AMC 10 · 2007 · #15
Grade 7 arithmeticPick an answer.
The mean is easy — it is (28+n)/5 for every n. The median is the hard half, because it is a different expression depending on whether n lands below the old middle, inside it, or above it. So the spine of the problem is Tool #2 (Make a Systematic List): carve the number line into stretches on which the median is a fixed expression, and make sure the carving leaves no gap and no overlap, because a missed stretch is a missed value of n and the question asks for the sum of all of them. Inside each stretch Tool #4 (Introduce a Variable) and Tool #13 (Convert to Algebra) reduce the condition to one linear equation with exactly one root. Tool #6 (Guess and Check) then does the step most quick write-ups skip. Each case assumed n lies in some range and used that assumption to compute the median; the number it spits out is only a real solution if it actually lands back inside the range it assumed. Without that return trip the casework proves necessity and never sufficiency.
Write the mean once, for good
The mean can be written once and reused.
The mean only needs the total, so shuffling where n sits cannot change it.
6.SP.B.5Introduce A VariableThe median is the middle of 6, n, 9
The median takes one of three values depending on where the number lands.
Adding one number can only nudge the middle by one slot, so 6 and 9 act as walls the median cannot pass.
Adding one number can only nudge the middle by one slot, so the two known neighbours act as walls.
▸ Why?
The median is whatever sits in the middle once the list is ordered, so it depends only on rank.
▸ Why?
Wherever the new number lands relative to those two, the ordering already fixes which value takes the middle.
Case n ＜ 6: the median is 6
The low case gives 2.
If the middle value is locked at 6, the total is forced to be 30, which pins n exactly.
7.EE.B.4Convert To AlgebraCase 6 ＜ n ＜ 9: the median is n
The middle case gives 7.
A number equals the mean exactly when the other numbers average to it too, so 28 has to split into four copies of n.
7.EE.B.4Convert To AlgebraCase n ＞ 9: the median is 9
The high case gives 17.
Pushing n past 9 leaves the middle stuck at 9, so only the total is still free to be tuned.
7.EE.B.4Convert To AlgebraSend each answer back to its own case
Each value falls inside its own case, so all three survive.
A case that names a number outside its own window has named nothing at all.
6.EE.B.5Guess And CheckRebuild the sets, then add
Adding gives 26, choice (E).
Writing out the five numbers and reading off both statistics is the cheapest possible proof that a candidate is real.
6.SP.B.5Guess And CheckWhen the median depends on where the new number lands, split the number line so no stretch is missed — then send every answer back to the case that produced it, because a case that names a number outside its own window has named nothing.
- Write the mean once, for good
- The median is the middle of 6, n, 9
- Case n ＜ 6: the median is 6
- Case 6 ＜ n ＜ 9: the median is n
- Case n ＞ 9: the median is 9
- Send each answer back to its own case
- Rebuild the sets, then add
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Same archetype — closest grade level first.
