AMC 10 · 2002 · #15
Grade 6 arithmeticPick an answer.
The question asks for the biggest a single number can be, which is exactly what Tool #14 (Extreme Principle) is built for: to push one number to its maximum, force every other number down to the smallest value the rules still allow. Tool #4 (Introduce a Variable) pins the picture down — call the largest L, so the smallest is L-8 from the range. Tool #6 (Guess and Check) does the final work: with a fixed total of 64, test the top answer choices and either build a valid collection or show one is impossible. The mode clue is the brake — it stops the small numbers from all being tiny, because 8 has to appear more than anything else.
Turn the four clues into number facts
Read the clues as facts: the eight numbers total 64, and largest minus smallest is 8.
Each average, mode, and range word is really a plain statement about the eight numbers, so writing them out turns the puzzle into arithmetic.
6.SP.B.5Introduce A VariableAim the largest number as high as possible
To lift the largest L, make the rest minimal; the range then fixes the smallest at L-8.
A fixed total is like a fixed budget: spending less on the other numbers leaves more room for the one you want to be large.
With the total fixed by the average, spending less on the other numbers is what leaves room for one large one.
▸ Why?
The average is the total shared out equally, so the total is the average times the count and cannot move.
▸ Why?
That fixed total is the eight numbers added together, so whatever the others do not take is exactly what the largest may have.
Test the biggest choice, 15
If L were 15, the other seven must total 49 and each is at least 7, forcing seven 7's — mode 7, not 8.
When seven numbers each at least 7 must total exactly 49, there is no slack left, so they are all pinned to 7.
6.EE.B.5Guess And CheckBuild a collection with largest 14
With L = 14 the set 6,6,6,8,8,8,8,14 satisfies mean, median, mode and range all 8.
Loading up on 8's secures the mode, and using the smallest legal values for the rest leaves exactly enough total for a 14.
6.SP.B.5Guess And CheckCompare and conclude
15 is impossible and 14 is achieved, so the largest possible integer is 14, choice (D).
The greatest possible value is the largest one you can actually construct without breaking any rule.
6.SP.B.5Extreme PrincipleTo make one number as big as possible under a fixed total, squeeze every other number down to the smallest the rules allow — then check the biggest choice actually builds.
- Turn the four clues into number facts
- Aim the largest number as high as possible
- Test the biggest choice, 15
- Build a collection with largest 14
- Compare and conclude
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