1. ## Decimal Representations...

1/27 = 0.037037037...
1/37 = 0.027027027...

What is the relationship between these two decimals? How would I find other pairs of decimals with the same relationship?

2. ## Re: Decimal Representations...

You can contemplate Whole Number, Single-Digit solutions (a,b,c,d) to this:

$\frac{1}{10\cdot a + b}\;=\;\frac{10\cdot c + d}{999}$

You suggested one such solution:

a = 2
b = d = 7
c = 3

3. ## Re: Decimal Representations...

Hello, thamathkid1729!

$\begin{array}{ccc}\dfrac{1}{27} \:=\: 0.037\:037\:037\hdots \\ \\[-3mm] \dfrac{1}{37} \:=\: 0.027\:027\:027\hdots \end{array}$

What is the relationship between these two decimals?
How would I find other pairs of decimals with the same relationship?

We have: . $\frac{1}{a} \;=\;\frac{b}{10^3} + \frac{b}{10^6} + \frac{b}{10^9} + \hdots \;=\;\frac{b}{10^3}\left(1 + \frac{1}{10^3} + \frac{1}{10^6} + \hdots\right)$

. . . . . . . . $\frac{1}{a} \;=\;\frac{b}{10^3}\left(\frac{1000}{999}\right) \;=\;\frac{b}{999} \quad\Rightarrow\quad ab \,=\,999$

We have: . $\frac{1}{b} \;=\;\frac{a}{10^3} + \frac{a}{10^6} + \frac{a}{10^9} + \hdots \quad\Rightarrow\quad ab \:=\:999$

Therefore:
$(a,b) \:=\:(1,999) \quad\Rightarrow\quad \begin{Bmatrix} \dfrac{1}{1} &=& 0.999\:999\:999\hdots \\ \\[-3mm] \dfrac{1}{999} &=& 0.001\:001\:001\hdots \end{Bmatrix}$

$(a,b) \:=\:(3,333) \quad\Rightarrow\quad \begin{Bmatrix}\dfrac{1}{3} &=& 0.333\:333\:333\hdots \\ \\[-3mm] \dfrac{1}{333} &=& 0.003\:003\:003\hdots \end{Bmatrix}$

$(a,b)\:=\:(9,111) \quad\Rightarrow\quad \begin{Bmatrix}\dfrac{1}{9} &=& 0.111\:111\:111\hdots \\ \\[-3mm] \dfrac{1}{111} &=& 0.009\:009\:009\hdots \end{Bmatrix}$

$(a,b) \:=\:(27,37) \quad\Rightarrow\quad \begin{Bmatrix}\dfrac{1}{27} &=& 0.037\:037\:037\hdots \\ \\[-3mm] \dfrac{1}{37} &=& 0.027\:027\:027\hdots \end{Bmatrix}$

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For four-digit decimal cycles: . $ab \,=\,9999$

Therefore:
$(a,b) \,=\,(1,9999) \quad\Rightarrow\quad \begin{Bmatrix}\dfrac{1}{1} &=& 0.9999\:9999\hdots \\ \\[-3mm] \dfrac{1}{9999} &=& 0.0001\:0001\hdots \end{Bmatrix}$

$(a,b) \,=\,(3,3333) \quad\Rightarrow\quad \begin{Bmatrix}\dfrac{1}{3} &=& 0.3333\:3333\hdots \\ \\[-3mm] \dfrac{1}{3333} &=& 0.0003\:0003 \hdots\end{Bmatrix}$

$(a,b) \,=\,(9,1111) \quad\Rightarrow\quad \begin{Bmatrix}\dfrac{1}{9} &=& 0.1111\:1111\hdots \\ \\[-3mm] \dfrac{1}{1111} &=& 0.0009\:0009\hdots \end{Bmatrix}$

$(a,b) \,=\,(11,909) \quad\Rightarrow\quad \begin{Bmatrix} \dfrac{1}{11} &=& 0.0909\:0909\hdots \\ \\[-3mm] \dfrac{1}{909} &=& 0.0011\:0011\hdots \end{Bmatrix}$

$(a,b) \,=\,(33,303) \quad\Rightarrow\quad \begin{Bmatrix}\dfrac{1}{33} &=& 0.0303\:0303\hdots \\ \\[-3mm] \dfrac{1}{303} &=& 0.0033\:0033\hdots \end{Bmatrix}$

$(a,b) \,=\,(99,101) \quad\Rightarrow\quad \begin{Bmatrix}\dfrac{1}{99} &=& 0.0101\:0101\hdots \\ \\[-3mm] \dfrac{1}{101} &=& 0.0099\:0099\hdots \end{Bmatrix}$

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For your amusement, a seven-digit cycle:

. . $\begin{array}{cccc}\dfrac{1}{2151} &=& 0.0004649\:0004649\hdots \\ \\[-3mm] \dfrac{1}{4649} &=& 0.0002151\:0002151\hdots \end{array}$