How do you convert 0.325 (25 being repeated) to a fraction?

2 Answers
Mar 3, 2016

#0.3bar(25) = 161/495#

Explanation:

A common notation to denote a repeating decimal is to put a bar over the repeating sequence. In this case, we would write the number in question as #0.3bar(25)#.

The trick we will use is to multiply by a power of #10# and subtract the original value to eliminate the repeating digits.

Let #x = 0.3bar(25)#

#=>100x = 32.5bar(25)#

#=>100x - x = 32.5bar(25)-0.3bar(25)#

#=>99x = 32.2 = 161/5#

#:.x = 161/(5*99) = 161/495#

Mar 3, 2016

Multiply by #(1000-10)# and rearrange to find:

#0.3bar(25) = 161/495#

Explanation:

#(1000-10)*0.3bar(25) = 325.bar(25) - 3.bar(25) = 322#

So, dividing both sides by #(1000-10)# we find:

#0.3bar(25) = 322/(1000-10) = 322/990 = 161/495#

Why #(1000-10)# ?

#1000-10 = 10 (100-1)#

The initial factor of #10# is to shift our starting number one place to the left so that the repeating portion starts just after the decimal point.

The #(100-1)# factor is to shift the number a couple more places to the left and subtract the original value. Since the repeating pattern is of length #2# this serves to cancel out the repeating part, leaving us with an integer.