### 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.