The population of a rural region is currently 100000 and is expected to fall by 2% per year for the next 10 years. Assuming this decline is compounded annually, what will be the expected population after 10 years?

Introduction / Context:
This question uses the compound interest framework to model population decline instead of financial growth. A decrease of 2% per year compounded annually means that each year the population becomes 98% of the previous year. Over ten years, this repeated percentage decrease leads to a significantly lower population. Such exponential decay models are common in demography, ecology, and many applied fields.

Given Data / Assumptions:

• Current population P0 = 100000.
• Annual rate of decline = 2% per year.
• Time period n = 10 years.
• Decline is compounded annually.
• Each year, population is multiplied by 1 - 0.02 = 0.98.

Concept / Approach:
The same formula used for compound interest growth can be used for compound decrease by using a factor less than 1. The population after n years, Pn, is given by Pn = P0 * (1 - d) ^ n, where d is the rate of decrease in decimal form. Here d = 0.02 and P0 = 100000. After computing the factor (0.98) ^ 10, we multiply this by 100000 to find the expected population after ten years. This demonstrates how percentage changes accumulate multiplicatively over time.

Step-by-Step Solution:
Step 1: Convert the rate of decline to decimal form: d = 2% = 0.02. Step 2: The yearly multiplier for the population is 1 - d = 1 - 0.02 = 0.98. Step 3: The population after 10 years is P10 = 100000 * (0.98) ^ 10. Step 4: Compute (0.98) ^ 2 = 0.9604. Step 5: Continue multiplying to get (0.98) ^ 10, which is approximately 0.81707. Step 6: Multiply by the initial population: P10 = 100000 * 0.81707 = 81707. Step 7: Therefore, the expected population after 10 years is about 81707.

Verification / Alternative check:
A rough check can be done by considering that a 2% annual decline over 10 years is roughly a 20% decline if it were linear, so the population might be expected to reduce to about 80000. With compounding, the decline is slightly more than the simple 20% calculation suggests but still in that vicinity. The computed value 81707 is a little more than 80% of 100000 but still close, which is consistent with the fact that compounding each year leads to a total decline slightly greater than 20% but not dramatically more. This confirms that the answer is reasonable.

Why Other Options Are Wrong:
The option 91707 is too high and corresponds to only an 8% decrease, which does not match a 2% decline compounded over 10 years. The option 61707 is too low and would require a much higher annual decline rate than 2%. The options 71707 and 70707 also imply a far larger overall drop than 2% per year, and they do not match the compound decay calculation. Only 81707 is consistent with multiplying by 0.98 for ten consecutive years.

Common Pitfalls:
Students sometimes mistakenly subtract 2% per year in a linear fashion and compute population as 100000 - 10 * 2% of 100000 instead of compounding. Others may accidentally use 1.02 rather than 0.98, which would represent growth rather than decline. Another error is to round intermediate results aggressively, resulting in a final answer that deviates significantly from the precise value. Correctly recognising that a decline is modelled by a multiplier less than 1 and consistently applying that multiplier over the full period is crucial.

Final Answer:
The expected population of the rural region after 10 years, with a 2% annual decrease compounded annually, is approximately 81707, which matches option A.