A conservation project begins in a remote settlement with a fixed number of residents. However, due to a government relocation plan, 25% of the population relocates to urban areas at the end of each year. Assuming no new settlers arrive and no one is born or dies, the population decreases by 25% annually.
Consider the following statements:
Statement I: After 5 years, the fraction of the original population remaining is (243/1024).
Statement II: The ratio of the population remaining at the end of the 3rd year to that at the end of the 4th year is 4/3.

Select the correct answer from the options below:
(a) Only Statement I is correct.
(b) Only Statement II is correct.
(c) Both Statements I and II are correct.
(d) None of the Statements is correct.

Question

A conservation project begins in a remote settlement with a fixed number of residents. However, due to a government relocation plan, 25% of the population relocates to urban areas at the end of each year. Assuming no new settlers arrive and no one is born or dies, the population decreases by 25% annually.
Consider the following statements:
Statement I: After 5 years, the fraction of the original population remaining is (243/1024).
Statement II: The ratio of the population remaining at the end of the 3rd year to that at the end of the 4th year is 4/3​.

Select the correct answer from the options below:
(a) Only Statement I is correct.
(b) Only Statement II is correct.
(c) Both Statements I and II are correct.
(d) None of the Statements is correct.

ONLiNE UPSC · Accepted Answer

Each year, 75% of the population remains. So after:
After 5 years:
Remaining population = (3/4)5 = 243/1024.
After 3 years:
Remaining population = (3/4)3 = 27/64​
After 4 years:
Remaining population = (3/4)4 = 81/256​
Required ratio = (27/64) / (81/256) = (27 &times; 256) / (64 &times; 81) = (27 &times; 4)/81 = 4/3
Statement I is correct.
Statement II is correct.

Hence, option c is correct.

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M Question 1	⏱ 0
A conservation project begins in a remote settlement with a fixed number of residents. However, due to a government relocation plan, 25% of the population relocates to urban areas at the end of each year. Assuming no new settlers arrive and no one is born or dies, the population decreases by 25% annually. Consider the following statements: Statement I: After 5 years, the fraction of the original population remaining is (243/1024). Statement II: The ratio of the population remaining at the end of the 3rd year to that at the end of the 4th year is 4/3. Select the correct answer from the options below: (a) Only Statement I is correct. (b) Only Statement II is correct. (c) Both Statements I and II are correct. (d) None of the Statements is correct.