Consider the fractions two-thirds, five-ninths, one over 16, and three-tenths. How many of these fractions convert to a recurring decimal? Circle your answer.
So to enable us to work out how many of these convert to recurring decimal, we need to first understand what does a recurring decimal mean. Well, a recurring decimal is a decimal which has a repeated sequence of numbers within it. So, for instance, if we had 0.11111 and it continued the same way, that would be a recurring decimal. Similarly, if we had a decimal that had 0.123123123, so the 123 was repeating, then this would also be a recurring decimal.
Okay, so to solve this, what we’re gonna do is consider each of our fractions in turn. So the first fraction we’re gonna consider is two-thirds. Well, we should actually know that two-thirds is equal to 0.666 or 0.6 recurring. And we should know this because we know that a third is equal to 0.3 recurring.
However, if we’d forgotten this or we weren’t sure, what we could do is work it out. And the way we’d work it out is we use the bus stop method. And that’s because two-thirds or two over three means two divided by three.
So first of all, we see how many threes go into two. Well, this will be zero times, and we’d have a remainder of two. And then we’d see how many threes go into 20 cause we carry that two. Well, three goes into 20 six times, and that’s because three multiplied by six is 18. And again, we’d have a remainder of two. So then again we’d have threes in 20, which is six, remainder two. And this was continue indefinitely.
So therefore, we could say that two-thirds or two over three was equal to 0.6 recurring. So therefore, we can say that our first fraction, two-thirds, is equal to 0.6 recurring, so it is a recurring decimal.
Now let’s move on to our second fraction. Again, with our second fraction, five-ninths, this is one of the values that we should try and remember and one that we should know because we should know that five over nine or five-ninths is equal to 0.5 recurring. And this is as one-ninth is equal to 0.1 recurring. So it’s one of the common ones that we should remember.
But again, if we don’t or if we’re not sure, we can use the bus stop method to work it out. So we see how many nines go into five. So it will be zero, then carry the five. Then it be n- how many nines go into 50, which would be five. That’s because nine fives are 45. That’d be remainder five, so we carry another five. So then again we get how many nines go into 50, which will be five remainder five. And then the result would be five because obviously nine multiplied by five is 45, et cetera. So we would find that it’s 0.5 recurring.
So that’s our second fraction dealt with because again it is a recurring decimal. Let’s move on to our third fraction. Well, our third fraction is one over 16 or one sixteenth. Well, if we think about one sixteenth as one over four squared, because four squared is 16, well then this is the same as one over four all squared or a quarter all squared. And that’s because one squared is one. And then we’ve also got four squared. Well, this will be equal to 0.25 squared. That’s cause 0.25 is the same as one-quarter when we represent it as a decimal.
Well, 0.25 multiplied by 0.25 would have a maximum of four decimal places because when we do a multiplication of a decimal, if we have two decimal places in the first number and two decimal places in the second number or second decimal, then we put that together and have a maximum of four decimal places in our answer.
So therefore, we can say that this is not going to be a recurring decimal and must be a terminating decimal. Again, it’s worth noting that we could use a bus stop method to work this out. So we’d see how many 16s go into one. Well, 16s don’t go into one, so it’ll be zero with a remainder of one. So then we see how many 16s go into 10. Well then this will be zero remainder 10. So then we’d have to see how many 16s go into 100, and this will be six remainder four. And that’s because six multiplied by 16 is 96. And then we’d see how many 16s go into 40, and that’s two, cause that’ll give us 32 remainder eight. And then finally how many 16s go into 80, which is five, so we get 0.0625. And this is a terminating decimal, so not a recurring decimal.
So now what we do is move on to our final fraction, which is three-tenths. Well, we know that three-tenths is equal to 0.3. So again, this is not a recurring decimal because 0.3 is a terminating decimal. So then when the question says how many of these fractions convert to a recurring decimal, we can say two-thirds is a recurring decimal because it’s 0.6 recurring and five-ninths is a recurring decimal because it’s 0.5 recurring. So therefore, there are two fractions that convert to a recurring decimal, and I’ve circled the number two.