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How to solve this word problem with an equation?

During an evening out, Dean paid a cab driver $20. He then spent $25 on dinner and half of his remaining money on a painting. He brought an umbrella fro $23.75 and had $42.15 left. How much money did Dean have at the beginning of the weekend?

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3 Answers

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.

:) 

Comments

Your textual description is perfect, but I don't agree with your equations.  To put it in a single equation, I would say:

        /------------------- Initial amount
       / /------------------ Cab
      / /  /---------------- Dinner
     / /  /   /------------- Half (painting)
    / /  /   /   /---------- Umbrella
   / /  /   /   /       /--- Amount left
((X-20-25)/2)-23.75 = 42.15

Solving this for X, we find that X=$176.80, same answer you arrived at.

If we let X be the amount of money that Dean had at the start, we can write a single equation like this:

        /------------------- Initial amount
       / /------------------ Cab
      / /  /---------------- Dinner
     / /  /   /------------- Half for Painting
    / /  /   /   /---------- Umbrella
   / /  /   /   /       /--- Amount left
((X-20-25)/2)-23.75 = 42.15

Solving this for X, we find that X=$176.80

Solve with an equation.

Let x = the total money at the beginning.

Balance right before paying for painting:

x - (20+25) = 2(23.75+42.15)

Solve for x,

x = $176.80

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