How to convert decimals to fractions

• May 13th 2010, 08:52 PM
florx
How to convert decimals to fractions
http://i44.tinypic.com/2qlfo91.jpg

How would we solve problems such as 16 and 17? These problems do not repeat in numbers such as 0.235235235.

Like for 16, how would we put that in geometric form first?
I tried 0.122 + 0.0012 + 0.000012 ect but cannot find out how to get a fraction that will give that answer.

• May 13th 2010, 10:15 PM
slovakiamaths
Solution13: Let x=0.232323............. (1)
multiply (1) with 100,we get
100x=23.232323............ (2)
subtract (1) from (2),we get
99x=23
x=23/99(Happy)
• May 14th 2010, 03:44 AM
Prove It
16.

$0.1222222\dots = \frac{1}{10} + \frac{2}{100} + \frac{2}{1000} + \frac{2}{10\,000} + \dots$

$= \frac{1}{10} + 2\sum_{k = 2}^{\infty}\frac{1}{10^k}$.

In this case, the geometric series has $a = \frac{1}{100}$ and $r = \frac{1}{10}$.

Since $S_{\infty} = \frac{a}{1 - r}$

$S_{\infty} = \frac{\frac{1}{100}}{1 - \frac{1}{10}}$

$= \frac{\frac{1}{100}}{\frac{9}{10}}$

$= \frac{10}{900}$

$= \frac{1}{90}$.

So $\frac{1}{10} + 2\sum_{k = 2}^{\infty}\frac{1}{10^k} = \frac{1}{10} + 2\left(\frac{1}{90}\right)$

$= \frac{9}{90} + \frac{2}{90}$

$= \frac{11}{90}$.

Therefore $0.122222\dots = \frac{11}{90}$.
• May 14th 2010, 03:50 AM
Prove It
17.

$0.478888\dots = \frac{47}{100} + \frac{8}{1000} + \frac{8}{10\,000} + \frac{8}{100\,000} + \dots$

$= \frac{47}{100} + 8\sum_{k = 3}^{\infty}\frac{1}{10^k}$

In this case, the geometric series has $a = \frac{1}{1000}$ and $r = \frac{1}{10}$.

Therefore $S_{\infty} = \frac{\frac{1}{1000}}{1 - \frac{1}{10}}$

$= \frac{\frac{1}{1000}}{\frac{9}{10}}$

$= \frac{10}{9000}$

$= \frac{1}{900}$.

Thus $\frac{47}{100} + 8\sum_{k = 3}^{\infty}\frac{1}{10^k} = \frac{47}{100} + 8\left(\frac{1}{900}\right)$

$= \frac{423}{900} + \frac{8}{900}$

$= \frac{431}{900}$.

Therefore $0.478888\dots = \frac{431}{900}$.