If I went on a trip with family but did business work during 1/2 the time can my 1/2 part of the trip be deducted as a business expense?
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From the IRS (emphasis added):
Travel expenses are the ordinary and necessary expenses of traveling away from home for your business, profession, or job. You can't deduct expenses that are lavish or extravagant, or that are for personal purposes.
You're traveling away from home if your duties require you to be away from the general area of your tax home for a period substantially longer than an ordinary day's work, and you need to get sleep or rest to meet the demands of your work while away.
So a trip to Italy with your family to "help your mental state" does not seem "ordinary and necessary", nor do your duties require it. It also seemed to be "for personal purposes" since it was with your family.
You can read the rest of the IRS article to get more details, but I think it would be tough to justify deducting even half of the expenses of the trip in an audit. Business expenses are scrutinized heavily and tend to be significant triggers of audits.
In addition to what was said already, I want to point out that some expenses might be deducible. This is particularly true if your boss asks you to do work during your travel. E.g:
Additionally, the time you spend working is not vacation time. So if you work 1/2 time, your vacation allowance should only be reduced by half, meaning you can stay twice as long [according to European laws that is, but I guess it's similar if you are employed in the US].