# 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
3 years, 9 months ago

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

Staff - 3 years, 9 months ago

!!!

- 3 years, 9 months ago

It is Fantastic! Thanks for sharing @Danish Ahmed

- 3 years, 9 months ago

nice

- 3 years, 9 months ago

Excellent

- 3 years, 9 months ago