The number of trees in a town is 17,640 after a 5% annual increase for two years. How many trees were there 2 years ago?

Introduction / Context:
This question examines percentage growth over multiple years, which is a classic example of compound increase. Instead of simple addition of percentages, each year the increase is applied to the updated value. In population growth, interest calculations, or value appreciation, we must treat these as multiplicative, not additive, processes. Here we are given the current number of trees and the annual growth rate, and we must find the original number of trees two years earlier.

Given Data / Assumptions:
Present number of trees in the town = 17,640.
Annual percentage increase in the number of trees = 5 percent per year.
The increase has taken place for 2 consecutive years at the same rate.
We need to find the number of trees 2 years ago, before these increases.

Concept / Approach:
A 5 percent increase means the quantity becomes 105 percent of its previous value, which is 1.05 times the earlier amount. After two years with the same annual increase, the overall multiplication factor is 1.05 * 1.05, that is 1.05^2. If the initial number of trees is N, then after two years the number becomes N * 1.05^2. We know this final number is 17,640. Therefore, N = 17,640 / (1.05^2). Evaluating this gives the original population of trees. This is the standard compound growth formula rearranged to recover the initial value.

Step-by-Step Solution:
Let the number of trees 2 years ago be N. After 1 year at 5 percent increase, trees become N * 1.05. After another year at 5 percent increase, trees become N * 1.05 * 1.05 = N * 1.05^2. We are given that after 2 years the number of trees is 17,640. So N * 1.05^2 = 17640. Compute 1.05^2: 1.05 * 1.05 = 1.1025. Therefore N = 17640 / 1.1025. 17640 / 1.1025 = 16000. Hence, there were 16,000 trees in the town 2 years ago.

Verification / Alternative check:
Verify using forward calculation. Start with 16,000 trees. After 1 year with 5 percent increase: 16,000 * 1.05 = 16,800. After the second year: 16,800 * 1.05 = 17,640. This matches the final value given in the question, so our initial value is correct. Any other starting value would not produce exactly 17,640 after two successive increases of 5 percent each.

Why Other Options Are Wrong:
If the initial number were 14,000, then after 2 years at 5 percent per year the final value would be 14,000 * 1.05^2, which is less than 17,640. Similarly, with 15,000 as the starting value, the final population would still be smaller than 17,640. If we take 18,000 as the starting value, the final population becomes 18,000 * 1.05^2, which exceeds 17,640. Only 16,000 gives the exact required final count.

Common Pitfalls:
A frequent mistake is to double the 5 percent and assume the total growth over two years is 10 percent, then divide 17,640 by 1.10. This ignores compounding and leads to an incorrect answer. Another error is to subtract twice 5 percent from 17,640 instead of reversing the growth properly. Always remember that repeated percentage increases or decreases must be handled with multiplication by growth factors, not simple addition or subtraction of percentages, especially when reversing the process to find an original amount.

Final Answer:
The town had 16,000 trees 2 years ago before the two successive 5 percent increases.