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?

Correct Option (D)

If 1/3rd of the population migrates each year, the proportion of the population remaining is 1 - 1/3 = 2/3 of the previous year's population. This process is repeated annually for six years, with no additional population growth.

The population remaining after the first year will be (2/3) of the initial population.

After the second year, the remaining population will be (2/3) × (2/3) = (2/3)² of the initial population.

Following this geometric progression, the population remaining after the sixth year will be (2/3)⁶ of the initial population.

Calculating the value: (2/3)⁶ = 2⁶ / 3⁶ = 64 / 729.

Therefore, 64/729th part of the initial population remains after the sixth year.

Incorrect Options:

Options A, B, and C are incorrect as they represent different powers or incorrect calculations of the remaining population fraction over the specified period.

• Option A (16/243rd part): This value does not correspond to the correct calculation of (2/3)⁶. For instance, 16/243 is equivalent to 2⁴/3⁵.
• Option B (32/243rd part): This value corresponds to (2/3)⁵, which would be the population remaining after the fifth year, not the sixth.
• Option C (32/729th part): This value corresponds to 2⁵/3⁶, indicating a miscalculation in either the numerator's or the denominator's power.