2005 AMC 10B Problem 1

P1-10~1 minPrint

Ratios, Percents & Averages Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?

Show solution without answering

Key insight

Find the price of one bar when buying (40 cents) and when selling (50 cents). Each bar earns 10 cents, and there are 1000 bars.

Solution

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

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.
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.
Profit. Money in minus money out: $500 - 400 = 100$ dollars.

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

Why this works

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.

Alternative approach

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(A)}\ 100}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)