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2005 AMC 10A Problem 1

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

While eating out, Mike and Joe each tipped their server $2\$2 . Mike tipped 10%10\% of his bill and Joe tipped 20%20\% of his bill. What was the difference, in dollars, between their bills?

  1. A)

    22

  2. B)

    44

  3. C)

    55

  4. D)

    1010

  5. E)

    2020

Answer

D

Key insight

A $2 tip at 10% means a $20 bill and at 20% means a $10 bill; the smaller percentage belongs to the larger bill.

Solution

Undo each percentage. If 10%10\% of Mike's bill is $2\$2, the whole bill is ten times that: $20\$20. If 20%20\% of Joe's bill is $2\$2, the whole bill is five times that: $10\$10.

The difference is 20−10=1020 - 10 = 10 dollars.

The answer is (D) 10\boxed{\textbf{(D)}\ 10}.

Why this works

"p%p\% of the bill is kk" is the equation p100⋅bill=k\frac{p}{100} \cdot \text{bill} = k, so the bill is k⋅100pk \cdot \frac{100}{p}. The same tip at a smaller percentage forces a larger bill, which is the direction students most often flip.

The trap

Reversing the relationship and thinking the 20% tipper had the bigger bill, or reporting a bill (10 or 20) instead of the difference.

Common mistakes

  • Reversing the relationship and thinking the 20% tipper had the bigger bill, or reporting a bill (10 or 20) instead of the difference.
  • Subtracting the percentages (20%−10%=10%20\% - 10\% = 10\%) and treating that as a dollar amount, which happens to land on the right choice for the wrong reason.

Techniques

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