Express the number `2.bar(7)` as a ratio of integers:  

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Express `2.bar(7)` as a ratio of integers:

(1) If you recognize `0.bar(7)` as `7/9` , then you have `2 7/9=25/9`

(2) If you don't recognize the fraction:

(a) `2.bar(7)=x`

`27.bar(7)=10x`  Subtract the first from the second to get:

`25=9x==>x=25/9`

(b) This is a geometric series:

`2+7/10+7/100+7/1000+...`

If we concentrate on...

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Express `2.bar(7)` as a ratio of integers:

(1) If you recognize `0.bar(7)` as `7/9` , then you have `2 7/9=25/9`

(2) If you don't recognize the fraction:

(a) `2.bar(7)=x`

`27.bar(7)=10x`  Subtract the first from the second to get:

`25=9x==>x=25/9`

(b) This is a geometric series:

`2+7/10+7/100+7/1000+...`

If we concentrate on the sum of fractions we have:

`S=7/10+7/100+7/1000+...+7/(10^n)+...`

Here the first term is `a=7/10` and the common ratio is `1/10` so the sum `S` is given by `a/(1-r)` so `S=(7/10)/(9/10)=7/9` .

Including the 2 we get `2.bar(7)=2+sum_(k=0)^(oo)(7/10)(1/10)^k=2+7/9=25/9`

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`2.bar(7)=25/9`

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