Express 0.28 recurring as a fraction in its simplest form.
So first of all, let’s have a look at the notation. So we’ve got 0.28 with a dot on it. And I said this is recurring. So what this means is that we have the number 0.2 and then eight, eight, eight and this continues indefinitely. So to enable us to turn this into a fraction, it’s the eight recurring or the repeated eight that we need to deal with. So to enable us to do this, we’re gonna use some algebra.
So we’re gonna let 𝑥 equal 0.28 recurring. So then, next, what we’re going to do is we’re gonna multiply this by 10 cause we’re gonna find 10𝑥. Well, if we multiply 𝑥 by 10 to get 10𝑥, we have to multiply the other side of the equation. So we multiply 0.28 recurring by 10 and that gives us 2.8 recurring. And that’s because each of our digits has moved one place value to the left. So if we take a look at what we’ve got now, can this help us-can this help us to eliminate our recurring decimal? Well, it can’t, not yet. So we need to do another stage.
Well, this next stage is to multiply by 10 again. So we’ve got 100𝑥 is equal to 28.8 recurring. And that’s again if we multiply by 10, we move each of the digits a place value to the left. So we get 28.8 recurring. Well, it’s here that we can see that we’ve got something useful because both in 10𝑥 and 100𝑥, we have .8 recurring. So this means that we’re gonna be able to eliminate this.
So what we can do next is subtract 10𝑥 from 100𝑥. So we have 100𝑥 minus 10𝑥 is equal to 28.8 recurring minus 2.8 recurring. So a quick note here: be careful not to fall into the trap of a common mistake and that is that students will subtract the 𝑥 from 100𝑥. But this won’t help because the 𝑥 is 0.28 recurring. So it’s not 0.8 recurring. So it won’t allow us to cancel.
So this is gonna give us 90𝑥 is equal to 26 and that’s because 100𝑥 minus 10𝑥 is 90𝑥 and 28.8 recurring minus 2.8 recurring is gonna be 26 cause if you subtract the .8 recurrings, they’ll cancel and if you subtract two away from 28, you get 26.
So now what we do is we divide each side of the equation by 90 and that’s cause we want a single 𝑥. When we do that, we get 𝑥 is equal to 26 over 90. Well, have we finished here? Because we said that 𝑥 was equal to 0.28 recurring and we were looking to express 0.28 recurring as a fraction. But well, we’ve done this: we’ve expressed it as a fraction. But it’s not in its simplest form. So what we need to do now is cancel down our fraction.
Well to do that, we need to find a factor that goes into both 26 and 90. Well, the only factor that goes into both of these is two, well apart from one. But one wouldn’t help us to cancel. So if we divide 26 by two, we’re gonna get 13 as our numerator. And then, if we divide 90 by two, we’re gonna get 45 as our denominator.
So therefore, we can say that 0.28 recurring expressed as a fraction in its simplest form is equal to 13 over 45.