To find out how long it will take for King County and Prince County to have the same population, we can set up an equation based on the given information.
Let \( t \) be the number of years it takes for the two populations to be equal.
King County:
- Initial population: 157,800
- Decrease in population per year: 800
The population of King County after \( t \) years is: \[ 157800 - 800t \]
Prince County:
- Initial population: 140,300
- Increase in population per year: 1,700
The population of Prince County after \( t \) years is: \[ 140300 + 1700t \]
Set the two equations equal to each other: \[ 157800 - 800t = 140300 + 1700t \]
Now, solve for \( t \):
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Combine like terms: \[ 157800 - 140300 = 1700t + 800t \] \[ 17400 = 2500t \]
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Divide both sides by 2500: \[ t = \frac{17400}{2500} \]
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Simplify the fraction: \[ t = \frac{1740}{250} = \frac{87}{12} \approx 7.25 \text{ years} \]
Thus, it will take approximately \( \frac{87}{12} \) years for King County and Prince County to have the same population, or about 7 years and 3 months.
The answer is: \[ \frac{87}{12} \text{ years} \]