I assume the question means to say that the population increased by 6.6666......%, and not by precisely 6.66% (if it means that, none of the answers are right).
As a fraction, 12.5% is 1/8 (it's half of 25%, so half of 1/4), and 6.66666...% is 2/30 (it's one tenth of 66.666...%, so one tenth of 2/3).
So when something increases by 12.5%, we are multiplying it by 1 + 1/8 = 9/8, when something decreases by 12.5% we are multiplying it by 1 - 1/8 = 7/8, and when something increases by 6.66...% we are multiplying it by 1 + 2/30 = 32/30 = 16/15. So overall, the population is being multiplied by
(9/8)(7/8)(16/15) = (3)(7/4)(1/5) = 21/20
and the increase is thus 21/20 - 1 = 1/20 = 5%.
Or we can avoid all that work - if something goes up by x%, then down by precisely x%, you always end up with something smaller than what you started with. That's because the second percent change, the percent drop, is taken from a larger value than the first. So here, when the population goes up 12.5% and then down 12.5%, the overall population has declined slightly. The final increase of 6.66...% therefore will not increase the population all the way to something 6.66...% above the initial value, so the right answer here must be less than 6.66...%, and the only candidate is 5%.
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