Validating approach for a question on "repeating decimal to fraction"

[cringes_squeak.0q]

Hi everyone, I have been quite stuck with this question from prep swift. I think I understood it but
couldn’t fully wrap my head around because it feels like I am still missing something. I wonder if this is an example of hard question in GRE?

I wonder if I could use the following comparison approach instead? Would it present a universal solution for similar problem or it only works for this particular question?

My proposed solution

W = # digit after decimal and where the digits after decimal is 0.abc

Illustrating with an example

Step 1: Assume that abc = 567

0.567bar = x

Step 2: Multiply by 1,000

567.567bar = 1000x

Step 3: minus both sides by x

567 = 1000x - x

Step 4: factor the RHS

567 = (10^3 - 1)x

Step 5: re-arrange

X = 567 / (10^3 - 1) or

X = a / (10^3 - 1)

Step 6: compare with the given answer and we know that w = 3 because the exponent represent the # of digit after decimal

Therefore, answer is 10^w - 1

[Leaderboard]

You’re just choosing numbers - that’s a perfectly valid strategy for the GRE. In fact we recommend it at the end of the solution, because the “abstract” solution may be hard to understand for students.