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Originally Posted by tinapbeana
well, if it were me i'd be ok leaving the money where it is as a general EF and targeted savings account (i.e. i've got 5k in there, 3k is EF 1k is car savings and 1k is trip savings). perhaps if your sons sat down with you and thought about goals they have for their money and amounts they'd like to reach. for instance, one might want a 2k EF and at least 2k going into a house fund, and then after that he might be comfortable with splitting all savings into 3rds between ef, house, and roth. once the ef reaches a specified point, then it would be house and roth...
for me this is a bit different that the question regarding your own budget (i.e. the vet fund). in my world i would have a vet fund be an amount that would cover shots, meds, etc that normally pop up but not emergency care (that would be EF). in that case i'd set a goal (i want 500 available if needed for vets) and once that's reached i'd funnel overages to another vehicle. if some of the vet money gets used, stop the funneling until it's back up to the desired level and start again...
please bear in mind i'm still on my first cuppa coffee if this doesn't make huge sense 
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At age 18 and 20, the "kids" have time few others do to "save". Meaning they could contribute less than you or I and have more in the end. To me, sitting on cash does not make much sense.
The sooner money is "put to work", the sooner you won't to work for money.
I do believe in having ~ 1 month of bills in cash... and having funds equal to 2-4 months available. But the 2-4 months do not have to be "liquid". Could be a credit card, stock account, CD or other.
Mid term purchases (like cars) change this somewhat (need more than 4-5 months cash on hand).
In addition "free cash flow" each month changes the equation. If I make 10k per month, and need only 4k for bills, the 6k "positive cash flow" is 1.5 months emergency cash.