How many times should you post on Facebook page per day?

This really depends on your brand and the amount of fans you have.
If you don't post enough, if run the risk of users not knowing that you're still alive.
If you post too much, you also run the risk of spamming your fans and irritating them constantly.
Not all news is relevant news! On average, two posts a day are substantial.
You should focus on posts that will gain interaction and showcase that you're communicating to your fans - two way communication is vital for a page.
Posting and not replying will not garner success - even if you post up to 5 times a day - the page won't see success if you're not communicating with your fans.
Depending on the type of page you are - your fans might be on Facebook in the morning, afternoon or evening - so testing the best times is up to you to see when you receive the largest amount of interaction.

I would suggest you post three times in a day one in the morning, one in the mid-day to catch people during their break and the last one in the evening to serve as take home.

I would suggest an average 3 - 5 posts per day.
You should track interactions (Likes, comments, shares) to know what is the best times to post.

Statistics show that around50% of all Facebook users check their page at least once a day, and they need to see that you are regularly posting new material. Some guides recommend posting at least once a day, but the core principles are to post when you have interesting content, and to judge how often your audience wants to hear from you.

Facebook just ONE post a day! Not more...