# Lesson Video: Converting Recurring Decimals to Fractions Mathematics • 7th Grade

In this video, we will learn how to convert a recurring decimal to a fraction or a mixed number.

17:23

### Video Transcript

In this video, we’ll learn how to convert a recurring decimal to a fraction or a mixed number. By this stage, you should feel confident in converting between terminating decimals and fractions such as 0.3, 0.71, and so on. The process for converting recurring decimals to fractions is a little different however.

We recall that a recurring decimal is a decimal number which has a repeating pattern of digits after the decimal point. And we represent this recurring part in one of two ways. For example, 0.3 with a dot above the three means the three recurs. It’s 0.33333 and so on. 0.25 with a dot above both the two and the five means both these numbers recur. It’s 0.252525 and so on. In both cases, though, it’s important to realize we can also use a bar to represent the recurring part as shown. 0.301 with a dot above the three and the one means everything between the three and the one inclusive recurs. So it’s 0.301301 and so on. Again, we can use a bar to represent this as shown.

Now, somewhat counterintuitively, a recurring decimal is an example of a rational number. That means every recurring decimal can be written as a fraction in the form 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers. They’re whole numbers. Our first example will demonstrate the process for converting from recurring decimals to fractions with a very simple recurring decimal with just one recurring digit.

Answer the following questions for the recurring decimal 0.4 recurring, that is, 0.44444 and so on. Let 𝑥 be equal to 0.4 recurring. Find an expression for 10𝑥. Subtract 𝑥 from 10𝑥 to find an expression for nine 𝑥. Find 𝑥.

By breaking the question down into three individual steps, it’s showing us the process for converting a simple recurring decimal into a fraction. We have a simple recurring decimal. It’s 𝑥 equals 0.4 recurring. That is, 𝑥 equals 0.444 and so on. It can be easier to write out a few digits of the recurring part so we can get an idea of the pattern. The question asks us to find an expression for 10𝑥. Well, to get to 10𝑥 from 𝑥, we’re clearly going to need to multiply it by 10. And since 𝑥 is equal to 0.4 recurring, it follows that we’ll find an expression for 10𝑥 by multiplying 0.4 recurring by 10.

𝑥 multiplied by 10 is 10𝑥 as required. And to multiply a decimal number by 10, we move the digits to the left exactly one space. So 10𝑥 is equal to 4.444 and so on. The expression for 10𝑥 is therefore 4.4 recurring. The second part of this question tells us to subtract 𝑥 from 10𝑥. In doing so, we’re going to subtract the entire equation for 𝑥 from the entire equation from 10𝑥. 10𝑥 minus 𝑥 is nine 𝑥 as required.

Now let’s look at our recurring decimals. Notice how the digits after the decimal point are identical in both numbers. So that means when we subtract each four, we end up getting zero. So we see that 4.4 recurring minus 0.4 recurring is simply four. And so our expression for nine 𝑥 is simply four. The third part of this question asks us to find 𝑥. What it’s really saying is solve this equation for 𝑥. Solve the equation nine 𝑥 equals four.

Well, to solve the equation, we’ll perform a series of inverse operations. Currently, nine 𝑥 means nine times 𝑥. So we’re going to divide both sides of our equation by nine. And so, when we do, we find 𝑥 is four divided by nine or four-ninths. 𝑥 is equal to four-ninths. Note that we originally defined 𝑥 to be equal to 0.4 recurring, but we’ve just written that 𝑥 is equal to four-ninths. That must mean that 0.4 recurring is four-ninths.

So let’s now have a look at an example which demonstrates how this process works for more complicated recurring decimals.

Express 0.75 recurring as a rational number in its simplest form.

We have a recurring decimal, 0.75 recurring. Remember, the seven and the five both recur. So it’s 0.757575 and so on. The question wants us to express this as a rational number. That’s a fraction 𝑎 over 𝑏, where 𝑎 and 𝑏 are both whole numbers; they’re integers. So what are the steps we need to follow to do so? Step one is to let 𝑥 be equal to the recurring decimal number. It’s often sensible to write out a few digits to get an idea of the pattern. So here 𝑥 is 0.757575 and so on. Our aim is to find another decimal number that has the exact same pattern of digits immediately after the decimal point.

