Question

0.7141414 repeating (14) decimal to a fraction ?

are both methods valid?

1000x=714.1414
10x=7.14
990x=707

707/990

Or

100x=71.41414
x = .71414
99x=70.7
70.7/99 x 10 both sides

707/990

Answer 1

Yes, both are valid, because they basically do the same thing. The first one works with numbers 10 times bigger than the second.
For a quick check, subtract 7 from 714 and divide by 990 (99 from 14 repeating, 0 from 7) to get 707/990.
Example,
0.7423232323...=(7423-74)/9900=7349/9900=0.74232323....