AMC 10 · 2011 · #7
Grade 7 rate-ratioPick an answer.
Tool #13 (Convert to Algebra): "mean 60" is a sentence, and turning it into x+y=120 is what exposes the link between the two numbers. Tool #15 (Organize Information in More Ways): rewriting x/y so that only y appears turns a two-unknown maximum into a one-unknown minimum, which is the whole difficulty of the problem. Tool #14 (Extreme Principle): once the ratio depends on a single variable and moves in one direction only, the answer sits at the boundary of the legal range for y — so the real work is finding which condition sets that boundary. Tool #6 (Guess and Check): a boundary value is only useful if a legal pair sits on it, so I test the candidate pair against every condition before reporting it.
Mean 60 means the sum is fixed
A fixed average fixes the sum.
A fixed mean for two numbers is just a fixed sum, and a fixed sum means one number completely decides the other.
6.SP.B.5Convert To AlgebraRewrite the ratio using y alone
The ratio rewrites with one letter.
Writing the ratio in terms of one variable shows there is only one thing you can actually change.
6.EE.A.3Organize Information In More WaysSmaller y always gives a bigger ratio
It always grows as that letter shrinks.
When the only variable lives in the denominator, the fraction is largest exactly where that variable is smallest.
When the only variable lives in the denominator, the fraction is largest exactly where that variable is smallest.
▸ Why?
Dividing by a smaller positive number is multiplying by a larger reciprocal, so the value climbs.
▸ Why?
Because that trade never turns around, the extreme of the fraction sits at the extreme of the variable.
The floor on y is 21, not 10
The real floor is 21, not ten.
With two lower limits on the same quantity, only the larger one is real.
7.EE.B.4Extreme PrincipleCheck the boundary pair and compute
That boundary pair gives 33/7, choice (B).
An extreme value only counts once you exhibit an actual case that hits it.
6.EE.B.5Guess And CheckWhen the sum is locked, one number decides the other, so a maximum with two unknowns is really a minimum with one — and the limit that stops you often comes from the number you were not watching.
- Mean 60 means the sum is fixed
- Rewrite the ratio using y alone
- Smaller y always gives a bigger ratio
- The floor on y is 21, not 10
- Check the boundary pair and compute