Write negative 2.7 recurring as a mixed number.
So the key to this question is this part here that I’ve circled in orange. It’s the seven with a line above it, which means seven recurring. Well, we say recurring number or recurring decimal. What does that actually mean?
Well, if we’ve got here two ways of writing it, we’ve got 1.3 with a line above it or 1.3 with a dot above the three. And these both mean the same thing. They both mean 1.333. And the three recurs indefinitely. And this same notation can also be used in different ways. So, for instance, here we’ve got 1.304 with a line above them all or 1.304 with a dot above the three and a dot above the four. And with these, we can see that what we’d get is 1.304304. And then, the 304 would repeat indefinitely.
So now, we’re clear what a recurring decimal is and what the notation means. Let’s look at how we’d write our number as a mixed number. Well, what I’m gonna do is deal with the negative two and the 0.7 recurring separately. So I’m gonna call 0.7 recurring 𝑥. And then what we’re gonna do is multiply each side of the equation by 10. So we’re gonna have 10𝑥 is equal to 7.7 recurring.
But why we’re doing this? Well, the reason I’m doing this is because I want to be able to eliminate the recurring part of our number. So to enable this to happen, what I’m going to do is I’m going to subtract our first equation from our second equation. So if I have 10𝑥 minus 𝑥 on the left-hand side of the equation and 7.7 recurring minus 0.7 recurring on the right-hand side of the equation.
Well, 10𝑥 minus 𝑥 would just give us nine 𝑥. And then, on the right-hand side of the equation, we’re just gonna get seven. That’s because we had 7.7 recurring minus 0.7 recurring. And thus, the seven recurrings actually disappear because if you have 0.7 recurring minus 0.7 recurring, it would just be zero. So we’re left with our seven.
So then, as we’re trying to find 𝑥 because we’re trying to find out what 0.7 recurring would be as a fraction, what we do is we divide each side of the equation by nine. And when we do that, we can see that 𝑥 is gonna be equal to seven over nine or seven-ninths. So therefore, we can say 0.7 recurring is the same as seven over nine or seven-ninths.
But does that mean that we’ve solved the problem? Well no, because if we look at the question, the question wants us to write negative 2.7 recurring as a mixed number. So we have to bring back the negative two. So therefore, when we bring back the negative two, we can say that negative 2.7 recurring as a mixed number is equal to negative two and seven-ninths. And that’s because a mixed number is a number that has an integer and also a fraction.
And we’ve got back to our negative two as our integer and seven-ninths as our fraction.