Don't mention it
This is a slightly more relaxed way of saying "you're welcome".
No worries
It is similar in sentiment to 'no problem', but is a little more relaxed and colloquial.
Not a problem
This is very common, especially in the US and Canada, where it is almost an automatic response. You can expect to hear it in all kinds of situations, such as at the grocery store, in a restaurant, or in conversation with friends.
My pleasure
This is probably the most formal and gracious option on the list. You are telling the other person that whatever favour you've just done for them, you truly enjoyed it and perhaps gained some benefit from it.
It was nothing
When you really want to assure someone that you didn't go to too much trouble for them. With the wrong tone of voice, it could sound like you're deflecting their gratitude.
I'm happy to help
If someone is thanking you for a service or a favour that required some exertion on your part but was very helpful to them, this is a good way to say "you're welcome". If the task was very simple for you, or did not have much impact on the other person, you might want to use one of the more informal options.
Not at all
This is also a very polite option, and you would normally use it after performing a service for someone. This option courteously suggests that thanks aren't necessary.
Sure
When you've just done a small favour: if someone thanks you for holding the door, say, or for picking up something that they've dropped.
Anytime
This is a nice response to use in a friendly conversation, because it implies that you would be happy to perform a similar favour again. It also has a relaxed and informal feel to it, while subtly letting the other person know that you're there for them.
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