### Video Transcript

Convert the decimal 0.2253
recurring to a fraction in its simplest form.

First, let’s just be clear about
exactly which digits are recurring. The first digit after the decimal
point, a two, doesn’t have a dot. So this digit is not recurring. The second and fourth digits after
the decimal point each have a dot over them. This means that they are the first
and last digits of the recurring section. So the two, the five, and the three
are all recurring. This means that the decimal written
out in longhand is 0.2253253253, and so on.

Now we’ve been asked to convert
this decimal to a fraction in its simplest form. And there’s a standard process that
we need to follow. First, we’ll just assign a letter
to this decimal. So we can choose to refer to it as
𝑥. What we’re going to do is multiply
this decimal by powers of 10, so that’s 10, 100, 1000, and so on, until we create
two numbers which have exactly the same pattern of repeating digits after their
decimal point.

To begin with, we can multiply both
sides by 10. 𝑥 multiplied by 10 gives 10𝑥. And to multiply a decimal by 10, we
keep the decimal point fixed and move all of the digits one place to the left. So we now have the decimal
2.253253253, and so on. Notice though that the pattern of
repeating digits after the decimal point does not agree between the two numbers. So we need to multiply by 10
again.

Now you can either think of this as
multiplying 10𝑥 by 10 or multiplying the original number 𝑥 by 100. And it gives 100𝑥 equals
22.532532532, and so on. Again, notice though that the
pattern of repeating digits after the decimal point for 100𝑥 doesn’t agree with the
pattern for either 10𝑥 or 𝑥. So we need to multiply by 10
again.

So now we have 1000𝑥 equals
225.325325325, and so on. We still haven’t created two
numbers which have an identical pattern of repeating digits after the decimal
point. So we need to multiply by 10
another time. This gives 10000𝑥 equals
2253.253253253, and so on. This time, if we look carefully, we
see that, for 10000𝑥, the pattern of repeating digits after the decimal point is
the same as it is for 10𝑥. In each case, it’s 253, 253, 253,
and so on.

This is what we were hoping to
achieve. And the reason for this is we can
now subtract our value for 10𝑥 from our value for 10000𝑥. I’ve written the value for 10𝑥 out
again directly below the value for 10000𝑥 for ease. On the left, we have 10000𝑥 minus
10𝑥, which is 9990𝑥.

Now we always start a column
subtraction from the right-hand side. So we’re starting after the decimal
point. But because all of the numbers
after the decimal point in 10𝑥 agree exactly with the corresponding numbers after
the decimal point in 10000𝑥, when we subtract, we just get an infinitely long
stream of zeros. This is why we set out trying to
find two multiples of 𝑥 which had an identical pattern of repeating digits after
the decimal point, because then we knew they would cancel when we subtract.

To the left of the decimal point,
we can perform our column subtraction as usual. But it’s actually just 2253
subtract two, which is 2251. We don’t need to include that
string of zeros. So we now have that 9990𝑥 is equal
to 2251.

To find the value of 𝑥 then, we
need to divide both sides of this equation by 9990. On the left, this will cancel with
the 9990 in the numerator. And on the right, we’ll have 2251
over 9990. So we found this recurring decimal
as a fraction.

Now the question does specify that
it needs to be a fraction in its simplest form. But in fact, this fraction is
already in its simplest form, as 2251 is a prime number, although probably not one
that you’re familiar with. 2251 isn’t a factor of 9990, which
means the fraction can’t be reduced any further. And so our final answer is that, as
a fraction in its simplest form, the recurring decimal 0.2253 recurring is equal to
2251 over 9990.