# Fractions

There exists a interesting relationship between $27$ and $37$

$\frac{1}{27}=0.037037037...$ and $\frac{1}{37}=0.027027027...$ (That is, each number forms the other number's repeating decimal.)

This can be explained as follows :

Say $\frac{1}{X}=0.YYY....$ If $Y$ is n digits long,

then the fraction equals $Y$ times $10^{-n}+10^{-2n}+10^{-3n}...= \frac{Y}{(10^n-1)}$.

Therefore, we need $XY=10^n-1$. In the above case, $27*37=10^3-1$; other examples are $3*3=10^1-1$

So $\frac{1}{3}=0.333...$

or $11*9=10^2-1$ so $\frac{1}{11}=0.090909...$ and $\frac{1}{9}=0.11111...$ and so on. Note by Danish Ahmed
5 years, 10 months ago

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Oh that's nice! Thanks for sharing this pattern that you noticed.

Staff - 5 years, 10 months ago

Excellent

- 5 years, 10 months ago

nice

- 5 years, 10 months ago

It is Fantastic! Thanks for sharing @Danish Ahmed

- 5 years, 10 months ago

!!!

- 5 years, 10 months ago