A hospital emergency room $($ER$)$ is currently organized so that all patients register through an initial check$-$in process. At his or her turn, each patient is seen by a doctor and then exits the process, either with a prescription or with admission to the hospital. Currently, $10$ people per hour arrive at the ER$,10\%$ of whom are admitted to the hospital. On average, $5$ patients are waiting to be registered and $10$ are waiting to see a doctor. The registration process takes, on average, $6$ minutes per patient. Among patients who receive prescriptions, the average time spent with a doctor is $10$ minutes. Among those admitted to the hospital, the average time is $30$ minutes.

a$.$ On average, how many patients are being examined by doctors?

b$.$ On average, how many patients are in the ER$?$

c$.$ On average, how long does a routine care patient $($who receives prescriptions$)$ stay in the ER$?$

d$.$ On average, how long does an emergency care patient $($who is admitted to the hospital$)$ stay in the ER$?$

e$.$ On average, how long does a patient stay in the ER$?$

Please be correct! Last time someone gave me all the wrong answers so I still didn$$t know how to do it$.$