Answer:
127/495
Step-by-step explanation:
Turning repeating decimals into fractions is a little like surgery: you need to make sure to "cut out" just the right part to let you simplify. In this case, it'd be very nice if we could deal with that repeating 0...565656... part. Let's start by giving our decimal a name:
[tex]x=0.25656...[/tex]
We're gonna split this into two parts: a number we can operate on, and "knife" to make the incision. For both, we'd like the only part to the right of the decimal to be the repeating part. To make our "knife," we'll multiply both sides of the equation by 10:
[tex]10x=2.5656...[/tex]
To make the number we'll be cutting from, we can shift the decimal point by any multiple of two more to the right. I'll keep things simple and stick with 2. This corresponds to multiplying the "knife" equation by 100 on both sides:
[tex]1000x=256.5656...[/tex]
Surgery time! To cut out the repeating bit, we can now just subtract the "knife" equation from the one above:
[tex]1000x-10x=256.5656...-2.5656...\\990x=254[/tex]
And just like that, it's gone! All that's left to do now is divide both sides by 990 and simplify:
[tex]x=254/990=127/495[/tex]
