AMC 10 · 2011 · #8

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

percentageratio-proportionconditional-probability complementary-counting ↑ Prerequisites: percentage

📏 Medium solution 💡 2 insights

Problem

Among the birds on a lake, 30% are geese, 25% are swans, 10% are herons, and 35% are ducks. Look only at the birds that are not swans, and find what percent of that smaller group are geese.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

The trap is that the 30% for geese is measured against ALL birds, but the question wants a percent measured against only the non-swan birds. So the real move is to change what counts as the whole. The non-swan birds are the complement of the swans, which is quick to find by subtracting. Pretending there are exactly 100 birds turns every percent into a plain count, so the final division is easy.

1STEP 1

Notice the whole changes

The percent is measured against only the non-swan birds, not all of them — finding that new whole is the whole job.

2STEP 2

Find the non-swan share

Swans are 25% of the flock, so the birds that are not swans make up 75% of all the birds.

100% - 25% = 75%

3STEP 3

Use 100 birds to make it concrete

Suppose the lake holds 100 birds; then 30 are geese and 75 are not swans.

geese = 30, non-swans = 75

4STEP 4

Compare geese to the non-swans

Divide geese by non-swans: $\frac{30}{75} = \frac{2}{5} = 0.4$ , that is 40% — choice (C).

30/75 = 2/5 = 0.4 = 40%

Answer

Removing the swans shrinks the whole from 100 to 75 while the 30 geese stay, so the geese must take up a bigger share than their original 30%. The answer 40% is indeed larger than 30%, which rules out 20% (A) and 30% (B). As a further check, the other non-swan groups give herons 10/75 ≈ 13% and ducks 35/75 ≈ 47%, and 40 + 13 + 47 rounds to 100%, so the non-swan shares add up correctly. That confirms choice (C).

💡Key takeaway

When a percent is 'out of' a smaller group, first find that group, then compare against it, not against everything.

Notice the whole changes
Find the non-swan share
Use 100 birds to make it concrete
Compare geese to the non-swans

Notice the whole changes

Find the non-swan share

Use 100 birds to make it concrete

Compare geese to the non-swans

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