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2005 AMC 10B Problem 1

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2005 AMC 10B Problem 1AMC 10 Step by Step
Name:                             Date: September 13, 2026
Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

A scout troop buys 10001000 candy bars at a price of five for 22 dollars. They sell all the candy bars at the price of two for 11 dollar. What was their profit, in dollars?

  1. A)

    100100

  2. B)

    200200

  3. C)

    300300

  4. D)

    400400

  5. E)

    500500

Answer

A

Key insight

Convert both prices to a per-bar rate (40 cents to buy, 50 cents to sell) so the profit is 10 cents per bar times 1000 bars.

Solution

Buying five bars for 22 dollars means each bar costs 25\frac{2}{5} dollar, so the whole purchase costs
1000⋅25=400 dollars.1000 \cdot \frac{2}{5} = 400 \text{ dollars}.

Selling two bars for 11 dollar means each bar brings in 12\frac{1}{2} dollar, so the total revenue is
1000⋅12=500 dollars.1000 \cdot \frac{1}{2} = 500 \text{ dollars}.

Profit is revenue minus cost: 500−400=100500 - 400 = 100.

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

Why this works

"Five for two dollars" and "two for one dollar" are rates; converting each to a price per single bar puts them on the same footing, after which profit is a subtraction. Whenever two group prices are compared, reduce both to a unit rate (or to a common batch size) first.

Alternative approach

Use a batch of 1010 bars: it costs 44 dollars and sells for 55 dollars, a gain of 11 dollar per 1010 bars. With 100100 such batches the profit is 100100.

The trap

Reporting the revenue (500) or the cost (400) instead of their difference.

Common mistakes

  • Reporting the revenue (500) or the cost (400) instead of their difference.
  • Inverting a rate, e.g. treating "five for 22 dollars" as 2.52.5 dollars per bar.

Techniques

Set up the equation/formula and compute; no special trick needed