AMC 10 · 2008 · #3
Grade 5 arithmeticPick an answer.
The $700,000 cap is shared among all 21 players. To hand one player as much as possible, push everyone else to the smallest amount the rules allow. Once the other 20 sit at the floor, whatever is left of the cap belongs to the last player.
Push the others to the minimum
One share is the cap minus everyone else's.
When a total is fixed, one part grows the most when every other part shrinks to its limit.
When the total is fixed, one part grows the most exactly when every other part shrinks to its limit.
▸ Why?
The parts add up to the whole, so whatever the others do not take is exactly what is left for the one.
▸ Why?
Whatever one part gives up another gains, so the total never moves and only the split does.
Cost of the other 20 players
Pushing the others to the floor costs 300,000.
Twenty equal groups of $15,000 is just repeated addition sped up by multiplication.
5.NBT.B.5Analyze The UnitsWhatever is left goes to one player
What is left is 400,000, choice (E).
The leftover budget after covering everyone else is exactly what one player can claim.
4.NBT.B.4Convert To AlgebraWhen a total is capped, make one share the biggest by shrinking every other share to its minimum.
- Push the others to the minimum
- Cost of the other 20 players
- Whatever is left goes to one player