The towns of Franklin and Chester post their populations on signs just outside of their towns. The signs are updated once a year at the beginning of the year.
During year 111, Franklin's sign read "\text{Franklin: Population of } 20{,}000Franklin: Population of 20,000start text, F, r, a, n, k, l, i, n, colon, space, P, o, p, u, l, a, t, i, o, n, space, o, f, space, end text, 20, comma, 000", while Chester's sign read "\text{Chester: Population of } 25{,}000Chester: Population of 25,000start text, C, h, e, s, t, e, r, colon, space, P, o, p, u, l, a, t, i, o, n, space, o, f, space, end text, 25, comma, 000". Each year, the populations grew. Specifically, Franklin's population grew by 5\%5%5, percent each year, and Chester's population grew by 500500500 people each year.
What is the first year in which Franklin's sign shows a larger number than Chester's sign?
To find the first year in which Franklin's population surpasses Chester's population, we need to determine when Franklin's population becomes greater than Chester's population.
Let's analyze the populations for each year:
Year 111:
Franklin's population: 20,000 (grows by 5% each year)
Chester's population: 25,000 (grows by 500 people each year)
To compare the populations, we can set up a simple inequality:
Franklin's population > Chester's population
Let's calculate the populations for the next few years and continue to compare them until Franklin's population exceeds Chester's population:
Year 112:
Franklin's population: 20,000 * 1.05 (5% growth) = 21,000
Chester's population: 25,000 + 500 = 25,500
Year 113:
Franklin's population: 21,000 * 1.05 = 22,050
Chester's population: 25,500 + 500 = 26,000
Year 114:
Franklin's population: 22,050 * 1.05 = 23,152.5 (rounded to 23,153)
Chester's population: 26,000 + 500 = 26,500
Year 115:
Franklin's population: 23,153 * 1.05 = 24,310.65 (rounded to 24,311)
Chester's population: 26,500 + 500 = 27,000
Year 116:
Franklin's population: 24,311 * 1.05 = 25,526.55 (rounded to 25,527)
Chester's population: 27,000 + 500 = 27,500
In Year 116, Franklin's population (25,527) becomes larger than Chester's population (27,500). Therefore, the first year in which Franklin's sign shows a larger number than Chester's sign is Year 116.
what a mess of redundant and uselessuselessuseless t-e-x-t !!
after x years,
Franklin is at 20000*1.05^x
Chester is at 25000+500x
so you need to solve
20000*1.05^x > 25000+500x
you will need to do this graphically, or using some numeric technique.
If your calculator can solve such an equation (or you use one of the many handy online calculators), you will find that
x ≈ 7.4
See the graphs at
https://www.wolframalpha.com/input/?i=20000*1.05%5Ex+%3E+25000%2B500x+for+-20+%3C+x+%3C+20