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