The way I see it is like this:
He ended with $42.15 -- the last thing that he did before that was buy the umbrella
$42.15 + $23.75 (umbrella) = $65.90
The next thing backwards in order is that he spent "half of his remaining money on a painting", so what he has left is one half what he had before he bought the painting.
$65.90 + $65.90 (cost of the painting) = $131.80
Before that, he spent $25 on dinner and $20 on a cab. So we add those all up:
$131.80 + $25 (dinner) + $20 (cab) = $176.80 (how much money he had at the start of the weekend)
To write it as a single equation:
$42.15 = -$23.75 + (-$25.00 + -$20.00)/2
end of the weekend = cost of umbrella + (cost of dinner + cost of cab)/cost of painting
Because the things to the right of the equal sign are costs, they are losses of money and therefore negative numbers. You don't have a cost for how much the painting is until you know how much he had before he bought the umbrella because it said that he spent "half of his remaining money on a painting", so he must have had twice the amount of money as how much he had before he bought the umbrella.