Convert this into fractional number step by step?

Question

3.41287548754875...

Convert the above number to a rational number?

I was reviewing some pre calculus on my own but couldn't figure this out.

Answer 1

$$0.412\overline{8754}=\frac{4128754-412}{9999000}$$

Edit: the numerator is the difference of the number build by the preperiod followed by the period, in our case $4128754$, and the preperiod, here $412$. For the denominator, write down as much nines as the period is long, here $9999$, followed by as much zeros as the preperiod is long, in our case $000$.

Answer 2

General method:

• Let $x$ denote the input number
• Let $|n|$ denote the number of decimal digits in $n$
• Split $x$ into the following parts:
  • $\color\red{A}=$ the integer part, i.e., $\lfloor{x}\rfloor$
  • $\color\green{B}=$ the fraction part's non-periodic prefix
  • $\color\orange{C}=$ the fraction part's periodic postfix

Then:

$$x=\frac{(10^{|B|+|C|}-10^{|B|})\color\red{A}+(10^{|C|}-1)\color\green{B}+\color\orange{C}}{10^{|B|+|C|}-10^{|B|}}$$

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For example, if $x=3.412\overline{8754}$:

• $A=3$
• $B=412$
• $C=8754$

Then:

$$x=\frac{(10^{3+4}-10^{3})\cdot\color\red{3}+(10^{4}-1)\cdot\color\green{412}+\color\orange{8754}}{10^{3+4}-10^{3}}=\frac{34125342}{9999000}$$