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

2 Answers
Mar 3, 2016

0.3bar(25) = 161/4950.3¯¯¯¯25=161495

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)0.3¯¯¯¯25.

The trick we will use is to multiply by a power of 1010 and subtract the original value to eliminate the repeating digits.

Let x = 0.3bar(25)x=0.3¯¯¯¯25

=>100x = 32.5bar(25)100x=32.5¯¯¯¯25

=>100x - x = 32.5bar(25)-0.3bar(25)100xx=32.5¯¯¯¯250.3¯¯¯¯25

=>99x = 32.2 = 161/599x=32.2=1615

:.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.