AMC 10 · 2011 · #5
Grade 6 rate-ratioPick an answer.
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.
Notice the whole changes
The whole being measured against has changed.
A percent always needs a 'whole' to sit on, and here the whole is switched from all birds to non-swan birds.
A percent always needs a whole to sit on, and here the whole is switched from all birds to the non-swans.
▸ Why?
A percent is a count out of a hundred of whatever it is applied to, so the base decides its meaning.
▸ Why?
Not being a swan is the opposite of being one, so the new whole is what is left after the swans are removed.
Find the non-swan share
What remains is 75 percent of the flock.
'Not swans' is the complement of swans, so subtract the swan share from the whole.
4.OA.A.3Change Focus Count The ComplementUse 100 birds to make it concrete
Taking a hundred birds makes it concrete.
With 100 birds a percent turns straight into a count, so there is no fraction bookkeeping to slip on.
6.RP.A.3Solve An Easier Related ProblemCompare geese to the non-swans
Comparing gives 40 percent, choice (C).
Reading '30 out of 75' as a fraction and then as a percent answers 'what part of the new whole.'
5.NF.B.3Analyze The UnitsWhen 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