In the series 4/9, 9/20, ....,39/86, the numerator of each fraction is increasing by 5 (9 - 4 = 5) and the denominator is increasing by 11 (20 - 9 = 11). If we apply the same pattern to find the missing fraction, we would add 5 to 9 (given numerator of second fraction) to get 14 (which is the numerator of the missing fraction) and add 11 to 20 (given denominator of second fraction) to get 31 (which is the denominator of the missing fraction). However, none of the provided options have 14/31. This means we have to apply some other pattern.

Option-2 i.e., 19/42 satisfies the series pattern, if we look closely to the given series, it`s actually the gap between the numerator and denominator of each fraction which is 5 (9 - 4 = 5, 20 - 9 = 11). So, the gap between 19 and 42 (42 - 19 = 23) is also 23 just like the next term 39/86 (86 - 39 = 47). So, the correct answer is option-2