In this video we're looking at how to convert recurring decimals into fractions. Recurring decimals are those whose digits recur, like 0.473, 473, 473 and so on. These sorts of decimals carry on recurring forever, and so instead of writing digits on forever, we just put a dot above the first and last digit that recur. So in this case...
where the 473 is recurring, we would write it as 0.473 with little dots above the 4 and the 3. So now we're happy with what recurring decimals are, let's have a go at this question here and convert it into a fraction. Before we start I want to point out that converting recurring decimals into fractions is actually quite complicated and takes a fair few steps. so we're going to work through it slowly one step at a time.
The first thing you want to do is name your decimal with a letter. For example we might call it r and so write r equals 0.473 recurring. Next you need to multiply both sides of the equation by 10 until you end up with one entire set of the recurring digits on the left of the decimal points. So we want to end up with a whole set of 473 on the left of the decimal point.
So if we multiply it by 10 once, we'll get 10r equals 4.734 recurring, times 10 again gives us 100r equals 47.347 recurring, and times 10 once more gives us 1000r equals 473.473 recurring. So we now have a full copy of the recurring digits to the left of the decimal point, and so we can stop multiplying by 10. And to give ourselves some more space, let's just rewrite this up here. Next, if we now subtract a single r, which remember has a value of 0.473 recurring, from that 1000r, we'll be left with 999r. but importantly we'll have completely removed the recurring decimal places from our equation, and so we'll only have 473 left. And because we only have whole numbers in this equation now, to get a single r by itself, all we have to do is divide both sides by 999, which gives us r equals 473 over 999. And remember because r is just a letter that represents 0.473 recurring, we're really writing 0.473 recurring is equal to 473 over 999. And so that's our answer.
We've converted that recurring decimal into a fraction. We just had to go through all these weird steps to get here. Now, sometimes you might have a recurring decimal where the recurring part doesn't come right after the decimal point, like in 0.23 recurring, where only the 3 recurs. In these cases, we have to use a slightly different technique.
We still name our decimal r, but this time we write two extra sums. In the first, we multiply it by 10 until one whole sets of the recurring decimals are to the left of the decimal point, just like we did last time. So here we'd end up having to multiply by 10 once to get 10r equals 2.3 recurring, and then again to get 100r equals 23.3 recurring.
For the second sum though, we only multiply it by 10 until the non-recurring decimals are to the left of the decimal point. So in this case it's just the 2 that we want on the left. meaning that we only need to multiply it by 10 once to get 10r equals 2.3 recurring.
Next, we subtract the smaller sum from the larger one. So 100r equals 23.3 recurring minus 10r equals 2.3 recurring. And for things like this, where you're subtracting one entire equation from another, you just do the left side first and then the right side. So 100R minus 10R is 90R.
and 23.3 recurring minus 2.3 recurring is 21, meaning that we end up with 90r equals 21. And importantly we've completely removed the recurring decimal part, which is the main aim. Last we just need to rearrange this to get the r by itself, so we can divide both sides by 90 to get r equals 21 over 90. and we should always check if we can simplify our answer, which in this case we can, by dividing top and bottom by 3 to get 7 over 30. And that's it, we've turned 0.23 recurring into a fraction. And just before we finish, I want to point out that this is a fairly difficult topic, so don't worry if you found all this a bit confusing.
Just have another watch of the video, and then have a go at some practice questions. and the more practice you do, the more sense it will make. That's everything for today though, so hope you found it useful, and we'll see you again soon.