2005 AMC 10B Problem 1

Profit means the money that comes in minus the money that goes out. So find both amounts.

1. Money spent. Five bars cost $2$ dollars. The troop buys $1000$ bars, which is $1000 \div 5 = 200$ groups of five. That costs $200 \cdot 2 = 400$ dollars.
2. Money taken in. Two bars sell for $1$ dollar. Selling $1000$ bars is $1000 \div 2 = 500$ groups of two, which brings in $500 \cdot 1 = 500$ dollars.
3. Profit. Money in minus money out: $500 - 400 = 100$ dollars.

The answer is $\boxed{\textbf{(A)}\ 100}$.

The buying price and the selling price are given for different group sizes (five and two), so you cannot compare them directly. Turning each one into a total for all $1000$ bars, or into a price for one bar, puts them on the same footing. Then profit is just one subtraction.

Compare the prices bar by bar. Buying: $2$ dollars for $5$ bars is $40$ cents a bar. Selling: $1$ dollar for $2$ bars is $50$ cents a bar. Each bar earns $10$ cents, and $1000$ bars earn $1000 \cdot 10 = 10000$ cents, which is $100$ dollars.

Buying five bars for $2$ dollars means each bar costs $\frac{2}{5}$ dollar, so the whole purchase costs
$$1000 \cdot \frac{2}{5} = 400 \text{ dollars}.$$

Selling two bars for $1$ dollar means each bar brings in $\frac{1}{2}$ dollar, so the total revenue is
$$1000 \cdot \frac{1}{2} = 500 \text{ dollars}.$$

Profit is revenue minus cost: $500 - 400 = 100$.

The answer is $\boxed{\textbf{(A)}\ 100}$.