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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?

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Olivia

  1. 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

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    oobleck

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