AMC 10 · 2008 · #5
Grade 6 countingPick an answer.
Tool #4 (Introduce a Variable) turns the money story into one clean equation, 3r+2c=50, which pins down exactly what must be true. From there the choices are still infinite-looking, so Tool #3 (Eliminate Possibilities) uses a parity fact to throw away every odd number of roses at once. Tool #14 (Extreme Principle) finds the biggest number of roses that still leaves room, capping the range. Then Tool #2 (Make a Systematic List) walks the survivors in order so none are missed and none are double-counted.
Turn the money into an equation
The spending becomes one equation in two counts.
Naming the two counts turns a vague shopping question into one exact equation you can test.
6.EE.B.6Introduce A VariableOnly an even number of roses works
Parity forces one count to be even.
Even total minus even carnation cost leaves an even amount for roses, and odd-priced roses hit an even amount only in even-sized groups.
An even total minus an even carnation cost leaves an even amount, which odd-priced roses reach only in even-sized bunches.
▸ Why?
An odd price repeated an odd number of times gives an odd total, which cannot match an even one.
▸ Why?
The money spent has to be exhausted exactly, so nothing may be left over at the end.
Find the largest number of roses that fits
The budget caps it at 16.
Push the number of roses to its limit to find where the money runs out.
6.EE.B.5Extreme PrincipleList the survivors and count them
Listing the survivors gives 9, choice (C).
Walk the allowed rose counts in order so you catch every combination exactly once.
4.OA.C.5Make A Systematic ListTurn the spending into one equation, use even-and-odd to knock out half the cases, then list the rest in order and count.
- Turn the money into an equation
- Only an even number of roses works
- Find the largest number of roses that fits
- List the survivors and count them