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Express the repeating decimal as the ratio of two integers. 0.09 = 0.0999_The ratio that represents the repeating decimal is (Type an integer or a simplified fracti...

Question

Express the repeating decimal as the ratio of two integers. 0.09 = 0.0999_The ratio that represents the repeating decimal is (Type an integer or a simplified fraction:)

Express the repeating decimal as the ratio of two integers. 0.09 = 0.0999_ The ratio that represents the repeating decimal is (Type an integer or a simplified fraction:)



Answers

The repeating decimal is expressed as a geometric series. Find the sum of the geometric series and write the decimal as the ratio of two integers. $$ 0 . \overline{9}=0.9+0.09+0.009+0.0009+\cdots $$

Hello. So 0.9 repeating. So there's many proofs that zero point in repeating is equal to one that closely one but is equal to one. Um But we can go ahead and look at this. Um As one is zero point in repeating this is equal to as a sum. This is 9/10 Um plus 900 Plus 9000s and so on. So this is going to be equal to the sum where we have n going from one to infinity of nine times 1/10 to the end. Okay we have a geometric series now so therefore we get our first term initial term A is going to be equal to 9/10 and we have that are common ratio. R is equal to 1/10. That's part A. And part B. Well since we have that are common ratio is less than one. The series converges and the some of the series is then going to be equal to the first term which is 9/10 and then divided by one minus the common ratio. That's one minus 1/10. That's going to give us just while 9/10 divided by one minus 1/10 is 9/10. So 9/10 divided by 9/10. That's well nine tens times 10 91 thing about it either way that is equal to equal to exactly one

To over nine needs to be converted into a decimal, so we'll start with 2.0 Put a nine on the outside there nine goes into 20 times nine times zero is zero Subtract were left with two. Bring down that 0 to 20 nine goes into 22 times. Nine times two is 18. Subtract. We're left with two. Bring down the zero to make. 20 nine goes into 22 times. Nine times two is 18. Subtract be left with two Bring down that zero make 20 and I can see that we have a pattern here. Thes twenties. Keep emerging and we're never going to get to the end of this. We could go on for an infinite amount of time trying to solve this. And we're just gonna get 0.2222 on and on forever. Um, and so there's a way we can represent this. We could just do use a bar over the first number there. So our final answer is going to be 0.2 with a bar on top of the two to indicate that the two repeats to infinity

You know that we have does a bunk tree committing that as a syrup on three plus the 103 plus thereupon zozo three plus on and it can be written down is the, uh is there a pawn three, in fact it out and then we get one. Plus, uh, I'm not going to do that. And then I will have this one will be one day running by, uh, three and then plus one day running by. It's not quite here. It will be that three armored 10 isn't between over 100 less tree over 1000 100,000 and so on. So we can run this town do that three times. Submission. I'm the one. I'm a 10 about off the ah. Okay. Take us from one to infinity so it doesn't go me that geometric series. Yeah,

Okay. So started noticing that we have 4747 and 47 So this refutes sold to break them up into those portions. Um, 00 for someone and in plus, Cyril points cereals are wrong. No. 047 So we can rewrite this as 47 over 100 and then our next one is 47 over, $10 in, and then I last one would be 47 over. Uh, look. See, this is 12 47/1 million. Um, so it's now some days are actually multi multiply this by, so we want to have a tsunami there of one million. So 100 times what gives us one million? That's, uh, $10 in julie gets, um, 47 times 10,000. Do we get for seven? 1234 over 1 10,002 123 Once you and in this term here, we want to multiply that we have 10,000 and you want to make it into Mylan. So it's won't supply it by 100. So we get 47 times 100 bets. 4700. You should want to 3123 So we have four 712 for those 471 seal and 47. So we get who are 74747 over one million. Okay, so this is our decimal as a fraction.


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