Jeri finds a pile of money with at least $\$200$. If she puts $\$80$ of the pile in her left pocket, gives away $\frac{1}{3}$ of the rest of the pile, and then puts the rest in her right pocket, she'll have more money than if she instead gave away $\$200$ of the original pile and kept the rest. What are the possible values of the number of dollars in the original pile of money? (Give your answer as an interval.)

 Aug 17, 2022

Let the amount of money Jeri owns be \(j\). Off the bat, we know \(j \geq 200\).


Represent the first part as \({2 \over 3}(j-80) + 80\). The second part represents how much money Jeri has in her right pocket, and the latter represents the amount of money in her left pocket. The second part is just \(j - 200\).


So we have the inequality: \({2 \over 3}(j-80) + 80 > j-200\)


Solving, we find \(j<680\), but because we know \(j \geq 200\), we have \(200 \leq j < 680\), which equals \(\color{brown}\boxed{[200, 680)}\) in interval notation. 

 Aug 17, 2022

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