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

Question

42724 views around the world

You can reuse this answer
Creative Commons License

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:

#0.41666666...xx3=1.25/3#

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

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