Skip to 0 minutes and 6 seconds PAULA KELLY: So we’re going to use some algebra to put these recurring decimals into a fraction. Now we’ve written this as 0.555, could go on infinitely. To make it a bit easier, we would say a spot about the five to indicate just the five reoccurs. Also we can use some algebra. So we’ll call this x. We’ll say x is going to equal 0.5. We’ll put the spot that just indicate just the five recurs OK, now as a fraction, we want to get rid of this recurring decimal. So for example, if we had our 10 lots of x, if we multiply this by 10, all our numbers would go one place larger. So we have now 5.5 recurring.

Skip to 0 minutes and 51 seconds Now we’re trying to get rid of this recurring decimal. If we had our 10x, and we subtracted our x, our 0.5 recurring, we can see now if we had 10x subtract an x, we’re left with 9x. 5.5 recurring, if we subtract the recurring part, the 5, we just have 5.

Skip to 1 minute and 18 seconds MICHAEL ANDERSON: So all the 5s that go on forever kind of cancel each other out because both of these numbers have that.

Skip to 1 minute and 23 seconds PAULA KELLY: Perfect. So we got rid of those. We’re still looking to get a fraction. So our final step, rather than multiplying by 9, we do our inverse. So we’re going to finish with x equals 5/9. And that’s the equivalent of our five recurring. And if you weren’t convinced, you can always check in a calculator. OK so we’ll have a look now at another example where we have a different of decimals recurring. So in this instance, we have now two recurring decimals. So it’s similar to how we did this before. We’ll call this x. We’ll save ourselves by writing, we’ll put 0.27.

Skip to 2 minutes and 3 seconds This time I’m going to use two spots, just to indicate the 2 and the 7 are going to recur. We’re looking again to get rid of our recurring decimals. So I mentioned before we could try multiplying by 10. If we multiply this by 10, all our numbers go one place larger. So we have 2.7272727. That doesn’t necessarily help us. So we could try instead 100x.

Skip to 2 minutes and 39 seconds This time all our numbers go two places larger. So now we have 27.27. And now we have our recurring decimal.

Skip to 2 minutes and 51 seconds MICHAEL ANDERSON: Because both are x and 100x both of the same values after the decimal point?

Skip to 2 minutes and 55 seconds PAULA KELLY: Perfect, yeah. Lovely. So if we use our 100x and our x, so we’ll have our x down here. We’ll call this 0.27. They recur. And similar how we did earlier, we’ll subtract these. So 100x subtract our x, we’re left at 99x.

Skip to 3 minutes and 18 seconds And then we can just take away our recurring decimals. So we’re left with just 27.

Skip to 3 minutes and 24 seconds MICHAEL ANDERSON: Oh nice.

Skip to 3 minutes and 25 seconds PAULA KELLY: Still we haven’t got a fraction, so our final step, rather multiply than by 99, let’s do our inverse. So we’ll finish with x is 27 over 99.

Skip to 3 minutes and 39 seconds Similarly with this, though, we can see we can simplify. They’re both multiples of 9. So we’re to end up with just 3/11.

Skip to 3 minutes and 48 seconds MICHAEL ANDERSON: Ah, OK.

Skip to 3 minutes and 48 seconds PAULA KELLY: OK.

Skip to 3 minutes and 55 seconds OK, for our final example of a recurring decimal, slightly more difficult this time because just one of our digits is recurring. Same process, though. We’ll call this x.

Skip to 4 minutes and 8 seconds This time we’re going to put a spot just above the 6 because only the 6 is going to recur, OK? Similarly, we’ll try with our 10x. Does this help us? We all move the numbers one place. So we have 10x is going to be 1.6 recurring. So it is useful to see you’ve isolated our the recurring decimal. Systematically, if we try 100x.

Skip to 4 minutes and 35 seconds All our numbers go two places larger. So we’re going to end up with 16.6 recurring. Now we want to still eliminate our recurring decimal. We can do it with our numbers here because we’ve isolated our recurring decimal.

Skip to 4 minutes and 52 seconds MICHAEL ANDERSON: I see, yeah.

Skip to 4 minutes and 53 seconds PAULA KELLY: So say I’m going to have 100x. This time I’m going to subtract our 10x. So if we draw place value, line these up. So you’ve got 1.63 recurring. With our subtraction, 100x subtract 10x gives us 90x. Our 16.6 recurring, if we subtract our 1.6 recurring, we’d just end up with 15.

Skip to 5 minutes and 19 seconds MICHAEL ANDERSON: Because the 16 take away is 15, and the recurring parts cancel each other out.

Skip to 5 minutes and 23 seconds PAULA KELLY: Perfect, yeah lovely. Same again, we haven’t quite got a fraction yet. We’ll do our inverse of multiplying by 90. So we’re going to finish with the x is 15/90.

Skip to 5 minutes and 37 seconds We know as well we can simplify this. We know we could do it with 15s. We could do it with 5s. If we start off past our 5s, we’re going to have our 3s. Then how many 5s go into here? We’ve got 18. Again, both these are multiples of 3. We can go even a further. And have a finish of 1/6th.

Skip to 5 minutes and 57 seconds MICHAEL ANDERSON: Well, so 1/6 is equivalent to 0.16 recurring?

Skip to 6 minutes and 1 second PAULA KELLY: Perfect, with just the 6 recurring.

Skip to 6 minutes and 3 seconds MICHAEL ANDERSON: Oh.

# Decimals as fractions: recurring decimals

In the last step we saw how terminating decimals can be expressed as a fraction. Now consider the following recurring decimals.

a. 0.4444444444444…

b. 0.23232323232323…

c. 0.166666666666666…

Do you think they can:

- always be expressed as a fraction?
- sometimes be expressed as a fraction?
- never be expressed as a fraction?

## Initial thoughts

Do you think the answer depends upon how the decimal recurs? If recurring decimals can be expressed as a fraction how do you think this is done? Discus your thoughts below before watching the video in which Paula and Michael consider these questions

## Teaching resource

We’ve selected a lesson, from the Mathematics Assessment Resource service (MARS), which develops the concept of translating between decimal and fraction notation, particularly when decimals are repeating. The main activity is to match together cards. There are pre-lesson and post-lesson formative assessment tasks. Detailed teacher notes give suggestions on questioning and how to use the resources. Full solutions are given for each of the sections.

## Problem worksheet

Now complete question 4 from this week’s worksheet.

As a reminder, the worksheet can be found in the first step of this week.

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