How do you convert 0.47 (47 being repeated) to a fraction?

1 Answer
Dec 19, 2017

#0.bar(47) = 47/99#

Explanation:

First note that we can indicate the repeating part of a decimal representation by placing a bar over the repeating pattern:

#0.47474747... = 0.bar(47)#

To find the equivalent fraction, first multiply by #(100-1)# to get an integer:

#(100-1) 0.bar(47) = 47.bar(47)-0.bar(47) = 47#

Then divide both ends by #(100-1)# to find:

#0.bar(47) = 47/(100-1) = 47/99#

This is in simplest form, since #47# and #99# have no common factor larger than #1#

Why #(100-1)# ?

Multiplying by #100# has the effect of shifting the number #2# places to the left, which is the length of the repeating pattern. Then the #-1# has the effect of subtracting the original, resulting in the cancellation of the recurring tail.

Alternative method

If you know that #1 = 0.bar(9)#, then you can say:

#0.bar(47) = 47/99 * 0.bar(99) = 47/99 * 1 = 47/99#