Suppose you have a part-time sales position over the winter break in a small clothing store that is part of a national chain. The store’s one full-time employee, with whom you have become friendly, hired you. Explain what you would do in the situations described below, and identify two internal control problems that exist in each situation.
1. You arrive at the store at 6 P.M. to take over the evening shift from the full- time employee who hired you. You notice that this person takes a coat from a rack, puts it on, and leaves by the back door. You are not sure if the coat is one that was for sale or if it belonged to the employee.
2. You are the only person in the store on a busy evening. At closing time, you total the cash register and the receipts and discover that the cash register is $20 short of cash. You consider replacing the $20 out of your pocket because you think you may have made a mistake and are afraid you might lose your job if the company thinks you took the money.p

Considering this is homework, tell us what you think. We may help after that.

I don't know how to do in this situation.

How would you check to see to whom the coat belonged? How do you balance the register?

Ethics questions don't always have right and wrong answers. You need to say what you think is right, and why!
Without going into why, I will say that I would tell the owner of the store about the coat, and the same day I would confront the employee. Plus, aside from ethics, you need to save your own skin by not being the one accused of stealing the coat later on.
I would not put $20 in. The mistake might be yours, but you learn from it. Keep careful track of when you give change, and learn how to stop a short change trick.

One thing you should always do is count the register when taking over a shift. Inventory tags are a good idea too.

Coats that are for sale should have a security tag that will set-off an alarm if you try to leave the store without purchasing the coat. Also the back door of the store should be an emergency exit that should sound an alarm when opened. Also employees should have a special area assigned that is for employees only for storing personal items like coats and bags like a break room or a locker room.

ma0641 is correct in that you should count your register when you take over a shift and make sure both you and the previous employee sign for the amount in the register. This way you will be responsible for the amount in the register when you take over the register and the previous employee will be responsible for the amount from his or her starting and ending their shift.

OP's quiz is under Ethics, not store security and how to be a good cashier.

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1. You arrive at the store at 6 P.M. to take over the evening shift from the full- time employee who hired you. You notice that this person takes a coat from a rack, puts it on, and leaves by the back door. You are not sure if the coat is one that was for sale or if it belonged to the employee.

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This question is ridiculous. Employees don't hang their coats on a sales rack. No retail store would ask employees to hang their personal property on the floor with items that are for sale. So if the employee took a coat from a sales rack, and didn't come to the cashier to pay for it, it's very obvious that this coat does not belong to them.

What to do about it? That's up to you. But this is not a question of whether the coat belongs to the employee, it's a matter of what you will do about seeing an employee stealing merchandise.

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2. You are the only person in the store on a busy evening. At closing time, you total the cash register and the receipts and discover that the cash register is $20 short of cash. You consider replacing the $20 out of your pocket because you think you may have made a mistake and are afraid you might lose your job if the company thinks you took the money.p

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When a new shift begins the register is cashed out, and the new cashier coming on for their shift counts their register. At the end of the night if you're short $20, it's your mistake, not someone else's. Replacing the money is ridiculous. Everyone makes mistakes. Some stores will give you more than one chance, some have a limit to how much you can be short on your register. But really, if you're short $20, maybe it's best to find another job. It doesn't sound like you're cut out to work in retail.

The issue, is they want to know what you would do.

So you be honest and say what you would do.

I am taking one currently with a lot harder ethical questions.

'xactly fr chuck

It might be posted under ethics but also tells to "identify internal control problems ".

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The issue, is they want to know what you would do.

So you be honest and say what you would do.

I am taking one currently with a lot harder ethical questions.

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Exactly. It's not about what any of us would do, it's about the OP, and their ethics, not ours. There's no way to answer this for the OP, he/she has to answer this them-self, based on their ethics.