To achieve this, our second step is to multiply by some power of 10. So 10, 100, or 1000, and so on. We want the digits immediately after the decimal point to match those of the original number. So that pattern 757575 and so on needs to sort of line up against the decimal point. We can see that there are two digits recurring after the decimal point. So we’re going to need to multiply by a power of 10 so that we move the digits two spaces to the left. Well, to achieve that, we clearly need to multiply by 10 squared or 100. So let’s do the same to the 𝑥. 𝑥 times 100 is 100𝑥. So we find 100𝑥 is equal to 75.75 recurring.

Notice now that in each equation, the digits immediately after the decimal point are identical. The third step is always to subtract the two numbers that have the same digits after the decimal point. So here we’re going to subtract the equation for 𝑥 from the equation for 100𝑥. 100𝑥 minus 𝑥 is 99𝑥. But what happens with our decimals? Notice that when we subtract, the recurring parts are going to cancel out. 0.75 recurring minus 0.75 recurring is zero. And so we end up simply working out 75 minus zero, which is 75. And that’s the whole purpose of performing these steps. We want the recurring bits, the bits after the decimal point, to disappear.

Step four is to solve our equation for 𝑥. We have 99𝑥 equals 75. And so we’ll divide both sides of this equation by 99. In doing so, we see 𝑥 is equal to 75 divided by 99 or 75 over 99. This isn’t yet in its simplest form though. We need to divide both the numerator and denominator of our fraction by three. And when we do, we find 𝑥 is equal to 25 over 33. Remember, we started by defining 𝑥 to be equal to 0.75 recurring. But we’ve just shown that 𝑥 is equal to 25 over 33. So in turn, we’re saying that 0.75 recurring must be the same as 25 over 33. And we’ve written it as a rational number in its simplest form.

We’ll now consider an example where three digits repeat.

Convert 0.354 recurring to a fraction.

Here, we have a recurring decimal. The digits three, five, and four all repeat. So we could say that it’s equal to 0.354354354 and so on. Let’s recall the steps required to convert a recurring decimal to a fraction. Step one, we let 𝑥 be equal to our recurring decimal. We’re going to write out a few digits just to get an idea of the pattern. 𝑥 is 0.354354 and so on. Our second step is try and create another decimal number whose digits after the decimal point are identical to the original.

To achieve this, we’re going to multiply by some power of 10. That’s 10, 100, 1000, and so on so that the digits immediately after the decimal point match those of our original number. The pattern needs to sort of line up against the decimal point. We said that we had three digits that recur. So we’re going to need to figure out which power of 10 we multiply by to ensure that these digits move to the left three spaces. We end up with 354.354 and so on. Well, to achieve this, we multiply by 10 cubed or 1000. So let’s do the same to the 𝑥. In doing so, we get 1000𝑥 is equal to 354.354 and so on.

Then, our third step is to subtract. We’re hoping that in doing so, we completely eliminate the digits after the decimal point or at least the recurring bits. So we’re going to subtract the entire equation for 𝑥 from the entire equation for 1000𝑥. Note at this stage that we could do that the other way around. We just end up with two negatives. 1000𝑥 minus 𝑥 is 999𝑥. Then we notice that subtracting 0.354 recurring from 0.354 recurring gives zero. And so the sum becomes 354 minus zero, which is just 354. Remember, we were looking to eliminate the bit after the decimal. And we’ve done so.

Our fourth and final step is to solve this equation for 𝑥. We have 999𝑥 equals 354. So to solve, we’re going to divide both sides of this equation by 999. So 𝑥 is 354 divided by 999, which we can simply write as a fraction. Now, in fact, it’s not in its simplest form. We can divide both the numerator and denominator of this fraction by three. And when we do, we find 𝑥 is equal to 118 over 333. Remember, we originally defined 𝑥 to be equal to 0.354 recurring. But we’ve just shown that 𝑥 is equal to 118 over 333. So, in turn, we’ve shown that 0.354 recurring as a fraction in its simplest form is 118 over 333.

Next, let’s consider what happens when just part of the decimal recurs.

Answer the following questions for the recurring decimal 0.265 recurring; that’s 0.2656565 and so on. Let 𝑥 be equal to 0.265 recurring. Find an expression for 10𝑥. Find an expression for 1000𝑥. Subtract 10𝑥 from 1000𝑥 to find an expression for 990𝑥. And find 𝑥.

The question has defined our recurring decimal to be equal to 𝑥. And it wants us to begin by finding an expression for 10𝑥. It can be useful to write out a few digits of the recurring number to get an idea of the pattern. To get from 𝑥 to 10𝑥, we’re going to need to multiply by 10. And so let’s do the exact same thing to our recurring number. That’s 2.656565 and so on. Remember when we multiply by 10, we move the digits to the left one space. And so the expression for 10𝑥 here is 2.65 recurring.

