide by 10, we do actually move all the digits right one space. And so the operation that’s occurring between column two and column three is a division by 10.

Now, if we look at the bottom of our screen, we see that this is the same as dividing by 1000. And we know that one way to divide by 1000 is to move all of the digits in our dividend right three spaces. Alternatively, we could just take all of the digits in the number 0.215 and move them right one space, since dividing by 1000 is the same, as we see, as dividing by 100 and then dividing by 10.

When we move each digit to the right one place, the two moves from the tenths column into the hundredths column. The one and the five both move down with it. And we add a zero in the tenths column as a placeholder. So 0.215 divided by 10 is the same as 21.5 divided by 1000. It’s 0.0215.

We’ll now recap the key points from our lesson. In this lesson, we recalled that multiplication and division are inverse operations to one another. This helped us to find rules for dividing by various powers of 10. We said that to divide by 10 itself, we move all of the digits right exactly one space. To divide by 100, we move those very same digits right two spaces. But also we saw that we could consider this as dividing by 10 and then dividing by 10 again and that if we wanted to divide by 1000 of course, we move all of the digits to the right three spaces. And this is in fact the same as dividing by 10, dividing by 10, and then dividing by 10 a third time.

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