2016 AMC 10A Problem 8

We know how the story ends (Fox has $0$ coins), so run it backwards.

1. One crossing does two things in this order: double the money, then pay $40$.
2. To undo a crossing, undo those two things in reverse order: first add the $40$ back, then halve.
3. After the third crossing Fox has $0$.
- Before the third crossing: $(0 + 40) \div 2 = 20$.
- Before the second crossing: $(20 + 40) \div 2 = 30$.
- Before the first crossing: $(30 + 40) \div 2 = 35$.
4. Check forwards: $35 \to 70 \to 30$, then $30 \to 60 \to 20$, then $20 \to 40 \to 0$. Fox is broke after three crossings, as the story says.

The answer is $\boxed{\textbf{(C)}\ 35}$.

When you know the final result and each step can be undone, working backwards replaces algebra with a few easy calculations. The key is to undo the last action first, just as you take off your shoes before your socks. Always check your answer by running the story forwards.

Use a letter. Start with $x$ coins.

• After crossing 1: $2x - 40$.
• After crossing 2: $2(2x - 40) - 40 = 4x - 120$.
• After crossing 3: $2(4x - 120) - 40 = 8x - 280$.

Setting $8x - 280 = 0$ gives $x = 35$.

Each crossing sends an amount $m$ to $2m - 40$. Undoing a crossing means adding $40$ and then halving.

Fox ends with $0$ coins after the third crossing.

• Before the third crossing: $(0 + 40)/2 = 20$.
• Before the second crossing: $(20 + 40)/2 = 30$.
• Before the first crossing: $(30 + 40)/2 = 35$.

Check forward: $35 \to 70 - 40 = 30 \to 60 - 40 = 20 \to 40 - 40 = 0$.

The answer is $\boxed{\textbf{(C)}\ 35}$.