Linear population change: City A has population 136000 decreasing by 2400 per year, and City B has population 84000 increasing by 1600 per year. After how many years will their populations be equal?
Aptitude
Simplification
Difficulty: Easy
Choose an option
Answer
Correct Answer: 13
Explanation
Introduction / Context:This problem models two populations changing linearly but in opposite directions. We set both expressions equal and solve for time. It is a straightforward linear equation in t (years).
Given Data / Assumptions:
- City A at t years: A(t) = 136000 - 2400*t
- City B at t years: B(t) = 84000 + 1600*t
- Equality A(t) = B(t)
Concept / Approach:Equate A(t) and B(t) to find the year where the populations match. Since both changes are linear with constant rates, a single solution exists if rates differ.
Step-by-Step Solution:
136000 - 2400t = 84000 + 1600t136000 - 84000 = 1600t + 2400t52000 = 4000tt = 52000 / 4000 = 13Verification / Alternative check:
A(13) = 136000 - 2400*13 = 136000 - 31200 = 104800B(13) = 84000 + 1600*13 = 84000 + 20800 = 104800 (matches)Why Other Options Are Wrong:
- 15, 18, 19: Substitution yields mismatched populations, so they are incorrect.
Common Pitfalls:
- Sign errors when moving terms (mixing +1600t and -2400t).
- Arithmetic errors dividing 52000 by 4000.
Final Answer:
13