Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top
  1. $.\overline{36}=\frac{36}{99}$
  2. $2.1\overline{36}=2\frac3{22}$

The part I do not understand however, is "you could used 1) to speed up the working of 2)" which is written in the book.

How would I use 1) to help me work out 2)?


share|cite|improve this question
$\frac3{22}=.1\overline{36}\text{ not } .1\overline{39}$ – lab bhattacharjee May 5 '13 at 15:03
my bad, both were .36 not .39. – user2121482 May 5 '13 at 15:09
up vote 3 down vote accepted


So, $$2.1\overline{39}=2+\frac1{10}+\frac1{10}\cdot\frac{36}{99}=2+\frac1{10}\cdot\left(1+\frac4{11}\right)=2+\frac1{10}\cdot\frac{15}{11}=2+\frac3{22}$$

share|cite|improve this answer
I've probably missed something here, but would that not = 2.239? – user2121482 May 5 '13 at 15:13
@user2121482, please find the edited answer – lab bhattacharjee May 5 '13 at 15:15
thanks. really confused at first (for example where the 4/11 came from), however it was my own bad since i had confused .39 with .36. – user2121482 May 5 '13 at 15:47
how did you get 15/11? – user2121482 May 5 '13 at 16:13
@user2121482, $$1+\frac4{11}=\frac{11+4}{11}=\frac{15}{11}$$ and cancelling out $9$ from the numerator & the denominator, $$\frac{36}{99}=\frac4{11}$$ – lab bhattacharjee May 5 '13 at 16:14
  1. $.\overline{36}=\frac{36}{99}$
  2. $2.1\overline{36}=2\frac3{22}$

part 1.

suppose $$\begin {equation}\dfrac pq=0.\overline{36}\end {equation}$$ this is eqn (1).

mulipltly this eqn with 100 $$\begin {equation}100\dfrac pq=36.\overline{36}\end {equation}$$ this is eqn (2).Now subtract (1) from (2). $$\begin {equation}100\dfrac pq-\frac pq=36.\overline{36}-0.\overline{36}\end {equation}$$ $$99\dfrac pq=36.0$$ $$\dfrac pq=\frac {36}{99}$$

part 2

$2.1\overline{36}=2+0.1\overline{36}$ $$\dfrac pq=0.1\overline{36}\implies 10\dfrac pq=1.\overline{36}$$this is eqn (1).

multilpy with 100 in eqn(1) $$1000\dfrac pq=136.\overline{36}$$ then subtract (1) from (2). $$990\dfrac pq=135.0$$ $$\dfrac pq=\dfrac {135}{990}\implies\dfrac {3}{22}$$

so $2.1\overline{36}\implies 2+0.1\overline{36}\implies 2+\dfrac{3}{22}\implies 2\dfrac3{22}$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.