Video Transcript
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.
