2013 AMC 10A Problem 5

1. Total cost. $105 + 125 + 175 = 405$ dollars.
2. Fair share. Split evenly among $3$ people, each should pay $405 \div 3 = 135$ dollars.
3. What Tom owes. Tom paid $105$, which is $135 - 105 = 30$ too little. He gives Sammy $30$, so $t = 30$.
4. What Dorothy owes. Dorothy paid $125$, which is $135 - 125 = 10$ too little. She gives Sammy $10$, so $d = 10$.
5. Check with Sammy. Sammy paid $175$, which is $175 - 135 = 40$ too much. He gets back $30 + 10 = 40$. It matches.
6. Answer. $t - d = 30 - 10 = 20$.

The answer is $\boxed{\textbf{(B)}\ 20}$.

"Split the cost evenly" means everyone ends up paying the average. Each person who paid too little makes up the gap between the average and what they paid. Checking that the person who overpaid gets back exactly the extra is a free way to catch mistakes.

You do not even need the fair share. Tom owes $135 - 105$ and Dorothy owes $135 - 125$. When you subtract, the $135$ cancels:
$$t - d = (135 - 105) - (135 - 125) = 125 - 105 = 20.$$
So $t - d$ is just how much more Dorothy paid than Tom.

The total cost is $105 + 125 + 175 = 405$ dollars, so each person's fair share is $405/3 = 135$ dollars.

Tom paid $105$, which is $135 - 105 = 30$ short, so $t = 30$. Dorothy paid $125$, which is $135 - 125 = 10$ short, so $d = 10$. (Sammy overpaid by $175 - 135 = 40 = t + d$, as expected.)

Therefore $t - d = 30 - 10 = 20$, and the answer is $\boxed{\textbf{(B)}\ 20}$.