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Express the number as a ratio of intergers.

$ 2. \overline {516} = 2.516516516 . . . $

$2.516516516 \cdots=\frac{838}{333}$

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Missouri State University

Harvey Mudd College

Idaho State University

Boston College

let's express this number as a ratio of imagers. So what we like is something of the form p over Q or peeing cure imagers. This means that pink your numbers of the form. Such is like negative two negative one zero one two and so on. So no decimals, no fractions, things like pie and so on. So here, let's rewrite this. Let's pull that. Who then we'LL write. This is point five one six, then point zero zero zero five one six and so on. So this time I have since I've already used up the first, there's three, and then there's a total of we take a step back here three more that six pots already occupied that corresponds to the three right here and then also these three. So that's six. So to get to the next five one six over here, I'Ll need six year olds in the front, and then I'LL put the five one six here and so on. So each time we always add three new zeros or, if you want a nicer way to say this, that's Amore moves three places too little left each time. So here, three to the left three to the left and so on. And so that's equivalent dividing by ten cute each time. So this is geometric and we even see what were multiplying by by Ra multiplying by our equals one over attend to the cube But to really see that we can come back over here, Write this out. Five, one, six over thousands. Let's thank you. But then here this will be six places. So I need tents of the six. So you could see we mult from this term to this term or even take a step back from first to second. Probably shouldn't start there because two is not part of the geometric series. So that the geometric series is in the process is there so we can see going from first to second multiplied by one over tense of the cube and from seconds of third also multiplied by one over ten. Cute. So this is our are We're dealing with the geometric series here, so we know the formula. The formula says you take the first term in the series and then you divide by one minus R. So I have to first term. We see that right over here. Five, one, six over, ten cubed and then one minus R, which is one over ten. Cute recall that our values over here. So two plus five, one six over ten cubed over. Then we have a thousand minus one. So that's nine nine nine over ten. Cute. And you could go ahead and cancel those. So then you'LL have two plus five one six over ninety nine and then you could go ahead and combine that to get a fraction. And that final answer should be a three eight over three, three, three. So here the last step would be to just get a common denominator after simple fine and then just combined those fractions. We have a three a over three, three three and that's precisely what we wanted. This is our ratio of imagers. So that's your final answer.