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2005 AMC 10A Problem 1

2005 AMC 10A Problem 1AMC 10 Step by Step

Name: Date: October 28, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

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

A) $2$
B) $4$
C) $5$
D) $10$
E) $20$

Answer

Key insight

The same 2 dollar tip is a smaller percent of a bigger bill. Work backward from each tip to each whole bill, then subtract.

Solution

Each tip is a known part of a bill, so work backward from the tip to the whole bill.

Mike. $10\%$ is one tenth. If one tenth of his bill is $\$2$ , the whole bill is $10 \times 2 = \$20$ .
Joe. $20\%$ is one fifth. If one fifth of his bill is $\$2$ , the whole bill is $5 \times 2 = \$10$ .
Difference. $20 - 10 = 10$ dollars.

Check: $10\%$ of $\$20$ is $\$2$ , and $20\%$ of $\$10$ is $\$2$ . Both match.

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

Why this works

A percent tells you what fraction of the whole you have. If you know the part and the fraction, multiply back up to get the whole. Notice the direction: the same $\$2$ is a smaller percent of a bigger bill, so Mike, who tipped the smaller percent, had the larger bill.

The trap

Thinking the person who tipped 20 percent had the bigger bill, or answering with one of the bills (10 or 20) instead of the difference between them.

Concise solution

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

The difference is $20 - 10 = 10$ dollars.

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

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)

Techniques

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