Thread: Converting Decimals to Fractions

1/34 to decimal=


Change to Fraction
x=.12361236


Changes to Fraction
x=.352171717


Please help

Thanks,

Steve

Originally Posted by skweres1

Change to Fraction
x=.12361236

For a repeating fraction take the part that repeats and put the same number of 9s as the denominator and cancel if appropriate.

$\displaystyle x = \frac{1236}{9999} = \frac{412}{3333}$

Originally Posted by skweres1

1/34 to decimal=
Change to Fraction
x=.12361236
Changes to Fraction
x=.352171717
Please help
Thanks,
Steve

x = 0.352171717
This is a repeating decimal:
Take it apart

0.352 + 0.000171717 = 0.352171717

as e^(i*pi) showed in his post:
$\displaystyle \dfrac{17}{99000} = 0.0001717171717171717...$


$\displaystyle \dfrac{352}{1000} + \dfrac{17}{99000}$ = 0.352171717

you can see that 352/1000 can be reduced
$\displaystyle \dfrac{352}{1000} = \dfrac{176}{500} = \dfrac{88}{250} = \dfrac{44}{125}$

so add the fractions
$\displaystyle \dfrac{44}{125} + \dfrac{17}{99000}$

need a common denominator
$\displaystyle \dfrac{44}{125} \times \dfrac{99000}{99000} \,+\, \dfrac{17}{99000}\times \dfrac{125}{125}$

$\displaystyle \dfrac{ 44 \cdot 99000 \, + \, 17 \cdot 125 }{ 125 \cdot 99000}$

$\displaystyle \dfrac{ 4356000 \, + \, 2125 }{ 12375000}$

$\displaystyle \dfrac{ 4358125 }{ 12375000}$
and that can be reduced:

divide numerator & denominator by 625 to get the fraction in lowest terms.

Spoiler:

$\displaystyle \dfrac{6973}{19800}$

The continued fraction algorithm is the method of choice for doing this type of conversions.
The above is not that algorithm.
Perhaps someone will explain/demonstrate the continued fraction algorithm for converting decimals to rationals.

if the digits are recurring indefinitely

x= 0.12361236...

10000x=1236.12361236...
-
x= 0.12361236...

9999x=1236

$\displaystyle x=\frac{1236}{9999}$


if the fraction doesnt repeat indefinitely

$\displaystyle 0.12361236=\frac{12361236}{100000000}$

Hello, Steve!

I will assume those are repeating decimals.

$\displaystyle \frac{1}{34}\text{ to decimal}$

Just divide it out until you get a repeating cycle . . .

$\displaystyle \frac{1}{34} \;=\;0.0 \overline{2941176970588235}\, \hdots$

$\displaystyle \text{Change to fraction: }\:x \:=\:0.12361236\hdots$

$\displaystyle \begin{array}{ccccc}\text{Multiply by 10,000:} & 10,000x &=& 1236.12361236\hdots \\
\text{Subtract }x\!: & \qquad\;\; x &=& \quad\;\; 0.12361236\hdots \\
\text{and we have:} & \;9999x &=& 1236\qquad\qquad\quad\;\; \end{array}
$

Therefore: .$\displaystyle x \;=\;\frac{1236}{9999} \;=\;\frac{412}{3333}$

$\displaystyle \text{Change to fraction: }\:x \:=\:0.362171717\hdots$

$\displaystyle \begin{array}{ccccc}\text{Multiply by 100,000:} & 100,\!000x &=& 36217.171717\hdots \\
\text{Multily by 1,000:} & \;\;\;1,\!000x &=& \quad 362.171717\hdots \\
\text{Subtract:} & \;99,\!000x &=& 35855\qquad\qquad\;\;
\end{array}$

Therefore: .$\displaystyle x \;=\;\frac{35,\!855}{99,\!000} \;=\;\frac{7,\!171}{19,\!800} $