The population of a town increases by 5% in the first year, by 6% in the second year, and by 8% in the third year. If the present population after these three increases is 120204, what was the population of the town 3 years ago?

Introduction / Context:

This question is a classic example of successive percentage increase applied to population growth. The population grows by different percentages each year. The final population is given and the task is to compute the population at the beginning of the three-year period. This is essentially a reverse successive percentage problem, similar to compound growth in finance.

Given Data / Assumptions:

• First year increase: 5%.
• Second year increase: 6%.
• Third year increase: 8%.
• Population after three years = 120204.
• We assume growth is applied sequentially on the latest population each year.

Concept / Approach:

Each yearly increase can be represented as a multiplication factor: 5% increase corresponds to a factor 1.05, 6% to 1.06, and 8% to 1.08. The total growth factor over three years is the product of these three factors. Let the initial population be P. Then P multiplied by the total growth factor equals the final population 120204. We then divide the final population by this growth factor to find P.

Step-by-Step Solution:

Verification / Alternative check:

Verify by forward computation using P = 100000. After first year: 100000 * 1.05 = 105000. After second year: 105000 * 1.06 = 111300. After third year: 111300 * 1.08 = 120204. The computed final population matches the given value exactly, confirming that the original population was 100000.

Why Other Options Are Wrong:

• 102000 and 101204: These values, when grown by 5%, 6%, and 8%, do not reach exactly 120204.
• 98000: This would produce a significantly smaller final population after three years.
• 105000: This is actually the population after the first 5% increase, not at the beginning of the period.

Common Pitfalls:

A common mistake is to add the percentages (5 + 6 + 8 = 19%) and treat it as a single increase of 19%, which is incorrect because percentage increases compound multiplicatively, not additively. Another error is to treat the final population as if it were obtained by simple interest, dividing repeatedly without using growth factors. Always express each increase as a multiplication factor and multiply them to get the overall factor.

Final Answer:

The population of the town 3 years ago was 100000.