AMC 10 · 2002 · #21
Grade 6 arithmeticPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
Mean 8 over 8 numbers gives total 64; median 8 pins the middle pair; unique mode 8 outranks every value; range 8 links top and bottom.
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
Call the largest number L. To keep the total at 64 while L grows, push the others as low as allowed; the range 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.
A fixed total is like a fixed budget: spending less on the others leaves more for the one you want large.
▸ 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.
Test the biggest choice, 15
If the largest were 15, the other seven each sit at 7 or more yet total only 49, forcing all to 7 — mode and median become 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
Take largest 14, smallest 6: the collection 6,6,6,8,8,8,8,14 sums to 64, has median 8, unique mode 8 (four 8's), and range 14-6=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 built, so the largest integer the collection can contain 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