Problem 55 Hard Difficulty

Express the number as a ratio of intergers.
$ 1.234 \overline {567} $

Answer

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Video Transcript

let's express this given number as a ratio of imagers. This means we're looking for a fraction of the form peor que or pee and cure imagers and these air numbers of the farm, like zero one two and so on. And also the negatives of these numbers. So no fractions, no, no decimals or pies or anything like that. So now what we can do here is just start off by pulling out that one. And then I also pull out this two, three, four because it only appears once. And then we see that this five, six, seven will start repeating. So let's start that in a different color. Maybe so for the five six seven. So I'll put the three zeros before this because these three these first response are already being occupied by the two three and the four. So there's no reason to put a non zero year, and then we have our five six seven. But then after that we have another five six seven. So put another three zeros for the same reason. Thes three zeros here are being occupied by these three numbers, so I just want to put zeros there and then we could even go on more time. This time I would put nine zeros because the first nine positions have been occupied or this seven eight nine has been occupied in the previous step by these three terms here. And we can see that starting inside of these green print disease that this is when the pattern stars. This is the more interesting part of the problem. Here. You can see that the decimal is always moving three places in the left. So this is a global to saying that this is geometric and that were multiplying by one over ten cubed each time we're multiplying sense for multiplying by the same thing each time. This is geometric. And since we know that it moves to re decimal places to the left, that's the same thing, is dividing by ten three times. So this tells us that what they are is. So now let's go ahead and start writing rewriting these fractions, I can write this function here is two three for over a thousand. So let me just write that over ten cued and then inside the apprentices, we have five, six, seven over tens of six five six seven. And then since we mall supplied by this each time three to the left Well, the nine there and so on. And here I should have put the dots to indicate that that some continues. So again, the first term I had put a six on the bottom there because we're going all the way out. This is six places after the decimal. So since this is a geometric series, we can go out and actually find us some here for this This lawyer, some inside the apprentices. We know that for a geometric series, the sun just equals you. Take the first term in the Siri's and then you just divide by one minus R So this lets me, right? This is one plus two, three, four over. Thank you. If there's just the first two terms from over here and then using this formula at the first term in the series, we can see that just by looking at it here. Five, six, seven, ten to the sixth. And then I divined by one minus. R and R is one over Thank Yu tw now, since I am running out of room here, let me go ahead and write this On the next page, we have one plus two thirty for over ten. Cute and then five six seven over. Sense of the six one minus one over thin. Cute. We could go ahead and just re write the first two, and then we can rewrite thiss, get a common denominator there, and then we have, after canceling appear. And then maybe we can add these two fractions first, eight, six, seven, nine. Thirty seven thousand and then get another common denominator here. So forty five thousand six hundred seventy nine over thirty seven thousand, and that's your final answer.