𝑥 equals 0.368 recurring. Write 𝑥 as a fraction in its simplest form.
Now, 0.368 recurring means 0.368888888 and so on going on forever. So we can write it down like this. And there are various different ways of approaching these questions. But they’re much easier if the decimal point is right up against all of the recurring
digits. So the recurring digits are immediately to the right of it.
So I’m actually first of all going to multiply 𝑥 by 100 to make the three and the
six appear to the left of the decimal place. Now, you don’t have to do that, but it just makes the math a little bit easier when
we get to the later part of the question.
Next, we’re going to multiply that by 10. There was one recurring digit. So I’m gonna multiply by 10 once, once for each recurring digit. And if I multiply 36.8888 by 10, I get 368.888 and so on. But remember because those eights are recurring, they’ll go on forever. So the fact that I had three eights here and four eights here isn’t relevant because
there’re in fact an infinite number of eights going on forever.
Now, we can notice 10 times 100𝑥 is 1000𝑥. So we can write 1000𝑥 is equal to 368.88888 and so on and so on. And now, I’m gonna subtract 100𝑥 from 1000𝑥. We wanna write out the value of 100𝑥. And we’re gonna be very careful to line up the decimal point and then make sure that
the digits with the same place values line up underneath each other. So that’s 36.88888 and so on.
Now, we can carry out the subtractions. 1000𝑥 minus 100𝑥 is 900𝑥. And over on the right-hand side, remember we start with the least significant
digit. So there’s gonna be an infinite number of eights: eight minus eight, eight minus
eight. They all will become zeros. Eight minus eight becomes a zero here, eight minus eight becomes a zero, and again
and again and again. So everything to the right-hand side of that decimal point is just gonna become a
zero. Then, eight minus six is two, six minus three is three, and three minus nothing is
three. So we found out that 900𝑥 is equal to 332.
Now, if I divide both sides of this equation by 900, then the 900s will cancel over
on the left-hand side, just leaving us with 𝑥. And we’ve got 𝑥 is equal to 332 over 900. So we’ve got 𝑥 as a fraction, but it’s not in its simplest form. Look I can divide the numerator and the denominator both by two. And half of 332 is 166 and half of 900 is 450. But wait! I can divide both of those by two. And that gives me 83 over 225. And they got no common factors. So that’s the fraction in its simplest form. So our answer is 𝑥 is 83 over 225 as a fraction in its simplest form.
Now, just before we go, let’s take a look at what would’ve happened if we hadn’t have
done this step by multiplying by 100 in order to get all our recurring digits right
up against the decimal point there.
We’d have started off with 𝑥 is equal to 0.3688888 and so on. We’d have just multiplied by 10 once because there’s one recurring decimal. And then, we’d have done 10𝑥 minus 𝑥 gives us nine 𝑥, which also leaves us with
3.688888 minus 0.368888 equals 3.32. So we’d have ended up with 𝑥 is equal to 3.32 over nine. But of course, we’re not allowed decimals in fractions. So we’d have had to multiply the numerator and the denominator by 100 to get an
equivalent fraction, which didn’t have decimals in the numerator. So that’ll be 332 over 900, which again cancels down to 83 over 225.
So with that method, we would have ended up with the same answer. But we had to deal with these decimal fractions here and fix that along the way. By multiplying by 100 at the beginning, we just avoided that problem down the