2005 AMC 10A Problem 1

P1-10~1 minPrint

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

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?

Show solution without answering

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.

Concise solution

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

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)