If you have $500 in cash and give $25 in change, tip out $125 to a stylist, and take $10.50 for lunch, how much cash remains?

To determine how much cash remains after the transactions, you start with $500. From this amount, you need to subtract each of the expenditures sequentially.

First, giving out $25 in change reduces the total to $475 ($500 - $25). Next, tipping out $125 to a stylist further decreases the amount to $350 ($475 - $125). Finally, taking $10.50 for lunch brings the cash down to $339.50 ($350 - $10.50).

Thus, after all the transactions, the total remaining cash is indeed $339.50. This calculation demonstrates a clear deduction of expenses from the initial cash amount, leading to the remaining cash balance.