AMC 10 · 2014 · #11
Grade 6 arithmeticlogicPick an answer.
The total of the eleven numbers is locked at 110. To push one entry as high as possible (Tool #14, Extreme Principle), every other entry must be pushed as low as the rules allow. Tool #4 (Introduce a Variable) sorts the list as a₁ ≤ … ≤ a₁₁ so the median pins a₆ = 9 and the mode is forced into the low slots. The only real choice is how many 8s to use: more 8s raises the mode's count (letting other numbers repeat and stay small) but costs more total, so Tool #6 (Guess and Check) tests the count. Tool #3 (Eliminate Possibilities) then reads the outcome against the five answer choices — which, satisfyingly, are exactly the values the different 8-counts produce.
Turn the three statistics into hard facts
The three statistics become hard facts.
Mean, median, and mode are just three different summaries of the same list — each one is really a rule the numbers must obey.
6.SP.B.5Introduce A VariableFix the total, shrink the rest
The total is fixed, so shrink everything else.
With a fixed pot of 110, making one number big is the same as making all the others small.
With a fixed pot to share, making one number big is the same as making all the others small.
▸ Why?
The total is the numbers added together, so what the others do not take is exactly what is left.
▸ Why?
Whatever one entry gives up another gains, so the total never moves and only the split does.
Build the cheap lower half
The lower half is built as cheaply as allowed.
Three 8s is enough to lead the vote, so the leftover slots can drop all the way to 1.
6.SP.B.5Guess And CheckBuild the cheap upper half
The upper half must respect the mode.
Keep the numbers just above 9, but never let any of them tie the three 8s.
6.NS.C.7Guess And CheckRead off the maximum
What is left over is 35, choice (E).
Everything below was squeezed to the legal minimum, so whatever is left over is forced into the last, biggest number.
6.NS.B.3Extreme PrincipleWhen the total is fixed, make one number giant by squeezing every other number down to the smallest value the mean, median, and mode rules will allow.
- Turn the three statistics into hard facts
- Fix the total, shrink the rest
- Build the cheap lower half
- Build the cheap upper half
- Read off the maximum