AMC 10 · 2011 · #8
Grade 6 rate-ratioLast 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?
Pick an answer.
AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: Geese make up 30% of all the birds; Swans make up 25% of all the birds; Herons make up 10% of all the birds; Ducks make up 35% of all the birds; The four percents cover every bird: 30 + 25 + 10 + 35 = 100
Unknowns: What percent of the non-swan birds are geese
Understand
Restated: 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.
Givens: Geese make up 30% of all the birds; Swans make up 25% of all the birds; Herons make up 10% of all the birds; Ducks make up 35% of all the birds; The four percents cover every bird: 30 + 25 + 10 + 35 = 100
Plan
Primary tool: #16 Change Focus / Count the Complement
Secondary: #9 Solve an Easier Related Problem, #8 Analyze the Units
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.
Execute — Answer: C
6.RP.A.3 Step 1 Notice the whole changes
- The question does not ask what percent of all the birds are geese; it asks what percent of the non-swan birds are geese.
- That means the group we compare against is no longer all the birds but only the ones that are not swans.
- Finding that new whole is the whole job.
💡 A percent always needs a 'whole' to sit on, and here the whole is switched from all birds to non-swan birds.
4.OA.A.3 Step 2 Find the non-swan share
- The non-swan birds are everything except the swans.
- Since swans are 25% of all the birds, the birds that are not swans are 100% - 25% = 75% of all the birds.
💡 'Not swans' is the complement of swans, so subtract the swan share from the whole.
6.RP.A.3 Step 3 Use 100 birds to make it concrete
- Suppose there are 100 birds in all.
- Then 30 of them are geese and 75 of them are not swans.
- The percent we want is the 30 geese compared with the 75 non-swan birds.
💡 With 100 birds a percent turns straight into a count, so there is no fraction bookkeeping to slip on.
5.NF.B.3 Step 4 Compare geese to the non-swans
- Divide the geese by the non-swan birds to get the share: 30 out of 75 is \frac{30}{75} = \frac{2}{5} = 0.4, which is 40%.
- So 40% of the birds that were not swans were geese, choice (C).
💡 Reading '30 out of 75' as a fraction and then as a percent answers 'what part of the new whole.'
6.RP.A.3 The question does not ask what percent of all the birds are geese; it asks what 4.OA.A.3 The non-swan birds are everything except the swans. Since swans are 25% of all t 6.RP.A.3 Suppose there are 100 birds in all. Then 30 of them are geese and 75 of them are 5.NF.B.3 Divide the geese by the non-swan birds to get the share: 30 out of 75 is \frac{3 Review
Reasonableness: 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 \approx 13\% and ducks 35/75 \approx 47\%, and 40 + 13 + 47 rounds to 100%, so the non-swan shares add up correctly. That confirms choice (C).
Alternative: Skip the count of 100 and work with the percents directly: the geese are 30% of all birds and the non-swans are 75% of all birds, so the answer is \frac{30\%}{75\%} = \frac{30}{75} = 40\%. The 'of all birds' cancels top and bottom, giving the same 40% without ever choosing a total.
CCSS standards used (min grade 6)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems, including percents (Recognizing that the percent is measured against a new whole (the non-swan birds) and reading percents as counts per 100)4.OA.A.3Solve multistep word problems using the four operations (Subtracting the swan share to get the non-swan share: 100% - 25% = 75%)5.NF.B.3Interpret a fraction as division of the numerator by the denominator (Reading 30 out of 75 as 30/75 and turning it into 40%)
⭐ When a percent is 'out of' a smaller group, first find that group, then compare against it, not against everything.
⭐ When a percent is 'out of' a smaller group, first find that group, then compare against it, not against everything.
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