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2011 AMC 10A Problem 8

2011 AMC 10A Problem 8AMC 10 Step by Step

Name: Date: October 28, 2026

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

Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?

A) $20$
B) $30$
C) $40$
D) $50$
E) $60$

Answer

Key insight

Pretend there are 100 birds. Leaving out the 25 swans leaves 75 birds, and 30 of them are geese, so the answer is 30 out of 75, which is 40%.

Solution

Percents are easiest to handle when the total is $100$ , so pretend there were exactly $100$ birds.

Then there were $30$ geese, $25$ swans, $10$ herons and $35$ ducks.
The question is only about the birds that are not swans. There are $100 - 25 = 75$ of those.
All $30$ geese are in that group of $75$ .
So the geese are $\dfrac{30}{75}$ of the non-swans. Simplify: $\dfrac{30}{75} = \dfrac{2}{5} = \dfrac{40}{100}$ , which is $40\%$ .

The answer is $\boxed{\textbf{(C)}\ 40}$ .

Why this works

A percent always means "out of some whole". Here the whole changes from "all the birds" to "the birds that are not swans". The number of geese stays at $30$ , but the number you divide by shrinks to $75$ . Pretending there are $100$ birds turns every percent into a plain count, so you can see exactly what to divide by what.

The trap

Answering 30% because the geese are 30% of all the birds. The question asks about only the birds that are not swans, which is a smaller group.

Concise solution

Suppose there were $100$ birds. Then there were $30$ geese, $25$ swans, $10$ herons and $35$ ducks.

The birds that were not swans number $100 - 25 = 75$ , and all $30$ geese are among them. So the required percent is
$\frac{30}{75} = \frac{2}{5} = 40\%.$

The answer is $\boxed{\textbf{(C)}\ 40}$ .

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 · Test small/specific values or special cases to find or verify the answer