Express 3.72 recurring as a common factor.
Okay, so in this question we want to convert our decimal into a common factor. But what does this decimal actually mean? Well, the lineup of the seven two, which can also be written as a dot dot, actually means that both the seven and the two are recurring. So in this instance, it would equal 3.727272, and this could continue on. So in this question, what we’re looking to do is actually convert this into a common factor, which will actually be more accurate because it’s not recurring.
Okay, so let’s have a go at doing that. Our first step is to make 𝑥 equal to 3.72 recurring. The next stage, it’s actually multiplying both sides of our equation by 100, which gives us 100𝑥 is equal to 372.72 recurring. But why do we multiply it by 100? We multiply it by 100 because we are actually looking to eliminate the recurring decimal parts of our numbers.
And in order to do that, we’d have to multiply it by 100 because if we multiply it by 10, you would get two values with different recurring decimals. So you get 3.72 recurring and 37.27 recurring. So they’re actually different, which will enable us to eliminate them, which is what we’re looking to do when trying to convert it into a common factor.
So when you’re doing these questions in the future, the tip to remember is remember to multiply by a factor of 10; that gives the two values with the same recurring decimal because it will allow you to eliminate them.
For the next stage, we actually want to eliminate our recurring decimal. And in order to do this, we’re actually gonna subtract 𝑥 from our 100𝑥, which gives us 100𝑥 minus 𝑥 is equal to 372.72 recurring minus 3.72 recurring, which gives us 99𝑥 because 100𝑥 minus 𝑥 is 99𝑥 is equal to 369. And it’s 369 because 372 minus three is 369. And then .72 recurring minus .72 recurring actually gives us zero, so it just leaves us with the 369.
So now if we divide both of our sides of the equation by 99, we’re left with 𝑥 equals 369 divided by 99. We can then simplify the fraction further by dividing the numerator and denominator both by nine, which gives us 𝑥 is equal to 41 divided by 11 or 41 over 11.
But what does this 𝑥 mean? Well, if you look back up at the top of our answer, we can see that 𝑥 is equal to 3.72 recurring. But we’ve also found that 𝑥 is equal to 41 over 11. So therefore, what we can say is that that we’ve expressed 3.70 recurring as a common factor because 3.72 recurring is the same as 41 over 11.