If there is a policy that 1/3rd of a population of a community has migrated every year from one place to some other place, what is the leftover population of that community after the sixth year, if there is no further growth in the population during this period?

(a) 16/243rd part of the population
(b) 32/243rd part of the population
(c) 32/729th part of the population
(d) 64/729th part of the population ✓ UPSC's answer

Why the answer is (d)

• If 1/3rd of the population migrates every year, then 2/3rd of the population remains at the end of each year.

• The leftover population after the first year is (2/3) of the initial population.

• The leftover population after the second year is (2/3) * (2/3) = (2/3)^2 of the initial population.

• Following this pattern, the leftover population after the sixth year is (2/3)^6 of the initial population.

• Calculating (2/3)^6 gives 2^6 / 3^6, which equals 64 / 729.

• Therefore, the leftover population is 64/729th part of the original population, corresponding to option (d).

Why the other options are wrong

(a) 16/243rd part of the population: 16/243 is equal to (2/3)^4, which represents the population after 4 years, not 6.
(b) 32/243rd part of the population: 32/243 is not a power of 2/3 with an integer exponent, as 32 is 2^5 and 243 is 3^5, but the numerator and denominator powers do not match the required 6th power structure correctly for this specific fraction value.
(c) 32/729th part of the population: 32/729 is incorrect because the numerator should be 2^6 = 64, not 32.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2017, held on 18 June 2017. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.