Annie usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for $$36$$ hours. If she is using it constantly, the battery will last for only $$10$$ hours. Since the last recharge, her phone has been on $$20$$ hours, and during that time she has used it for $$120$$ minutes. If she only keeps using the phone $3$ hours more and then leaves the phone on, how many more hours will the battery last after she leaves the phone on? (Adapted from $$2004$$ AMC $$8$$ Problem, Question \#$$12$$) 
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When not being used, the cell phone uses up $$\dfrac{1}{36}$$ of its battery per hour. When being used, the cell phone uses up $$\dfrac{1}{10}$$ of its battery per hour. Since Annie\textquotesingle s phone has been on for $$20$$ hours, of those $$18$$ simply on and $2$ being used, $$18\times\left(\dfrac{1}{36}\right)+2\times\left(\dfrac{1}{10}\right)=\dfrac{7}{10}$$ of its battery has been used up. To drain the remaining $$\dfrac{3}{10}$$, the phone can last for $$(\frac{3}{10}-3\times\frac1{10})\div \frac1{36}=0$$ more hours without being used. 