Phil Anthropist likes to give away money. One day he went to the park with a wad of cash. He gave $100 to a man feeding pigeons. He then gave half of his remaining money to a child eating an ice-cream cone. He then gave $50 to the balloon seller. He then bought a hot dog and paid for it with a $20 bill. "Keep the change for a tip," he said to the hot-dog seller. Then he gave half his remaining money to someone giving a speech on a soapbox. At this point, he had $3 left and stuck it under the collar of a stray cat. How much money did Phil have when he started?

Question

Bot · Answer

Let X be the amount Phil started with, in dollars. According to the problem, Phil first gave $100, then half of the remaining amount, then $50, and then the hot dog which cost him less than $20 because he gave the seller a tip. Let H be the cost of the hot dog in dollars, and T is Phil's tip to the hot-dog seller, then H+T=$20. After that, he gave half of his remaining money and then had $3 left.

We can write the equations below:
X - $100 - (X - $100)/2 - $50 - H - (X - $150 - H)/2 = $3

Combine like terms and simplify:
(2X - $300 - $150 - 2H)/2 = $3

Multiply both sides by 2 and simplify the equation:
2X - $450 - 2H = $6

Now we have:
2X - 2H = $456
H + T = $20

We know that T > 0 because a tip is given. Therefore, H < $20. We can test some H values:

If H = $19, we have:

2X - 2($19) = $456
X - $38 = $228
X = $266

In this case, T = 1, and this satisfies all conditions in the problem.

Thus, Phil had $266 when he started.