This is an ambiguous question because the word quick is being used grammatically incorrectly. The question should be, 'If a shutter speed is 1/100, what is twice as LONG .... or twice as FAST?' If you ask what is twice as long or half as long as 1/100, the answers would be 1/50 and 1/200 respectively. However, twice as fast or half as fast would be 1/200 and 1/50 respectively. Twice as quick means nothing because the adverb quickly or quicker would more often be used - i.e. 'He can get to school twice as quickly on a bike than he can walking' which implies that the distance covered is twice as quick by bike than on foot......or alternatively 'He can get to school quicker on a bike than walking.' which implies that cycling to school is simply quicker than walking but not necessarily twice as quick. No wonder children of today are sadly lacking grammatical skills!
1/50 is half the speed of 1/100, it takes exactlytwice as much time, so in order for it to be "twice as quick" you have to go for something faster, which would be 1/200 (And, for those few who are reading this, that would be the correct answer).