How do you convert 0.416 (6 being repeated) to a fraction?

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Answer 1 · 2016-02-20T14:08:20

$5/12$

The following is the standard procedure to solve the problem:

$0.41bar(6)=0.41+0.00bar(6)$

if $x=0.0066666....$ :

$1000x=6.6666....$

and $100x=0.666666$

Then, if we subtract 1000x and 100x, we obtain:

$900x=6$

$x=6/900$

So

$0.41bar(6)=0.41+6/900= 41/100+6/900$

Now multiply 41/100 by 9/9 for equalizing the denominators:

$369/900+6/900=375/900$

Now, for simplication, decompose the numerator and denominator in prime factors:

$375/900=(3xx5^3)/(2^2xx3^2xx5^2)=5/(2^2xx3)=5/12$

Now, the simpler:

If you have something that ends in 33333.... or 666666., you just have to multiply by three to remove the repetition:

Now, you have a number 1.25 which is easily recognizable as 5/4:

$(5/4)/3=5/12$