Next, we need to find an expression for 1000𝑥. Well, this time to get from 𝑥 to 1000𝑥, we’re going to need to multiply by 1000. So we’ll do the same to our recurring decimal. This time, when we do so, the digits move to the left three spaces. So 1000𝑥 is 265.6565 and so on. 1000𝑥 is therefore 265.65 recurring. Notice that we now have two numbers whose digits after the decimal point are identical. And so we’re ready for the next part of this question. We’re going to subtract 10𝑥 from 1000𝑥. And in turn, we’re going to subtract their decimals.

So we’re going to work out 265.65 recurring minus 2.65 recurring. When we subtract these numbers, we notice that the recurring part gives us zero. 0.65 recurring minus 0.65 recurring is zero. 265 minus two is 263. And 1000𝑥 minus 10𝑥 is 990𝑥. And so our expression of 990𝑥 is 263. The very final part of this question says to find 𝑥. In other words, we’re going to solve our equation 990𝑥 equals 263. To do so, we need to divide through by 990. 𝑥 is therefore equal to 263 divided by 990, which we can write as a fraction as shown.

Now remember, we originally defined 𝑥 to be equal to 0.265 recurring. But we’ve just shown that it’s equal to 263 over 990. And that means the fraction equivalent of the recurring decimal 0.265 recurring must be 263 over 990.

We’re going to combine everything we’ve learned so far into our final example. At this stage, you might wish to pause the video and attempt to follow the steps we’ve covered up until now.

Convert 0.347 recurring to a fraction.

We have a recurring decimal with a bar just above the seven. So that tells us the seven is the only bit that recurs. It’s 0.34777 and so on. So let’s recall the steps that we take to convert a recurring decimal to a fraction. Our first step is to define 𝑥. We let 𝑥 be equal to our recurring decimal. And at this stage, it can be helpful to write out a few digits of the recurring part just to get an idea of the pattern. Then, our second step is to multiply this by some power of 10 so that the digits after the decimal point match.

We do have a little bit of a problem here. The only bits that recur are the seven. So we’re actually going to need to do this twice. We want to create two numbers whose digits after the decimal point match. Well here, that’s going to be seven recurring. So let’s begin by multiplying by some power of 10 so that the digits move twice to the left, in other words, so that we get 34.7 recurring. Well, the only way to achieve this is to multiply by 100. So let’s multiply 𝑥 by 100. 𝑥 times 100 is 100𝑥, so we get 100𝑥 equals 34.7 recurring.

Let’s do this again. Now we could multiply our original number by something. However, if we look carefully, we notice that if we multiply 34.7 recurring by 10, the digits move to the left one space. And we’ll still end up with 0.7 recurring. 34.7 recurring times 10 is 347.7 recurring. And 100𝑥 times 10 is 1000𝑥. Notice that this is the same as multiplying our original value for 𝑥 by 1000. That would’ve moved the digits three spaces to the left.

Now that we’ve created two numbers whose digits after the decimal point perfectly match, we subtract these two numbers. In other words, we’re going to subtract the entire equation for 100𝑥 from the equation for 1000𝑥. In doing so, we notice that the recurring part of the decimal disappears. 0.7 recurring minus 0.7 recurring is zero. And so we simply need to work out 347 minus 34. That’s 313. Similarly, 1000𝑥 minus 100𝑥 is 900𝑥. So we have an equation for 𝑥. It’s 900𝑥 equals 313.

Our fourth and final step will always be to solve this equation for 𝑥. We perform inverse operations to do so. We’re going to divide both sides of this equation by 900. 𝑥 is therefore equal to 313 divided by 900, which we can write as a fraction as shown. Note that we originally defined 𝑥 to be equal to 0.347 recurring. But we’ve now shown that 𝑥 is equal to 313 over 900. This must mean that in its fractional form, 0.347 recurring is 313 over 900.

In this video, we’ve seen that all recurring decimals are rational numbers. That is, they can be written as a fraction 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers. They’re whole numbers. When doing so, our first step is to begin by defining 𝑥. We let 𝑥 be equal to our recurring decimal. Next, we multiply by some power of 10 to generate two numbers whose digits after the decimal point exactly match. This will sometimes need to be performed more than once to achieve this. Our third step is to subtract these values. And finally, we solve our equation for 𝑥.